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SSC/Stability/BiPolytropes/RedGiantToPN

2026-03-30T17:46:07Z

70.180.56.157: /* The EFC98 Sequence Plot */

__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==

<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.) On each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. We have shown ''analytically'' that the oscillation frequency of the fundamental-mode of radial oscillation is precisely zero for each one of these maximum-mass models. 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.

<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.''

----

<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> — 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=

==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>

<ul>
<li>
<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>

Alternatively, [[#Sequence_Plots|as derived above]],
<table border="0" align="center" cellpadding="5">
<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>

</li>
</ul>
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>.

<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]].

===What Indicates Dynamical Instability?===

=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

2026-03-30T17:43:20Z

70.180.56.157: /* What Indicates Dynamical Instability? */

__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==

<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.) On each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass. We have shown ''analytically'' that the oscillation frequency of the fundamental-mode of radial oscillation is precisely zero for each one of these maximum-mass models. 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.

<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.''

----

<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> — 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=

==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>

<ul>
<li>
<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>

Alternatively, [[#Sequence_Plots|as derived above]],
<table border="0" align="center" cellpadding="5">
<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>

</li>
</ul>
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>.

<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>

===What Indicates Dynamical Instability?===

=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

2025-11-13T19:12:56Z

70.180.56.157: /* Fixed Total Mass */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=

==Preface==

<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>\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>

====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 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)^{-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>\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</td>
<td align="center"> </td>
<td align="center">Abscissa</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>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
</math>
</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>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 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)^{-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>

===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>

=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

2025-11-13T19:11:38Z

70.180.56.157: /* Fixed Total Mass */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=

==Preface==

<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>\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>

====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 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)^{-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>\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</td>
<td align="center"> </td>
<td align="center">Abscissa</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>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
</math>
</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>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 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)^{-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>

====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>

=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

2025-11-13T19:07:59Z

70.180.56.157: /* Fixed Total Mass */

__FORCETOC__
=Main Sequence to Red Giant to Planetary Nebula=

==Preface==

<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>\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>

====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 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)^{-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>\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</td>
<td align="center"> </td>
<td align="center">Abscissa</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>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta_i^{-5}
</math>
</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>



==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>

=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 }}

ParabolicDensity/Axisymmetric/Structure

2024-08-30T23:26:46Z

70.180.56.157: /* 6th Try */

__FORCETOC__

=Parabolic Density Distribution=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/GravPot|Part I:   Gravitational Potential]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Spheres/Structure|Part II:   Spherical Structures]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Axisymmetric/Structure|Part III:   Axisymmetric Equilibrium Structures]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[ParabolicDensity/Triaxial/Structure|Part IV:   Triaxial Equilibrium Structures (Exploration)]]
 
</td>
</tr>
</table>

==Axisymmetric (Oblate) Equilibrium Structures==

===Setup===

Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

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

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>\rho_c \biggl[ 1 - \biggl( \frac{x^2 + y^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \, ,</math>
</td>
</tr>
</table>
that is, axisymmetric (<math> a_m = a_\ell</math>, i.e., oblate) configurations with ''parabolic density distributions''. Much of our presentation, here, is drawn from our separate, detailed description of what we will refer to as [[ThreeDimensionalConfigurations/FerrersPotential|Ferrers potential]].

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
This can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and,

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>
Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that,

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

<tr>
<td align="right">
<math>\frac{H}{H_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>h(\xi_1) \, .</math>
</td>
</tr>
</table>
If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that,

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

<tr>
<td align="right">
<math>h(\xi_1)</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_2 \xi_1^2 + h_4 \xi_1^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>
</table>

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

===Gravitational Potential===
As we have detailed in [[ThreeDimensionalConfigurations/FerrersPotential|an accompanying discussion]], for an oblate-spheroidal configuration — that is, when <math>a_s < a_m = a_\ell</math> — the gravitational potential may be obtained from the expression,

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

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
+ \frac{1}{6} \biggl(3A_{11}x^4 + 3A_{22}y^4 + 3A_{33}z^4 \biggr)
\, ,
</math>
</td>
</tr>
</table>

where, in the present context, we can rewrite this expression as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell (x^2 + y^2) + A_s z^2 \biggr]
+ \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} x^2z^2 + A_{\ell s} y^2z^2\biggr]
+ \frac{1}{6} \biggl[3A_{\ell \ell} x^4 + 3A_{\ell \ell}y^4 + 3A_{ss}z^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} \varpi^2 z^2 \biggr]
+ \frac{1}{2} \biggl[A_{\ell \ell} (x^4 + y^4) + A_{ss}z^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \frac{A_{\ell \ell}}{2} \biggl[(x^2 + y^2)^2\biggr]
+ \frac{1}{2} \biggl[ A_{ss}z^4 \biggr]
+ \biggl[ A_{\ell s} \varpi^2 z^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \frac{A_{\ell \ell}}{2} \biggl[\varpi^4\biggr]
+ \frac{1}{2} \biggl[ A_{ss}z^4 \biggr]
+ \biggl[ A_{\ell s} \varpi^2 z^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

====Index Symbol Expressions====
The expression for the zeroth-order normalization term <math>(I_{BT})</math>, and the relevant pair of 1<sup>st</sup>-order index symbol expressions are:

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

<tr>
<td align="right"><math>I_\mathrm{BT}</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
A_\ell
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>A_s</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ,
</math>
</td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (36)</font><br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], <font color="#00CC00">§4.5, Eqs. (48) & (49)</font>
</div>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, .
</math>
</div>

The relevant [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Index_Symbols_of_the_2nd_Order|2<sup>nd</sup>-order index symbol]] expressions are:

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

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{3}{2} a_\ell^2 A_{ss} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2)
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
We can crosscheck this last expression by [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|drawing on a shortcut expression]],
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>A_{\ell s}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{A_\ell - A_s}{(a_\ell^2 - a_s^2)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ a_\ell^2 A_{\ell s}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^2}\biggl\{
A_s - A_\ell
\biggr\}
</math>
</td>
</tr>

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

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{
\biggl[ 2 - 2(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr]
-
\biggl[ (1-e^2)^{1/2} \frac{\sin^{-1}e}{e} - (1-e^2) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\}
\, .
</math>
</td>
</tr>
</table>

====Meridional Plane Equi-Potential Contours====
Here, we follow closely our separate discussion of equipotential surfaces for [[Apps/MaclaurinSpheroids#norotation|Maclaurin Spheroids, assuming no rotation]].

=====Configuration Surface=====
In the meridional <math>(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration — identified by the thick, solid-black curve below, in Figure 1 — is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho}{\rho_c} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>1 - \biggl[\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2} \biggr] = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>1 </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ z^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>a_s^2\biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr] = a_\ell^2 (1-e^2) \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{z}{a_\ell}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\pm ~(1-e^2)^{1 / 2} \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]^{1 / 2} \, ,</math>
</td>
<td align="right">        for <math>~0 \le \frac{| \varpi |}{a_\ell} \le 1 \, .</math></td>
</tr>
</table>

=====Expression for Gravitational Potential=====
Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{eff}</math> = constant, equipotential surface is defined by the expression,


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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} \equiv \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(\pi G\rho_c a_\ell^2)} + \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
- \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

Letting,
<div align="center"><math>\zeta \equiv \frac{z^2}{a_\ell^2}</math>,</div>
we can rewrite this expression for <math>\phi_\mathrm{choice}</math> as,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \zeta
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
- \frac{1}{2} A_{ss} a_\ell^2 \zeta^2
- A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\zeta
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \zeta^2
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\zeta
+
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
\, .
</math>
</td>
</tr>
</table>

=====Potential at the Pole=====
At the pole, <math>(\varpi, z) = (0, a_s)</math>. Hence,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{mid} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \cancelto{0}{\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)}\biggr]\biggl(\frac{a_s^2}{a_\ell^2}\biggr)
+
A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)}
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_s \biggl(\frac{a_s^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2 \, .
</math>
</td>
</tr>
</table>

=====General Determination of Vertical Coordinate (ζ)=====
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Given values of the three parameters, <math>e</math>, <math>\varpi</math>, and <math>\phi_\mathrm{choice}</math>, this last expression can be viewed as a quadratic equation for <math>\zeta</math>. Specifically,

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

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\alpha \zeta^2 + \beta\zeta + \gamma \, ,
</math>
</td>
</tr>
</table>
where,

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

<tr>
<td align="right">
<math>\alpha</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
\frac{1}{2} A_{ss} a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{3}\biggl\{
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\beta</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr) - A_s
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\}
\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)
-
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\gamma</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{8e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1 / 2} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
\, .
</math>
</td>
</tr>
</table>
The solution of this quadratic equation gives,

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

<tr>
<td align="right">
<math>\zeta</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta \pm \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
</td>
</tr>
</table>

Should we adopt the ''superior'' (positive) sign, or is it more physically reasonable to adopt the ''inferior'' (negative) sign? As it turns out, <math>\beta</math> is intrinsically negative, so the quantity, <math>-\beta</math>, is positive. Furthermore, when <math>\gamma</math> goes to zero, we need <math>\zeta</math> to go to zero as well. This will only happen if we adopt the ''inferior'' (negative) sign. Hence, the physically sensible root of this quadratic relation is given by the expression,

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

<tr>
<td align="right">
<math>\zeta</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta - \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
</td>
</tr>
</table>



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

Here we present a quantitatively accurate depiction of the shape of the (Ferrers) gravitational potential that arises from oblate-spheroidal configurations having a parabolic density distribution. We closely follow the discussion of [[Apps/MaclaurinSpheroids#Example_Equi-gravitational-potential_Contours|equi-gravitational potential contours that arise in (uniform-density) Maclaurin spheroids]]. In order to facilitate comparison with Maclaurin spheroids, we will focus on a model with …
<table border="0" align="center" width="80%">
<tr>
<td align="center"><math>\frac{a_s}{a_\ell} = 0.582724 \, ,</math></td>
<td align="center"><math>e = 0.81267 \, ,</math></td>
<td align="center"> </td>
</tr>
<tr>
<td align="center"><math>A_\ell = A_m = 0.51589042 \, ,</math></td>
<td align="center"><math>A_s = 0.96821916 \, ,</math></td>
<td align="center"><math>I_\mathrm{BT} = 1.360556 \, ,</math></td>
</tr>
<tr>
<td align="center"><math>a_\ell^2 A_{\ell \ell} = 0.3287756 \, ,</math></td>
<td align="center"><math>a_\ell^2 A_{s s} = 1.5066848 \, ,</math></td>
<td align="center"><math>a_\ell^2 A_{\ell s} = 0.6848975 \, .</math></td>
</tr>
</table>

[<font color="red">NOTE:</font>   Along the Maclaurin spheroid sequence, this is the eccentricity that marks bifurcation to the Jacobi ellipsoid sequence — see the first model listed in Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and/or see Tables 1 and 2 of [[ThreeDimensionalConfigurations/JacobiEllipsoids|our discussion of the Jacobi ellipsoid sequence]]. It is unlikely that this same eccentricity has a comparably special physical relevance along the sequence of spheroids having parabolic density distributions.]

The largest value of the gravitational potential that will arise inside (actually, on the surface) of the configuration is at <math>(\varpi, z) = (1, 0)</math>. That is, when,


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

<tr>
<td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{max} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_\ell
- \frac{1}{2} A_{\ell \ell} a_\ell^2 = 0.3515026 \, .
</math>
</td>
</tr>
</table>
So we will plot various equipotential surfaces having, <math>0 < \phi_\mathrm{choice} < \phi_\mathrm{choice}|_\mathrm{max} </math>, recognizing that they will each cut through the equatorial plane <math>(z = 0)</math> at the radial coordinate given by,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \cancelto{0}{\zeta^2}
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\cancelto{0}{\zeta}
+
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} A_{\ell \ell} a_\ell^2 \chi^2
- A_\ell \chi
+ \phi_\mathrm{choice} \, ,
</math>
</td>
</tr>
</table>
where,
<div align="center"><math>\chi \equiv \frac{\varpi^2}{a_\ell^2} \, .</math></div>
The solution to this quadratic equation gives,

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

<tr>
<td align="right">
<math>\chi_\mathrm{eqplane} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{A_{\ell \ell} a_\ell^2}\biggl\{
A_\ell \pm \biggl[A_\ell^2 - 2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}\biggr]^{1 / 2}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{A_\ell}{A_{\ell \ell} a_\ell^2}\biggl\{
1 - \biggl[1 - \frac{2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}}{A_\ell^2}\biggr]^{1 / 2}
\biggr\}
\, .
</math>
</td>
</tr>
</table>
Note that, again, the physically relevant root is obtained by adopting the ''inferior'' (negative) sign, as has been done in this last expression.

=====Equipotential Contours that Lie Entirely Within Configuration=====
For all <math>0 < \phi_\mathrm{choice} \le \phi_\mathrm{choice} |_\mathrm{mid}</math>, the equipotential contour will reside entirely within the configuration. In this case, for a given <math>\phi_\mathrm{choice}</math>, we can plot points along the contour by picking (equally spaced?) values of <math>\chi_\mathrm{eqplane} \ge \chi \ge 0</math>, then solve the above quadratic equation for the corresponding value of <math>\zeta</math>.

In our example configuration, this means … (to be finished)

===Hydrostatic Balance (Algebraic Condition)===
Following our [[Apps/MaclaurinSpheroids#Equilibrium_Structure|separate discussion of the equilibrium structure]] of Maclaurin spheroids, and given that our solution of the Poisson equation fixes the expression for <math>\Phi_\mathrm{grav} </math>, the algebraic expression ensuring hydrostatic balance is,

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

<tr>
<td align="right">
<math>H(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B - \biggl[ \Phi_\mathrm{grav}(\varpi, z) + \Psi(\varpi, z) \biggr] \, ,
</math>
</td>
</tr>
</table>
where, <math>\Psi</math> is the centrifugal potential. <font color="red">NOTE:</font>   Generally when modeling axisymmetric astrophysical systems (see our [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying discussion of ''simple'' rotation profiles]]) it is assumed that <math>\Psi</math> does not functionally depend on <math>z</math>. Here, our other constraints — for example, demanding that the configuration have a parabolic density distribution — may force us to adopt a <math>z</math>-dependent rotation profile.

Here, we know that the adopted parabolic density distribution gives rise to a gravitational potential of the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

Hence,

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

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B - \Phi_\mathrm{grav}(\varpi, z) - H(\varpi, z)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
-
H_c h(\xi_1) \, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
We presume that the enthalpy profile, as well as the density profile, can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and,

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>
Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that,

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

<tr>
<td align="right">
<math>\frac{H}{H_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>h(\xi_1) \, .</math>
</td>
</tr>
</table>
If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that,

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

<tr>
<td align="right">
<math>h(\xi_1)</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_2 \xi_1^2 + h_4 \xi_1^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>
</table>

</td></tr></table>
Adopting this last expression for the enthalpy, we have,

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

<tr>
<td align="right">
<math>\frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
-
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

At the pole of the configuration — that is, when <math>(\varpi, z) = (0, a_s)</math> — this statement of hydrostatic balance becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)} + A_s \biggl( \frac{a_s^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)}
+ A_{ss} a_\ell^2 \biggl(\frac{a_s^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^2} + \biggl(\frac{a_s}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^4} + \biggl(\frac{a_s}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
-
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
\, .
</math>
</td>
</tr>
</table>

For centrally condensed configurations, it is astrophysically reasonable to assume that <math>\Psi(\varpi, z)</math> is of the form such that the centrifugal potential goes to zero when <math>\varpi \rightarrow 0</math>. Adopting that assumption here means that the Bernoulli constant has the value,

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

<tr>
<td align="right">
<math>C_B</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<td align="left">
<math>
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
\, .
</math>
</td>
</tr>
</table>
Plugging this expression for <math>C_B</math> back into the general statement of hydrostatic balance gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\biggl[A_s (1-e^2)-A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1} - 1\biggr]
+
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} -1
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2(1-e^2)^{-1}\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr]
+
h_4 a_s^4 (1-e^2)^{-2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2}
\biggr]
\biggr\}
</math>
</td>
</tr>
</table>

Let's set …
<div align="center">
<math>H_c h_0 = \pi G \rho_c a_\ell^2 \, ;</math>      
<math>h_2 = \frac{A_s(1-e^2)}{a_s^2} \, ;</math>      
<math>h_4 = - \frac{ A_{ss}a_\ell^2 (1-e^2)^2 }{ 2a_s^4 } \, .</math>
</div>
This gives,

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

<tr>
<td align="right">
<math>\frac{\Psi(\varpi, z)}{\pi G \rho_c a_\ell^2}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) \biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
- A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+ \biggl[
\frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+
\frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
\biggr]
+ A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
-
A_{ss} a_\ell^2 \biggl[ \biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+ \frac{1}{2}\biggl\{
A_{\ell \ell} a_\ell^2 - A_{ss} a_\ell^2 (1-e^2)^{2}
\biggr\} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl\{ A_{\ell s}a_\ell^2
-
A_{ss} a_\ell^2 (1-e^2) \biggr\}\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\, .
</math>
</td>
</tr>
</table>

===2<sup>nd</sup> Try===

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">Keep in Mind, from Above</div>

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

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\rho_c \biggl[ 1 - \biggl(\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\, ,
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>

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

From our presentation of [[AxisymmetricConfigurations/PGE#Eulerian_Formulation_(CYL.)|the Eulerian formulation of the Euler equation in cylindrical coordinates]], we see that in steady-state axisymmetric flows, the two relevant equilibrium conditions are,

<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>~{\hat{e}}_\varpi</math>:    </td>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}
</math>
</td>
</tr>
<tr>
<td align="right"><math>~{\hat{e}}_z</math>:    </td>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math>
</td>
</tr>
</table>

====Vertical Component====
We will focus, first, on the vertical component. Specifically, since both <math>\rho</math> and <math>\Phi_\mathrm{grav}</math> are known, the vertical gradient of the (unknown) scalar pressure is
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{\partial P}{\partial z}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \rho ~ \frac{\partial}{\partial z} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>
</table>

Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell)} \cdot \frac{\partial P}{\partial z}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial z} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \chi^4
+ A_{ss} a_\ell^2 \zeta^4
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl\{
- 2A_s \zeta
+ \biggl[
2A_{ss} a_\ell^2 \zeta^3
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta
+ A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta
+ A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>
</table>

where (unlike above) we are using the dimensionless lengths, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. Continuing to streamline this function, we have,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \chi^2\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \zeta^2\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \biggl[
A_{\ell s}a_\ell^2 \chi^4 \zeta - A_s \chi^2 \zeta + A_{ss} a_\ell^2 \chi^2 \zeta^3
\biggr]
- \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_s \zeta^3 + A_{ss} a_\ell^2 \zeta^5
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2 - A_s \biggr]\zeta
+ A_{ss} a_\ell^2 \zeta^3
+ \biggl[
A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
- A_{ss} a_\ell^2 \chi^2 \zeta^3
+ \biggl[
A_s \zeta^3 - A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_{ss} a_\ell^2 \zeta^5
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2 - A_s
+
A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
+
\biggl[ A_{ss} a_\ell^2 - A_{ss} a_\ell^2 \chi^2 + A_s(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} \biggr]\zeta^3
- A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
+
\biggl\{
[A_s(1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\chi^2
\biggr\}\zeta^3
- A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5
\, .
</math>
</td>
</tr>
</table>
So, let's see what happens if we assume that the pressure has the form,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>p_0 + p_2 \zeta^2 + p_4\zeta^4 + p_6\zeta^6
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P_\mathrm{vert}}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>2p_2 \zeta + 4p_4\zeta^3 + 6p_6\zeta^5 \, ,</math>
</td>
</tr>
</table>
in which case,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
p_0 + \frac{1}{2}\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4
\biggr] \zeta^2
+
\frac{1}{4}\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4
+
\frac{1}{6}\biggl[
- A_{ss} a_\ell^2 (1-e^2)^{-1}
\biggr]\zeta^6
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

<font color="red">REMINDER:</font>
From [[#2nd_Try|above]] …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\, .
</math>
</td>
</tr>
</table>

And, in the case of the spherically symmetric equilibrium configuration, the [[SSC/Structure/OtherAnalyticModels#Pressure|pressure distribution]] derived by {{ Prasad49 }} has the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P}{P_c}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[1 + \biggl(\frac{\rho}{\rho_c}\biggr)\biggr]
\, .
</math>
</td>
</tr>
</table>
In the context of rotationally flattened configurations, therefore, we might expect the (vertical) pressure distribution to be of the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P}{P_c}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl\{
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
-\chi^2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
-
\zeta^2(1-e^2)^{-1}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl\{
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
+ \biggl[-\chi^2 + \chi^4 + \chi^2\zeta^2(1-e^2)^{-1}\biggr]
+
\biggl[- \zeta^2(1-e^2)^{-1} + \chi^2\zeta^2(1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr]
\biggr\}
</math>
</td>
</tr>
</table>

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

====Radial Component====

Start with,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>- \frac{j^2 \rho}{\varpi^3}
+
\frac{\partial P}{\partial \varpi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \rho ~ \frac{\partial}{\partial \varpi} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>
</table>

Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>- \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr] \frac{j^2 \rho}{\varpi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\frac{\partial P}{\partial \varpi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\rho ~ \frac{\partial}{\partial \varpi} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \chi} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \chi} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \chi^4
+ A_{ss} a_\ell^2 \zeta^4
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\} \, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">EXACT!</font></div>

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\}
</math>
</td>
</tr>
</table>

</td></tr></table>
Continuing to streamline this function, we have,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}
-
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}\chi^2
-
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}(1-e^2)^{-1}\zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3
-
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi^3 - 2 A_{\ell \ell} a_\ell^2 \chi^5
+
\biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2 - 2 A_{\ell \ell} a_\ell^2 \chi^3 (1-e^2)^{-1}\zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+\biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2
+ \biggl[2 A_{\ell \ell} a_\ell^2
-2A_{\ell s}a_\ell^2 \zeta^2 + 2A_\ell - 2 A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
- 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2
+ A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
+ 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell
-A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
- 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
\, .
</math>
</td>
</tr>
</table>

====Determine Specific Angular Momentum Distribution====

Now, from our analysis of the vertical component, we determined that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{12P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
12p_0 + 6\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4
\biggr] \zeta^2
+
3\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4
+
2\biggl[
- A_{ss} a_\ell^2 (1-e^2)^{-1}
\biggr]\zeta^6
\, .
</math>
</td>
</tr>
</table>

<span id="RadialDerivative">The radial derivative of this function is</span>,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[ \frac{12}{(2\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
6\biggl[
2(A_{\ell s}a_\ell^2 + A_s ) \zeta^2 \chi - 4A_{\ell s}a_\ell^2\zeta^2 \chi^3
\biggr]
+
6\biggl\{ - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ] \zeta^4 \chi \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3 \, .
</math>
</td>
</tr>
</table>

We hypothesize that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi}
-
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\biggl[\frac{\partial P}{\partial \chi}\biggr]_\mathrm{rad}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
- 2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2
+ A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
- 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell
-A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
+ 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>2\biggl\{
\biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4 \biggr] \chi
- 2A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl[ A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2
- A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
+ \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell
+ A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(2\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4
+ A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2
- A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell
+ A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2- 2A_{\ell s}a_\ell^2 \zeta^2\biggr] \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ A_\ell(1-e^2)^{-1} +(A_{\ell s}a_\ell^2 + A_s - A_{\ell s}a_\ell^2 - A_\ell)\zeta^2
+ \frac{1}{2}\biggl[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}
+ 2A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell)
+ (A_{\ell s}a_\ell^2 + A_{\ell \ell} a_\ell^2 - 2A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ A_\ell(1-e^2)^{-1} + (A_s - A_\ell)\zeta^2
+ \frac{1}{2}\biggl[- A_{ss} a_\ell^2
+ A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi
+ \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell)
+ (A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>
</table>
Now, from [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|our layout of relevant index symbol expressions]], let's see if the coefficients of various ζ-dependent terms go to zero.

<font color="red">FIRST:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
A_{s\ell}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{A_s - A_\ell}{(a_s^2 - a_\ell^2)} = \frac{A_s - A_\ell}{a_\ell^2 e^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~
A_{s \ell}a_\ell^2 e^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(A_s - A_\ell)
</math>
</td>
</tr>

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

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

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^2}\biggl[
3 - e^2 - 3(1-e^2)^{1 / 2}\frac{\sin^{-1}e}{e}
\biggr]
\, ;
</math>
</td>
</tr>
</table>

<font color="red">SECOND:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
3A_{s s}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_s^2} - 2A_{s \ell}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{3}{2}A_{s s}a_\ell^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{a_\ell^2}{a_s^2} - A_{s \ell}a_\ell^2 = (1 - e^2)^{-1} - A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ - A_{s s}a_\ell^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \biggl[ - A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1}
+ A_{\ell s}a_\ell^2 (1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\frac{1}{3(1-e^2)}\biggl[
2A_{s\ell}a_\ell^2 (1-e^2) - 2
+ 3A_{\ell s}a_\ell^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{3(1-e^2)}\biggl[A_{s\ell}a_\ell^2 (5-2e^2) - 2 \biggr]\, ;
</math>
</td>
</tr>
</table>

<font color="red">THIRD:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
3A_{\ell \ell}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_\ell^2} - A_{\ell \ell} - A_{s\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ 4A_{\ell \ell}a_\ell^2</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2 - A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
(A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} - \frac{5}{4}A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{4}\biggl[2 - 5A_{s\ell}a_\ell^2\biggr] \, .
</math>
</td>
</tr>
</table>

===3<sup>rd</sup> Try===
From the [[#Radial_Component|above, "2<sup>nd</sup> Try" discussion of the radial component]], we can write the following "EXACT!" relation,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">EXACT!</font></div>

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\} \, .
</math>
</td>
</tr>
</table>

</td></tr></table>
Now, our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math> suggests that the left-hand-side of this expression should be of the form,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
LHS  
<math>
\equiv \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>\sim</math></td>
<td align="left">
<math>
c_2\zeta^2 + c_4\zeta^4 \, ,
</math>
</td>
</tr>
</table>
where it is understood that the coefficients, <math>c_2</math> and <math>c_4</math>, are both functions of <math>\chi</math>. This should be compared with the "EXACT!" expression for the RHS after multiplying through by the expression for the dimensionless density, that is,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
RHS
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\biggr] \cdot \biggl\{
\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi \zeta^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(1 - \chi^2)\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
- \biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr](1-e^2)^{-1}\zeta^2
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4
\, .
</math>
</td>
</tr>
</table>

Because we are not expecting to see a term that is independent of <math>\zeta</math>, this suggests that the term inside the large square brackets must be zero. This leads to an expression for the distribution of specific angular momentum of the form,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<table border="0" align="center" cellpadding="8">

<tr><td align="center" colspan="3"><font color="red">EXCELLENT !!</font></td></tr>

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{j^2 }{(\pi G \rho_c a_\ell^4)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6
\, .
</math>
</td>
</tr>
</table>

According to our [[AxisymmetricConfigurations/SolutionStrategies#Specifying_Radial_Rotation_Profile_in_the_Equilibrium_Configuration|accompanying discussion of ''Simple'' rotation profiles]], the corresponding centrifugal potential is given by the expression,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>\Psi</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \int \frac{j^2(\varpi)}{\varpi^3} d\varpi
=
- (\pi G \rho_c a_\ell^2) \int \frac{1}{\chi^3} \biggl[2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6\biggr]d\chi
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \int \biggl[2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr]d\chi
=
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, .
</math>
</td>
</tr>
</table>
(Here, we ignore the integration constant because it will be folded in with the Bernoulli constant.)

