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SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-11T00:41:54Z

Joel2: /* Temporary Excel Interpolations */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

====Temporary Excel Interpolations====
<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="6"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
<td align="center"><math>M_\mathrm{tot}</math></td>
<td align="center"><math>\log_{10}(\rho_\mathrm{max})</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
<td align="center" colspan="2">(Fixed <math>P_i</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6684629814</td>
<td align="right">12.03999149</td>
<td align="right">0.092175036</td>
<td align="right">3.46986909</td>
<td align="right">0.712724159</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.4459132276</td>
<td align="right">13.67957562</td>
<td align="right">0.091291571</td>
<td align="right">3.50879154</td>
<td align="right">0.688526899</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0482530437</td>
<td align="right">17.09391244</td>
<td align="right">0.086279818</td>
<td align="right">3.69798999</td>
<td align="right">0.58112284</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7170001608</td>
<td align="right">20.48027265</td>
<td align="right">0.079651055</td>
<td align="right">3.91920968</td>
<td align="right">0.484075667</td>
</tr>

<tr>
<td align="right">0.74</td>
<td align="right">0.6365283705</td>
<td align="right">21.40307774</td>
<td align="right">0.0777495</td>
<td align="right">3.97973335</td>
<td align="right">0.464464039</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-11T00:36:15Z

Joel2: /* Temporary Excel Interpolations */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

====Temporary Excel Interpolations====
<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="6"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
<td align="center"><math>M_\mathrm{tot}</math></td>
<td align="center"><math>\log_{10}(\rho_\mathrm{max})</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
<td align="center" colspan="2">(Fixed <math>P_i</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6684629814</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.4459132276</td>
<td align="right">13.67957562</td>
<td align="right">0.091291571</td>
<td align="right">3.50879154</td>
<td align="right">0.688526899</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0482530437</td>
<td align="right">17.09391244</td>
<td align="right">0.086279818</td>
<td align="right">3.69798999</td>
<td align="right">0.58112284</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7170001608</td>
<td align="right">20.48027265</td>
<td align="right">0.079651055</td>
<td align="right">3.91920968</td>
<td align="right">0.484075667</td>
</tr>

<tr>
<td align="right">0.74</td>
<td align="right">0.6365283705</td>
<td align="right">21.40307774</td>
<td align="right">0.0777495</td>
<td align="right">3.97973335</td>
<td align="right">0.464464039</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T23:46:02Z

Joel2: /* Temporary Excel Interpolations */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

====Temporary Excel Interpolations====
<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="6"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
<td align="center"><math>M_\mathrm{tot}</math></td>
<td align="center"><math>R</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
<td align="center" colspan="2">(Fixed <math>P_i</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6671</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.455</td>
<td align="right">13.679</td>
<td align="right">0.091144</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0482530437</td>
<td align="right">17.09391244</td>
<td align="right">0.086279818</td>
<td align="right">3.69798999</td>
<td align="right">0.58112284</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7170001608</td>
<td align="right">20.48027265</td>
<td align="right">0.079651055</td>
<td align="right">3.91920968</td>
<td align="right">0.484075667</td>
</tr>

<tr>
<td align="right">0.74</td>
<td align="right">0.6365283705</td>
<td align="right">21.40307774</td>
<td align="right">0.0777495</td>
<td align="right">3.97973335</td>
<td align="right">0.464464039</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T23:37:13Z

Joel2: /* Temporary Excel Interpolations */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

====Temporary Excel Interpolations====
<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="6"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
<td align="center"><math>M_\mathrm{tot}</math></td>
<td align="center"><math>R</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
<td align="center" colspan="2">(Fixed <math>P_i</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6671</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.455</td>
<td align="right">13.679</td>
<td align="right">0.091144</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0613</td>
<td align="right">17.093</td>
<td align="right">0.08612</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7170001608</td>
<td align="right">20.48027265</td>
<td align="right">0.079651055</td>
<td align="right">3.91920968</td>
<td align="right">0.484075667</td>
</tr>

