Editing SSC/Stability/BiPolytropes/RedGiantToPN (section)

=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:&nbsp; Background &amp; Objective]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt2|Part II:&nbsp; ]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt3|Part III:&nbsp; ]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/RedGiantToPN/Pt4|Part IV:&nbsp;]]
&nbsp;
  </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: &nbsp;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>.)  

&nbsp;<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 Sch&ouml;nberg-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>. 

<!--
<font color="red">'''The principal question is:'''</font>  ''Along bipolytropic sequences, are maximum-mass models (identified by the solid green circular markers in Fig. 2) associated with the onset of dynamical instabilities?''</span> For more details, look [[SSC/Stability/BiPolytropes/51Models#Structure|here]].
-->

<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>  &#8212; as well as <math>G</math> and <math>K_c</math> &#8212; constant along an equilibrium sequence, mass will scale as &hellip;
<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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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>  &#8212; as well as <math>G</math> and <math>K_c</math> &#8212; constant along an equilibrium sequence, mass will scale as &hellip;
<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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">&nbsp;</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">&nbsp;</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 &hellip;
<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">&nbsp;</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">
&nbsp;
  </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">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </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  &hellip;
<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>

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

<tr>
  <td align="right">
<math> \Rightarrow ~~~
m_3 (2-m_3)  (1 - m_3)\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[ 2(1-m_3)^2 - 1   \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\ell_i}\bigg\{
\frac{\pi}{2} + m_3 \ell_i (1+\ell_i^2)^{-1} 
+ \tan^{-1}\biggl[\frac{[ 1+ (1 - m_3)\ell_i^2]}{m_3 \ell_i} \biggr] \biggr\} 
</math>
  </td>
</tr>
END HIDE -->

</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" &hellip;
<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: &nbsp; 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" &hellip;

<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 &nbsp;<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> &#8212; as well as <math>K_c</math> and <math>G</math> &#8212; 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 &#8212; rather than <math>\rho_0</math> &#8212; 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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">&nbsp;</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===
<!--
In order to build a sequence along which <math>M_\mathrm{tot}</math> is held fixed, we must set
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>C_\mathrm{M} \equiv A\eta_s</math>
  </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]^{1 / 2} 
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr]
=~\mathrm{const.}
</math>
  </td>
</tr>
</table>
This means,

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

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

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

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

==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">; &nbsp;&nbsp;&nbsp;</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">; &nbsp;&nbsp;&nbsp;</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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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> &#8212; test values shown (in parentheses) assuming <math>\mu_e/\mu_c = 1.0</math> and <math>\xi_i = 0.5</math> &#8212; 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">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </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 &#8212; again, from our accompanying table of [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|parameter values]] &#8212;
<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">&nbsp; &nbsp; &nbsp;</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">&nbsp; &nbsp; &nbsp;</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">&nbsp; &nbsp; &nbsp;</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">&nbsp; &nbsp; &nbsp;</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">&nbsp; &nbsp; &nbsp;</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">&nbsp; &nbsp; &nbsp;</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">&nbsp; &nbsp; &nbsp;</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>