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

It also means that the RHS expression simplifies to the form,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
RHS
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4 \, .
</math>
</td>
</tr>
</table>

This should be compared to our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math>, namely,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(2A_{\ell s}a_\ell^2 + 2A_s )\chi\zeta^2- 4A_{\ell s}a_\ell^2 \chi^3\zeta^2 - \biggl[A_{\ell s}a_\ell^2 (1-e^2)^{-1} + A_{ss} a_\ell^2\biggr]\chi\zeta^4

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

===4<sup>th</sup> Try===

In our [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|accompanying discussion of Ferrers Potential]], we have derived the expression for the gravitational potential inside (and on the surface of) a triaxial ellipsoid with a parabolic density distribution. Specifically, for
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\rho(\mathbf{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr]
\, ,</math>
</td>
</tr>
</table>
[[ThreeDimensionalConfigurations/FerrersPotential#GravFor1|we find]],
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2} \biggl(A_{11}x^4 + A_{22}y^4 + A_{33}z^4 \biggr)
\, .
</math>
</td>
</tr>
</table>
In this [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|same accompanying discussion]], we plugged this expression for the gravitational potential into the Poisson equation and demonstrated that it properly generates the expression for the parabolic density distribution. For the axisymmetric configuration being considered here — with the short axis aligned with <math>c = a_3 = a_s</math> — these two relations become,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr]
=
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \frac{\varpi^2}{a_\ell^2} - A_s \frac{z^2}{a_\ell^2}
+ (A_{\ell s}a_\ell^2 )\frac{ \varpi^2z^2 }{a_\ell^4} + \frac{1}{2}(A_{s s} a_\ell^2) \frac{z^4}{a_\ell^4}
+ \frac{A_{\ell \ell}a_\ell^2}{2} \biggl[ \frac{(x^4 + 2 x^2y^2 + y^4 )}{a_\ell^4} \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, .
</math>
</td>
</tr>
</table>
where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. (This matches the [[#Gravitational_Potential|expression derived above]].)

----

Discuss scalar relationship between the enthalpy <math>(H)</math> and the effective potential.

As has been detailed in [[AxisymmetricConfigurations/SolutionStrategies#Technique|an accompanying discussion of solution techniques]], a configuration will be in dynamic equilibrium if,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla\biggl[ H + \Phi_\mathrm{grav} + \Psi \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
0
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ H + \Phi_\mathrm{grav} + \Psi
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
constant
<math>
= C_B
</math>
</td>
</tr>
</table>

Given that, in our particular case, we have analytic expressions for <math>\Phi_\mathrm{grav}(\chi,\zeta)</math> and for <math>\Psi(\chi,\zeta)</math>, we deduce that, to within a constant, the enthalpy distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{\Phi_\mathrm{grav}}{{(\pi G\rho_c a_\ell^2)}} - \frac{\Psi}{{(\pi G\rho_c a_\ell^2)}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
-
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(A_{\ell s}a_\ell^2 )\chi^2 \biggr]
</math>
</td>
</tr>
</table>
Now, according to our [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|related discussion of index symbols]],

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>3A_{s s}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_s^2} - 2A_{\ell s}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ 3A_{s s}a_\ell^2</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2(1-e^2)^{-1} - 2A_{\ell s}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~2(A_{\ell s}a_\ell^2)\chi^2 </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Examining the radial derivative …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \chi}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \chi} \biggl\{
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[-3(A_{s s} a_\ell^2) + 2(1-e^2)^{-1} \biggr]\zeta^2\chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2(A_{\ell s} a_\ell^2)\zeta^2\chi
\, .
</math>
</td>
</tr>
</table>
<font color="red">YES !!!</font> This matches the "radial" pressure-gradient, below.

Now, examining the vertical derivative …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \zeta}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \zeta} \biggl\{
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \zeta} \biggl\{
- A_s \zeta^2
+ \frac{1}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^4 + [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_s \zeta
+
\biggl[2(A_{s s} a_\ell^2) \zeta^3
+ [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_s \zeta
+
\biggl[2(A_{s s} a_\ell^2) \zeta^3
+ 2(A_{\ell s} a_\ell^2) \chi^2\zeta \biggr]
</math>
</td>
</tr>
</table>
<font color="red">HURRAY !!!</font> This matches the "vertical" pressure-gradient, below.

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

----

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\}
</math>
</td>
</tr>
</table>
Plug in …

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3
\, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
2A_{\ell s}a_\ell^2 \zeta^2 \chi
\biggr\}
</math>
</td>
</tr>
</table>



Hence, examination of the radial component leads to the following suggested expression for the pressure:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
- \chi^2
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
- \zeta^2(1-e^2)^{-1}
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{P}{(\pi G \rho_c^2 a_\ell^2)}
</math>
</td>
<td align="center"><math>\sim</math></td>
<td align="left">
<math>
\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
- \frac{1}{2}\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^4\biggr]
- \zeta^2(1-e^2)^{-1} \biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 1 - \frac{\chi^2}{2}
- \zeta^2(1-e^2)^{-1} \biggr]
\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
\, .
</math>
</td>
</tr>
</table>

While examination of the vertical component leads to the following suggested expression for the pressure:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \frac{\chi^2}{2} - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
- \frac{\chi^2}{2}\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
</math>
</td>
</tr>
</table>

===Tentative Summary===

====Known Relations====

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

<tr>
<td align="left"><font color="orange"><b>Density:</b></font></td>
<td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Specific Angular Momentum:</b></font></td>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Centrifugal Potential:</b></font></td>
<td align="right">
<math>
\frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Enthalpy:</b></font></td>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Radial Pressure Gradient:</b></font></td>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
2A_{\ell s}a_\ell^2 \zeta^2 \chi
\biggr\}
</math>
</td>
</tr>
</table>

where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are:

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

<tr>
<td align="right"><math>I_\mathrm{BT}</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
A_\ell
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>A_s</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{3}{2} a_\ell^2 A_{ss} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2)
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\} \, ,
</math>
</td>
</tr>
</table>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, .
</math>
</div>

====Examine Behavior of Enthalpy====
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_s}\biggr)^2 \biggr]^{1 / 2}
=
a_s\biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>

====Try to Construct Pressure Distribution====

Drawing from the expression for the vertical pressure gradient, namely,

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

<tr>
<td align="right">
<math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]\biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
- \chi^2
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
- \zeta^2(1-e^2)^{-1}
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
+
\biggl[-2A_{\ell s}a_\ell^2 \chi^4\zeta + 2A_s \chi^2\zeta - 2A_{ss} a_\ell^2\chi^2 \zeta^3 \biggr]
+
\biggl[-2A_{\ell s}a_\ell^2 \chi^2\zeta^3(1-e^2)^{-1} + 2A_s \zeta^3(1-e^2)^{-1} - 2A_{ss} a_\ell^2 \zeta^5(1-e^2)^{-1} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \chi^2 - 2A_s -2A_{\ell s}a_\ell^2 \chi^4 + 2A_s \chi^2 \biggr]\zeta
+
\biggl[ - 2A_{ss} a_\ell^2\chi^2 + 2A_{ss} a_\ell^2 -2A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} + 2A_s (1-e^2)^{-1} \biggr]\zeta^3
+
\biggl[ - 2A_{ss} a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^5
\, .
</math>
</td>
</tr>
</table>
try the following pressure expression:

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

<tr>
<td align="right">
<math>\frac{P}{(\pi G\rho_c^2 a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
f_0
+ f_2 \biggl(\frac{\xi_1}{a_s} \biggr)^2
+ f_4 \biggl(\frac{\xi_1}{a_s} \biggr)^4
+ f_6 \biggl(\frac{\xi_1}{a_s} \biggr)^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
f_0
+ f_2 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]
+ f_4 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^2
+ f_6 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
f_0
+ f_2 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]
+ f_4 \biggl[\chi^4 + 2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ f_6 \biggl[\chi^4 + 2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr]
\biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
f_0
+ f_2 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]
+ f_4 \biggl[\chi^4 + 2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ f_6
\biggl[\chi^6 + 3\chi^4\zeta^2 (1-e^2)^{-1} + 3\chi^2\zeta^4(1-e^2)^{-2}
+
\zeta^6(1-e^2)^{-3} \biggr]
\, .
</math>
</td>
</tr>
</table>
The vertical derivative of this expression is,

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

<tr>
<td align="right">
<math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)}\biggr] \frac{\partial P}{\partial \zeta} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial }{\partial \zeta}\biggl\{
f_2 \biggl[\zeta^2 (1-e^2)^{-1}\biggr]
+ f_4 \biggl[2\chi^2\zeta^2 (1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr]
+ f_6
\biggl[3\chi^4\zeta^2 (1-e^2)^{-1} + 3\chi^2\zeta^4(1-e^2)^{-2}
+
\zeta^6(1-e^2)^{-3} \biggr]
\biggr\}

</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
f_2 \biggl[2\zeta (1-e^2)^{-1}\biggr]
+ f_4 \biggl[4\chi^2\zeta (1-e^2)^{-1} + 4\zeta^3(1-e^2)^{-2}\biggr]
+ f_6
\biggl[6\chi^4\zeta (1-e^2)^{-1} + 12\chi^2\zeta^3(1-e^2)^{-2}
+
6\zeta^5(1-e^2)^{-3} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\biggl[2f_2 (1-e^2)^{-1} + 4f_4\chi^2 (1-e^2)^{-1} + 6f_6\chi^4 (1-e^2)^{-1} \biggr]\zeta
+ \biggl[ 4f_4 (1-e^2)^{-2} + 12f_6\chi^2(1-e^2)^{-2}\biggr]\zeta^3
+ \biggl[6f_6 (1-e^2)^{-3} \biggr]\zeta^5
\biggr\} \, .
</math>
</td>
</tr>
</table>
Matching <math>\zeta^5</math> terms gives,

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

<tr>
<td align="right">
<math>6f_6 (1-e^2)^{-3} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_{ss} a_\ell^2 (1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ f_6 </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{1}{3}A_{ss} a_\ell^2 (1-e^2)^{2}
\, .
</math>
</td>
</tr>
</table>
Matching <math>\zeta^3</math> terms gives,

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

<tr>
<td align="right">
<math>4f_4 (1-e^2)^{-2} + 12f_6\chi^2(1-e^2)^{-2} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_{ss} a_\ell^2\chi^2 + 2A_{ss} a_\ell^2 -2A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} + 2A_s (1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 4f_4 (1-e^2)^{-2} + 12 \biggl[- \frac{1}{3}A_{ss} a_\ell^2 (1-e^2)^{2} \biggr] \chi^2(1-e^2)^{-2} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
[2A_{ss} a_\ell^2 + 2A_s (1-e^2)^{-1}] - 2A_{ss} a_\ell^2\chi^2 -2A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 4f_4 (1-e^2)^{-2} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
[2A_{ss} a_\ell^2 + 2A_s (1-e^2)^{-1}]
+ \biggl[2A_{ss} a_\ell^2 -2A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr] \chi^2 \, .</math>
</td>
</tr>
</table>

Matching <math>\zeta^1</math> terms gives,

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

<tr>
<td align="right">
<math>2f_2 (1-e^2)^{-1} + 4f_4\chi^2 (1-e^2)^{-1} + 6f_6\chi^4 (1-e^2)^{-1} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2A_{\ell s}a_\ell^2 \chi^2 - 2A_s -2A_{\ell s}a_\ell^2 \chi^4 + 2A_s \chi^2
</math>
</td>
</tr>
</table>

===5<sup>th</sup> Try===

We should leave untouched the ''form'' of the expression for the centrifugal potential, but let its coefficient values remain unspecified. The enthalpy function will therefore remain flexible, and, in tern, so will the components of the pressure gradient. We should adjust these new coefficients in such a way that the gradient of the pressure is everywhere perpendicular to the surface of a constant-density contour; this means that the P-constant contours will be identical to the density-constant contours.

====Modifiable Relations====

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

<tr>
<td align="left"><font color="orange"><b>Density:</b></font></td>
<td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="purple"><b>Specific Angular Momentum:</b></font></td>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2j_1 \chi - 2 j_3 \chi^3
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="purple"><b>Centrifugal Potential:</b></font></td>
<td align="right">
<math>
\frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2}\biggl[j_3 \chi^4 -2j_1 \chi^2 \biggr]\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="purple"><b>Enthalpy:</b></font></td>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
- \frac{1}{2}\biggl[j_3 \chi^4 -2j_1 \chi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot
\biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="left"><font color="purple"><b>Radial Pressure Gradient:</b></font></td>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot
\biggl\{
\biggl[ 2j_1 - 2A_\ell +
2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi
+
\biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3
\biggr\}
</math>
</td>
</tr>
</table>

where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are:

====Desired Slopes of Normal Vectors====

A vector that is normal to the surface of a constant-density (oblate-spheroidal) contour has the following components:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\partial}{\partial \chi}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \chi}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
=
-2\chi
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\partial}{\partial \zeta}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \zeta}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
=
-2\zeta (1-e^2)^{-1}
\, .</math>
</td>
</tr>
</table>
Hence, the slope, <math>m</math>, of this normal vector is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>m = \biggl\{\frac{\partial}{\partial \zeta}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]\biggr\}
\biggl\{\frac{\partial}{\partial \chi}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]\biggr\}^{-1}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{ -2\zeta (1-e^2)^{-1}}{-2\chi}
=
\frac{\zeta}{\chi(1-e^2)} \, .
</math>
</td>
</tr>
</table>
Now, if the constant-pressure contours are to lie precisely on top of our constant-density contours, the normals have to have the same slopes. This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\partial P}{\partial \zeta}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\zeta}{\chi(1-e^2)} \biggl[\frac{\partial P}{\partial \chi}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \chi(1-e^2)\biggl\{
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\zeta
\biggl\{
\biggl[ 2j_1 - 2A_\ell +
2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi
+
\biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
2A_{\ell s}a_\ell^2 (1-e^2) \chi^3\zeta - 2A_s(1-e^2) \chi\zeta
+ 2A_{ss} a_\ell^2 (1-e^2)\chi \zeta^3
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 2j_1 - 2A_\ell ) \chi \zeta + 2A_{\ell s}a_\ell^2 \chi \zeta^3
+
(2A_{\ell \ell} a_\ell^2 - 2j_3 )\chi^3 \zeta
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[ 2A_{\ell s}a_\ell^2 (1-e^2)
-
(2A_{\ell \ell} a_\ell^2 - 2j_3 ) \biggr] \chi^3 \zeta
+ \biggl[- 2A_s(1-e^2) - ( 2j_1 - 2A_\ell ) \biggr]\chi\zeta
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ 2A_{\ell s}a_\ell^2 - 2A_{ss} a_\ell^2 (1-e^2) \biggr] \chi \zeta^3
</math>
</td>
</tr>
</table>

<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
Note …

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

<tr>
<td align="right">
<math>3A_{ss}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{2}{a_s^2} - 2A_{\ell s}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{2}{a_\ell^2(1-e^2)} - 2A_{\ell s}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ 3(1-e^2) (A_{ss} a_\ell^2)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2 - 2(1-e^2) (A_{\ell s}a_\ell^2)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \mathrm{RHS}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{ 2A_{\ell s}a_\ell^2 - 2A_{ss} a_\ell^2 (1-e^2) \biggr\} \chi \zeta^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{ 2A_{\ell s}a_\ell^2 - \frac{2}{3}\biggl[
2 - 2(1-e^2) (A_{\ell s}a_\ell^2)\biggr]
\biggr\} \chi \zeta^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{ \biggl[2 + \frac{4}{3}(1-e^2)\biggr] (A_{\ell s}a_\ell^2)
-\frac{4}{3}
\biggr\} \chi \zeta^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{2}{3}\biggl[ (5-2e^2) (A_{\ell s}a_\ell^2) - 2
\biggr] \chi \zeta^3
\, .
</math>
</td>
</tr>
</table>

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

In order for the <math>\chi^3\zeta</math> term on the LHS to be zero, we should set …

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

<tr>
<td align="right">
<math>0 </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ 2A_{\ell s}a_\ell^2 (1-e^2)
-
2A_{\ell \ell} a_\ell^2 + 2j_3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ j_3 </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)
\, ;
</math>
</td>
</tr>

</table>
and in order for the <math>\chi\zeta</math> term on the LHS to be zero, we should set …

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

<tr>
<td align="right">
<math>0 </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[- 2A_s(1-e^2) - ( 2j_1 - 2A_\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ j_1 </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
A_\ell - A_s(1-e^2)
\, .
</math>
</td>
</tr>
</table>

====Desired Slopes of Tangent Vectors====
Alternatively, if the constant-pressure contours are to lie precisely on top of our constant-density contours, the tangent vectors have to have slopes given by <math>-1/m</math>. This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\partial P}{\partial \zeta}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{1}{m}\biggl[\frac{\partial P}{\partial \chi}\biggr]
=
-\frac{\chi(1-e^2)}{\zeta} \biggl[\frac{\partial P}{\partial \chi}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \zeta \biggl\{
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-\chi(1-e^2)
\biggl\{
\biggl[ 2j_1 - 2A_\ell +
2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi
+
\biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl\{
\biggl[ A_{\ell s}a_\ell^2 \chi^2 - A_s \biggr]\zeta^2
+ A_{ss} a_\ell^2 \zeta^4
\biggr\}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- (1-e^2)
\biggl\{
\biggl[ j_1 - A_\ell +
A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi^2
+
\biggl[ A_{\ell \ell} a_\ell^2 - j_3 \biggr]\chi^4
\biggr\}
</math>
</td>
</tr>
</table>

===6<sup>th</sup> Try===

====Euler Equation====

From, for example, [[PGE/Euler#in_terms_of_velocity:_2|here]] we can write the,

<div align="center">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

{{Template:Math/EQ_Euler02}}
</div>
In steady-state, we should set <math>\partial\vec{v}/\partial t = 0</math>. There are various ways of expressing the nonlinear term on the LHS; from [[PGE/Euler#in_terms_of_the_vorticity:|here]], for example, we find,
<div align="center">
<math>
(\vec{v}\cdot\nabla)\vec{v} = \frac{1}{2}\nabla(\vec{v}\cdot\vec{v}) - \vec{v}\times(\nabla\times\vec{v})
= \frac{1}{2}\nabla(v^2) + \vec{\zeta}\times \vec{v} ,
</math>
</div>
where,
<div align="center">
<math>
\vec\zeta \equiv \nabla\times\vec{v}
</math>
</div>
is commonly referred to as the [https://en.wikipedia.org/wiki/Vorticity vorticity].

====Axisymmetric Configurations====

From, for example, [[AxisymmetricConfigurations/PGE#CYLconvectiveOperator|here]], we appreciate that, quite generally, for axisymmetric systems when written in cylindrical coordinates,

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

<tr>
<td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\hat{e}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial\varpi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi} \biggr]
+ \hat{e}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi} + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi} \biggr]
+ \hat{e}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial\varpi} + v_z \frac{\partial v_z}{\partial z} \biggr] \, .
</math>
</td>
</tr>
</table>
We seek steady-state configurations for which <math>v_\varpi =0</math> and <math>v_z = 0</math>, in which case this expression simplifies considerably to,

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

<tr>
<td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\hat{e}_\varpi \biggl[ - \frac{v_\varphi v_\varphi}{\varpi} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\hat{e}_\varpi \biggl[ - \frac{j^2}{\varpi^3} \biggr]
\, ,
</math>
</td>
</tr>
</table>
where, in this last expression we have replaced <math>v_\varphi</math> with the specific angular momentum, <math>j \equiv \varpi v_\varphi = (\varpi^2 \dot\varphi)</math>, which is a [[AxisymmetricConfigurations/PGE#Conservation_of_Specific_Angular_Momentum_(CYL.)|conserved quantity in dynamically evolving systems]]. NOTE:   Up to this point in our discussion, <math>j</math> can be a function of both coordinates, that is, <math>j = j(\varpi, z)</math>.

As has been highlighted [[AxisymmetricConfigurations/PGE#RelevantCylindricalComponents|here]] for example — for the axisymmetric configurations under consideration — the <math>\hat{e}_\varpi</math> and <math>\hat{e}_z</math> components of the Euler equation become, respectively,</span>
<table border="1" align="center" cellpadding="10"><tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>{\hat{e}}_\varpi</math>:    </td>
<td align="right">
<math>
- \frac{j^2}{\varpi^3}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr]
</math>
</td>
</tr>
<tr>
<td align="right"><math>{\hat{e}}_z</math>:    </td>
<td align="right">
<math>
0
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math>
</td>
</tr>
</table>
</td></tr></table>

====Strategy====

<font color="red">STEP 1:</font>   For the problem being tackled here, we start by recognizing that when considering hydrostatic balance in the <math>\hat{e}_z</math> direction, we have analytically known expressions for both <math>\rho(\varpi, z)</math> and <math>\partial\Phi/\partial z</math>. This means, therefore, that we can construct an analytical expression for the vertical component of the pressure gradient, specifically,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\frac{\partial P}{\partial z}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
- \rho \cdot \frac{\partial \Phi}{\partial z} \, .
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>
</table>

<font color="red">STEP 2:</font>   Because we want the meridional-plane, constant-pressure contours to align with the meridional-plane, constant density contours, we can determine the radial component of the pressure gradient by forcing the slope of the tangent vector to match the tangent vector of the density contour.

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

<tr>
<td align="right">
<math>\frac{\partial P}{\partial \zeta}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{1}{m}\biggl[\frac{\partial P}{\partial \chi}\biggr]
=
-\frac{\chi(1-e^2)}{\zeta} \biggl[\frac{\partial P}{\partial \chi}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)} \biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)}
\biggl\{
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
\biggr\}
\, .
</math>
</td>
</tr>
</table>

<font color="red">STEP 3:</font>   Via the radial component of the hydrostatic balance expression, we can determine analytically the distribution of specific angular momentum.
<table border="0" align="center" cellpadding="8">
<tr>
<td align="right">
<math>
\frac{j^2}{\varpi^3}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)^{-1} \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
-
\frac{\partial}{\partial \chi} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>
</table>

====Implication====

Hence,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
-\frac{\zeta}{\chi(1-e^2)}
\cdot
\frac{\partial}{\partial \zeta} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
-
\frac{\partial}{\partial \chi} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)}
\biggr\}
\, .
</math>
</td>
</tr>
</table>

Now, given that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, ,
</math>
</td>
</tr>
</table>
we see that the pair of partial derivative expressions are:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\partial}{\partial \zeta} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2\biggl[(A_{s s} a_\ell^2) \zeta^3
+ (A_{\ell s}a_\ell^2 )\chi^2 \zeta
- A_s \zeta
\biggr]
\, ;
</math>
</td>

<tr>
<td align="right">
<math>\frac{\partial}{\partial \chi} \biggl[
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2\biggl[
(A_{\ell \ell} a_\ell^2) \chi^3 + (A_{\ell s}a_\ell^2 )\chi \zeta^2 - A_\ell \chi\biggr]
\, .
</math>
</td>
</tr>
</table>

=See Also=

{{ SGFfooter }}

ParabolicDensity/Axisymmetric/Structure

2024-08-21T18:48:32Z

70.180.56.157: /* Tentative Summary */

__FORCETOC__

=Parabolic Density Distribution=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/GravPot|Part I:   Gravitational Potential]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Spheres/Structure|Part II:   Spherical Structures]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Axisymmetric/Structure|Part III:   Axisymmetric Equilibrium Structures]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[ParabolicDensity/Triaxial/Structure|Part IV:   Triaxial Equilibrium Structures (Exploration)]]
 
</td>
</tr>
</table>

==Axisymmetric (Oblate) Equilibrium Structures==

===Setup===

Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

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

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>\rho_c \biggl[ 1 - \biggl( \frac{x^2 + y^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \, ,</math>
</td>
</tr>
</table>
that is, axisymmetric (<math> a_m = a_\ell</math>, i.e., oblate) configurations with ''parabolic density distributions''. Much of our presentation, here, is drawn from our separate, detailed description of what we will refer to as [[ThreeDimensionalConfigurations/FerrersPotential|Ferrers potential]].