<tr>
<td align="right">0.74</td>
<td align="right">0.6365283705</td>
<td align="right">21.40307774</td>
<td align="right">0.0777495</td>
<td align="right">3.97973335</td>
<td align="right">0.464464039</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T23:32:01Z

Joel2: /* Temporary Excel Interpolations */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

====Temporary Excel Interpolations====
<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="6"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
<td align="center"><math>M_\mathrm{tot}</math></td>
<td align="center"><math>R</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
<td align="center" colspan="2">(Fixed <math>P_i</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6671</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.455</td>
<td align="right">13.679</td>
<td align="right">0.091144</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0613</td>
<td align="right">17.093</td>
<td align="right">0.08612</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7141</td>
<td align="right">20.480</td>
<td align="right">0.0797</td>
<td align="right">3.9192</td>
<td align="right">0.4828</td>
</tr>

<tr>
<td align="right">0.74</td>
<td align="right">0.6365283705</td>
<td align="right">21.40307774</td>
<td align="right">0.0777495</td>
<td align="right">3.97973335</td>
<td align="right">0.464464039</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T20:20:16Z

Joel2: /* Temporary Excel Interpolations */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

====Temporary Excel Interpolations====
<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="6"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
<td align="center"><math>M_\mathrm{tot}</math></td>
<td align="center"><math>R</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
<td align="center" colspan="2">(Fixed <math>P_i</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6671</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.455</td>
<td align="right">13.679</td>
<td align="right">0.091144</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0613</td>
<td align="right">17.093</td>
<td align="right">0.08612</td>
<td align="right">0</td>
<td align="right">0</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7141</td>
<td align="right">20.480</td>
<td align="right">0.0797</td>
<td align="right">3.9192</td>
<td align="right">0.4828</td>
</tr>

<tr>
<td align="right">0.74</td>
<td align="right">0.6364</td>
<td align="right">21.403</td>
<td align="right">0.07775</td>
<td align="right">3.980</td>
<td align="right">0.4644</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T19:45:11Z

Joel2: /* Fixed Interface Pressure Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

====Temporary Excel Interpolations====
<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="4"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6671</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.455</td>
<td align="right">13.679</td>
<td align="right">0.091144</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0613</td>
<td align="right">17.093</td>
<td align="right">0.08612</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7141</td>
<td align="right">20.480</td>
<td align="right">0.0797</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T18:14:25Z

Joel2: /* Fixed Interface Pressure Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="4"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6671</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.455</td>
<td align="right">13.679</td>
<td align="right">0.091144</td>
</tr>

<tr>
<td align="right">0.8</td>
<td align="right">1.0613</td>
<td align="right">17.093</td>
<td align="right">0.08612</td>
</tr>

<tr>
<td align="right">0.75</td>
<td align="right">0.7141</td>
<td align="right">20.480</td>
<td align="right">0.0797</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T17:59:47Z

Joel2: /* Fixed Interface Pressure Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

<font color="red">HERE</font>

<table border="1" align="center" cellpadding="5">

<tr>
<td align="center" colspan="4"><b>Properties of Turning-Points Along Sequences Having Various  <math>\mu_e/\mu_c</math></b></td>
</tr>

<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="2"><math>\xi_i</math></td>
<td align="center"><math>P_i</math></td>
<td align="center"><math>R</math></td>
</tr>

<tr>
<td align="center" colspan="2">(Fixed <math>M_\mathrm{tot}</math>)</td>
</tr>

<tr>
<td align="right">1.000</td>
<td align="right">1.6671</td>
<td align="right">12.040</td>
<td align="right">0.0922</td>
</tr>

<tr>
<td align="right">0.9</td>
<td align="right">1.455</td>
<td align="right">13.679</td>
<td align="right">0.091144</td>
</tr>
</table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-10T16:07:20Z

Joel2: /* Fixed Interface Pressure Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>

<tr>
<td align="left" colspan="3">NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, <math>\xi_i = 1.668462981</math>. In each case, this vertical line intersects a key turning point along the model sequence.
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-08T22:41:37Z