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
This can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and,

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>
Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that,

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

<tr>
<td align="right">
<math>\frac{H}{H_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>h(\xi_1) \, .</math>
</td>
</tr>
</table>
If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that,

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

<tr>
<td align="right">
<math>h(\xi_1)</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_2 \xi_1^2 + h_4 \xi_1^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>
</table>

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

===Gravitational Potential===
As we have detailed in [[ThreeDimensionalConfigurations/FerrersPotential|an accompanying discussion]], for an oblate-spheroidal configuration — that is, when <math>a_s < a_m = a_\ell</math> — the gravitational potential may be obtained from the expression,

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

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
+ \frac{1}{6} \biggl(3A_{11}x^4 + 3A_{22}y^4 + 3A_{33}z^4 \biggr)
\, ,
</math>
</td>
</tr>
</table>

where, in the present context, we can rewrite this expression as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell (x^2 + y^2) + A_s z^2 \biggr]
+ \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} x^2z^2 + A_{\ell s} y^2z^2\biggr]
+ \frac{1}{6} \biggl[3A_{\ell \ell} x^4 + 3A_{\ell \ell}y^4 + 3A_{ss}z^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} \varpi^2 z^2 \biggr]
+ \frac{1}{2} \biggl[A_{\ell \ell} (x^4 + y^4) + A_{ss}z^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \frac{A_{\ell \ell}}{2} \biggl[(x^2 + y^2)^2\biggr]
+ \frac{1}{2} \biggl[ A_{ss}z^4 \biggr]
+ \biggl[ A_{\ell s} \varpi^2 z^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \frac{A_{\ell \ell}}{2} \biggl[\varpi^4\biggr]
+ \frac{1}{2} \biggl[ A_{ss}z^4 \biggr]
+ \biggl[ A_{\ell s} \varpi^2 z^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

====Index Symbol Expressions====
The expression for the zeroth-order normalization term <math>(I_{BT})</math>, and the relevant pair of 1<sup>st</sup>-order index symbol expressions are:

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

<tr>
<td align="right"><math>I_\mathrm{BT}</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
A_\ell
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>A_s</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ,
</math>
</td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (36)</font><br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], <font color="#00CC00">§4.5, Eqs. (48) & (49)</font>
</div>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, .
</math>
</div>

The relevant [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Index_Symbols_of_the_2nd_Order|2<sup>nd</sup>-order index symbol]] expressions are:

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

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{3}{2} a_\ell^2 A_{ss} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2)
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
We can crosscheck this last expression by [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|drawing on a shortcut expression]],
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>A_{\ell s}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{A_\ell - A_s}{(a_\ell^2 - a_s^2)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ a_\ell^2 A_{\ell s}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^2}\biggl\{
A_s - A_\ell
\biggr\}
</math>
</td>
</tr>

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

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{
\biggl[ 2 - 2(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr]
-
\biggl[ (1-e^2)^{1/2} \frac{\sin^{-1}e}{e} - (1-e^2) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\}
\, .
</math>
</td>
</tr>
</table>

====Meridional Plane Equi-Potential Contours====
Here, we follow closely our separate discussion of equipotential surfaces for [[Apps/MaclaurinSpheroids#norotation|Maclaurin Spheroids, assuming no rotation]].

=====Configuration Surface=====
In the meridional <math>(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration — identified by the thick, solid-black curve below, in Figure 1 — is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho}{\rho_c} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>1 - \biggl[\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2} \biggr] = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>1 </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ z^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>a_s^2\biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr] = a_\ell^2 (1-e^2) \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{z}{a_\ell}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\pm ~(1-e^2)^{1 / 2} \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]^{1 / 2} \, ,</math>
</td>
<td align="right">        for <math>~0 \le \frac{| \varpi |}{a_\ell} \le 1 \, .</math></td>
</tr>
</table>

=====Expression for Gravitational Potential=====
Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{eff}</math> = constant, equipotential surface is defined by the expression,


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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} \equiv \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(\pi G\rho_c a_\ell^2)} + \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
- \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

Letting,
<div align="center"><math>\zeta \equiv \frac{z^2}{a_\ell^2}</math>,</div>
we can rewrite this expression for <math>\phi_\mathrm{choice}</math> as,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \zeta
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
- \frac{1}{2} A_{ss} a_\ell^2 \zeta^2
- A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\zeta
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \zeta^2
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\zeta
+
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
\, .
</math>
</td>
</tr>
</table>

=====Potential at the Pole=====
At the pole, <math>(\varpi, z) = (0, a_s)</math>. Hence,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{mid} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \cancelto{0}{\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)}\biggr]\biggl(\frac{a_s^2}{a_\ell^2}\biggr)
+
A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)}
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_s \biggl(\frac{a_s^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2 \, .
</math>
</td>
</tr>
</table>

=====General Determination of Vertical Coordinate (ζ)=====
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Given values of the three parameters, <math>e</math>, <math>\varpi</math>, and <math>\phi_\mathrm{choice}</math>, this last expression can be viewed as a quadratic equation for <math>\zeta</math>. Specifically,

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

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\alpha \zeta^2 + \beta\zeta + \gamma \, ,
</math>
</td>
</tr>
</table>
where,

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

<tr>
<td align="right">
<math>\alpha</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
\frac{1}{2} A_{ss} a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{3}\biggl\{
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\beta</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr) - A_s
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\}
\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)
-
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\gamma</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{8e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1 / 2} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
\, .
</math>
</td>
</tr>
</table>
The solution of this quadratic equation gives,

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

<tr>
<td align="right">
<math>\zeta</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta \pm \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
</td>
</tr>
</table>

Should we adopt the ''superior'' (positive) sign, or is it more physically reasonable to adopt the ''inferior'' (negative) sign? As it turns out, <math>\beta</math> is intrinsically negative, so the quantity, <math>-\beta</math>, is positive. Furthermore, when <math>\gamma</math> goes to zero, we need <math>\zeta</math> to go to zero as well. This will only happen if we adopt the ''inferior'' (negative) sign. Hence, the physically sensible root of this quadratic relation is given by the expression,

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

<tr>
<td align="right">
<math>\zeta</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta - \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
</td>
</tr>
</table>



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

Here we present a quantitatively accurate depiction of the shape of the (Ferrers) gravitational potential that arises from oblate-spheroidal configurations having a parabolic density distribution. We closely follow the discussion of [[Apps/MaclaurinSpheroids#Example_Equi-gravitational-potential_Contours|equi-gravitational potential contours that arise in (uniform-density) Maclaurin spheroids]]. In order to facilitate comparison with Maclaurin spheroids, we will focus on a model with …
<table border="0" align="center" width="80%">
<tr>
<td align="center"><math>\frac{a_s}{a_\ell} = 0.582724 \, ,</math></td>
<td align="center"><math>e = 0.81267 \, ,</math></td>
<td align="center"> </td>
</tr>
<tr>
<td align="center"><math>A_\ell = A_m = 0.51589042 \, ,</math></td>
<td align="center"><math>A_s = 0.96821916 \, ,</math></td>
<td align="center"><math>I_\mathrm{BT} = 1.360556 \, ,</math></td>
</tr>
<tr>
<td align="center"><math>a_\ell^2 A_{\ell \ell} = 0.3287756 \, ,</math></td>
<td align="center"><math>a_\ell^2 A_{s s} = 1.5066848 \, ,</math></td>
<td align="center"><math>a_\ell^2 A_{\ell s} = 0.6848975 \, .</math></td>
</tr>
</table>

[<font color="red">NOTE:</font>   Along the Maclaurin spheroid sequence, this is the eccentricity that marks bifurcation to the Jacobi ellipsoid sequence — see the first model listed in Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and/or see Tables 1 and 2 of [[ThreeDimensionalConfigurations/JacobiEllipsoids|our discussion of the Jacobi ellipsoid sequence]]. It is unlikely that this same eccentricity has a comparably special physical relevance along the sequence of spheroids having parabolic density distributions.]

The largest value of the gravitational potential that will arise inside (actually, on the surface) of the configuration is at <math>(\varpi, z) = (1, 0)</math>. That is, when,


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

<tr>
<td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{max} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_\ell
- \frac{1}{2} A_{\ell \ell} a_\ell^2 = 0.3515026 \, .
</math>
</td>
</tr>
</table>
So we will plot various equipotential surfaces having, <math>0 < \phi_\mathrm{choice} < \phi_\mathrm{choice}|_\mathrm{max} </math>, recognizing that they will each cut through the equatorial plane <math>(z = 0)</math> at the radial coordinate given by,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \cancelto{0}{\zeta^2}
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\cancelto{0}{\zeta}
+
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} A_{\ell \ell} a_\ell^2 \chi^2
- A_\ell \chi
+ \phi_\mathrm{choice} \, ,
</math>
</td>
</tr>
</table>
where,
<div align="center"><math>\chi \equiv \frac{\varpi^2}{a_\ell^2} \, .</math></div>
The solution to this quadratic equation gives,

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

<tr>
<td align="right">
<math>\chi_\mathrm{eqplane} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{A_{\ell \ell} a_\ell^2}\biggl\{
A_\ell \pm \biggl[A_\ell^2 - 2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}\biggr]^{1 / 2}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{A_\ell}{A_{\ell \ell} a_\ell^2}\biggl\{
1 - \biggl[1 - \frac{2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}}{A_\ell^2}\biggr]^{1 / 2}
\biggr\}
\, .
</math>
</td>
</tr>
</table>
Note that, again, the physically relevant root is obtained by adopting the ''inferior'' (negative) sign, as has been done in this last expression.

=====Equipotential Contours that Lie Entirely Within Configuration=====
For all <math>0 < \phi_\mathrm{choice} \le \phi_\mathrm{choice} |_\mathrm{mid}</math>, the equipotential contour will reside entirely within the configuration. In this case, for a given <math>\phi_\mathrm{choice}</math>, we can plot points along the contour by picking (equally spaced?) values of <math>\chi_\mathrm{eqplane} \ge \chi \ge 0</math>, then solve the above quadratic equation for the corresponding value of <math>\zeta</math>.

In our example configuration, this means … (to be finished)

===Hydrostatic Balance (Algebraic Condition)===
Following our [[Apps/MaclaurinSpheroids#Equilibrium_Structure|separate discussion of the equilibrium structure]] of Maclaurin spheroids, and given that our solution of the Poisson equation fixes the expression for <math>\Phi_\mathrm{grav} </math>, the algebraic expression ensuring hydrostatic balance is,

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

<tr>
<td align="right">
<math>H(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B - \biggl[ \Phi_\mathrm{grav}(\varpi, z) + \Psi(\varpi, z) \biggr] \, ,
</math>
</td>
</tr>
</table>
where, <math>\Psi</math> is the centrifugal potential. <font color="red">NOTE:</font>   Generally when modeling axisymmetric astrophysical systems (see our [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying discussion of ''simple'' rotation profiles]]) it is assumed that <math>\Psi</math> does not functionally depend on <math>z</math>. Here, our other constraints — for example, demanding that the configuration have a parabolic density distribution — may force us to adopt a <math>z</math>-dependent rotation profile.

Here, we know that the adopted parabolic density distribution gives rise to a gravitational potential of the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

Hence,

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

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B - \Phi_\mathrm{grav}(\varpi, z) - H(\varpi, z)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
-
H_c h(\xi_1) \, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
We presume that the enthalpy profile, as well as the density profile, can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and,

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>
Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that,

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

<tr>
<td align="right">
<math>\frac{H}{H_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>h(\xi_1) \, .</math>
</td>
</tr>
</table>
If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that,

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

<tr>
<td align="right">
<math>h(\xi_1)</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_2 \xi_1^2 + h_4 \xi_1^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>
</table>

</td></tr></table>
Adopting this last expression for the enthalpy, we have,

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

<tr>
<td align="right">
<math>\frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
-
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

At the pole of the configuration — that is, when <math>(\varpi, z) = (0, a_s)</math> — this statement of hydrostatic balance becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)} + A_s \biggl( \frac{a_s^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)}
+ A_{ss} a_\ell^2 \biggl(\frac{a_s^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^2} + \biggl(\frac{a_s}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^4} + \biggl(\frac{a_s}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
-
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
\, .
</math>
</td>
</tr>
</table>

For centrally condensed configurations, it is astrophysically reasonable to assume that <math>\Psi(\varpi, z)</math> is of the form such that the centrifugal potential goes to zero when <math>\varpi \rightarrow 0</math>. Adopting that assumption here means that the Bernoulli constant has the value,

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

<tr>
<td align="right">
<math>C_B</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<td align="left">
<math>
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
\, .
</math>
</td>
</tr>
</table>
Plugging this expression for <math>C_B</math> back into the general statement of hydrostatic balance gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\biggl[A_s (1-e^2)-A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1} - 1\biggr]
+
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} -1
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2(1-e^2)^{-1}\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr]
+
h_4 a_s^4 (1-e^2)^{-2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2}
\biggr]
\biggr\}
</math>
</td>
</tr>
</table>

Let's set …
<div align="center">
<math>H_c h_0 = \pi G \rho_c a_\ell^2 \, ;</math>      
<math>h_2 = \frac{A_s(1-e^2)}{a_s^2} \, ;</math>      
<math>h_4 = - \frac{ A_{ss}a_\ell^2 (1-e^2)^2 }{ 2a_s^4 } \, .</math>
</div>
This gives,

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

<tr>
<td align="right">
<math>\frac{\Psi(\varpi, z)}{\pi G \rho_c a_\ell^2}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) \biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
- A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+ \biggl[
\frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+
\frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
\biggr]
+ A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
-
A_{ss} a_\ell^2 \biggl[ \biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+ \frac{1}{2}\biggl\{
A_{\ell \ell} a_\ell^2 - A_{ss} a_\ell^2 (1-e^2)^{2}
\biggr\} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl\{ A_{\ell s}a_\ell^2
-
A_{ss} a_\ell^2 (1-e^2) \biggr\}\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\, .
</math>
</td>
</tr>
</table>

===2<sup>nd</sup> Try===

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">Keep in Mind, from Above</div>

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

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\rho_c \biggl[ 1 - \biggl(\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\, ,
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>

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

From our presentation of [[AxisymmetricConfigurations/PGE#Eulerian_Formulation_(CYL.)|the Eulerian formulation of the Euler equation in cylindrical coordinates]], we see that in steady-state axisymmetric flows, the two relevant equilibrium conditions are,

<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>~{\hat{e}}_\varpi</math>:    </td>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}
</math>
</td>
</tr>
<tr>
<td align="right"><math>~{\hat{e}}_z</math>:    </td>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math>
</td>
</tr>
</table>

====Vertical Component====
We will focus, first, on the vertical component. Specifically, since both <math>\rho</math> and <math>\Phi_\mathrm{grav}</math> are known, the vertical gradient of the (unknown) scalar pressure is
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{\partial P}{\partial z}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \rho ~ \frac{\partial}{\partial z} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>
</table>

Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell)} \cdot \frac{\partial P}{\partial z}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial z} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \chi^4
+ A_{ss} a_\ell^2 \zeta^4
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl\{
- 2A_s \zeta
+ \biggl[
2A_{ss} a_\ell^2 \zeta^3
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta
+ A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta
+ A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>
</table>

where (unlike above) we are using the dimensionless lengths, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. Continuing to streamline this function, we have,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \chi^2\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \zeta^2\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \biggl[
A_{\ell s}a_\ell^2 \chi^4 \zeta - A_s \chi^2 \zeta + A_{ss} a_\ell^2 \chi^2 \zeta^3
\biggr]
- \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_s \zeta^3 + A_{ss} a_\ell^2 \zeta^5
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2 - A_s \biggr]\zeta
+ A_{ss} a_\ell^2 \zeta^3
+ \biggl[
A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
- A_{ss} a_\ell^2 \chi^2 \zeta^3
+ \biggl[
A_s \zeta^3 - A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_{ss} a_\ell^2 \zeta^5
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2 - A_s
+
A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
+
\biggl[ A_{ss} a_\ell^2 - A_{ss} a_\ell^2 \chi^2 + A_s(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} \biggr]\zeta^3
- A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
+
\biggl\{
[A_s(1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\chi^2
\biggr\}\zeta^3
- A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5
\, .
</math>
</td>
</tr>
</table>
So, let's see what happens if we assume that the pressure has the form,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>p_0 + p_2 \zeta^2 + p_4\zeta^4 + p_6\zeta^6
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P_\mathrm{vert}}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>2p_2 \zeta + 4p_4\zeta^3 + 6p_6\zeta^5 \, ,</math>
</td>
</tr>
</table>
in which case,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
p_0 + \frac{1}{2}\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4
\biggr] \zeta^2
+
\frac{1}{4}\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4
+
\frac{1}{6}\biggl[
- A_{ss} a_\ell^2 (1-e^2)^{-1}
\biggr]\zeta^6
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

<font color="red">REMINDER:</font>
From [[#2nd_Try|above]] …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\, .
</math>
</td>
</tr>
</table>

And, in the case of the spherically symmetric equilibrium configuration, the [[SSC/Structure/OtherAnalyticModels#Pressure|pressure distribution]] derived by {{ Prasad49 }} has the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P}{P_c}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[1 + \biggl(\frac{\rho}{\rho_c}\biggr)\biggr]
\, .
</math>
</td>
</tr>
</table>
In the context of rotationally flattened configurations, therefore, we might expect the (vertical) pressure distribution to be of the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P}{P_c}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl\{
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
-\chi^2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
-
\zeta^2(1-e^2)^{-1}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl\{
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
+ \biggl[-\chi^2 + \chi^4 + \chi^2\zeta^2(1-e^2)^{-1}\biggr]
+
\biggl[- \zeta^2(1-e^2)^{-1} + \chi^2\zeta^2(1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr]
\biggr\}
</math>
</td>
</tr>
</table>

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

====Radial Component====

Start with,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>- \frac{j^2 \rho}{\varpi^3}
+
\frac{\partial P}{\partial \varpi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \rho ~ \frac{\partial}{\partial \varpi} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>
</table>

Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>- \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr] \frac{j^2 \rho}{\varpi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\frac{\partial P}{\partial \varpi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\rho ~ \frac{\partial}{\partial \varpi} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \chi} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \chi} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \chi^4
+ A_{ss} a_\ell^2 \zeta^4
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\} \, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">EXACT!</font></div>

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\}
</math>
</td>
</tr>
</table>

</td></tr></table>
Continuing to streamline this function, we have,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}
-
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}\chi^2
-
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}(1-e^2)^{-1}\zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3
-
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi^3 - 2 A_{\ell \ell} a_\ell^2 \chi^5
+
\biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2 - 2 A_{\ell \ell} a_\ell^2 \chi^3 (1-e^2)^{-1}\zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+\biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2
+ \biggl[2 A_{\ell \ell} a_\ell^2
-2A_{\ell s}a_\ell^2 \zeta^2 + 2A_\ell - 2 A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
- 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2
+ A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
+ 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell
-A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
- 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
\, .
</math>
</td>
</tr>
</table>

====Determine Specific Angular Momentum Distribution====

Now, from our analysis of the vertical component, we determined that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{12P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
12p_0 + 6\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4
\biggr] \zeta^2
+
3\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4
+
2\biggl[
- A_{ss} a_\ell^2 (1-e^2)^{-1}
\biggr]\zeta^6
\, .
</math>
</td>
</tr>
</table>

<span id="RadialDerivative">The radial derivative of this function is</span>,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[ \frac{12}{(2\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
6\biggl[
2(A_{\ell s}a_\ell^2 + A_s ) \zeta^2 \chi - 4A_{\ell s}a_\ell^2\zeta^2 \chi^3
\biggr]
+
6\biggl\{ - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ] \zeta^4 \chi \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3 \, .
</math>
</td>
</tr>
</table>

We hypothesize that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi}
-
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\biggl[\frac{\partial P}{\partial \chi}\biggr]_\mathrm{rad}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
- 2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2
+ A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
- 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell
-A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
+ 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>2\biggl\{
\biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4 \biggr] \chi
- 2A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl[ A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2
- A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
+ \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell
+ A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(2\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4
+ A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2
- A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell
+ A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2- 2A_{\ell s}a_\ell^2 \zeta^2\biggr] \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ A_\ell(1-e^2)^{-1} +(A_{\ell s}a_\ell^2 + A_s - A_{\ell s}a_\ell^2 - A_\ell)\zeta^2
+ \frac{1}{2}\biggl[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}
+ 2A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell)
+ (A_{\ell s}a_\ell^2 + A_{\ell \ell} a_\ell^2 - 2A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ A_\ell(1-e^2)^{-1} + (A_s - A_\ell)\zeta^2
+ \frac{1}{2}\biggl[- A_{ss} a_\ell^2
+ A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi
+ \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell)
+ (A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>
</table>
Now, from [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|our layout of relevant index symbol expressions]], let's see if the coefficients of various ζ-dependent terms go to zero.

<font color="red">FIRST:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
A_{s\ell}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{A_s - A_\ell}{(a_s^2 - a_\ell^2)} = \frac{A_s - A_\ell}{a_\ell^2 e^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~
A_{s \ell}a_\ell^2 e^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(A_s - A_\ell)
</math>
</td>
</tr>

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

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

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^2}\biggl[
3 - e^2 - 3(1-e^2)^{1 / 2}\frac{\sin^{-1}e}{e}
\biggr]
\, ;
</math>
</td>
</tr>
</table>

<font color="red">SECOND:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
3A_{s s}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_s^2} - 2A_{s \ell}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{3}{2}A_{s s}a_\ell^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{a_\ell^2}{a_s^2} - A_{s \ell}a_\ell^2 = (1 - e^2)^{-1} - A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ - A_{s s}a_\ell^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \biggl[ - A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1}
+ A_{\ell s}a_\ell^2 (1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\frac{1}{3(1-e^2)}\biggl[
2A_{s\ell}a_\ell^2 (1-e^2) - 2
+ 3A_{\ell s}a_\ell^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{3(1-e^2)}\biggl[A_{s\ell}a_\ell^2 (5-2e^2) - 2 \biggr]\, ;
</math>
</td>
</tr>
</table>

<font color="red">THIRD:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
3A_{\ell \ell}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_\ell^2} - A_{\ell \ell} - A_{s\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ 4A_{\ell \ell}a_\ell^2</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2 - A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
(A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} - \frac{5}{4}A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{4}\biggl[2 - 5A_{s\ell}a_\ell^2\biggr] \, .
</math>
</td>
</tr>
</table>

===3<sup>rd</sup> Try===
From the [[#Radial_Component|above, "2<sup>nd</sup> Try" discussion of the radial component]], we can write the following "EXACT!" relation,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">EXACT!</font></div>

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\} \, .
</math>
</td>
</tr>
</table>

</td></tr></table>
Now, our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math> suggests that the left-hand-side of this expression should be of the form,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
LHS  
<math>
\equiv \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>\sim</math></td>
<td align="left">
<math>
c_2\zeta^2 + c_4\zeta^4 \, ,
</math>
</td>
</tr>
</table>
where it is understood that the coefficients, <math>c_2</math> and <math>c_4</math>, are both functions of <math>\chi</math>. This should be compared with the "EXACT!" expression for the RHS after multiplying through by the expression for the dimensionless density, that is,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
RHS
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\biggr] \cdot \biggl\{
\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi \zeta^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(1 - \chi^2)\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
- \biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr](1-e^2)^{-1}\zeta^2
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4
\, .
</math>
</td>
</tr>
</table>

Because we are not expecting to see a term that is independent of <math>\zeta</math>, this suggests that the term inside the large square brackets must be zero. This leads to an expression for the distribution of specific angular momentum of the form,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<table border="0" align="center" cellpadding="8">

<tr><td align="center" colspan="3"><font color="red">EXCELLENT !!</font></td></tr>

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{j^2 }{(\pi G \rho_c a_\ell^4)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6
\, .
</math>
</td>
</tr>
</table>

According to our [[AxisymmetricConfigurations/SolutionStrategies#Specifying_Radial_Rotation_Profile_in_the_Equilibrium_Configuration|accompanying discussion of ''Simple'' rotation profiles]], the corresponding centrifugal potential is given by the expression,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>\Psi</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \int \frac{j^2(\varpi)}{\varpi^3} d\varpi
=
- (\pi G \rho_c a_\ell^2) \int \frac{1}{\chi^3} \biggl[2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6\biggr]d\chi
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \int \biggl[2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr]d\chi
=
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, .
</math>
</td>
</tr>
</table>
(Here, we ignore the integration constant because it will be folded in with the Bernoulli constant.)

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

It also means that the RHS expression simplifies to the form,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
RHS
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4 \, .
</math>
</td>
</tr>
</table>

This should be compared to our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math>, namely,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(2A_{\ell s}a_\ell^2 + 2A_s )\chi\zeta^2- 4A_{\ell s}a_\ell^2 \chi^3\zeta^2 - \biggl[A_{\ell s}a_\ell^2 (1-e^2)^{-1} + A_{ss} a_\ell^2\biggr]\chi\zeta^4

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

===4<sup>th</sup> Try===

In our [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|accompanying discussion of Ferrers Potential]], we have derived the expression for the gravitational potential inside (and on the surface of) a triaxial ellipsoid with a parabolic density distribution. Specifically, for
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\rho(\mathbf{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr]
\, ,</math>
</td>
</tr>
</table>
[[ThreeDimensionalConfigurations/FerrersPotential#GravFor1|we find]],
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2} \biggl(A_{11}x^4 + A_{22}y^4 + A_{33}z^4 \biggr)
\, .
</math>
</td>
</tr>
</table>
In this [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|same accompanying discussion]], we plugged this expression for the gravitational potential into the Poisson equation and demonstrated that it properly generates the expression for the parabolic density distribution. For the axisymmetric configuration being considered here — with the short axis aligned with <math>c = a_3 = a_s</math> — these two relations become,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr]
=
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \frac{\varpi^2}{a_\ell^2} - A_s \frac{z^2}{a_\ell^2}
+ (A_{\ell s}a_\ell^2 )\frac{ \varpi^2z^2 }{a_\ell^4} + \frac{1}{2}(A_{s s} a_\ell^2) \frac{z^4}{a_\ell^4}
+ \frac{A_{\ell \ell}a_\ell^2}{2} \biggl[ \frac{(x^4 + 2 x^2y^2 + y^4 )}{a_\ell^4} \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, .
</math>
</td>
</tr>
</table>
where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. (This matches the [[#Gravitational_Potential|expression derived above]].)

----

Discuss scalar relationship between the enthalpy <math>(H)</math> and the effective potential.