Joel2: /* Fixed Interface Pressure Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">(Interface Pressure<math>)^{1 / 3}</math> vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-08T22:37:18Z

Joel2: /* Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">External Pressure vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed total mass will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-08T22:36:16Z

Joel2: /* Fixed Interface Pressure Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="3">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">External Pressure vs. Radius<br />(Fixed Total Mass)</td>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4} \biggl(\frac{2}{\pi}\biggr) A^2\eta_s^2</math>
<br />vs.<br />
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^3 \biggl(\frac{\pi}{2^3}\biggr)^{1 / 2} \frac{1}{A^2\eta_s}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-08T22:21:50Z

Joel2: /* Fixed Interface Pressure Sequence Plots */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<td align="center" colspan="2">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math><br />
(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr> <td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>

<tr>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\frac{\eta_s}{\sqrt{2\pi}}</math>
</td>
<td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s</math>
<br />vs.<br />
<math>\log_\mathrm{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5/2}\theta_i^{-5} \biggr]</math>
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-05T03:22:19Z

Joel2: /* Preface */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>.

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<th align="center" colspan="2">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math>
</th>
</tr>
<tr>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr> <td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-05T03:02:53Z

Joel2: /* Preface */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center"><font size="-1">'''Figure 2: Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
</td>
</tr>
</table>
Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>. Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface.

<font color="red">'''HERE'''</font>

Along each sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<th align="center" colspan="2">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math>
</th>
</tr>
<tr>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr> <td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

{{ SGFfooter }}

SSC/Stability/BiPolytropes/RedGiantToPN

2026-04-05T02:52:41Z

Joel2: /* Preface */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN|Part I:  Background & Objective]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:  ]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:  ]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV: ]]
 
</td>
</tr>
</table>

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

<table border="0" align="left" cellpadding="10"><tr><td align="center">
<table border="1" align="left" cellpadding="2">
<tr><td align="center">
[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
</td></tr>
<tr><td align="center">
[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
</td></tr>
</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]]. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)

 <br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font> ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>. Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface.

<font color="red">'''HERE'''</font>

Along each sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

indexes, as labeled, over the range <math>1 \le n \le 6</math>.



<table border="1" align="center" cellpadding="3">
<tr>
<td align="center" rowspan="1">
'''Figure 2: Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
</td>
<td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
</td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
</td>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass-density
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
Pressure (energy-density)
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m \ell^{-1} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
Newtonian gravitational constant, <math>G</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{3} t^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
The core's polytropic constant, <math>K_c</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
The envelope's polytropic constant, <math>K_e</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
m^{-1} \ell^{5} t^{-2}
</math>
</td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2}
~\biggl[ m^{-1} \ell^{3} t^{-2} \biggr]^{-3/2}
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5} \biggr]
~\biggl[ \ell^{39/10 - 9/2 + 3/5 } \biggr]
~\biggl[ t^{-3 + 3 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math> — as well as <math>G</math> and <math>K_c</math> — constant along an equilibrium sequence, mass will scale as …
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
Mass
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]
\biggl[m^{-1} \ell^{5} t^{-2}\biggr]
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr]
\biggl[ t^{-10 - 2 + 12}\biggr]
~\biggr\}^{1 / 4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[ m^{+1} \biggr]
~\biggl[ \ell^{0 } \biggr]
~\biggl[ t^{0} \biggr]
\, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Original Model Construction==

===Fixed Central Density===
From [[SSC/Structure/BiPolytropes/Analytic51/Pt2|Examples]], we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} = M^*_\mathrm{core} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} = M^*_\mathrm{tot} \biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} = r_\mathrm{core}^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr] \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R = r_s^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}
\frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>
</table>

where, rewriting the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>
we find,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell_i^2)^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c}\biggr)\sqrt{3}\theta_i^2 \xi_i
= m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
=
\biggl[ \frac{1+\ell_i^2}{m_3 \ell_i}\biggr] - \ell_i
=
\frac{1}{m_3 \ell_i} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i^2 (1 + \Lambda_i^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3^2 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr)^2
\cdot
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ (1+\Lambda_i^2)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i^{-2}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{1}{m_3^2} \biggl[ \frac{(1+\ell_i^2)^2}{\ell_i^2} \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
=
\frac{(1+\ell_i^2)}{m_3^2 \ell_i^2}
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i)
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===

====Equilibrium Sequence Expressions====

From the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}
\biggr]
\, .</math>
</td>
</tr>
</table>

Hence, keeping <math>K_c</math> and <math>K_e</math> constant, we have,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c^{3/2} G^{-3/2}\biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} (\xi_i \theta_i)^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4}\biggr]
\biggl[
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>r_\mathrm{core} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_e}{K_c} \biggr]^{-5 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3/2}\biggr] \theta_i^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{5}K_e^{-3} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \, .
</math>
</td>
</tr>
</table>
This last expression shows that <font color="red"><b>if <math>K_c</math> and <math>K_e</math> are both held fixed, then the interface pressure, <math>P_i</math>, will be constant</b></font> along the sequence of equilibrium models.

Note also:

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2}
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^4
\biggr\}
\biggl\{
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}
\frac{\sqrt{3}\xi_i^3 \theta_i^4}{A\eta_s}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>q \equiv \frac{r_\mathrm{core}}{R} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1 / 2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \theta_i^2
\biggr\}
\biggl\{
\biggl[K_e G^{-1} \biggr]^{1/2}
\frac{\eta_s}{\sqrt{2\pi}}
\biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)
\frac{ \sqrt{3}\xi_i \theta_i^2 }{\eta_s}
\, .
</math>
</td>
</tr>
</table>

====Fixed Interface Pressure Sequence Plots====
A plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>R~\biggl[K_e G^{-1} \biggr]^{-1/2}</math> at <font color="red"><b>fixed interface pressure</b></font> will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>R</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi}} \, .</math>
</td>
</tr>
</table>

Alternatively, a plot of <math>M_\mathrm{tot}~\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}</math> versus <math>\log_{10}(\rho_0) ~\biggl[ \frac{K_e}{K_c} \biggr]^{5 / 4}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate: <math>M_\mathrm{tot}</math></td>
<td align="center"> </td>
<td align="center">Abscissa: <math>\log_{10}(\rho_0)</math></td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2}
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\log_{10}\biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}\biggr]
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8"><tr><td align="left">

The expression for <math>dM_\mathrm{tot}/d\ell_i</math> is …
<table align="center" cellpadding="8">

<tr>
<td align="right">
<math>\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl[
A\eta_s
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3/2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3/2} \biggl(\frac{2}{\pi}\biggr)^{-1 / 2}
\biggl[K_c^{5}K_e G^{-6}\biggr]^{-1 / 4}
\frac{dM_\mathrm{tot}}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d\eta_i}{d\ell_i}
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{\frac{d}{d\ell_i}
\biggl[ \tan^{-1}(\Lambda_i) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

The extremum in <math>M_\mathrm{tot}</math> occurs when the LHS of this expression is zero, that is, when …
<table align="center" cellpadding="8">