As has been detailed in [[AxisymmetricConfigurations/SolutionStrategies#Technique|an accompanying discussion of solution techniques]], a configuration will be in dynamic equilibrium if,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla\biggl[ H + \Phi_\mathrm{grav} + \Psi \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
0
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ H + \Phi_\mathrm{grav} + \Psi
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
constant
<math>
= C_B
</math>
</td>
</tr>
</table>

Given that, in our particular case, we have analytic expressions for <math>\Phi_\mathrm{grav}(\chi,\zeta)</math> and for <math>\Psi(\chi,\zeta)</math>, we deduce that, to within a constant, the enthalpy distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{\Phi_\mathrm{grav}}{{(\pi G\rho_c a_\ell^2)}} - \frac{\Psi}{{(\pi G\rho_c a_\ell^2)}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
-
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(A_{\ell s}a_\ell^2 )\chi^2 \biggr]
</math>
</td>
</tr>
</table>
Now, according to our [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|related discussion of index symbols]],

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>3A_{s s}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_s^2} - 2A_{\ell s}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ 3A_{s s}a_\ell^2</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2(1-e^2)^{-1} - 2A_{\ell s}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~2(A_{\ell s}a_\ell^2)\chi^2 </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Examining the radial derivative …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \chi}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \chi} \biggl\{
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[-3(A_{s s} a_\ell^2) + 2(1-e^2)^{-1} \biggr]\zeta^2\chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2(A_{\ell s} a_\ell^2)\zeta^2\chi
\, .
</math>
</td>
</tr>
</table>
<font color="red">YES !!!</font> This matches the "radial" pressure-gradient, below.

Now, examining the vertical derivative …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \zeta}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \zeta} \biggl\{
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \zeta} \biggl\{
- A_s \zeta^2
+ \frac{1}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^4 + [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_s \zeta
+
\biggl[2(A_{s s} a_\ell^2) \zeta^3
+ [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_s \zeta
+
\biggl[2(A_{s s} a_\ell^2) \zeta^3
+ 2(A_{\ell s} a_\ell^2) \chi^2\zeta \biggr]
</math>
</td>
</tr>
</table>
<font color="red">HURRAY !!!</font> This matches the "vertical" pressure-gradient, below.

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

----

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\}
</math>
</td>
</tr>
</table>
Plug in …

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3
\, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
2A_{\ell s}a_\ell^2 \zeta^2 \chi
\biggr\}
</math>
</td>
</tr>
</table>



Hence, examination of the radial component leads to the following suggested expression for the pressure:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
- \chi^2
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
- \zeta^2(1-e^2)^{-1}
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{P}{(\pi G \rho_c^2 a_\ell^2)}
</math>
</td>
<td align="center"><math>\sim</math></td>
<td align="left">
<math>
\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
- \frac{1}{2}\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^4\biggr]
- \zeta^2(1-e^2)^{-1} \biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 1 - \frac{\chi^2}{2}
- \zeta^2(1-e^2)^{-1} \biggr]
\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
\, .
</math>
</td>
</tr>
</table>

While examination of the vertical component leads to the following suggested expression for the pressure:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \frac{\chi^2}{2} - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
- \frac{\chi^2}{2}\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
</math>
</td>
</tr>
</table>

===Tentative Summary===

====Known Relations====

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

<tr>
<td align="left"><font color="orange"><b>Density:</b></font></td>
<td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Specific Angular Momentum:</b></font></td>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Centrifugal Potential:</b></font></td>
<td align="right">
<math>
\frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Enthalpy:</b></font></td>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Radial Pressure Gradient:</b></font></td>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
2A_{\ell s}a_\ell^2 \zeta^2 \chi
\biggr\}
</math>
</td>
</tr>
</table>

where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are:

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

<tr>
<td align="right"><math>I_\mathrm{BT}</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
A_\ell
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>A_s</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{3}{2} a_\ell^2 A_{ss} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2)
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\} \, ,
</math>
</td>
</tr>
</table>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, .
</math>
</div>

====Examine Behavior of Enthalpy====
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_s}\biggr)^2 \biggr]^{1 / 2}
=
a_s\biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>

====Try to Construct Pressure Distribution====

Drawing from the expression for the vertical pressure gradient, namely,

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

<tr>
<td align="right">
<math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr] \, ,
</math>
</td>
</tr>
</table>
try the following pressure expression:

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

<tr>
<td align="right">
<math>\frac{P}{(\pi G\rho_c^2 a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
f_0
+ f_2 \biggl(\frac{\xi_1}{a_s} \biggr)^2
+ f_4 \biggl(\frac{\xi_1}{a_s} \biggr)^4
+ f_6 \biggl(\frac{\xi_1}{a_s} \biggr)^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
f_0
+ f_2 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]
+ f_4 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^2
+ f_6 \biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^3
\, .
</math>
</td>
</tr>
</table>
The vertical derivative of this expression is,

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

<tr>
<td align="right">
<math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)}\biggr] \frac{\partial P}{\partial \zeta} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\zeta
</math>
</td>
</tr>
</table>

=See Also=

{{ SGFfooter }}

ParabolicDensity/Axisymmetric/Structure

2024-08-21T18:36:28Z

70.180.56.157: /* Try to Construct Pressure Distribution */

__FORCETOC__

=Parabolic Density Distribution=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/GravPot|Part I:   Gravitational Potential]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Spheres/Structure|Part II:   Spherical Structures]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Axisymmetric/Structure|Part III:   Axisymmetric Equilibrium Structures]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[ParabolicDensity/Triaxial/Structure|Part IV:   Triaxial Equilibrium Structures (Exploration)]]
 
</td>
</tr>
</table>

==Axisymmetric (Oblate) Equilibrium Structures==

===Setup===

Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

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

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>\rho_c \biggl[ 1 - \biggl( \frac{x^2 + y^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \, ,</math>
</td>
</tr>
</table>
that is, axisymmetric (<math> a_m = a_\ell</math>, i.e., oblate) configurations with ''parabolic density distributions''. Much of our presentation, here, is drawn from our separate, detailed description of what we will refer to as [[ThreeDimensionalConfigurations/FerrersPotential|Ferrers potential]].

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
This can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and,

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>
Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that,

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

<tr>
<td align="right">
<math>\frac{H}{H_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>h(\xi_1) \, .</math>
</td>
</tr>
</table>
If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that,

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

<tr>
<td align="right">
<math>h(\xi_1)</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_2 \xi_1^2 + h_4 \xi_1^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>
</table>

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

===Gravitational Potential===
As we have detailed in [[ThreeDimensionalConfigurations/FerrersPotential|an accompanying discussion]], for an oblate-spheroidal configuration — that is, when <math>a_s < a_m = a_\ell</math> — the gravitational potential may be obtained from the expression,

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

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
+ \frac{1}{6} \biggl(3A_{11}x^4 + 3A_{22}y^4 + 3A_{33}z^4 \biggr)
\, ,
</math>
</td>
</tr>
</table>

where, in the present context, we can rewrite this expression as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell (x^2 + y^2) + A_s z^2 \biggr]
+ \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} x^2z^2 + A_{\ell s} y^2z^2\biggr]
+ \frac{1}{6} \biggl[3A_{\ell \ell} x^4 + 3A_{\ell \ell}y^4 + 3A_{ss}z^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \biggl[ A_{\ell \ell} x^2y^2 + A_{\ell s} \varpi^2 z^2 \biggr]
+ \frac{1}{2} \biggl[A_{\ell \ell} (x^4 + y^4) + A_{ss}z^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \frac{A_{\ell \ell}}{2} \biggl[(x^2 + y^2)^2\biggr]
+ \frac{1}{2} \biggl[ A_{ss}z^4 \biggr]
+ \biggl[ A_{\ell s} \varpi^2 z^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_\ell^2
- \biggl[A_\ell \varpi^2 + A_s z^2 \biggr]
+ \frac{A_{\ell \ell}}{2} \biggl[\varpi^4\biggr]
+ \frac{1}{2} \biggl[ A_{ss}z^4 \biggr]
+ \biggl[ A_{\ell s} \varpi^2 z^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

====Index Symbol Expressions====
The expression for the zeroth-order normalization term <math>(I_{BT})</math>, and the relevant pair of 1<sup>st</sup>-order index symbol expressions are:

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

<tr>
<td align="right"><math>I_\mathrm{BT}</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
A_\ell
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>A_s</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ,
</math>
</td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (36)</font><br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], <font color="#00CC00">§4.5, Eqs. (48) & (49)</font>
</div>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, .
</math>
</div>

The relevant [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Index_Symbols_of_the_2nd_Order|2<sup>nd</sup>-order index symbol]] expressions are:

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

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{3}{2} a_\ell^2 A_{ss} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2)
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
We can crosscheck this last expression by [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|drawing on a shortcut expression]],
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>A_{\ell s}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{A_\ell - A_s}{(a_\ell^2 - a_s^2)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ a_\ell^2 A_{\ell s}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^2}\biggl\{
A_s - A_\ell
\biggr\}
</math>
</td>
</tr>

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

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{
\biggl[ 2 - 2(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr]
-
\biggl[ (1-e^2)^{1/2} \frac{\sin^{-1}e}{e} - (1-e^2) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\}
\, .
</math>
</td>
</tr>
</table>

====Meridional Plane Equi-Potential Contours====
Here, we follow closely our separate discussion of equipotential surfaces for [[Apps/MaclaurinSpheroids#norotation|Maclaurin Spheroids, assuming no rotation]].

=====Configuration Surface=====
In the meridional <math>(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration — identified by the thick, solid-black curve below, in Figure 1 — is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho}{\rho_c} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>1 - \biggl[\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2} \biggr] = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>1 </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ z^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left" colspan="2">
<math>a_s^2\biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr] = a_\ell^2 (1-e^2) \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{z}{a_\ell}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\pm ~(1-e^2)^{1 / 2} \biggl[1 - \frac{\varpi^2}{a_\ell^2} \biggr]^{1 / 2} \, ,</math>
</td>
<td align="right">        for <math>~0 \le \frac{| \varpi |}{a_\ell} \le 1 \, .</math></td>
</tr>
</table>

=====Expression for Gravitational Potential=====
Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{eff}</math> = constant, equipotential surface is defined by the expression,


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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} \equiv \frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(\pi G\rho_c a_\ell^2)} + \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
- \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

Letting,
<div align="center"><math>\zeta \equiv \frac{z^2}{a_\ell^2}</math>,</div>
we can rewrite this expression for <math>\phi_\mathrm{choice}</math> as,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \zeta
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
- \frac{1}{2} A_{ss} a_\ell^2 \zeta^2
- A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\zeta
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \zeta^2
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\zeta
+
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
\, .
</math>
</td>
</tr>
</table>

=====Potential at the Pole=====
At the pole, <math>(\varpi, z) = (0, a_s)</math>. Hence,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{mid} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \cancelto{0}{\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)}\biggr]\biggl(\frac{a_s^2}{a_\ell^2}\biggr)
+
A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)}
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_s \biggl(\frac{a_s^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{ss} a_\ell^2 \biggl(\frac{a_s^2}{a_\ell^2}\biggr)^2 \, .
</math>
</td>
</tr>
</table>

=====General Determination of Vertical Coordinate (ζ)=====
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Given values of the three parameters, <math>e</math>, <math>\varpi</math>, and <math>\phi_\mathrm{choice}</math>, this last expression can be viewed as a quadratic equation for <math>\zeta</math>. Specifically,

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

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\alpha \zeta^2 + \beta\zeta + \gamma \, ,
</math>
</td>
</tr>
</table>
where,

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

<tr>
<td align="right">
<math>\alpha</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
\frac{1}{2} A_{ss} a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{3}\biggl\{
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\beta</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr) - A_s
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^4}\biggl\{(3-e^2) - 3(1-e^2)^{1 / 2} \frac{\sin^{-1}e}{e} \biggr\}
\biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)
-
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\gamma</math>
</td>
<td align="center"><math>\equiv</math></td>
<td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\phi_\mathrm{choice}
+
\frac{1}{8e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1 / 2} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
\, .
</math>
</td>
</tr>
</table>
The solution of this quadratic equation gives,

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

<tr>
<td align="right">
<math>\zeta</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta \pm \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
</td>
</tr>
</table>

Should we adopt the ''superior'' (positive) sign, or is it more physically reasonable to adopt the ''inferior'' (negative) sign? As it turns out, <math>\beta</math> is intrinsically negative, so the quantity, <math>-\beta</math>, is positive. Furthermore, when <math>\gamma</math> goes to zero, we need <math>\zeta</math> to go to zero as well. This will only happen if we adopt the ''inferior'' (negative) sign. Hence, the physically sensible root of this quadratic relation is given by the expression,

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

<tr>
<td align="right">
<math>\zeta</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2\alpha}\biggl\{ - \beta - \biggl[\beta^2 - 4\alpha\gamma \biggr]^{1 / 2}\biggr\}
\, .
</math>
</td>
</tr>
</table>



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

Here we present a quantitatively accurate depiction of the shape of the (Ferrers) gravitational potential that arises from oblate-spheroidal configurations having a parabolic density distribution. We closely follow the discussion of [[Apps/MaclaurinSpheroids#Example_Equi-gravitational-potential_Contours|equi-gravitational potential contours that arise in (uniform-density) Maclaurin spheroids]]. In order to facilitate comparison with Maclaurin spheroids, we will focus on a model with …
<table border="0" align="center" width="80%">
<tr>
<td align="center"><math>\frac{a_s}{a_\ell} = 0.582724 \, ,</math></td>
<td align="center"><math>e = 0.81267 \, ,</math></td>
<td align="center"> </td>
</tr>
<tr>
<td align="center"><math>A_\ell = A_m = 0.51589042 \, ,</math></td>
<td align="center"><math>A_s = 0.96821916 \, ,</math></td>
<td align="center"><math>I_\mathrm{BT} = 1.360556 \, ,</math></td>
</tr>
<tr>
<td align="center"><math>a_\ell^2 A_{\ell \ell} = 0.3287756 \, ,</math></td>
<td align="center"><math>a_\ell^2 A_{s s} = 1.5066848 \, ,</math></td>
<td align="center"><math>a_\ell^2 A_{\ell s} = 0.6848975 \, .</math></td>
</tr>
</table>

[<font color="red">NOTE:</font>   Along the Maclaurin spheroid sequence, this is the eccentricity that marks bifurcation to the Jacobi ellipsoid sequence — see the first model listed in Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and/or see Tables 1 and 2 of [[ThreeDimensionalConfigurations/JacobiEllipsoids|our discussion of the Jacobi ellipsoid sequence]]. It is unlikely that this same eccentricity has a comparably special physical relevance along the sequence of spheroids having parabolic density distributions.]

The largest value of the gravitational potential that will arise inside (actually, on the surface) of the configuration is at <math>(\varpi, z) = (1, 0)</math>. That is, when,


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

<tr>
<td align="right">
<math>\phi_\mathrm{choice}\biggr|_\mathrm{max} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
A_\ell
- \frac{1}{2} A_{\ell \ell} a_\ell^2 = 0.3515026 \, .
</math>
</td>
</tr>
</table>
So we will plot various equipotential surfaces having, <math>0 < \phi_\mathrm{choice} < \phi_\mathrm{choice}|_\mathrm{max} </math>, recognizing that they will each cut through the equatorial plane <math>(z = 0)</math> at the radial coordinate given by,

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

<tr>
<td align="right">
<math>\phi_\mathrm{choice} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{1}{2} A_{ss} a_\ell^2 \cancelto{0}{\zeta^2}
+ \biggl[ A_s - A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 }{a_\ell^2}\biggr)\biggr]\cancelto{0}{\zeta}
+
A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
- \frac{1}{2} A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} A_{\ell \ell} a_\ell^2 \chi^2
- A_\ell \chi
+ \phi_\mathrm{choice} \, ,
</math>
</td>
</tr>
</table>
where,
<div align="center"><math>\chi \equiv \frac{\varpi^2}{a_\ell^2} \, .</math></div>
The solution to this quadratic equation gives,

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

<tr>
<td align="right">
<math>\chi_\mathrm{eqplane} </math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{A_{\ell \ell} a_\ell^2}\biggl\{
A_\ell \pm \biggl[A_\ell^2 - 2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}\biggr]^{1 / 2}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{A_\ell}{A_{\ell \ell} a_\ell^2}\biggl\{
1 - \biggl[1 - \frac{2A_{\ell \ell} a_\ell^2 \phi_\mathrm{choice}}{A_\ell^2}\biggr]^{1 / 2}
\biggr\}
\, .
</math>
</td>
</tr>
</table>
Note that, again, the physically relevant root is obtained by adopting the ''inferior'' (negative) sign, as has been done in this last expression.

=====Equipotential Contours that Lie Entirely Within Configuration=====
For all <math>0 < \phi_\mathrm{choice} \le \phi_\mathrm{choice} |_\mathrm{mid}</math>, the equipotential contour will reside entirely within the configuration. In this case, for a given <math>\phi_\mathrm{choice}</math>, we can plot points along the contour by picking (equally spaced?) values of <math>\chi_\mathrm{eqplane} \ge \chi \ge 0</math>, then solve the above quadratic equation for the corresponding value of <math>\zeta</math>.

In our example configuration, this means … (to be finished)

===Hydrostatic Balance (Algebraic Condition)===
Following our [[Apps/MaclaurinSpheroids#Equilibrium_Structure|separate discussion of the equilibrium structure]] of Maclaurin spheroids, and given that our solution of the Poisson equation fixes the expression for <math>\Phi_\mathrm{grav} </math>, the algebraic expression ensuring hydrostatic balance is,

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

<tr>
<td align="right">
<math>H(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B - \biggl[ \Phi_\mathrm{grav}(\varpi, z) + \Psi(\varpi, z) \biggr] \, ,
</math>
</td>
</tr>
</table>
where, <math>\Psi</math> is the centrifugal potential. <font color="red">NOTE:</font>   Generally when modeling axisymmetric astrophysical systems (see our [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying discussion of ''simple'' rotation profiles]]) it is assumed that <math>\Psi</math> does not functionally depend on <math>z</math>. Here, our other constraints — for example, demanding that the configuration have a parabolic density distribution — may force us to adopt a <math>z</math>-dependent rotation profile.

Here, we know that the adopted parabolic density distribution gives rise to a gravitational potential of the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

Hence,

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

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B - \Phi_\mathrm{grav}(\varpi, z) - H(\varpi, z)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
-
H_c h(\xi_1) \, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
We presume that the enthalpy profile, as well as the density profile, can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]]. In particular, defining, <math>q \equiv a_\ell/a_s</math> and,

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>
Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that,

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

<tr>
<td align="right">
<math>\frac{H}{H_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>h(\xi_1) \, .</math>
</td>
</tr>
</table>
If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that,

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

<tr>
<td align="right">
<math>h(\xi_1)</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>h_2 \xi_1^2 + h_4 \xi_1^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
+
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>
</table>

</td></tr></table>
Adopting this last expression for the enthalpy, we have,

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

<tr>
<td align="right">
<math>\frac{h(\xi_1)}{h_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]
-
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

At the pole of the configuration — that is, when <math>(\varpi, z) = (0, a_s)</math> — this statement of hydrostatic balance becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)} + A_s \biggl( \frac{a_s^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)}
+ A_{ss} a_\ell^2 \biggl(\frac{a_s^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^2} + \biggl(\frac{a_s}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^4} + \biggl(\frac{a_s}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
-
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
\, .
</math>
</td>
</tr>
</table>

For centrally condensed configurations, it is astrophysically reasonable to assume that <math>\Psi(\varpi, z)</math> is of the form such that the centrifugal potential goes to zero when <math>\varpi \rightarrow 0</math>. Adopting that assumption here means that the Bernoulli constant has the value,

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

<tr>
<td align="right">
<math>C_B</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<td align="left">
<math>
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
\, .
</math>
</td>
</tr>
</table>
Plugging this expression for <math>C_B</math> back into the general statement of hydrostatic balance gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\Psi(\varpi, z)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}
- A_s (1-e^2)
+ \frac{1}{2} A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr]
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\biggl[A_s (1-e^2)-A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- A_{ss} a_\ell^2 (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1} - 1\biggr]
+
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} -1
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2(1-e^2)^{-1}\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr]
+
h_4 a_s^4 (1-e^2)^{-2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2}
\biggr]
\biggr\}
</math>
</td>
</tr>
</table>

Let's set …
<div align="center">
<math>H_c h_0 = \pi G \rho_c a_\ell^2 \, ;</math>      
<math>h_2 = \frac{A_s(1-e^2)}{a_s^2} \, ;</math>      
<math>h_4 = - \frac{ A_{ss}a_\ell^2 (1-e^2)^2 }{ 2a_s^4 } \, .</math>
</div>
This gives,

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

<tr>
<td align="right">
<math>\frac{\Psi(\varpi, z)}{\pi G \rho_c a_\ell^2}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2}
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) \biggr]
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{
- A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+ \biggl[
\frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
+
\biggl\{
A_s\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) \biggr]
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+
\frac{A_{\ell \ell} a_\ell^2}{2} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
-
\frac{A_{ss} a_\ell^2}{2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2}
\biggr]
+ A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
-
A_{ss} a_\ell^2 \biggl[ \biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ A_s (1-e^2) - A_\ell \biggr] \biggl(\frac{\varpi^2}{a_\ell^2}\biggr)
+ \frac{1}{2}\biggl\{
A_{\ell \ell} a_\ell^2 - A_{ss} a_\ell^2 (1-e^2)^{2}
\biggr\} \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ \biggl\{ A_{\ell s}a_\ell^2
-
A_{ss} a_\ell^2 (1-e^2) \biggr\}\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\, .
</math>
</td>
</tr>
</table>

===2<sup>nd</sup> Try===

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">Keep in Mind, from Above</div>

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

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c a_\ell^2)}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\rho_c \biggl[ 1 - \biggl(\frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\, ,
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>

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

From our presentation of [[AxisymmetricConfigurations/PGE#Eulerian_Formulation_(CYL.)|the Eulerian formulation of the Euler equation in cylindrical coordinates]], we see that in steady-state axisymmetric flows, the two relevant equilibrium conditions are,

<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>~{\hat{e}}_\varpi</math>:    </td>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}
</math>
</td>
</tr>
<tr>
<td align="right"><math>~{\hat{e}}_z</math>:    </td>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math>
</td>
</tr>
</table>

====Vertical Component====
We will focus, first, on the vertical component. Specifically, since both <math>\rho</math> and <math>\Phi_\mathrm{grav}</math> are known, the vertical gradient of the (unknown) scalar pressure is
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{\partial P}{\partial z}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \rho ~ \frac{\partial}{\partial z} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>
</table>

Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell)} \cdot \frac{\partial P}{\partial z}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial z} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr] \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \biggl(\frac{\varpi^4}{a_\ell^4}\biggr)
+ A_{ss} a_\ell^2 \biggl(\frac{z^4}{a_\ell^4}\biggr)
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \zeta} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \chi^4
+ A_{ss} a_\ell^2 \zeta^4
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl\{
- 2A_s \zeta
+ \biggl[
2A_{ss} a_\ell^2 \zeta^3
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta
+ A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta
+ A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>
</table>

where (unlike above) we are using the dimensionless lengths, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. Continuing to streamline this function, we have,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \chi^2\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \zeta^2\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta - A_s \zeta + A_{ss} a_\ell^2 \zeta^3
\biggr]
- \biggl[
A_{\ell s}a_\ell^2 \chi^4 \zeta - A_s \chi^2 \zeta + A_{ss} a_\ell^2 \chi^2 \zeta^3
\biggr]
- \biggl[
A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_s \zeta^3 + A_{ss} a_\ell^2 \zeta^5
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2 - A_s \biggr]\zeta
+ A_{ss} a_\ell^2 \zeta^3
+ \biggl[
A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
- A_{ss} a_\ell^2 \chi^2 \zeta^3
+ \biggl[
A_s \zeta^3 - A_{\ell s}a_\ell^2 \chi^2\zeta^3 - A_{ss} a_\ell^2 \zeta^5
\biggr](1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
A_{\ell s}a_\ell^2 \chi^2 - A_s
+
A_s \chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
+
\biggl[ A_{ss} a_\ell^2 - A_{ss} a_\ell^2 \chi^2 + A_s(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \chi^2(1-e^2)^{-1} \biggr]\zeta^3
- A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4 \biggr]\zeta
+
\biggl\{
[A_s(1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\chi^2
\biggr\}\zeta^3
- A_{ss} a_\ell^2(1-e^2)^{-1} \zeta^5
\, .
</math>
</td>
</tr>
</table>
So, let's see what happens if we assume that the pressure has the form,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>p_0 + p_2 \zeta^2 + p_4\zeta^4 + p_6\zeta^6
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{(2\pi G\rho_c^2 a_\ell^2)} \cdot \frac{\partial P_\mathrm{vert}}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>2p_2 \zeta + 4p_4\zeta^3 + 6p_6\zeta^5 \, ,</math>
</td>
</tr>
</table>
in which case,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
p_0 + \frac{1}{2}\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4
\biggr] \zeta^2
+
\frac{1}{4}\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4
+
\frac{1}{6}\biggl[
- A_{ss} a_\ell^2 (1-e^2)^{-1}
\biggr]\zeta^6
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

<font color="red">REMINDER:</font>
From [[#2nd_Try|above]] …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\, .
</math>
</td>
</tr>
</table>

And, in the case of the spherically symmetric equilibrium configuration, the [[SSC/Structure/OtherAnalyticModels#Pressure|pressure distribution]] derived by {{ Prasad49 }} has the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P}{P_c}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)^2 \biggl[1 + \biggl(\frac{\rho}{\rho_c}\biggr)\biggr]
\, .
</math>
</td>
</tr>
</table>
In the context of rotationally flattened configurations, therefore, we might expect the (vertical) pressure distribution to be of the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P}{P_c}</math>
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl\{
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
-\chi^2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
-
\zeta^2(1-e^2)^{-1}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\sim</math>
</td>
<td align="left">
<math>
\biggl[2 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
\biggl\{
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1}\biggr]
+ \biggl[-\chi^2 + \chi^4 + \chi^2\zeta^2(1-e^2)^{-1}\biggr]
+
\biggl[- \zeta^2(1-e^2)^{-1} + \chi^2\zeta^2(1-e^2)^{-1} + \zeta^4(1-e^2)^{-2}\biggr]
\biggr\}
</math>
</td>
</tr>
</table>

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

====Radial Component====

Start with,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>- \frac{j^2 \rho}{\varpi^3}
+
\frac{\partial P}{\partial \varpi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \rho ~ \frac{\partial}{\partial \varpi} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>
</table>

Multiply thru by <math>1/(\pi G \rho_c^2 a_\ell)</math>:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>- \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr] \frac{j^2 \rho}{\varpi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\frac{\partial P}{\partial \varpi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell)} \biggr]\rho ~ \frac{\partial}{\partial \varpi} \biggl\{
\Phi_\mathrm{grav}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \frac{\partial}{\partial \chi} \biggl\{
\frac{\Phi_\mathrm{grav}}{(-\pi G \rho_c a_\ell^2)}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \cdot \frac{\partial}{\partial \chi} \biggl\{
\frac{1}{2} I_\mathrm{BT}
- \biggl[A_\ell \chi^2 + A_s \zeta^2 \biggr]
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2 \chi^4
+ A_{ss} a_\ell^2 \zeta^4
+ 2A_{\ell s}a_\ell^2 \chi^2\zeta^2
\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ 1 - \biggl[ \chi^2 + \zeta^2(1-e^2)^{-1}\biggr] \biggr\} \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\} \, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">EXACT!</font></div>

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\}
</math>
</td>
</tr>
</table>

</td></tr></table>
Continuing to streamline this function, we have,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}
-
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}\chi^2
-
\biggl\{\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3 \biggr\}(1-e^2)^{-1}\zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi+ 2 A_{\ell \ell} a_\ell^2 \chi^3
-
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi^3 - 2 A_{\ell \ell} a_\ell^2 \chi^5
+
\biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2 - 2 A_{\ell \ell} a_\ell^2 \chi^3 (1-e^2)^{-1}\zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+\biggl[2A_\ell - 2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi(1-e^2)^{-1} \zeta^2
+ \biggl[2 A_{\ell \ell} a_\ell^2
-2A_{\ell s}a_\ell^2 \zeta^2 + 2A_\ell - 2 A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
- 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center"><math>=</math></td>
<td align="left">
<math>
2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2
+ A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
+ 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell
-A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
- 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
\, .
</math>
</td>
</tr>
</table>

====Determine Specific Angular Momentum Distribution====

Now, from our analysis of the vertical component, we determined that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{12P_\mathrm{vert}}{(2\pi G\rho_c^2 a_\ell^2)} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
12p_0 + 6\biggl[
- A_s + (A_{\ell s}a_\ell^2 + A_s )\chi^2 - A_{\ell s}a_\ell^2 \chi^4
\biggr] \zeta^2
+
3\biggl\{[A_s (1-e^2)^{-1} + A_{ss} a_\ell^2] - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ]\chi^2 \biggr\}\zeta^4
+
2\biggl[
- A_{ss} a_\ell^2 (1-e^2)^{-1}
\biggr]\zeta^6
\, .
</math>
</td>
</tr>
</table>

<span id="RadialDerivative">The radial derivative of this function is</span>,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[ \frac{12}{(2\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
6\biggl[
2(A_{\ell s}a_\ell^2 + A_s ) \zeta^2 \chi - 4A_{\ell s}a_\ell^2\zeta^2 \chi^3
\biggr]
+
6\biggl\{ - [A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} ] \zeta^4 \chi \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3 \, .
</math>
</td>
</tr>
</table>

We hypothesize that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi}
-
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\biggl[\frac{\partial P}{\partial \chi}\biggr]_\mathrm{rad}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
- 2 \biggl[ - A_\ell(1-e^2)^{-1} + A_{\ell s}a_\ell^2 \zeta^2
+ A_\ell\zeta^2 - A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
- 2\biggl[A_{\ell \ell} a_\ell^2 + A_\ell
-A_{\ell s}a_\ell^2 \zeta^2 - A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
+ 2 A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>2\biggl\{
\biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4 \biggr] \chi
- 2A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl[ A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2
- A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
+ \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell
+ A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2\biggr] \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ - \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(2\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (A_{\ell s}a_\ell^2 + A_s )\zeta^2 + \frac{1}{2}[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}]\zeta^4
+ A_\ell(1-e^2)^{-1} - A_{\ell s}a_\ell^2 \zeta^2
- A_\ell\zeta^2 + A_{\ell s}a_\ell^2 \zeta^4(1-e^2)^{-1} \biggr] \chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl[ -A_{\ell \ell} a_\ell^2 - A_\ell
+ A_{\ell s}a_\ell^2 \zeta^2 + A_{\ell \ell} a_\ell^2 \zeta^2- 2A_{\ell s}a_\ell^2 \zeta^2\biggr] \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ A_\ell(1-e^2)^{-1} +(A_{\ell s}a_\ell^2 + A_s - A_{\ell s}a_\ell^2 - A_\ell)\zeta^2
+ \frac{1}{2}\biggl[- A_{ss} a_\ell^2 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}
+ 2A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
+ \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell)
+ (A_{\ell s}a_\ell^2 + A_{\ell \ell} a_\ell^2 - 2A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl\{ A_\ell(1-e^2)^{-1} + (A_s - A_\ell)\zeta^2
+ \frac{1}{2}\biggl[- A_{ss} a_\ell^2
+ A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]\zeta^4 \biggr\} \chi
+ \biggl\{ (-A_{\ell \ell} a_\ell^2 - A_\ell)
+ (A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2) \zeta^2 \biggr\} \chi^3
+ A_{\ell \ell} a_\ell^2 (1-e^2)^{-1} \chi^5
</math>
</td>
</tr>
</table>
Now, from [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|our layout of relevant index symbol expressions]], let's see if the coefficients of various ζ-dependent terms go to zero.

<font color="red">FIRST:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
A_{s\ell}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \frac{A_s - A_\ell}{(a_s^2 - a_\ell^2)} = \frac{A_s - A_\ell}{a_\ell^2 e^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~
A_{s \ell}a_\ell^2 e^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(A_s - A_\ell)
</math>
</td>
</tr>

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

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

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{e^2}\biggl[
3 - e^2 - 3(1-e^2)^{1 / 2}\frac{\sin^{-1}e}{e}
\biggr]
\, ;
</math>
</td>
</tr>
</table>

<font color="red">SECOND:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
3A_{s s}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_s^2} - 2A_{s \ell}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{3}{2}A_{s s}a_\ell^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{a_\ell^2}{a_s^2} - A_{s \ell}a_\ell^2 = (1 - e^2)^{-1} - A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ - A_{s s}a_\ell^2
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \biggl[ - A_{ss} a_\ell^2 + A_{\ell s}a_\ell^2 (1-e^2)^{-1} \biggr]
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{3}A_{s\ell}a_\ell^2 - \frac{2}{3}(1 - e^2)^{-1}
+ A_{\ell s}a_\ell^2 (1-e^2)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\frac{1}{3(1-e^2)}\biggl[
2A_{s\ell}a_\ell^2 (1-e^2) - 2
+ 3A_{\ell s}a_\ell^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{3(1-e^2)}\biggl[A_{s\ell}a_\ell^2 (5-2e^2) - 2 \biggr]\, ;
</math>
</td>
</tr>
</table>

<font color="red">THIRD:</font>
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
3A_{\ell \ell}</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_\ell^2} - A_{\ell \ell} - A_{s\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ 4A_{\ell \ell}a_\ell^2</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2 - A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
(A_{\ell \ell} a_\ell^2 - A_{\ell s}a_\ell^2)</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2} - \frac{5}{4}A_{s\ell}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{4}\biggl[2 - 5A_{s\ell}a_\ell^2\biggr] \, .
</math>
</td>
</tr>
</table>

===3<sup>rd</sup> Try===
From the [[#Radial_Component|above, "2<sup>nd</sup> Try" discussion of the radial component]], we can write the following "EXACT!" relation,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">EXACT!</font></div>

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\} \, .
</math>
</td>
</tr>
</table>

</td></tr></table>
Now, our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math> suggests that the left-hand-side of this expression should be of the form,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
LHS  
<math>
\equiv \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>\sim</math></td>
<td align="left">
<math>
c_2\zeta^2 + c_4\zeta^4 \, ,
</math>
</td>
</tr>
</table>
where it is understood that the coefficients, <math>c_2</math> and <math>c_4</math>, are both functions of <math>\chi</math>. This should be compared with the "EXACT!" expression for the RHS after multiplying through by the expression for the dimensionless density, that is,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
RHS
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\biggr] \cdot \biggl\{
\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi \zeta^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(1 - \chi^2)\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"> </td>
<td align="left">
<math>
- \biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr](1-e^2)^{-1}\zeta^2
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4
\, .
</math>
</td>
</tr>
</table>

Because we are not expecting to see a term that is independent of <math>\zeta</math>, this suggests that the term inside the large square brackets must be zero. This leads to an expression for the distribution of specific angular momentum of the form,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<table border="0" align="center" cellpadding="8">

<tr><td align="center" colspan="3"><font color="red">EXCELLENT !!</font></td></tr>

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 2 A_{\ell \ell} a_\ell^2 \chi^3
- 2A_\ell \chi +
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{j^2 }{(\pi G \rho_c a_\ell^4)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6
\, .
</math>
</td>
</tr>
</table>

According to our [[AxisymmetricConfigurations/SolutionStrategies#Specifying_Radial_Rotation_Profile_in_the_Equilibrium_Configuration|accompanying discussion of ''Simple'' rotation profiles]], the corresponding centrifugal potential is given by the expression,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>\Psi</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \int \frac{j^2(\varpi)}{\varpi^3} d\varpi
=
- (\pi G \rho_c a_\ell^2) \int \frac{1}{\chi^3} \biggl[2A_\ell \chi^4 - 2 A_{\ell \ell} a_\ell^2 \chi^6\biggr]d\chi
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- \int \biggl[2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr]d\chi
=
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, .
</math>
</td>
</tr>
</table>
(Here, we ignore the integration constant because it will be folded in with the Bernoulli constant.)

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

It also means that the RHS expression simplifies to the form,

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
RHS
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4 \, .
</math>
</td>
</tr>
</table>

This should be compared to our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math>, namely,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 + 2A_s )\zeta^2 - A_{ss} a_\ell^2 \zeta^4 - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
(2A_{\ell s}a_\ell^2 + 2A_s )\chi\zeta^2- 4A_{\ell s}a_\ell^2 \chi^3\zeta^2 - \biggl[A_{\ell s}a_\ell^2 (1-e^2)^{-1} + A_{ss} a_\ell^2\biggr]\chi\zeta^4

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

===4<sup>th</sup> Try===

In our [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|accompanying discussion of Ferrers Potential]], we have derived the expression for the gravitational potential inside (and on the surface of) a triaxial ellipsoid with a parabolic density distribution. Specifically, for
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\rho(\mathbf{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr]
\, ,</math>
</td>
</tr>
</table>
[[ThreeDimensionalConfigurations/FerrersPotential#GravFor1|we find]],
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2} \biggl(A_{11}x^4 + A_{22}y^4 + A_{33}z^4 \biggr)
\, .
</math>
</td>
</tr>
</table>
In this [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|same accompanying discussion]], we plugged this expression for the gravitational potential into the Poisson equation and demonstrated that it properly generates the expression for the parabolic density distribution. For the axisymmetric configuration being considered here — with the short axis aligned with <math>c = a_3 = a_s</math> — these two relations become,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \biggl( \frac{\varpi^2}{a_\ell^2} + \frac{z^2}{a_s^2}\biggr) \biggr]
=
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \frac{\varpi^2}{a_\ell^2} - A_s \frac{z^2}{a_\ell^2}
+ (A_{\ell s}a_\ell^2 )\frac{ \varpi^2z^2 }{a_\ell^4} + \frac{1}{2}(A_{s s} a_\ell^2) \frac{z^4}{a_\ell^4}
+ \frac{A_{\ell \ell}a_\ell^2}{2} \biggl[ \frac{(x^4 + 2 x^2y^2 + y^4 )}{a_\ell^4} \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, .
</math>
</td>
</tr>
</table>
where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>. (This matches the [[#Gravitational_Potential|expression derived above]].)

----

Discuss scalar relationship between the enthalpy <math>(H)</math> and the effective potential.

As has been detailed in [[AxisymmetricConfigurations/SolutionStrategies#Technique|an accompanying discussion of solution techniques]], a configuration will be in dynamic equilibrium if,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla\biggl[ H + \Phi_\mathrm{grav} + \Psi \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
0
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ H + \Phi_\mathrm{grav} + \Psi
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
constant
<math>
= C_B
</math>
</td>
</tr>
</table>

Given that, in our particular case, we have analytic expressions for <math>\Phi_\mathrm{grav}(\chi,\zeta)</math> and for <math>\Psi(\chi,\zeta)</math>, we deduce that, to within a constant, the enthalpy distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{\Phi_\mathrm{grav}}{{(\pi G\rho_c a_\ell^2)}} - \frac{\Psi}{{(\pi G\rho_c a_\ell^2)}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
-
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(A_{\ell s}a_\ell^2 )\chi^2 \biggr]
</math>
</td>
</tr>
</table>
Now, according to our [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|related discussion of index symbols]],

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>3A_{s s}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{2}{a_s^2} - 2A_{\ell s}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ 3A_{s s}a_\ell^2</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2(1-e^2)^{-1} - 2A_{\ell s}a_\ell^2
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~2(A_{\ell s}a_\ell^2)\chi^2 </math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Examining the radial derivative …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \chi}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \chi} \biggl\{
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[-3(A_{s s} a_\ell^2) + 2(1-e^2)^{-1} \biggr]\zeta^2\chi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2(A_{\ell s} a_\ell^2)\zeta^2\chi
\, .
</math>
</td>
</tr>
</table>
<font color="red">YES !!!</font> This matches the "radial" pressure-gradient, below.

Now, examining the vertical derivative …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{1}{(\pi G\rho_c a_\ell^2)} \frac{\partial H}{\partial \zeta}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \zeta} \biggl\{
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\partial}{\partial \zeta} \biggl\{
- A_s \zeta^2
+ \frac{1}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^4 + [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta^2 \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_s \zeta
+
\biggl[2(A_{s s} a_\ell^2) \zeta^3
+ [2(1-e^2)^{-1} - 3 (A_{s s} a_\ell^2)] \chi^2\zeta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 2A_s \zeta
+
\biggl[2(A_{s s} a_\ell^2) \zeta^3
+ 2(A_{\ell s} a_\ell^2) \chi^2\zeta \biggr]
</math>
</td>
</tr>
</table>
<font color="red">HURRAY !!!</font> This matches the "vertical" pressure-gradient, below.

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

----

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\}
</math>
</td>
</tr>
</table>
Plug in …

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3
\, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2 \chi^3
+
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
2A_{\ell s}a_\ell^2 \zeta^2 \chi
\biggr\}
</math>
</td>
</tr>
</table>



Hence, examination of the radial component leads to the following suggested expression for the pressure:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
- \chi^2
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
- \zeta^2(1-e^2)^{-1}
\biggl[ 2A_{\ell s}a_\ell^2 \zeta^2 \chi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \frac{P}{(\pi G \rho_c^2 a_\ell^2)}
</math>
</td>
<td align="center"><math>\sim</math></td>
<td align="left">
<math>
\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
- \frac{1}{2}\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^4\biggr]
- \zeta^2(1-e^2)^{-1} \biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ 1 - \frac{\chi^2}{2}
- \zeta^2(1-e^2)^{-1} \biggr]
\biggl[ A_{\ell s}a_\ell^2 \zeta^2 \chi^2\biggr]
\, .
</math>
</td>
</tr>
</table>

While examination of the vertical component leads to the following suggested expression for the pressure:

<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[1 - \frac{\chi^2}{2} - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
- \frac{\chi^2}{2}\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
</math>
</td>
</tr>
</table>

===Tentative Summary===

====Known Relations====

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

<tr>
<td align="left"><font color="orange"><b>Density:</b></font></td>
<td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td>
<td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}
- A_\ell \chi^2 - A_s \zeta^2
+ \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2) \chi^4 \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Specific Angular Momentum:</b></font></td>
<td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
2A_\ell \chi - 2 A_{\ell \ell} a_\ell^2 \chi^3
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Centrifugal Potential:</b></font></td>
<td align="right">
<math>
\frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2 \biggr]\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Enthalpy:</b></font></td>
<td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- A_s \zeta^2
+ \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) (\zeta^2 - 3\chi^2) + 2(1-e^2)^{-1}\chi^2 \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td>
<td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr]
</math>
</td>
</tr>

<tr>
<td align="left"><font color="orange"><b>Radial Pressure Gradient:</b></font></td>
<td align="right">
<math>
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)} \biggr]\frac{\partial P}{\partial \chi}
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
2A_{\ell s}a_\ell^2 \zeta^2 \chi
\biggr\}
</math>
</td>
</tr>
</table>

where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are:

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

<tr>
<td align="right"><math>I_\mathrm{BT}</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
A_\ell
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>A_s</math> </td>
<td align="center"><math>=</math> </td>
<td align="left">
<math>
\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{3}{2} a_\ell^2 A_{ss} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{( 4e^2 - 3 )}{e^4(1-e^2)}
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2)
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr]
\biggr\} \, ,
</math>
</td>
</tr>
</table>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2 \biggr]^{1 / 2} \, .
</math>
</div>

====Examine Behavior of Enthalpy====
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\xi_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_s}\biggr)^2 \biggr]^{1 / 2}
=
a_s\biggl[\chi^2 + \zeta^2 (1-e^2)^{-1}\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
</td>
</tr>
</table>

====Try to Construct Pressure Distribution====

Drawing from the expression for the vertical pressure gradient, namely,

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

<tr>
<td align="right">
<math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta
+ 2A_{ss} a_\ell^2 \zeta^3
\biggr] \, ,
</math>
</td>
</tr>
</table>
try the following pressure expression:

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

<tr>
<td align="right">
<math>\frac{P}{(\pi G\rho_c^2 a_\ell^2)} </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\biggl[f_0 \chi^2\zeta^2 + f_1 \zeta^2 \biggr]
</math>
</td>
</tr>
</table>
The vertical derivative of this expression is,

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

<tr>
<td align="right">
<math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)}\biggr] \frac{\partial P}{\partial \zeta} </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
\frac{\partial }{\partial \zeta}
\biggl[A_{\ell s}a_\ell^2 \chi^2\zeta^2 - A_s \zeta^2 \biggr]
+
\biggl[A_{\ell s}a_\ell^2 \chi^2\zeta^2 - A_s \zeta^2 \biggr]
\frac{\partial }{\partial \zeta}
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[\frac{\rho}{\rho_c} \biggr]
\biggl[2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta + 2A_{ss} a_\ell^2 \zeta^3 \biggr]
+
\biggl[A_{\ell s}a_\ell^2 \chi^2\zeta^2 - A_s \zeta^2 + \frac{1}{2}A_{ss} a_\ell^2 \zeta^4 \biggr]
\biggl[ - 2\zeta(1-e^2)^{-1} \biggr]
</math>
</td>
</tr>
</table>

=See Also=

{{ SGFfooter }}

ParabolicDensity/Axisymmetric/Structure

2024-08-06T21:41:51Z

70.180.56.157: /* Meridional Plane Equi-Potential Contours */

ParabolicDensity/Axisymmetric/Structure

2024-08-06T21:34:06Z

70.180.56.157: /* Meridional Plane Equi-Potential Contours */

SSC/Virial/PolytropesSummary

2024-07-15T21:21:40Z

70.180.56.157: Created page with "__FORCETOC__  =Virial Equilibrium of Adiabatic Spheres (Summary)= The summary presented here has been drawn from our accompanying detailed analysis of the structure of pressure-truncated polytropes. ==Detailed Force-Balanced Solution== As has been discussed in detail in another chapter, {{ Horedt70fu..."

__FORCETOC__


=Virial Equilibrium of Adiabatic Spheres (Summary)=

The summary presented here has been drawn from our
[[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying detailed analysis of the structure of pressure-truncated polytropes]].

==Detailed Force-Balanced Solution==

As has been [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|discussed in detail in another chapter]], {{ Horedt70full }}, {{ Whitworth81full }} and {{ Stahler83full }} have separately derived what the equilibrium radius, <math>~R_\mathrm{eq}</math>, is of a polytropic sphere that is embedded in an external medium of pressure, <math>~P_e</math>. Their solution of the detailed force-balanced equations provides a pair of analytic expressions for <math>~R_\mathrm{eq}</math> and <math>~P_e</math> that are parametrically related to one another through [[SSC/Structure/Polytropes#Lane-Emden_Equation|the Lane-Emden function]], <math>~\theta</math>, and its radial derivative. For example — see our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|related discussion for more details]] — from {{ Horedt70 }} we obtain the following pair of equations:
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = r_a \cdot \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggl[ \frac{4\pi}{(n+1)^n}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_\mathrm{e}}{P_\mathrm{norm}} = p_a \cdot \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)} \, ,
</math>
</td>
</tr>
</table>
</div>
where we have introduced the normalizations,

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

<tr>
<td align="right">
<math>~R_\mathrm{norm}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P_\mathrm{norm}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>

</table>
</div>

In the expressions for <math>~r_a</math> and <math>~p_a</math>, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, <math>~\xi_1</math>, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the ''isolated polytrope,'' but at the radial coordinate, <math>~\tilde\xi</math>, where the internal pressure of the isolated polytrope equals <math>~P_e</math> and at which the ''embedded'' polytrope is to be truncated. The coordinate, <math>~\tilde\xi</math>, therefore identifies the surface of the embedded — or, pressure-truncated — polytrope. We also have taken the liberty of attaching the subscript "limit" to <math>~M</math> in both defining relations because it is clear that {{ Horedt70 }} intended for the normalization mass to be the mass of the pressure-truncated object, not the mass of the associated ''isolated'' (and untruncated) polytrope.

From these previously published works, it is not obvious how — or even ''whether'' — this pair of parametric equations can be combined to directly show how the equilibrium radius depends on the value of the external pressure. Our examination of the free-energy of these configurations and, especially, an application of the viral theorem shows this direct relationship. Foreshadowing these results, we note that,

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

<tr>
<td align="right">
<math>~\biggl[ \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr) \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4\biggr]_\mathrm{Horedt} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}} \biggr]
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, ;
</math>
</td>
</tr>
</table>
</div>
or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = GM_\mathrm{tot}^2</math>, this can be rewritten as,

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

<tr>
<td align="right">
<math>~\biggl[ \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}} \, .
</math>
</td>
</tr>
</table>
</div>

==Free Energy Function and Virial Theorem==

The variation with size of the normalized free energy, <math>~\mathfrak{G}^*</math>, of pressure-truncated adiabatic spheres is described by the following,
<div align="center" id="FreeEnergyExpression">
<font color="#770000">'''Algebraic Free-Energy Function'''</font><br />

<math>
\mathfrak{G}^* =
-3\mathcal{A} \chi^{-1} +~ \frac{1}{(\gamma - 1)} \mathcal{B} \chi^{3-3\gamma} +~ \mathcal{D}\chi^3 \, .
</math>
</div>
In this expression, the size of the configuration is set by the value of the dimensionless radius, <math>~\chi \equiv R/R_\mathrm{norm}</math>; as is clarified, below, the values of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, characterize the relative importance, respectively, of the gravitational potential energy and the internal thermal energy of the configuration; <math>~\gamma</math> is the exponent (from the adopted equation of state) that identifies the adiabat along which the configuration heats or cools upon expansion or contraction; and the relative importance of the imposed external pressure is expressed through the free-energy expression's third constant coefficient, specifically,
<div align="center">
<math>~\mathcal{D} \equiv \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, .</math>
</div>
When examining a range of physically reasonable configuration sizes for a given choice of the constants <math>~(\gamma, \mathcal{A}, \mathcal{B}, \mathcal{D})</math>, a plot of <math>~\mathfrak{G}^*</math> versus <math>~\chi</math> will often reveal one or two extrema. Each extremum is associated with an equilibrium radius, <math>~\chi_\mathrm{eq} \equiv R_\mathrm{eq}/R_\mathrm{norm}</math>.

[[File:AdabaticBoundedSpheres_Virial.jpg|thumb|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]
Equilibrium radii may also be identified through an algebraic relation that originates from the scalar virial theorem — a theorem that, itself, is derivable from the free-energy expression by setting <math>~\partial\mathfrak{G}^*/\partial\chi = 0</math>. In our
[[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying detailed analysis of the structure of pressure-truncated polytropes]], we use the virial theorem to show that the equilibrium radii that are identified by extrema in the free-energy function always satisfy the following,
<div align="center" id="ConciseVirial">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} = \frac{(\Chi_\mathrm{ad}^{4-3\gamma} - 1)}{\Chi_\mathrm{ad}^4} \, ,
</math>
</div>
where, after setting <math>~\gamma = (n+1)/n</math>,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~\Pi_\mathrm{ad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\mathcal{D} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)}
\, ,
</math>         and,
</td>
</tr>

<tr>
<td align="right">
<math>~\Chi_\mathrm{ad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\chi_\mathrm{eq} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .
</math>
</td>
</tr>

</table>
</div>

The curves shown in the accompanying "pressure-radius" diagram trace out this derived virial-theorem function for six different values of the adiabatic exponent, <math>~\gamma</math>, as labeled. They show the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, that is required to construct a nonrotating, self-gravitating, adiabatic sphere with a dimensionless equilibrium radius <math>~\Chi_\mathrm{ad}</math>. The mathematical solution becomes unphysical wherever the pressure becomes negative.