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
m_3 \ell_i (1+\ell_i^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1+\Lambda_i^2)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2} \biggl\{
\frac{d}{d\ell_i} \biggl[ m_3 \biggl( \frac{\ell_i}{1+\ell_i^2}\biggr) \biggr]
\biggr\}
+
2\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\frac{d}{d\ell_i} \biggl\{ \ell_i (1 + \ell_i^2)^{-1}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{1 / 2}
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr] \cdot \frac{d}{d\ell_i}\biggl\{ \ell_i^{-1} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]^{- 1 / 2}
~\frac{d}{d\ell_i} \biggl\{ (1+\ell_i^2)^{-1}
\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl\{ (1 + \ell_i^2)^{-1} - 2\ell_i^2 (1 + \ell_i^2)^{-2}
\biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\biggl[ \frac{1}{1+\Lambda_i^2}\biggr]
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i (1+\ell_i^2)^{-2}\biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)^{-1}\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2)^3 }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
\frac{2}{m_3}\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ 1+\ell_i^2 }\biggr]
\frac{m_3^2 \ell_i^2}{(1+\ell_i^2)} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{-1}
\biggl\{ -\ell_i^{-2} \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3) \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~(1+\ell_i^2)^{-2}\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2 }{ (1+\ell_i^2) }\biggr]
\biggl\{ (1 + \ell_i^2) - 2\ell_i^2 \biggr\}
+
2m_3
\biggl\{ - \biggl[ 1+ (1 - m_3)\ell_i^2 \biggr] + \biggl[ 2(1 - m_3)\ell_i^2 \biggr]\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
-2\ell_i \biggl[1 + (1-m_3)^2 \ell_i^2 \biggr]
+
(1+\ell_i^2)\biggl[2\ell_i(1-m_3)^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
2 m_3 \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ \frac{(1 - \ell_i^2)}{(1+\ell_i^2) }\biggr]
+
2m_3
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-~\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{
\biggl[-2 \ell_i -2 (1-m_3)^2 \ell_i^3 \biggr]
+
\biggl[2\ell_i(1-m_3)^2 \biggr]
+
\biggl[2(1-m_3)^2 \ell_i^3 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
\biggl[ (1 - \ell_i^2)\biggr]
+
(1 + \ell_i^2)\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~\biggl\{ 1 - (1-m_3)^2 \biggr\} 2 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{2 m_3 }{(1+\ell_i^2) }\biggl\{
\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
- \ell_i^2\biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]
+
\biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
+
\ell_i^2 \biggl[ - 1 + (1 - m_3)\ell_i^2 \biggr]
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) 2m_3 \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
(1+\ell_i^2)^{-1}\biggl\{
1 + (1-m_3)^2 \ell_i^2
-\ell_i^2 - (1-m_3)^2 \ell_i^4
- 1 + (1 - m_3)\ell_i^2
- \ell_i^2 + (1 - m_3)\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
~(2 - m_3) \ell_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{1}{(1+\ell_i^2)(2 - m_3) \ell_i}\biggl\{
\biggl[ m_3 - 3 \biggr]m_3\ell_i^2
+ \biggl[1 - m_3 \biggr]m_3\ell_i^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math> \Rightarrow ~~~
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
</math>
</td>
</tr>



</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>.
</td></tr></table>

----

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100]]Example data values drawn from worksheet "SequenceMuRatio100" …
<div align="left">
<math>\Delta \xi = (9.01499598 - 0.05)/99 = 0.0905551</math>
</div>

<table border="1" align="center" cellpadding="3">
<tr>
<td align="center"><math>n_\mathrm{grid}</math></td>
<td align="center"><math> \xi_i=0.05 + (n_\mathrm{grid} - 1)\cdot\Delta\xi</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="center">1</td>
<td align="left">0.05</td>
<td align="left">0.9995836</td>
<td align="left">1.00124818</td>
<td align="left">3.141592582</td>
<td align="left">2.510</td>
<td align="left">0.0009044</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">2</td>
<td align="left">0.140555</td>
<td align="left">0.9967235</td>
<td align="left">1.0097655</td>
<td align="left">3.141580334</td>
<td align="left">2.531</td>
<td align="left">0.0071264</td>
<td align="left">1.253</td>
</tr>
<tr>
<td align="center">18</td>
<td align="left">1.5894375</td>
<td align="left">0.7367887</td>
<td align="left">1.539943947</td>
<td align="left">2.821678456</td>
<td align="left">3.467</td>
<td align="left">0.663285301</td>
<td align="left">1.126</td>
</tr>
<tr>
<td align="center">19</td>
<td align="left">1.6799927</td>
<td align="left">0.7178117</td>
<td align="left">1.566601145</td>
<td align="left">2.775921455</td>
<td align="left" bgcolor="yellow">3.470</td>
<td align="left">0.719947375</td>
<td align="left">1.107</td>
</tr>
<tr>
<td align="center">20</td>
<td align="left">1.7705478</td>
<td align="left">0.6992927</td>
<td align="left">1.591530391</td>
<td align="left">2.728957898</td>
<td align="left">3.465</td>
<td align="left">0.7767049</td>
<td align="left">1.089</td>
</tr>
<tr>
<td align="center">100</td>
<td align="left">9.0149598</td>
<td align="left">0.1886798</td>
<td align="left">1.973119305</td>
<td align="left">0.841461698</td>
<td align="left">1.325</td>
<td align="left">3.62137</td>
<td align="left">0.336</td>
</tr>
</table>
</td></tr></table>

<table border="1" cellpadding="3" align="center">
<tr>
<th align="center" colspan="2">
Equilibrium Sequences of <math>(n_c, n_e) = (5, 1)</math> BiPolytropes Having <math>\mu_e/\mu_c = 1.0</math>
</th>
</tr>
<tr>
<td align="center" colspan="1">Mass vs. Radius<br />(Fixed Interface Pressure)</td>
<td align="center" colspan="1">Mass vs. Central Density<br />(Fixed Interface Pressure)</td>
</tr>
<tr> <td align="center" colspan="1">
[[File:MuRatio100MassVsRadiusA.png|350px|Total Mass vs Radius]]
</td>
<td align="center" colspan="1">
[[File:MuRatio100MassVsCentralDensityA.png|350px|Total Mass vs Central Density]]
</td>
</tr>
</table>

<table border="0" align="center" cellpadding="8"><tr><td align="left">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund]]Data values drawn from worksheet "MuRatio100Fund" …

<table border="1" align="center" cellpadding="3">
<tr>
<th align="center" colspan="7">
Properties of the Marginally Unstable Model
</th>
</tr>
<tr>
<td align="center"><math> \xi_i</math></td>
<td align="center"><math> \theta_i</math></td>
<td align="center"><math> A</math></td>
<td align="center"><math> \eta_s</math></td>
<td align="center"><math> M_\mathrm{tot}</math></td>
<td align="center"><math> \log10\rho_0</math></td>
<td align="center"><math> R</math></td>
</tr>
<tr>
<td align="left">1.6639103</td>
<td align="left">0.7211498</td>
<td align="left">1.561995126</td>
<td align="left">2.784147185</td>
<td align="left" bgcolor="yellow">3.4698598</td>
<td align="left">0.709872477</td>
<td align="left">1.1107140</td>
</tr>
</table>
</td></tr></table>

===Fixed Total Mass===

====Equilibrium Sequence Expressions====

Again, drawing from [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|previous Examples]] in which <math>\rho_0</math> — as well as <math>K_c</math> and <math>G</math> — is held fixed, equilibrium models obey the relations,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
M_\mathrm{tot}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M^*_\mathrm{tot} \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
=
\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R^* \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr]
=
\biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
P^*_i \biggl[ K_c \rho_0^{6/5} \biggr]
=
\biggl[ K_c \rho_0^{6/5} \biggr] ~\theta_i^6
\, .
</math>
</td>
</tr>

</table>

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than <math>\rho_0</math> — is held fixed. We find that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
\rho_0^{1/5}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ R
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{1/2} G^{-1/2} \biggr] \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\sqrt{2\pi}~\theta_i^2}
~
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^{-2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{\pi}{2}\biggr) \frac{\theta_i^2}{A^2 \eta_s^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c \theta_i^6
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i}
\biggr\}^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_c
\biggl\{
\biggl[ K_c^{3/2} G^{-3/2} M_\mathrm{tot}^{-1} \biggr]^6
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
\, .
</math>
</td>
</tr>
</table>

Note as well that,

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>
P_i \biggl(\frac{4\pi}{3} R^3\biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
~\biggl\{
\biggl[ K_c^{-5/2} G^{5/2} M_\mathrm{tot}^{2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{10} G^{-9} M_\mathrm{tot}^{-6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \eta_s^3
~\biggl\{
\biggl[ K_c^{-15/2} G^{15/2} M_\mathrm{tot}^{6} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \biggl(\frac{2}{\pi}\biggr)^{3}~
\biggl(\frac{\pi}{2^3}\biggr)^{3 / 2}
~\biggl[ K_c^{5/2} G^{-3/2} \biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} \eta_s^3
</math>
</td>
</tr>
</table>