If we multiply the above free=energy function through by an appropriate combination of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, and make the substitution, <math>~\gamma \rightarrow (n+1)/n</math>, it also takes on a particularly simple form featuring the newly defined dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, and the newly identified dimensionless radius, <math>~\Chi \equiv \chi(\mathcal{B}/\mathcal{A})^{n/(n-3)}</math>. Specifically, we obtain the,
<div align="center" id="RenormalizedFreeEnergyExpression">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />

<math>
\mathfrak{G}^{**} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} =
-3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, .
</math>
</div>

==Relationship to Detailed Force-Balanced Models==
===Structural Form Factors===

In our [[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying detailed analysis]], we demonstrate that the expressions given above for the free-energy function and the virial theorem are correct in sufficiently strict detail that they can be used to precisely match — and assist in understanding — the equilibrium of embedded polytropes whose structures have been determined from the set of detailed force-balance equations. In order to draw this association, it is only necessary to realize that, very broadly, the constant coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, in the above [[SSC/Virial/PolytropesSummary#FreeEnergyExpression|algebraic free-energy expression]] are expressible in terms of three [[SSCpt1/Virial#Structural_Form_Factors|structural form factors]], <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_W</math>, and <math>~\tilde\mathfrak{f}_A</math>, as follows:
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathcal{B}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)
\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A
= \frac{4\pi}{3}
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3(n+1)/n} \biggr]_\mathrm{eq}
\cdot \tilde\mathfrak{f}_A \, ;
</math>
</td>
</tr>
</table>
</div>
and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write …

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \tilde\theta^' \biggr]^2 + \tilde\theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>

<font color="red"><b>
January 13, 2015:
</b></font>
As is noted in our [[SSC/Virial/PolytropesEmbeddedOutline#Third_Effort|accompanying outline of work]], I no longer believe that <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math> have the same expressions as in isolated polytropes. Hence, all of the material that follows is suspect and needs to be reworked.

{{ SGFworkInProgress }}

After plugging these nontrivial expressions for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> into the righthand sides of the above equations for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> and, simultaneously, using Horedt's detailed force-balanced expressions for <math>~r_a</math> and <math>~p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>~P_e/P_\mathrm{norm}</math> in these same equations, we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Pi_\mathrm{ad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Chi_\mathrm{ad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
</td>
</tr>
</table>
</div>
where the newly identified, key physical parameter,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_\mathrm{ad} </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} \, .</math>
</td>
</tr>
</table>
</div>

It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[SSC/Virial/PolytropesSummary#ConciseVirial|virial theorem presented above]].

===Physical Meaning of Parameter <math>~\eta_\mathrm{ad}</math>===

As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>~\tilde\mathfrak{f}_A</math>. It is worth pointing out what physical quantities are associated with these two terms.

At any radial location within a polytropic configuration, the [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden function]], <math>~\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,
<div align="center">
<math>\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
</div>
Remembering that, at any location within the configuration, the pressure is related to the density via the polytropic equation of state,
<div align="center">
<math>P = K\rho^{(n+1)/n} \, ,</math>
</div>
we see that,
<div align="center">
<math>\frac{P}{P_c} = \theta^{n+1} \, .</math>
</div>
Hence, the quantity, <math>~\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>~\tilde\mathfrak{f}_A</math>, is the ratio, <math>~(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere. But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,
<div align="center">
<math>\tilde\theta^{n+1} = \frac{P_e}{P_c} \, .</math>
</div>

In our [[User:Tohline/SSC/Virial/Polytropes#Strategy2|accompanying detailed analysis]], we have employed the virial theorem expression to demonstrate that the first term in our definition of <math>~\tilde\mathfrak{f}_A</math> provides a measure the configuration's normalized central pressure. Specifically, we show that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[3 (n+1) (\tilde\theta^')^2]^{-1} \, .</math>
</td>
</tr>
</table>
</div>
We conclude, therefore, that quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>(5-n) \tilde\mathfrak{f}_A </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4} + (5-n) \frac{P_e}{P_c}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4} \biggl[1 + \eta_\mathrm{ad} \biggr] \, ,
</math>
</td>
</tr>
</table>
</div>
and that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_\mathrm{ad} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \, .</math>
</td>
</tr>
</table>
</div>

===Desired Pressure-Radius Relation===
It is now clear from our review, above, of [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution|Horedt's detailed force-balanced solution]], that
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi (5-n)}{3}\biggl[\frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\eta_\mathrm{ad} \, .</math>
</td>
</tr>
</table>
</div>
Hence, the pair of parametric equations obtained via a solution of the detailed force-balanced equations satisfy our, slightly rearranged,
<div align="center" id="ConciseVirial3">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = \Chi_\mathrm{ad}^{(n-3)/n} - 1 \, .
</math>
</div>
More to the point, it is now clear that this virial theorem expression provides the direct relationship between the configuration's dimensionless equilibrium radius as defined by Horedt, <math>~r_a</math>, and the dimensionless applied external pressure as defined by Horedt, <math>~p_a</math>, that was not apparent from the original pair of parametric relations. Horedt's parameters, <math>~r_a</math> and <math>~p_a</math>, can be directly associated to our parameters, <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math>, via two new normalizations, <math>~r_n</math> and <math>~p_n</math>, defined through the relations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Chi_\mathrm{ad} = \frac{r_a}{r_n}</math>
</td>
<td align="center">
     and     
</td>
<td align="left">
<math>~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .</math>
</td>
</tr>
</table>
</div>
Specifically in terms of the coefficients in the free-energy expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_n^{n-3}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{(n+1)^n}{4\pi} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^n
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{-n} \, ,
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~p_n^{n-3}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{3^{n-3}}{(4\pi)^4 (n+1)^{3(n+1)}} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^{-3(n+1)}
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{4n} \, ;
</math>
</td>
</tr>
</table>
</div>
while, in terms of the structural form factors,

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

<tr>
<td align="right">
<math>~r_n^{n-3}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{1}{3} \biggl[ \frac{(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n \mathfrak{f}_M^{1-n}
\, ,
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~p_n^{n-3}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{1}{(4\pi)^8} \biggl[ \frac{3\cdot 5^3}{(n+1)^3} \cdot \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W^3} \biggr]^{n+1} \mathfrak{f}_A^{4n}
\, .
</math>
</td>
</tr>
</table>
</div>



==Discussion==

===Model Sequences===
[[File:AdabaticBoundedSpheres_Virial.jpg|thumb|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]
After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), <math>~\gamma = (n+1)/n</math>, the functional form of the virial theorem expression, <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math>, is known and, hence, the equilibrium model sequence can be plotted. Half-a-dozen such model sequences are shown in the figure near the beginning of this discussion. Each curve can be viewed as mapping out a single-parameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, <math>~\eta_\mathrm{ad}</math>, over the physically relevant range, <math>0 \le \eta_\mathrm{ad} < \infty</math>. To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_\mathrm{ad} </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} = \biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Pi_\mathrm{ad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Chi_\mathrm{ad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
</td>
</tr>
</table>
</div>

====When <math>~\eta_\mathrm{ad}</math> is zero====
For the types of systems that are presently most relevant to astrophysical discussions, the key parameter, <math>~\eta_\mathrm{ad}</math>, can be zero for one of two reasons: Either <math>~n=5</math>; or <math>~\tilde\theta \rightarrow \theta_{\xi_1} = 0</math>. In the latter case, all curves converge on the same point, that is, <math>~(\Chi_\mathrm{ad}, \Pi_\mathrm{ad}) = (1, 0)</math>. This corresponds to the case of no external medium <math>~(P_e = 0)</math> and, hence, the associated equilibrium configuration is the familiar [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|''isolated'' polytropic sphere]]. As can be deduced from our above discussion of the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial|algebraic expression of the virial theorem]], because <math>~\Chi_\mathrm{ad} = 1</math>, the equilibrium radius of such a configuration is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \chi_\mathrm{eq}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>
As is demonstrated in an [[User:Tohline/SSC/Virial/Polytropes#Strategy2|accompanying discussion]] and also [[User:Tohline/SSC/Virial/PolytropesSummary#Physical_Meaning_of_Parameter|mentioned above]], after inserting the relevant expressions for the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, this provides the key relationship between the mass, equilibrium radius, and central pressure of an isolated polytrope, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{[4\pi (n+1) (\theta^')^2]_{\xi_1}} \, .</math>
</td>
</tr>
</table>
</div>
As we have [[User:Tohline/SSC/Virial/Polytropes#Central_and_Mean_Pressure|reviewed elsewhere]] — see also our [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|detailed discussion of isolated polytropes]] — this is a familiar relationship, appearing prominently in Chapter IV (p. 99, equations 80 and 81) of [[User:Tohline/Appendix/References|Chandrasekhar [C67]]] in association with his discussion of the dimensionless coefficient, <math>~W_n</math>, and the central pressure of polytropes.

In the former case — that is, in the case where <math>~\eta_\mathrm{ad} \rightarrow 0</math> because the chosen polytropic index is, <math>~n=5</math> — it must be the case that <math>~\Chi_\mathrm{ad} = 1</math> along the entire sequence (see the green curve labeled <math>~\gamma = (n+1)/n = 6/5</math> in the accompanying figure). This means that the expression for the central pressure,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{[4\pi (n+1) (\tilde\theta^')^2]} \, ,</math>
</td>
</tr>
</table>
</div>
does not explicitly depend on the size of the applied external pressure. But the central pressure ''does'' depend on the radial location at which the configuration is truncated, via the parameter <math>~\tilde\theta^'</math>, which is evaluated at <math>~\tilde\xi</math>, rather than at <math>~\xi_1</math>.

===Stability===

Analysis of the free-energy function allows us to not only ascertain the equilibrium radius of isolated polytropes and pressure-truncated polytropic configurations, but also the relative stability of these configurations. We begin by repeating the,
<div align="center" id="RenormalizedFreeEnergyExpression2">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />

<math>
\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, .
</math>
</div>
The first and second derivatives of <math>~\mathfrak{G}^{**}</math>, with respect to the dimensionless radius, <math>~\Chi</math>, are, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial\mathfrak{G}^{**}}{\partial\Chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3 \Chi^{-2} -3\Chi^{-(n+3)/n} + 3\Pi_\mathrm{ad} \Chi^2 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-6 \Chi^{-3} + \frac{3(n+3)}{n} \Chi^{-(2n+3)/n} + 6\Pi_\mathrm{ad} \Chi \, .</math>
</td>
</tr>
</table>
</div>
As alluded to, above, equilibrium radii are identified by values of <math>~\Chi</math> that satisfy the equation, <math>\partial\mathfrak{G}^{**}/\partial\Chi = 0</math>. Specifically, marking equilibrium radii with the subscript "ad", they will satisfy the
<div align="center" id="ConciseVirial2">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
</math>
</div>
Dynamical stability then depends on the sign of the second derivative of <math>~\mathfrak{G}^{**}</math>, evaluated at the equilibrium radius; specifically, configurations will be stable if,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}\biggr|_{\Chi_\mathrm{ad}}</math>
</td>
<td align="center">
<math>~></math>
</td>
<td align="left">
<math>~0 \, ,</math>      (stable)
</td>
</tr>
</table>
</div>
and they will be unstable if, upon evaluation at the equilibrium radius, the sign of the second derivative is less than zero. Hence, isolated polytropes as well as pressure-truncated polytropic configurations will be stable if,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~< </math>
</td>
<td align="left">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ - 2 + \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~< </math>
</td>
<td align="left">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl\{ \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2[\Chi_\mathrm{ad}^{(n-3)/n} -1] - 2\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~< </math>
</td>
<td align="left">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ \frac{3(n+1)}{n} \Chi_\mathrm{ad}^{(n-3)/n} - 4\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~\Chi_\mathrm{ad}</math>
</td>
<td align="center">
<math>~> </math>
</td>
<td align="left">
<math>~\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, .</math>      (stable)
</td>
</tr>
</table>
</div>
Reference to this stability condition proves to be simpler if we define the limiting configuration size as,
<div align="center">
<math>~\Chi_\mathrm{min} \equiv \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, ,</math>
</div>
and write the stability condition as,
<div align="center">
<math>~\Chi_\mathrm{ad} > \Chi_\mathrm{min} \, .</math>      (stable)
</div>

When examining the equilibrium sequences found in the upper-righthand quadrant of the figure at the top of this page — each corresponding to a different value of the polytropic index, <math>~n > 3</math> or <math>~n < 0</math> — we find that <math>~\Chi_\mathrm{min}</math> corresponds to the location along each sequence where the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, reaches a maximum. (Keeping in mind that the virial theorem defines each of these sequences, this statement of fact can be checked by identifying where the condition, <math>~\partial\Pi_\mathrm{ad}/\partial\Chi_\mathrm{ad} = 0</math>, occurs according to the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial2|algebraic expression of the virial theorem]].) Hence, we conclude that, along each sequence, no equilibrium configurations exist for values of the dimensionless external pressure that are greater than,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Pi_\mathrm{max}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\Chi_\mathrm{min}^{-4} \biggl[ \Chi_\mathrm{min}^{(n-3)/n} - 1 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{3(n+1)}{4n} \biggr]^{4n/(n-3)} \biggl[\frac{4n}{3(n+1)} - 1 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{ \biggl[ \frac{3(n+1)}{4n} \biggr]^{4n} \biggl[\frac{n-3}{3(n+1)} \biggr]^{n-3} \biggr\}^{1/(n-3)}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~\Pi_\mathrm{max}^{n-3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(4n)^{-4n}~[3(n+1)]^{3(n+1)} ~(n-3)^{n-3} \, .</math>
</td>
</tr>
</table>
</div>

In the context of a general examination of the free-energy of pressure-truncated polytropes, it is worth noting that this limit on the external pressure also establishes a limit on the coefficient, <math>~\mathcal{D}</math>, that appears in the free energy function. Specifically, we will not expect to find any extrema in the free energy if,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{D} > \mathcal{D}_\mathrm{max}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~(n-3) \biggl\{ \biggl[ \frac{\mathcal{B}}{4n} \biggr]^{4n}~\biggl[ \frac{3(n+1)}{\mathcal{A}} \biggr]^{3(n+1)} ~\biggr\}^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>

Finally, it is worth noting that the point along each equilibrium sequence that is identified by the coordinates, <math>~(\Chi_\mathrm{min}, \Pi_\mathrm{max})</math> always corresponds to,
<div align="center">
<math>~\eta_\mathrm{ad} = \eta_\mathrm{crit} \equiv \frac{n-3}{3(n+1)} \, .</math>
</div>

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Summary
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\eta_\mathrm{crit}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \frac{n-3}{3(n+1)} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Pi_\mathrm{max}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~(n-3) \biggl\{~\frac{ [3(n+1)]^{3(n+1)} }{(4n)^{4n}} \biggr\}^{1/(n-3)} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Chi_\mathrm{min} </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}
</math>
</td>
</tr>
</table>

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

===Try Polytropic Index of 4===
====Groundwork====
In an effort to more fully understand what can be learned from an examination of the free-energy, let's play with <math>~n=4</math> polytropic models. First, let's plot <math>~\mathfrak{G}^{**}(\Chi)</math> using a specific, trial value of the coefficient, <math>~\Pi_\mathrm{ad}</math>, keeping in mind that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_\mathrm{crit}\biggr|_{n=4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{15} = 0.066667 \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Chi_\mathrm{min}\biggr|_{n=4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{16}{15} \biggr)^4 = 1.294538 \, .</math>
</td>
</tr>
</table>
</div>

At the top of the table, shown below, we display a plot of the,
<div align="center" id="RenormalizedFreeEnergyExpression2">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />

<math>
\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, ,
</math>
</div>
where we have set <math>~n = 4</math>, and <math>~\Pi_\mathrm{ad} = 0.01</math>. Reading quantities off of the plot, the left and right extrema identify equilibria having the following approximate dimensionless radii: <math>~\Chi_\mathrm{left} \approx 1.03</math> and <math>~\Chi_\mathrm{right} \approx 2.13</math>. Upon closer examination (plots not shown), we have determined that, <math>~\Chi_\mathrm{left} \approx 1.0494</math> and <math>~\Chi_\mathrm{right} \approx 2.13905</math>. In accordance with our stability analysis, these values of <math>~\Chi_\mathrm{ad}</math> fall on either side of the demarcation value, <math>~\Chi_\mathrm{min} = (16/15)^4</math>, with the one on the left being a local maximum in the free energy — indicating an unstable equilibrium — while the one on the right is a local minimum — indicating a stable equilibrium. Next, let's check to see if both extrema satisfy the,
<div align="center" id="ConciseVirial2">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
</math>
</div>
For the unstable equilibrium configuration, we calculate,
<div align="center">
<math>\Pi_\mathrm{ad} \approx [(1.0494)^{1/4} - 1]/(1.0494)^4 = 1.000024 \times 10^{-2}</math>;
</div>
while, for the stable equilibrium we calculate,
<div align="center">
<math>\Pi_\mathrm{ad} \approx [(2.13905)^{1/4} - 1]/(2.13905)^4 = 1.000018 \times 10^{-2}</math>.
</div>
Because we inserted a value of <math>~\Pi_\mathrm{ad} = 0.01</math> into the free-energy expression, we conclude that, as desired, both identified extrema satisfy the virial relation to the measured accuracy. These parameter values, and the corresponding values of many other related physical parameters are summarized in the following table, along with the algebraic relations that were used to calculate them.

====First Table====

<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="4">
[[File:TryN4Pi0.01.png|450px|Dimensionless Free-Energy Curve]]
</th>
</tr>
<tr>
<th align="center" colspan="4">
Determined from Plot of Renormalized Free-Energy

with <math>~(n, \Pi_\mathrm{ad}) = (4, 0.01)</math>
</th>
</tr>
<tr>
<th align="center"> </th>
<th align="center"> </th>
<th align="center" width="25%">Maximum</th>
<th align="center" width="25%">Minimum</th>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~\Chi</math> </td>
<td align="center"> <math>~1.0494</math> </td>
<td align="center"> <math>~2.13905</math> </td>
</tr>
<tr>
<th align="center" colspan="4">
Immediate Implications from Virial Theorem
</th>
</tr>
<tr>
<th align="center"><math>~\Chi^{1/4} - 1</math></th>
<td align="center"> <math>~\eta_\mathrm{ad}</math> </td>
<td align="center"> <math>~0.012128</math></td>
<td align="center"> <math>~0.20936</math> </td>
</tr>
<tr>
<th align="center"><math>~(\Chi^{1/4} - 1)\cdot \Chi^{-4}</math></th>
<td align="center"> <math>~\Pi_\mathrm{ad}</math> </td>
<td align="center"> <math>~1.000024 \times 10^{-2}</math></td>
<td align="center"> <math>~1.000018 \times 10^{-2}</math> </td>
</tr>
<tr>
<th align="center" colspan="4">
Associated Detailed Force-Balanced Model Parameters

obtained via interpolation of tabulated numbers on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]
</th>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~\tilde\xi</math> (approx.) </td>
<td align="center"> <math>~4.81</math></td>
<td align="center"><math>~1.624</math></td>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~\tilde\theta</math> (approx.) </td>
<td align="center"> <math>~0.251</math></td>
<td align="center"><math>~0.709</math></td>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
<td align="center"><math>~0.0727</math></td>
<td align="center"> <math>~0.239</math></td>
</tr>
<tr>
<th align="center"> <math>~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2}</math></th>
<td align="center"> <math>~\eta</math> (check) </td>
<td align="center"><math>~0.0126</math></td>
<td align="center"> <math>~0.2091</math></td>
</tr>
<tr>
<th align="center" colspan="4">
and, hence, Implied Structural Form Factors & Coefficients <math>~\mathcal{B}</math> & <math>~\mathcal{A}</math>
</th>
</tr>
<tr>
<th align="center"> <math>~3(-\tilde\theta^')/\tilde\xi</math></th>
<td align="center"> <math>~\mathfrak{f}_M</math></td>
<td align="center"> <math>~0.0453</math></td>
<td align="center"><math>~0.4415</math></td>
</tr>
<tr>
<th align="center"> <math>~5[3(-\tilde\theta^')/\tilde\xi]^2</math></th>
<td align="center"> <math>~\mathfrak{f}_W</math></td>
<td align="center"> <math>~0.01028</math></td>
<td align="center"><math>~0.975</math></td>
</tr>
<tr>
<th align="center"> <math>~15(-\tilde\theta^')^2 + \tilde\theta^5</math></th>
<td align="center"> <math>~\mathfrak{f}_A</math></td>
<td align="center"> <math>~0.08028</math></td>
<td align="center"><math>~1.036</math></td>
</tr>
<tr>
<th align="center"> <math>~\biggl(\frac{3}{4\pi} \biggr)^{1/4} \mathfrak{f}_M^{-5/4} \cdot \mathfrak{f}_A</math></th>
<td align="center"> <math>~\mathcal{B}</math></td>
<td align="center"> <math>~2.682</math></td>
<td align="center"><math>~2.0122</math></td>
</tr>
<tr>
<th align="center"> <math>~\frac{\tilde\mathfrak{f}_W}{5 \tilde\mathfrak{f}_M^2} </math></th>
<td align="center"> <math>~\mathcal{A}</math></td>
<td align="center"> <math>~1</math></td>
<td align="center"><math>~1</math></td>
</tr>
<tr>
<th align="center" colspan="4">
Given <math>~\Pi_\mathrm{ad}</math>, <math>~\Chi</math>, and <math>~\mathcal{B}</math>, we obtain
</th>
</tr>
<tr>
<th align="center"> <math>~\frac{3}{4\pi}\mathcal{D} = \frac{3}{4\pi} \Pi_\mathrm{ad} \mathcal{B}^{16} </math></th>
<td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
<td align="center"> <math>~1.71 \times 10^4</math></td>
<td align="center"><math>~1.72 \times 10^2</math></td>
</tr>
<tr>
<th align="center"> <math>~\Chi \mathcal{B}^{-4}</math></th>
<td align="center"> <math>~\chi_\mathrm{eq}</math></td>
<td align="center"> <math>~0.0203</math></td>
<td align="center"><math>~0.1305</math></td>
</tr>
<tr>
<th align="center" colspan="4">
Compare with Horedt's Equilibrium Parameters obtained from DFB Models
</th>
</tr>
<tr>
<th align="center"><math>\biggl[ \biggl( \frac{5^3}{4\pi} \biggr)
\tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5}
</math>
</th>
<td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
<td align="center"> <math>~1.76 \times 10^4</math></td>
<td align="center"><math>~1.73 \times 10^2</math></td>
</tr>
<tr>
<th align="center"><math>
\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}
</math>
</th>
<td align="center"> <math>~\chi_\mathrm{eq}</math></td>
<td align="center"> <math>~0.0203</math></td>
<td align="center"><math>~0.130</math></td>
</tr>
</table>

Now, we are convinced that both extrema identify perfectly valid equilibrium configurations. However, in the context of astrophysics, the two identified equilibria are not connected to one another in any meaningful way. In particular, two of the free-energy coefficients, <math>~\mathcal{B}</math> and <math>~\mathcal{D}</math>, have different values in the two cases; and, by inference, the normalized external pressure, <math>~P_e/P_\mathrm{norm}</math>, is different in the two cases. So the plotted free-energy curve does not represent a "constant pressure" evolutionary trajectory. How do we identify two equilibria that are associated with the same normalized external pressure? And how do we identify the free-energy "evolutionary trajectory" that connects the two states?

====Second Table====

Here, we have decided to look for a stable equilibrium state that is bounded by the same external pressure as the ''unstable'' state that has been identified in the above figure and table. Rather than going straight to the free-energy expression in search of the desired stable configuration, we cheated a bit. Using the properties of an <math>~n=4</math> polytrope, as tabulated on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)], in conjunction with the algebraic expression found in the next-to-last row of the above table, namely,
<div align="center">
<math>
\frac{P_e}{P_\mathrm{norm}} = \biggl[ \biggl( \frac{5^3}{4\pi} \biggr) \tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5} \, ,
</math>
</div>
we examined how <math>~P_e</math> varies with <math>~\tilde\xi</math>. We found that <math>~P_e/P_\mathrm{norm} = 1.71\times 10^4</math> at <math>~\tilde\xi = 2.6</math>, which is almost identical to the value of the normalized external pressure that we determined was associated with the unstable equilibrium state (at <math>~\tilde\xi = 4.81</math>) above. As is illustrated by the figure and table that follows, we determined that the stable equilibrium state associated with this normalized external pressure is the minimum that occurs on the free energy curve having parameters, <math>~(n, \Pi_\mathrm{ad}) = (4, 0.02369)</math>.