====Sequence Plots====
A plot of <math>P_i\biggl[ K_c^{-10} G^{9} M_\mathrm{tot}^{6} \biggr]</math> versus <math>R^3\biggl[ K_c^{5/2} G^{-5/2} M_\mathrm{tot}^{-2} \biggr]^{3}</math> at fixed interface pressure will be generated via the relations,
<table align="center" cellpadding="8">
<tr>
<td align="center">Ordinate</td>
<td align="center"> </td>
<td align="center">Abscissa</td>
</tr>

<tr>
<td align="center">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \biggl(\frac{2}{\pi}\biggr)^{3} A^6\eta_s^6
</math>
</td>
<td align="center">
'''vs'''
</td>
<td align="center">
<math>
\biggl\{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{3}
~
\biggl(\frac{\pi}{2^3}\biggr)^{1 / 2}
\frac{1}{A^2 \eta_s}
\biggr\}^3
</math>
</td>
</tr>
</table>

[[File:MuRatio100PressureVsVolumeA.png|350px|center|Pressure vs Volume]][[File:MuRatio100nuVqA.png|350px|center|nu vs q]]

===Hidden Text===


==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>M_r^*</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
</tr>
</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, we can rewrite the "normalized" expressions as follows:

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2} \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4}\biggr]^{-2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl\{K_c^{1/2} G^{-1/2}
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \biggl( \frac{K_e}{K_c} \biggr)^{1 / 2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
r^* \biggl[ K_e^{1/2} G^{-1/2} \biggr]~ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i \, .
</math>
</td>
</tr>
</table>

===Fixed Interface Pressure===
Start with the model relation,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>P_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] P_i^*</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[K_c \rho_0^{6/5}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
</td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{6/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{-6}_i \biggl( \frac{K_e}{K_c} \biggr)^{-3 / 2} \, .</math>
</td>
</tr>
</table>

===Fixed Total Mass===

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
</td>
</tr>
</table>

Inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0^{4/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \rho_0^{1/5}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \, .</math>
</td>
</tr>
</table>

Hence, for a given specification of the interface location, <math>\xi_i</math> — test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> — the desired expression for the central density is,

<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\rho_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ K_e^{-5} K_c^5 \biggr]^{1 / 4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i \, ;</math>
</td>
</tr>
</table>
and, drawing the expression for the normalized total mass from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]], namely,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_\mathrm{tot}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2}\rho_0^{-1/5} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggr]
\biggl\{\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1 / 2} \theta^{-1}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1 / 4} \biggr\}^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>M_r^* \biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_c^{3/2} G^{-3/2} \biggl( \frac{K_e}{K_c} \biggr)^{1 / 4} \biggr]
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{1 / 2} \theta_i
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[K_e K_c^{-5}G^{-6} \biggr]^{1 / 4}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ,
</math>
</td>
</tr>
</table>
where — again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] —
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2} \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96077)</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)\sqrt{3} ~\theta_i^2 \xi_i \, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.79941)</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(0.96225)</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i (1 + \Lambda_i^2)^{1 / 2}\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(1.10940)</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\eta_i + \frac{\pi}{2} + \tan^{-1}( \Lambda_i)\, ;</math>
</td>
<td align="center">     </td>
<td align="left">(3.13637)</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{[K_e K_c^{-5}G^{-6} ]^{1 / 4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3 / 2}
\biggl(\frac{2}{\pi}\biggr)^{1/2} A\eta_s
\, ;
</math>
</td>
<td align="center">     </td>
<td align="left">(2.77623)</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho_0}{[K_e^{-5}K_c^{5}]^{1/4}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i\, .
</math>
</td>
<td align="center">     </td>
<td align="left">(1.22153)</td>
</tr>
</table>