<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="4">
[[File:TryN4Pi0.0237.png|450px|Dimensionless Free-Energy Curve]]
</th>
</tr>
<tr>
<th align="center" colspan="4">
Determined from Plot of Renormalized Free-Energy

with <math>~(n, \Pi_\mathrm{ad}) = (4, 0.02369)</math>
</th>
</tr>
<tr>
<th align="center"> </th>
<th align="center"> </th>
<th align="center" width="25%">Maximum</th>
<th align="center" width="25%">Minimum</th>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~\Chi</math> </td>
<td align="center"> <math>~1.274</math> </td>
<td align="center"> <math>~1.317</math> </td>
</tr>
<tr>
<th align="center" colspan="4">
Immediate Implications from Virial Theorem
</th>
</tr>
<tr>
<th align="center"><math>~\Chi^{1/4} - 1</math></th>
<td align="center"> <math>~\eta_\mathrm{ad}</math> </td>
<td align="center"> <math>~0.0624</math></td>
<td align="center"> <math>~0.0713</math> </td>
</tr>
<tr>
<th align="center"><math>~(\Chi^{1/4} - 1)\cdot \Chi^{-4}</math></th>
<td align="center"> <math>~\Pi_\mathrm{ad}</math> </td>
<td align="center"> <math>~0.02369</math></td>
<td align="center"> <math>~0.02369 </math> </td>
</tr>
<tr>
<th align="center" colspan="4">
Associated Detailed Force-Balanced Model Parameters

obtained via interpolation of tabulated numbers on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]
</th>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~\tilde\xi</math> (approx.) </td>
<td align="center"> ---- </td>
<td align="center"><math>~2.6</math></td>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~\tilde\theta</math> (approx.) </td>
<td align="center">---- </td>
<td align="center"><math>~0.5048</math></td>
</tr>
<tr>
<th align="center"> </th>
<td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
<td align="center"> ---- </td>
<td align="center"> <math>~0.175</math></td>
</tr>
<tr>
<th align="center"> <math>~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2}</math></th>
<td align="center"> <math>~\eta</math> (check) </td>
<td align="center"> ---- </td>
<td align="center"> <math>~0.0714</math></td>
</tr>
<tr>
<th align="center" colspan="4">
and, hence, Implied Structural Form Factors & Coefficients <math>~\mathcal{B}</math> & <math>~\mathcal{A}</math>
</th>
</tr>
<tr>
<th align="center"> <math>~3(-\tilde\theta^')/\tilde\xi</math></th>
<td align="center"> <math>~\mathfrak{f}_M</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~0.2019</math></td>
</tr>
<tr>
<th align="center"> <math>~5[3(-\tilde\theta^')/\tilde\xi]^2</math></th>
<td align="center"> <math>~\mathfrak{f}_W</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~0.2039</math></td>
</tr>
<tr>
<th align="center"> <math>~15(-\tilde\theta^')^2 + \tilde\theta^5</math></th>
<td align="center"> <math>~\mathfrak{f}_A</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~0.4922</math></td>
</tr>
<tr>
<th align="center"> <math>~\biggl(\frac{3}{4\pi} \biggr)^{1/4} \mathfrak{f}_M^{-5/4} \cdot \mathfrak{f}_A</math></th>
<td align="center"> <math>~\mathcal{B}</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~2.542</math></td>
</tr>
<tr>
<th align="center"> <math>~\frac{\tilde\mathfrak{f}_W}{5 \tilde\mathfrak{f}_M^2} </math></th>
<td align="center"> <math>~\mathcal{A}</math></td>
<td align="center"> <math>~1</math></td>
<td align="center"><math>~1</math></td>
</tr>
<tr>
<th align="center" colspan="4">
Given <math>~\Pi_\mathrm{ad}</math>, <math>~\Chi</math>, and <math>~\mathcal{B}</math>, we obtain
</th>
</tr>
<tr>
<th align="center"> <math>~\frac{3}{4\pi}\mathcal{D} = \frac{3}{4\pi} \Pi_\mathrm{ad} \mathcal{B}^{16} </math></th>
<td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~1.71 \times 10^4</math></td>
</tr>
<tr>
<th align="center"> <math>~\Chi \mathcal{B}^{-4}</math></th>
<td align="center"> <math>~\chi_\mathrm{eq}</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~0.0316</math></td>
</tr>
<tr>
<th align="center" colspan="4">
Compare with Horedt's Equilibrium Parameters obtained from DFB Models
</th>
</tr>
<tr>
<th align="center"><math>\biggl[ \biggl( \frac{5^3}{4\pi} \biggr)
\tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5}
</math>
</th>
<td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~1.71 \times 10^4</math></td>
</tr>
<tr>
<th align="center"><math>
\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}
</math>
</th>
<td align="center"> <math>~\chi_\mathrm{eq}</math></td>
<td align="center"> ---- </td>
<td align="center"><math>~0.0316</math></td>
</tr>
</table>

====Summary====

The algebraic free-energy function associated with pressure-truncated <math>~n=4</math> polytropes is,
<div align="center">
<math>
\mathfrak{G}^*\biggr|_{n=4} =
-3\mathcal{A} \chi^{-1} +~ 4\mathcal{B} \chi^{-3/4} +~ \mathcal{D}\chi^3 \, ,
</math>
</div>
and the corresponding ''renormalized'' free-energy function is,
<div align="center">
<math>
\mathfrak{G}^{**}\biggr|_{n=4} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} =
-3 \Chi^{-1} +~ 4\Chi^{-3/4} +~ \Pi_\mathrm{ad}\Chi^3 \, .
</math>
</div>
As has been demonstrated, above, the two equilibrium states that are supported by the same external pressure of, <math>~P_e/P_\mathrm{norm} = 1.71 \times 10^4</math>, are associated with extrema found in the following free-energy curves: The ''unstable'' equilibrium appears as a relative ''maximum'' in the free-energy curves having the coefficient values,
<div align="center">
<math>~\Pi_\mathrm{ad} = 0.01</math>      or     
<math>(\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.682, 7.16\times 10^4) \, .</math>
</div>
The ''stable'' equilibrium appears as a relative ''minimum'' in the free-energy curves having the coefficient values,
<div align="center">
<math>~\Pi_\mathrm{ad} = 0.02369</math>      or     
<math>(\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.542, 7.16\times 10^4) \, .</math>
</div>

<table border="1" cellpadding="8" align="center" width="75%">
<tr>
<th align="center">
Configurations Sharing the Same External Pressure
</th>
</tr>
<tr>
<td align="left">
'''<font color="maroon">ASIDE:</font>''' In retrospect, it is obvious that pairs of truncated equilibrium configurations of a given polytropic index that are bounded by the same external pressure — and, hence, that may share a ''physical'' evolutionary connection — will share the same value of Horedt's dimensionless pressure,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
~p_a
</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

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

The implication is that a single free-energy curve with ''constant'' coefficients cannot connect the two equilibrium states. There are certainly two separate equilibrium states that can be supported by the specified external pressure, but these two states exhibit somewhat different values of the structural form factors, which leads to different values of the coefficient, <math>~\mathcal{B}</math>. The righthand plot in the following figure shows how <math>~\mathcal{B}</math> varies with the applied external pressure in <math>~n=4</math> polytropes.

<table border="1" align="center" cellpadding="8">
<tr>
<th align="center">
Variation of Various Physical Parameters along the Sequence of Pressure-Truncated <math>~n=4</math> Polytropes

[Structural data obtained from the table provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]]

</th>
</tr>
<tr><td align="center">
[[File:SecondN4Parameters.png|750px|Parameters for n = 4 Embedded Polytropes]]
</td></tr>
<tr>
<td align="left">
'''<font color="maroon">Left:</font>''' This log-log plot displays the variation with applied external pressure, <math>~p_a</math> (increasing to the right along the horizontal axis), of the renormalized pressure, <math>~\Pi_\mathrm{ad}</math> (light blue diamonds), the renormalized equilibrium radius, <math>~\Chi_\mathrm{ad}</math> (light green triangles), and the key physical parameter, <math>~\eta_\mathrm{ad}</math> (maroon circles). As the diagram illustrates, each parameter is double-valued, demonstrating that, for any choice of the dimensionless external pressure (as long as the pressure is less than a well-defined limiting value), there are two available equilibrium states. Along all three curves, parameter values associated with the ''stable'' equilibrium are traced by the ''upper'' portion of the curve. The red vertical line has been drawn at the value of <math>~{p_a} = 0.176</math>, corresponding to the external pressure <math>~(P_e/P_\mathrm{norm} = 1.71\times 10^4)</math> examined in the above two tables. This red line intersects the <math>~\Pi(p_a)</math> curve at <math>~\Pi = 0.01</math> (unstable state examined above) and at <math>~\Pi = 0.02369</math> (stable state examined above).

'''<font color="maroon">Right:</font>''' This plot (linear scale on both axes) shows how <math>~\mathcal{B}</math> (curve outlined by light blue diamonds) varies with the applied external pressure, <math>~P_e/P_\mathrm{norm}</math>, in <math>~n=4</math> polytropes. The curve bends back on itself, showing that at any value of <math>~P_e</math>, below some limiting value, two equilibrium configurations exist and they have different values of <math>~\mathcal{B}</math>. The vertical red line identifies the value of the external pressure <math>~(P_e/P_\mathrm{norm} = 1.71\times 10^4)</math> that has been used as an example in the above two tables to illustrate how a pair of ''physically associated'' equilibrium states can be identified. This red line intersects the displayed curve at <math>~\mathcal{B} = 2.682</math> (unstable state examined above) and at <math>~\mathcal{B} = 2.542</math> (stable state examined above).
</td>
</tr>
</table>

====Curiosity====

[[File:PiVersusPa.png|thumb|300px|Pressure vs. pressure plot]]
The figure displayed here, on the right, is a magnification of a segment of the <math>~\Pi(p_a)</math> curve (light blue diamonds) shown in the lefthand panel of the preceding figure, although here we have used a linear, rather than a log, scale on both axes. The quantity being plotted along both axes is the external pressure, but normalized in different ways. The quantity, <math>~p_a</math> (horizontal axis), provides a direct measure of the physical external (hence, also, surface) pressure, while the quantity, <math>~\Pi</math> (vertical axis), is the external pressure ''renormalized'' by a specific combination of the free-energy coefficients. Our stability analysis has been conducted assuming that the free-energy coefficients — which are expressible in terms of structural form factors — are constants, that is, they do not vary with the size of the configuration. Hence, it is the limiting value of <math>~\Pi_\mathrm{ad}</math>, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ,</math>
</td>
</tr>
</table>
</div>
that identifies the demarcation between stable and unstable states. This limiting value is identified by the horizontal red-dashed line in the figure; and the relevant demarcation point appears where this tangent line touches the curve. According to our stability analysis, equilibrium configurations to the left of this demarcation point are stable while configurations to the right are unstable.

In the context of our discussion of the lefthand diagram in the preceding figure — see especially the relevant figure caption — we claimed that, for each physically allowed value of the external pressure, <math>~p_a</math>, the parameter, <math>~\Pi</math>, was double-valued and that configurations along the ''upper'' segment of its curve were stable. After studying a magnification of this parameter curve near its turning point, a bit of clarification is required. It appears as though equilibrium models lying along the short ''upper'' segment of the curve that falls between the demarcation/tangent point at <math>~\Pi_\mathrm{max}</math> and the maximum value of <math>~p_a</math> are unstable. This means that, even though two equilibrium configurations can be constructed at each value of <math>~p_a</math> in this region near and including the turning point, ''both'' configurations are dynamically unstable. We conclude, therefore, that stable configurations only exist for values of <math>~p_a</math> that are less than the value associated with <math>~\Pi_\mathrm{max}</math>.

==Mass-Radius Relation==
Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated polytropes. In effect, we have viewed the problem through the same lens as did [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and, separately, [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)], defining variable normalizations in terms of the polytropic constant, <math>~K</math>, and the configuration mass, <math>~M_\mathrm{tot}</math>, which were both assumed to be held fixed throughout the analysis. Here we switch to the approach championed by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)], defining variable normalizations in terms of <math>~K</math> and <math>~P_e</math>, and examining the ''mass-radius'' relationship of pressure-truncated polytropes.

===Detailed Force-Balanced Solution===
As has been summarized in our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying review]] of detailed force-balanced models of pressure-truncated polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{M_\mathrm{tot}}{M_\mathrm{SWS} }
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \tilde\theta^{(n-3)/2} (- \tilde\xi^2 \tilde\theta^') \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{(n-1)/2} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>M_\mathrm{SWS} \equiv
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
</div>
<div align="center">
<math>
R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .
</math>
</div>

===Mapping from Above Discussion===
Looking back on the definitions of <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> that we introduced in connection with our initial [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial|concise algebraic expression of the virial theorem]], we can write,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~P_e </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~P_\mathrm{norm} \biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~R_\mathrm{eq} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~R_\mathrm{norm} \Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)}
\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>

</table>
</div>
The first of these two expressions can be flipped around to give an expression for <math>~M_\mathrm{tot}</math> in terms of <math>~P_e</math> and, then, as normalized to <math>~M_\mathrm{SWS}</math>. Specifically,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~ M_\mathrm{tot}^{2(n+1)}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~M_\mathrm{SWS}^{2(n+1)} \biggl( \frac{n}{n+1} \biggr)^{3(n+1)}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow~~~ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\biggl( \frac{n}{n+1} \biggr)^{3/2}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)/[2(n+1)]} \biggl[ \frac{\mathcal{B}^{2n/(n+1)}}{\mathcal{A}^{3/2}} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

This means, as well, that we can rewrite the equilibrium radius as,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~R_\mathrm{eq}^{n-3} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
\biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
\biggl( \frac{G}{K} \biggr)^n \biggl\{
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
\biggr\}^{(n-1)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
\biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{(n-1)/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl( \frac{G}{K} \biggr)^n
\biggl\{
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
\biggr\}^{(n-1)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl\{ \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{2n(n+1)}
\biggl[ \frac{\mathcal{B}^{4n(n-1)}}{\mathcal{A}^{3(n+1)(n-1)}} \biggr]\biggr\}^{1/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl\{ \biggl( \frac{G}{K} \biggr)^{2n(n+1)}
\biggl[ \frac{K^{4n(n-1)}}{G^{3(n+1)(n-1)}P_e^{(n-3)(n-1)} } \biggr]
\biggr\}^{1/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
\biggl[ G^{(3-n)(n+1)} K^{2n(n-3)} P_e^{(n-3)(1-n)} \biggr]^{1/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~R_\mathrm{SWS}^{n-3} \biggl( \frac{n}{n+1} \biggr)^{(n-3)/2}
\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\biggl( \frac{n}{n+1} \biggr)^{1/2}
\Chi_\mathrm{ad} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-1)/[2(n+1)]}
\biggl[ \frac{\mathcal{B}^{n/(n+1)}}{\mathcal{A}^{1/2}} \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
Flipping both of these expressions around, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Pi_\mathrm{ad} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} \, ,
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Chi_\mathrm{ad} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{n+1}{n} \biggr)^{1/2}
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr]
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{n+1}{n} \biggr)^{1/2}
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr]
\biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{(1-n)/[2(n+1)(n-3)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)} \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, our earlier derived [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial3|compact expression for the virial theorem]] becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)} \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^{(n-3)/n}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
-~ \frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)}
\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)} \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^4 </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/n} \biggl( \frac{n}{n+1} \biggr)
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
-~ \frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^4 \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{-2} \biggl( \frac{n}{n+1} \biggr)
\frac{1}{\mathcal{A}} \, .
</math>
</td>
</tr>
</table>
</div>

Or, rearranged,

<div align="center" id="CompactStahlerVirial">
<table border="1" cellpadding="10" align="center">

<tr>
<td align="right">
<math>\frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^4 -
\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
+~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2} = 0 \, .
</math>
</td>
</tr>
</table>
</div>

After adopting modified length- and mass-normalizations, <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>, such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}^{3/2}}{\mathcal{B}^{2n/(n+1)}} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \, ,</math>
</td>
</tr>
</table>
</div>
we obtain the
<div align="center" id="ConciseVirialMR">
<font color="#770000">'''Virial Theorem in terms of Mass and Radius'''</font><br />

<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}
+ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2 = 0
\, .
</math>
</div>
This analytic function is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following table.



<table border="1" align="center" cellpadding="3" width="80%">
<tr>
<td align="center" rowspan="3">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
</td>
<td align="center">
Figure 17 extracted<sup>&dagger;</sup>from p. 184 of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S S. W. Stahler (1983)]<p></p>
"''The Equilibria of Rotating Isothermal Clouds.     II. Structure and Dynamical Stability''"<p></p>
ApJ, vol. 268, pp. 165-184 © [http://aas.org/ American Astronomical Society]
</td>
</tr>
<tr>
<td align="center">

[[Image:AAAwaiting01.png|400px|center|Stahler (1983) Figure 17 (edited)]]
</td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Figure displayed here, as a digital image, has been modified from the original publication only via the addition of the word "<font color="red">SCHEMATIC</font>".</td></tr>
</table>

Now that we have this very general, yet concise, algebraic expression for the mass-radius relationship of all pressure-truncated polytropes, let's replace the new "mod" normalizations with Stahler's original normalizations, <math>~R_\mathrm{SWS}</math> and <math>~M_\mathrm{SWS}</math>, to understand more completely how this general expression should be viewed in relation to the parametric relations provided by solutions of the detailed force-balanced models. We will henceforth use the notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{X}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathcal{Y}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, .</math>
</td>
</tr>
</table>
</div>
In the virial theorem expression we will make the following replacements:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathcal{X} \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}^{3/2}}{\mathcal{B}^{2n/(n+1)}} \, . </math>
</td>
</tr>
</table>
</div>
Next, we recognize that, in order to graphically display the mass-radius relation derived from the virial theorem in the <math>~\mathcal{X}-\mathcal{Y}</math> plane, we must write out the expressions for the free-energy coefficients. After setting <math>~M_\mathrm{limit}/M_\mathrm{tot} = 1</math> in the [[User:Tohline/SSC/Virial/PolytropesSummary#Structural_Form_Factors|above summary expressions]], we obtain,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} = \frac{1}{5-n} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathcal{B}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} =
\frac{1}{3(5-n) ( 4\pi )^{1/n}} \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
</td>
</tr>
</table>
</div>
In an effort not to be caught dividing by zero while investigating the specific case of <math>~n=5</math> polytropes, we will adopt as shorthand notation,
<div align="center">
<math>\mathfrak{b}_n \equiv \biggl[ (4\pi)^{1/n} (5-n)\mathcal{B}\biggr]
= \biggl[ (n+1) (\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
</div>
Hence, the replacements become,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} </math>
</td>
<td align="center">
<math>~~~ \rightarrow ~~~</math>
</td>
<td align="left">
<math>~\mathcal{X} \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2}
(5-n)^{(n-1)/[2(n+1)]} (4\pi)^{1/(n+1)} \mathfrak{b}_n^{-n/(n+1)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathcal{X} (5-n)^{(n-1)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{1/2} (4\pi)^{1/2} \cdot 3^{(1-n)/[2(n+1)]} \cdot
\mathfrak{b}_n^{-n/(n+1)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} </math>
</td>
<td align="center">
<math>~~~ \rightarrow ~~~</math>
</td>
<td align="left">
<math>~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2}
(5-n)^{(n-3)/[2(n+1)]} (4\pi)^{2/(n+1)} \mathfrak{b}_n^{-2n/(n+1)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathcal{Y} (5-n)^{(n-3)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{3/2} (4\pi)^{1/2} \cdot 3^{(3-n)/[2(n+1)]} \cdot \mathfrak{b}_n^{-2n/(n+1)} \, ,
</math>
</td>
</tr>
</table>
</div>
and, in particular, the cross term in the virial theorem expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl( \frac{n+1}{n} \biggr)^{2} (4\pi)^{(n-1)/n}
\cdot \mathfrak{b}_n^{(1-3n)/(n+1)} \, .
</math>
</td>
</tr>
</table>
</div>

Via these replacements the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialMR|concise and general Virial Theorem expression]] derived above morphs into the,
<div align="center" id="ConciseVirialXY">
<font color="#770000">'''Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\mathfrak{b}_n</math>'''</font><br />

<math>
k_\xi \biggl\{
\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr)
\biggr\} = 0 \, ,
</math>
</div>
where the leading coefficient is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_\xi </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 4\pi
\biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl(\frac{n+1}{n} \biggr)^2 \mathfrak{b}_n^{-4n/(n+1)} \, .
</math>
</td>
</tr>
</table>
</div>

In carrying out this last derivation we could be accused of reinventing the wheel, as the expression inside the curly braces is simply <math>~(5-n)</math> times the [[User:Tohline/SSC/Virial/PolytropesSummary#CompactStahlerVirial|virial expression presented inside an outlined box, above]], just before we introduced the modified normalization parameters, <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>.

===Relating and Reconciling Two Mass-Radius Relationships for n = 5 Polytropes===

Now, let's examine the case of pressure-truncated, <math>~n=5</math> polytropes. As we have discussed in the context of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|detailed force-balanced models]], [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] has deduced that all <math>~n=5</math> equilibrium configurations obey the mass-radius relationship,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, ,</math>
</td>
</tr>
</table>
</div>
where, as [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|reviewed above]], the mass and radius normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, may be treated as constants once the parameters <math>~K</math> and <math>~P_e</math> are specified. In contrast to this, the mass-radius relationship that we have just derived ''from the virial theorem'' for pressure-truncated, <math>~n=5</math> polytropes is,
<div align="center">
<math>
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{2/5} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{6/5}
+ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 = 0 \, ,
</math>
</div>
where the mass and radius normalizations,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M_\mathrm{mod}\biggr|_{n=5}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~M_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi} \biggr)^{5/3} \biggl[ \frac{2\cdot 5\pi}{3^2 \mathcal{A}} \biggr]^{3/2}
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~R_\mathrm{mod}\biggr|_{n=5}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~R_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi}\biggr)^{5/6} \biggl[ \frac{2\cdot 5\pi}{3^2\mathcal{A}} \biggr]^{1/2} \, ,</math>
</td>
</tr>
</table>

depend, not only on <math>~K</math> and <math>~P_e</math> via the definitions of <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, but also on the structural form factors via the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. While these two separate mass-radius relationships are similar, they are not identical. In particular, the middle term involving the cross-product of the mass and radius contains different exponents in the two expressions. It is not immediately obvious how the two different polynomial expressions can be used to describe the same physical sequence.

This apparent discrepancy is reconciled as follows: The structural form factors — and, hence, the free-energy coefficients — vary from equilibrium configuration to equilibrium configuration. So it does not make sense to discuss ''evolution along the sequence'' that is defined by the second of the two polynomial expressions. If you want to know how a given system's equilibrium radius will change ''as its mass changes'', the first of the two polynomials will do the trick. However, the equilibrium radius of ''a given system'' can be found by looking for extrema in the free-energy function while holding the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, constant; more importantly, the relative stability ''of a given equilibrium system'' can be determined by analyzing the behavior of the system's free energy ''while holding the free-energy coefficients constant''. Dynamically stable versus dynamically unstable configurations can be readily distinguished from one another along the sequence that is defined by the second polynomial expression; they cannot be readily distinguished from one another along the sequence that is defined by the first polynomial expression. It is useful, therefore, to determine how to map a configuration's position on one of the sequences to the other.

====Plotting Stahler's Relation====

[[File:CorrectedStahlerN5.png|thumb|300px|Pressure vs. pressure plot]]Switching, again, to the shorthand notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{X}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathcal{Y}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, ,</math>
</td>
</tr>
</table>
</div>
the equilibrium mass-radius relation defined by the first of the two polynomial expressions can be plotted straightforwardly in either of two ways. One way is to recognize that the polynomial is a quadratic equation whose solution is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{Y}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{5}{2} \mathcal{X} \biggl\{ 1 \pm \biggl[ 1 - \biggl( \frac{2^4\cdot \pi}{3\cdot 5} \biggr) \mathcal{X}^2 \biggr]^{1/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
In the figure shown here on the right — see also the bottom panel of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler1983Fig17|Figure 2 in our accompanying discussion of detailed force-balance models]] — Stahler's mass-radius relation has been plotted using the solution to this quadratic equation; the green segment of the displayed curve was derived from the ''positive'' root while the segment derived from the ''negative'' root is shown in orange. The two curve segments meet at the maximum value of the normalized equilibrium radius, namely, at
<div align="center">
<math>\mathcal{X}_\mathrm{max} \equiv \biggl[ \frac{3\cdot 5}{2^4 \pi} \biggr]^{1/2} \approx 0.54627 \, .</math>
</div>
We note that, when <math>~\mathcal{X} = \mathcal{X}_\mathrm{max}</math>, <math>~\mathcal{Y} = (5\mathcal{X}_\mathrm{max}/2) \approx 1.36569</math>. Along the entire sequence, the maximum value of <math>~\mathcal{Y}</math> occurs at the location where <math>~d\mathcal{Y}/d\mathcal{X} = 0</math> along the segment of the curve corresponding to the ''positive'' root. This occurs along the upper segment of the curve where <math>~\mathcal{X}/\mathcal{X}_\mathrm{max} = \sqrt{3}/2</math>, at the location,
<div align="center">
<math>\mathcal{Y}_\mathrm{max} \equiv \biggl[ \frac{3^3 \cdot 5^2}{2^6 } \biggr]^{1/2} \mathcal{X}_\mathrm{max}
= \biggl[ \frac{3^4 \cdot 5^3}{2^{10} \pi } \biggr]^{1/2} \approx 1.77408 \, .</math>
</div>

The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relations. Drawing partly from our [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|above discussion]] and partly from a separate discussion where we provide a [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of the properties of pressure-truncated <math>~n=5</math> polytropes]], these are,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\mathcal{X}\biggr|_{n=5}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{2} =
\biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\tilde\xi^2/3}{(1+\tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\mathcal{Y}\biggr|_{n=5}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \tilde\theta (- \tilde\xi^2 \tilde\theta^') =
\biggl[ \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\tilde\xi^2/3)^3}{(1+\tilde\xi^2/3)^{4}} \biggr]^{1/2} \, .
</math>
</td>
</tr>
</table>
</div>
The entire sequence will be traversed by varying the Lane-Emden parameter, <math>~\tilde\xi</math>, from zero to infinity. Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is,
<div align="center">
<math>
~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = \frac{1}{5^{1/2}} \biggl[ 2^5\pi - 15 + 2^3\pi^{1/2}(2^4\pi-15)^{1/2} \biggr]^{1/2}
\approx 5.8264 \, .
</math>
</div>
Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,
<div align="center">
<math>
~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3 \, .
</math>
</div>

====Plotting the Virial Theorem Relation====

The relevant relation is obtained by plugging <math>~n = 5</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation derived above]], repeated here for clarity:

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

<div align="center">
<math>
\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr)
= 0
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_n = \biggl[ (n+1) (-\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n}
</math>
</div>

</td></tr>
</table>
We will begin by plugging <math>~n = 5</math> into these expressions everywhere except for the coefficient <math>~(5-n)</math>, which we will leave unresolved, for the time being, in order to better appreciate the interplay of various terms. We obtain,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~
\frac{6}{5} \mathcal{Y}^2 - \biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(n-5)\biggl[ \biggl(\frac{4\pi}{3} \biggr) \mathcal{X}^4 - \biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} \biggr] \, ,
</math>
</td>
</tr>
</table>
</div>
where, if they are to be assigned values that are actually associated with a particular detailed force-balance model having truncation radius, <math>~\tilde\xi</math>,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~
\mathfrak{b}_I^5
</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl[ (n+1) (-\tilde\theta^')^2 \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5}
= 2^5\cdot 3^5 \biggl[ (-\tilde\theta^')^4 {\tilde\xi}^6 \biggr]
= 2^5 \cdot 3 \cdot \tilde\xi^{10} (1+\tilde\xi^2/3)^{-6}\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~
\mathfrak{b}_{II}^5
</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{1}{3} ~ \tilde\theta^{n+1} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5}
= \frac{1}{3^5} ~ \tilde\theta^{30} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{6}
= 3(1+\tilde\xi^2/3)^{-6} \, .
</math>
</td>
</tr>
</table>
</div>

Now, if we plug <math>~n=5</math> into the remaining unresolved <math>~(n-5)</math> coefficient, the righthand side goes to zero and the mass-radius relationship provided by the virial theorem becomes,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~\frac{6}{5} \mathcal{Y}^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y}^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\mathcal{X}^{2} \biggl[ \biggl( \frac{5^5}{2^5\cdot 3^5}\biggr) \frac{\mathfrak{b}^5_{I}}{4\pi} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\mathcal{X}^{1/2} \biggl( \frac{5^5 \mathfrak{b}^5_{I}}{2^7\cdot 3^5 \pi}\biggr)^{1/4} \, . </math>
</td>
</tr>
</table>
</div>
Hence, for a given value of the structural form factor(s) — which implies a specific value of the constant coefficient, <math>~\mathfrak{b}_I</math> — the scalar virial theorem defines a relationship where the normalized mass <math>~(\mathcal{Y})</math> varies as the square root of the normalized radius <math>~(\mathcal{X})</math>. On the other hand, if we demand that the expression inside the square brackets on the righthand side of the virial theorem relation go to zero on its own — without relying on the leading coefficient to knock it zero — the mass-radius relationship provided by the virial theorem becomes,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~\frac{4\pi}{3} ~ \mathcal{X}^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y}^6 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\mathcal{X}^{18} \biggl[ \biggl( \frac{4\pi}{3}\biggr)^5 \frac{4\pi}{\mathfrak{b}^5_{II}} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>4\pi \mathcal{X}^{3} ( 3^5\mathfrak{b}^5_{II} )^{-1/6} \, . </math>
</td>
</tr>
</table>
</div>



===Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes===

For pressure-truncated <math>~n=4</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\mathcal{X}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{3/2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\mathcal{Y}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \tilde\theta^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
</math>
</td>
</tr>
</table>
</div>
On the other hand, the polynomial that results from plugging <math>~n=4</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,

<div align="center">
<math>
\frac{4\pi}{3} \mathcal{X}^4 - \biggl[ \frac{\mathcal{X} \mathcal{Y}^{5}}{4\pi}\biggr]^{1/4} \mathfrak{b}_{n=4} + \frac{5}{4} \mathcal{Y}^2
= 0 \, ,
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_{n=4} = \biggl[ 5 (-\tilde\theta^')^2 + \frac{1}{3} \tilde\theta^{5} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{5/4} \, .
</math>
</div>

[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unity. Its value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>. A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[User:Tohline/SSC/Virial/PolytropesSummary#Summary|above parameter summary figure]].]