==Building on Earlier Eigenfunction Details==

In the heading of [[SSC/Stability/BiPolytropes/Pt3#Fig6|Figure 6 from our accompanying presentation]] of the properties of marginally unstable oscillation modes in <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,
<table border="0" align="center" cellpadding="8">
<tr>
<th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br />
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
</th>
</tr>
</table>

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram. We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>. For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>. Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>. [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
<li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
<li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
<li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - ν Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show — via a plot in the <math>(q, \nu)</math> diagram — how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
</td>
</tr>

<tr>
<td align="left" colspan="2">
'''Figure 1:''' Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
</td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2} \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>  Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
</td>
</tr>
<tr>
<td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
</td>
<td align="center">
<math>\xi_i</math>
</td>
<td align="center">
<math>\theta_i</math>
</td>
<td align="center">
<math>\eta_i</math>
</td>
<td align="center">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>A</math>
</td>
<td align="center">
<math>\eta_s</math>
</td>
<td align="center">
LHS
</td>
<td align="center">
RHS
</td>
<td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
</td>
<td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
</td>
<td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{3}</math>
</td>
<td align="center">
<math>\infty</math> </td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">0.0 </td>
<td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
<td align="center">
0.33
</td>
<td align="right">
24.00496 </td>
<td align="right">
0.0719668 </td>
<td align="right">
0.0710624 </td>
<td align="right">
0.2128753 </td>
<td align="right">
0.0726547 </td>
<td align="right">
1.8516032 </td>
<td align="right">
-223.8157 </td>
<td align="right">
-223.8159 </td>
<td align="right">
0.038378833 </td>
<td align="right">
0.52024552 </td>
</tr>

<tr>
<td align="center">
0.316943
</td>
<td align="right">
10.744571 </td>
<td align="right">
0.1591479 </td>
<td align="right">
0.1493938 </td>
<td align="right">
0.4903393 </td>
<td align="right">
0.1663869 </td>
<td align="right">
2.1760793 </td>
<td align="right">
-31.55254 </td>
<td align="right">
-31.55254 </td>
<td align="right">
0.068652714 </td>
<td align="right">
0.382383875 </td>
</tr>

<tr>
<td align="center">
0.3090
</td>
<td align="right">
8.8301772 </td>
<td align="right">
0.1924833 </td>
<td align="right">
0.1750954 </td>
<td align="right">
0.6130669 </td>
<td align="right">
0.2053811 </td>
<td align="right">
2.2958639 </td>
<td align="right">
-18.47809 </td>
<td align="right">
-18.47808 </td>
<td align="right">
0.076265588 </td>
<td align="right">
0.331475715 </td>
</tr>

<tr>
<td align="center">
<math>\frac{1}{4}</math>
</td>
<td align="right">
4.9379256 </td>
<td align="right">
0.3309933 </td>
<td align="right">
0.2342522 </td>
<td align="right">
1.4179907 </td>
<td align="right">
0.4064595 </td>
<td align="right">
2.761622 </td>
<td align="right">
-2.601255 </td>
<td align="right">
-2.601257 </td>
<td align="right">
0.084824137 </td>
<td align="right">
0.139370157 </td>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>. Note that,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
<td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
<td align="center" width="25%><math>\xi_i</math></td>
<td align="center" width="25%"><math>\nu</math></td>
<td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
<td align="center"><math>\frac{1}{4} </math></td>
<td align="center"><math>4.9379256</math></td>
<td align="center"><math>0.139370157</math></td>
<td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>  From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
</td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
<td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
<td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">1</td>
<td align="center" rowspan="1" colspan="1">1.6686460157</td>
<td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
<td align="center" rowspan="1" colspan="1">2.2792811317</td>
<td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>

=Related Discussions=
<ul>
<li>[[SSC/Stability/InstabilityOnsetOverview#Fig1|Instability Onset Overview]]</li>
<li>Analytic <math>(n_c, n_e) = (5, 1)</math>
<ul>
<li>[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_1)|Part 1]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|Part 2]]</li>
</ul>
</li>
</ul>

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