In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequence. The "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>.

<table border="1" cellpadding="8" align="center">
<tr>
<th align="center">
Comparing Two Separate Mass-Radius Relations for Pressure-Truncated ''n = 4'' Polytropes
</th>
</tr>
<tr><td align="center">
[[File:CompareN4SequencesRevised.png|750px|Comparison of Two Mass-Radius Relations]]
</td></tr>
</table>

According to [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt's (1986)]] tabulated data, the surface of an isolated <math>~(P_e = 0)</math>, spherically symmetric, <math>~n=4</math> polytrope occurs at the dimensionless (Lane-Emden) radius, <math>~\xi_1 = 14.9715463</math>. In both panels of the above figure, this ''isolated'' configuration is identified by the discrete (blue diamond) point at the origin, that is, at <math>~(\mathcal{X}, \mathcal{Y}) = (0, 0)</math>. As we begin to examine pressure-truncated models and <math>~\tilde\xi</math> is steadily decreased from <math>~\xi_1</math>, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation. A maximum mass of <math>~\mathcal{Y} \approx 2.042</math> (corresponding to a radius of <math>~\mathcal{X} \approx 0.4585</math>) is reached ''from the left'' as <math>~\tilde\xi</math> drops to a value of approximately <math>~3.4</math>. As <math>~\tilde\xi</math> continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554)</math> — corresponding to <math>~\tilde\xi \approx 2.0</math> — then decreasing in radius until, once again, the origin is reached, but this time because <math>~\tilde\xi</math> drops to zero.

If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>~\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>. Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.

When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of <math>~\mathfrak{b}_{n=4} = 4.8926</math> (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where <math>~\tilde\xi = 2.8</math>, for the reason just discussed. After constructing the plot, it became clear that the two curves also intersect at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67)</math> — also highlighted by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi \approx 6.0</math>. This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, <math>~\tilde\xi</math>, that assures precise agreement between the two different mass-radius expressions.

As is detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Stability|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center">
<math>\eta_\mathrm{crit}\biggr|_{n=4}~=~\frac{1}{15} \, ;</math>
        
<math>\Pi_\mathrm{max}\biggr|_{n=4}~=~\frac{15^{15}}{16^{16}} \, ;</math>
     and     
<math>\Chi_\mathrm{min}\biggr|_{n=4}~=~\biggl( \frac{16}{15} \biggr)^4 \, .</math>
</td>
</tr>
</table>
</div>
In the context of the above figure, independent of the chosen value of <math>~\mathfrak{b}_{n=4}</math>, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem. In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of <math>~\mathcal{Y}</math>; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of <math>~\mathcal{X}</math>. We have deduced (details of the derivation not shown) that, for pressure-truncated <math>~n=4</math> polytropes, the coordinates of this critical point in Stahler's <math>~\mathcal{X}-\mathcal{Y}</math> plane depends on the choice of <math>~\mathfrak{b}_{n=4}</math> as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{X}_\mathrm{crit}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\pi^{-1/2} 2^{-16/5} (3\mathfrak{b}_{n=4})^{4/5} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathcal{Y}_\mathrm{crit}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\pi^{-1/2} 2^{-22/5} (3\mathfrak{b}_{n=4})^{8/5} \, .</math>
</td>
</tr>
</table>
</div>

In practice, for a given plot of the type displayed in the above figure — that is, for a given choice of the structural parameter, <math>\mathfrak{b}_{n=4}</math> — it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations. If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable. On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable. We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is ''stable''; however, both configurations identified by filled black circles in the righthand panel are ''unstable''.

It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence. One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable. But the righthand panel of our above figure contradicts this expectation. While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies ''to the right'' of the maximum-mass point along the blue "Stahler" sequence. This finding is related to [[User:Tohline/SSC/Virial/PolytropesSummary#Curiosity|the curiosity raised earlier]] in our discussion of the structural properties of pressure-truncated, <math>~n=4</math> polytropes.

===Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes===

For pressure-truncated <math>~n=3</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\mathcal{X}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\mathcal{Y}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^3}{4\pi} \biggr)^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
</math>
</td>
</tr>
</table>
</div>
On the other hand, the polynomial that results from plugging <math>~n=3</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,

<div align="center">
<math>
\frac{2^3 \pi}{3} \mathcal{X}^4 - \biggl[ \frac{\mathcal{Y}^{4}}{4\pi}\biggr]^{1/3} \mathfrak{b}_{n=3} + \frac{4}{3} \mathcal{Y}^2
= 0 \, ,
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_{n=3} = \biggl[ 4 (-\tilde\theta^')^2 + \frac{2}{3} \tilde\theta^{4} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{4/3} \, .
</math>
</div>

==Summary Comments==
From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration. Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem. In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure <math>~(P_e)</math> constant, but also assuming that the configuration's structural form factors are invariable.

This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors. In support of this argument, we point to the paper by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)] which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[User:Tohline/Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|chapter of this H_Book details the Goldreich & Weber derivation]]. Goldreich & Weber use [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941, ApJ, 94, 245)], who has evaluated radial modes, and of [http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)], who has obtained eigenvalues of some low-order nonradial modes.)

In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors. In order to test this underlying assumption, following [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)], it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation. Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability. Better yet, the ''sign'' of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.

=Serious Concern=

==Statement of Concern==
Throughout our discussion of embedded (pressure-truncated) polytropes — both on this "summary" page and in an [[User:Tohline/SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying chapter]] where critical background derivations are presented — we have used expressions for the structural form factors that include an overall leading factor of <math>~(5-n)^{-1}</math>. For clarity, the form factors that we have used [[User:Tohline/SSC/Virial/Polytropes#Summary|for ''isolated'' polytropes]] is reprinted on the lefthand side of the following table while the ones that we have used [[User:Tohline/SSC/Virial/Polytropes#PTtable|for ''pressure-truncated'' polytropes]] is reprinted on the righthand side of the table.

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>
</td>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} + \tilde\Theta^{n+1}
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>
</div>
This factor seemed destined to become a nuisance in the specific case of <math>~n=5</math> polytropic structures. But we did not let its appearance in these expressions deter us from using a free-energy analysis to study the equilibrium and stability of spherical polytropes because, after all, the factor of <math>~(5-n)^{-1}</math> appears in [[User:Tohline/Appendix/References|Chandrasekhar's [C67]]] expression for the gravitational potential energy of ''isolated'' polytropes — see his Equation (90), p. 101. In retrospect, its appearance in the structural form factors for ''isolated'' polytropes did not prove to be a problem because, via a free-energy and virial theorem analysis, the [[User:Tohline/SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|derived expression for the configuration's equilibrium radius]] depends on the ratio of <math>~f_W</math> to <math>~f_A</math>, so the awkward factor of <math>~(5-n)^{-1}</math> cancels out.

However, in our discussion of ''pressure-truncated'' <math>~n=5</math> polytropic structures, the factor of <math>~(5-n)^{-1}</math> did not conveniently cancel out at the appropriate time and we were forced to carry out some logical contortions [[User:Tohline/SSC/Virial/PolytropesSummary#Plotting_the_Virial_Theorem_Relation|as we tried to compare the mass-radius relation obtained from the virial theorem]] to Stahler's mass-radius relation, which was derived from detailed force-balance arguments. This leads us to seriously question whether our, rather casually derived, expressions for the structural form factors in ''pressure-truncated'' polytropes are correct.

==Further Evaluation of n = 5 Polytropic Structures==
Throughout most of this subsection, we will adopt the shorthand notation,
<table align="center" border="1" cellpadding="10">
<tr><td align="center">
<math>~\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~~~\Rightarrow ~~~~~ \ell^2 = \frac{\tilde\xi^2}{3} \, .</math>
</td></tr>
</table>
This will not only simplify the appearance of some expressions, it will facilitate direct comparison with an expression for the free-energy coefficient, <math>~\mathcal{A}</math>, that has been derived in a [[User:Tohline/SSC/Virial/FormFactors#Structural_Form_Factors|companion chapter]] following a different train of logic and with an expression for the normalized gravitational potential energy that has been derived via a brute-force integration in association with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|discussion of bipolytropic configurations]] where the variable, <math>~x_i</math>, has the same definition as <math>~\ell</math>.

===Free-Energy Expression===

From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Free_Energy_Expression|general review of the topic]], to within an additive constant, the free-energy of a nonrotating, pressure-truncated polytrope comes from the sum of three principal energy terms, namely,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_e V \, .
</math>
</div>
Furthermore, as has been shown in our extended [[User:Tohline/SphericallySymmetricConfigurations/Virial#Gathering_it_all_Together|introductory discussion of free energy]], the corresponding ''normalized'' free energy (applied to <math>~n=5</math> or <math>~\gamma_g = 6/5</math> configurations) is,
<div align="center">
<math>
\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}} =
-3A\chi^{-1} + 5B \chi^{-3/5} +~ D\chi^3 \, ,
</math>
</div>
where,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~\chi</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{R}{R_\mathrm{norm}} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~A</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{4\pi} \biggr)^{1/5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{6/5} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{6/5}} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~D</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .
</math>
</td>
</tr>
</table>
</div>

Also note that the relevant normalizations are,
<div align="center">
<math>~E_\mathrm{norm} \equiv \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \, ;</math>       
<math>~R_\mathrm{norm} \equiv \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \, ;</math>       
<math>~P_\mathrm{norm} \equiv \frac{K^{10}}{G^9 M_\mathrm{tot}^{6}} \, .</math>       
</div>

===Virial Theorem===
The traditional expression for the virial theorem in this context is,
<div align="center">
<math>
~2S_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{E_\mathrm{norm}} = 0 \, .
</math>
</div>
From our [[User:Tohline/VE#Adiabatic_Systems|introductory discussion of the thermodynamic energy reservoir]], we know that, for <math>~\gamma_g=6/5</math> configurations,
<div align="center">
<math>
~S_\mathrm{therm} = \frac{3}{2}(\gamma_g-1) \mathfrak{S}_\mathrm{therm} = \frac{3}{10} \mathfrak{S}_\mathrm{therm}\, .
</math>
</div>
So, making this substitution and recognizing that <math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3</math>, the (normalized) virial theorem expression becomes,
<div align="center">
<math>
~\frac{3}{5}\mathfrak{S}_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{P_\mathrm{norm}R_\mathrm{norm}^3} = 0 \, .
</math>
</div>

Furthermore, by comparing terms in the first free-energy expression, above, with the second (normalized) free-energy expression, we see that,
<div align="center">
<math>~W_\mathrm{grav}^* \equiv \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \rightarrow -3A\chi^{-1} \, ;</math>       
<math>~\mathfrak{S}_\mathrm{therm}^* \equiv \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}} \rightarrow 5B\chi^{-3/5} \, ;</math>       
<math>~\frac{P_e V}{E_\mathrm{norm}} \rightarrow D\chi^{3} \, .</math>       
</div>
Hence, the normalized virial theorem may be written as,
<div align="center">
<math>
3B \chi^{-3/5} -3A\chi^{-1} -~ 3D\chi^3 =0\, .
</math>
</div>

For sake of consistency, let's check this by holding the coefficients <math>~A</math>, <math>~B</math>, and <math>~D</math> constant and setting <math>~d\mathfrak{G}^*/d\chi</math> equal to zero:
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~\frac{d\mathfrak{G}^*}{d\chi} = - 3B \chi^{-8/5} + 3A\chi^{-2} +~ 3D\chi^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ - \chi^{-1} \biggl[ 3B \chi^{-3/5} - 3A\chi^{-1} -~ 3D\chi^3 \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that the expression inside the square brackets sums to zero, which identically matches the (normalized) traditional virial theorem expression. Excellent!

===Borrowing from Bipolytrope Discussion===
In an accompanying chapter that presents the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|detailed force-balanced models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]] we explicitly show that, for configurations with the correct equilibrium radius, the virial theorem is satisfied. In the case of bipolytropes, which are not embedded in an external medium, the relevant normalized virial theorem states that,
<div align="center">
<math>
~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} + (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{env} = 0 \, .
</math>
</div>

In the bipolytrope, the (truncated) <math>~n=5</math> core is confined by an <math>~n=1</math> envelope; in addition to demanding that the relevant virial theorem be satisfied, there is also a constraint that the pressure at the inner edge of the envelope be equal to the pressure at the (truncated) outer edge of the core. As we have just discussed, for a (truncated) <math>~n=5</math> polytrope that is confined by a hot, tenuous external medium instead of by an enveloping envelope, the relevant normalized virial theorem is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} - \frac{3P_e V_\mathrm{eq}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3P_e }{(K^5/G^3)^{1/2}} \biggl( \frac{4\pi}{3} R_\mathrm{eq}^3 \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^3 \, ,</math>
</td>
</tr>
</table>
</div>
where, [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|as discussed/defined in an accompanying chapter of this H_Book]], we have adopted the normalization radius, <math>~R_\mathrm{SWS}</math>, first introduced by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Steven W. Stahler (1983)]. For <math>~n=5</math> configurations, its definition is,
<div align="center">
<math>
R_\mathrm{SWS}\biggr|_{n=5} = \biggl( \frac{2\cdot 3}{5} \biggr)^{1/2} \biggl[ \frac{(K^{5}/G^3)^{1/2}}{P_\mathrm{e}}\biggr]^{1/3} \, .
</math>
</div>
As has [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|also been discussed in the accompanying chapter]], we can deduce from Stahler's detailed force-balanced models that the equilibrium radius of embedded, <math>~n=5</math> polytropes is given in terms of the dimensionless, ''truncated'' Lane-Emden radius, <math>~\tilde\xi</math> — and our corresponding variable, <math>\ell</math> — by the expression,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^2
= \biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, upon careful evaluation of the thermal energy and gravitational potential energy of truncated <math>~n=5</math> polytropes, we should find that,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2} \biggl[ \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggr]^3 </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \ell^3 (1 + \ell^2)^{-3} \, .
</math>
</td>
</tr>
</table>
</div>
Well, it just so happens that, in our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#twoSplusWcore|accompanying chapter that presents the detailed force-balanced models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]], we explicitly carried out the volume integrals defining these two key components of the free energy expression with the results being,
<div align="center">
<table border="0" cellpadding="4">

<tr>
<td align="right">
<math>~(2S^* + W^*)_\mathrm{core}</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \biggl[ x_i ^3 (1 + x_i^2)^{-3} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Realizing that the variable, <math>~x_i</math>, in that context is the same as <math>~\ell</math>, in the present context, we see that the two separately derived results are identical to one another.

===Determining Expressions for Free-Energy Coefficients===
We should be able to convert the separately derived expression for <math>~W_\mathrm{grav}^*</math> into an expression for the free-energy coefficient, <math>~A</math>, in equilibrium configurations. As [[User:Tohline/SSC/Virial/PolytropesSummary#Virial_Theorem|noted above]], for a fixed value of <math>~A</math>,
<div align="center">
<math>~W_\mathrm{grav}^* ~~\rightarrow ~~ -3A\chi^{-1} \, .</math>
</div>
Therefore, in an equilibrium configuration, we can write,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~W_\mathrm{grav}^*
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
-3A\chi_\mathrm{eq}^{-1} = -3A \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)
= -3A \biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{eq}}\biggr)\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}}\biggr)</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\Rightarrow ~~~ A
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
-\frac{1}{3} W_\mathrm{grav}^* \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .</math>
</td>
</tr>
</table>
</div>
Now, from immediately above, we know that,
<div align="center">
<table border="0" cellpadding="4">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1} \, ;
</math>
</td>
</tr>
</table>
</div>
and, from our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|accompanying discussion of the free-energy of bipolytropic configurations]], we know that,
<div align="center">
<table border="0" cellpadding="4">

<tr>
<td align="right">
<math>~W^*_\mathrm{core}</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}
\biggl[ x_i\biggl(x_i^4 - \frac{8}{3}x_i^2 - 1\biggr) (1 + x_i^2)^{-3} + \tan^{-1}(x_i) \biggr] \, . </math>
</td>
</tr>
</table>
</div>
So, again realizing that <math>~x_i</math> and <math>~\ell</math> are interchangeable, we have,
<div align="center">
<table border="0" cellpadding="4">

<tr>
<td align="right">
<math>~A</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ \frac{1}{3}
\biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3} + \tan^{-1}(\ell) \biggr]
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3} + \tan^{-1}(\ell) \biggr]
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .
</math>
</td>
</tr>
</table>
</div>
Finally, we need to determine an expression for the ratio, <math>~R_\mathrm{SWS}/R_\mathrm{norm}</math>. Drawing the definition of <math>~R_\mathrm{norm}</math> from [[User:Tohline/SphericallySymmetricConfigurations/Virial#Normalizations|our introductory chapter on the virial equilibrium]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~R_\mathrm{norm}\biggr|_{\gamma = 6/5}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{2-6/5} \biggr]^{1/(4-18/5)}
= \biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{4/5} \biggr]^{5/2}
= \biggl( \frac{G}{K}\biggr)^{5/2} M_\mathrm{tot}^{2} \, .
</math>
</td>
</tr>
</table>
</div>
From our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of Stahler's derived mass & radius relationships for truncated, <math>~n=5</math> polytropes]] we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M_\mathrm{limit}^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
M_\mathrm{SWS}^2 \biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
In addition, from our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|review of Stahler's defined normalizations]], we see that,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~M_\mathrm{SWS}^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^3 \cdot 3^3}{5^3} \biggr) G^{-3} K^{10/3} P_e^{-1/3} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
and,    <math>~R_\mathrm{SWS}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2 \cdot 3}{5} \biggr)^{1/2} G^{-1/2} K^{5/6} P_e^{-1/3} \, ,
</math>
</td>
</tr>
</table>
</div>
which, when combined to cancel <math>~P_e</math> gives,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~M_\mathrm{SWS}^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^6 \cdot 3^6}{5^6} \biggr)^{1/2} G^{-3} K^{10/3}
\biggl( \frac{5}{2 \cdot 3} \biggr)^{1/2} G^{1/2} K^{-5/6} R_\mathrm{SWS}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} R_\mathrm{SWS} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M_\mathrm{limit}^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}
\biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] R_\mathrm{SWS}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \frac{1}{R_\mathrm{SWS}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \frac{1}{M_\mathrm{limit}^2} \, .
</math>
</td>
</tr>
</table>
</div>
In combination with the expression for <math>~R_\mathrm{norm}</math>, then, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{R_\mathrm{norm}}{R_\mathrm{SWS}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \biggl(\frac{M_\mathrm{tot}}{M_\mathrm{limit}}\biggr)^{2} \, ,
</math>
</td>
</tr>
</table>
</div>
which means that, for truncated <math>~n=5</math> polytropes, the expression for the free-energy coefficient is,
<div align="center">
<table border="0" cellpadding="4">

<tr>
<td align="right">
<math>~A</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \biggl( \frac{\pi^2}{2\cdot 3^7 \cdot 5} \biggr)^{1/2}
\ell (1 + \ell^2)^{-1} \cdot \ell^{-6} (1+\ell^2)^{4}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3} + \tan^{-1}(\ell) \biggr]
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
2^{-4} \ell^{-5} (1+\ell^2)^{3}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3} + \tan^{-1}(\ell) \biggr]
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, .
</math>
</td>
</tr>
</table>
</div>
Finally, drawing from our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|accompanying derivation of expressions for the structural form factors in this case]], we know that,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, ,
</math>
</td>
</tr>
</table>
</div>
which gives,
<div align="center">
<table border="0" cellpadding="4">

<tr>
<td align="right">
<math>~A</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\frac{\ell}{2^{4} }
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3} + \tan^{-1}(\ell) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

This exactly matches the expression for the free-energy coefficient, <math>~\mathcal{A}</math>, that we derived separately in conjunction with our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|derivation of expressions for the structural form factors]].

==Take Care Comparing Gravitational Potential Energies==
Does this derived relation for the coefficient, <math>~A</math>, make sense? Well, we've derived the relation by comparing two separate expressions for the gravitational potential energy that were normalized in slightly different ways, so the leading numerical coefficient may not be correct. We need to repeat the derivation, checking the relative normalizations carefully. But before doing this, let's determine what we ''expected'' the relation to be, based on the expressions for the structural form factors that we have been using.

From the [[User:Tohline/SSC/Virial/PolytropesSummary#Serious_Concern|lead-in paragraphs of this subsection]], we have previously assumed that,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}
= \frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \biggl\{ \frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \biggr\}
\biggl\{ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \biggr\}^{-2} = \frac{1}{(5-n)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \, .
</math>
</td>
</tr>

</table>
</div>
According to the line of reasoning presented above, the coefficient, <math>~A</math>, is related to the gravitational potential energy via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-3A\chi^{-1} E_\mathrm{norm} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-3A \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \frac{1}{R}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-3A \biggl(\frac{GM_\mathrm{tot}^2}{R} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Expressions_for_Various_Energy_Terms|introductory layout of the free-energy function for polytropes]] — see, also, p. 64, Equation (12) of [[User:Tohline/Appendix/References|Chandrasekhar [C67]]] — the gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M_r</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\int_0^r 4\pi r^2 \rho dr \, .</math>
</td>
</tr>
</table>
</div>
Now, independent of the chosen normalization, if we use <math>~M_\mathrm{tot}</math> to represent the total mass of an ''isolated'' <math>~n=5</math> polytrope, then from [[User:Tohline/SSC/Structure/PolytropesEmbedded#Review_2|an earlier review]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M_\mathrm{tot}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_c^{-1/5} \, ,</math>
</td>
</tr>
</table>
</div>
and we can write, in terms of the Lane-Emden dimensionless radius, <math>~\xi</math>,
<div align="center">
<math>
\frac{M_r}{M_\mathrm{tot}} = \xi^3 (3 + \xi^2)^{-3/2} \, .
</math>
</div>
===Virial Chapter===
Now, in our discussion of the virial equilibrium of embedded polytropes, we used the normalizations specified above and wrote,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- 4\pi GM_\mathrm{tot} R_\mathrm{norm}^2 \rho_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} \biggl[\frac{M_r}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^*
</math>
</td>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- E_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} 3\biggl[\frac{M_r}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^* \, .
</math>
</td>
</tr>
</table>
</div>
We can replace <math>~r^* \equiv r/R_\mathrm{norm}</math> with <math>~\xi \equiv r/a_5</math> by recognizing that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M_\mathrm{tot}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \rho_c^{-2/5} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr) M_\mathrm{tot}^2 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~a_5</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{3K}{2\pi G} \biggr)^{1/2} \rho_c^{-2/5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{3K}{2\pi G} \biggr)^{1/2} \biggl( \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr) M_\mathrm{tot}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^2 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~R_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{a_5}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \, .
</math>
</td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r^*</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .
</math>
</td>
</tr>
</table>
</div>

Also,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} M_\mathrm{tot}^{-5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl[ \biggl( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}\biggr]^{-5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl( \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)^{5/2} \rho_c
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3^2}{2^4\pi^2} \biggr)^{1/2} \biggl( \frac{\pi^5 }{2^5 \cdot 3^{20}} \biggr)^{1/2} \rho_c
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\pi^3 }{2^9 \cdot 3^{18}} \biggr)^{1/2} \rho_c \, .
</math>
</td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \frac{\rho}{\rho_c}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} \, .
</math>
</td>
</tr>
</table>
</div>
So the energy integral becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- 3 \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr) \biggl( \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2}
\int_0^{\tilde\xi} \biggl[\xi^3 (3 + \xi^2)^{-3/2} \biggr] \xi \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \biggl( \frac{2^3 \cdot 3^{6}}{\pi } \biggr)^{1/2}
\int_0^{\tilde\xi} \xi^4 \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-4} d\xi \, .
</math>
</td>
</tr>
</table>
</div>
This needs to be compared with the <math>~W_\mathrm{grav}^*</math> integral that we previously have handled in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|the chapter discussing bipolytrope models]].

=See Also=

{{ SGFfooter }}

VE/RiemannEllipsoids

2024-07-02T22:01:44Z

70.180.56.157: /* Riemann S-Type Ellipsoids */

__FORCETOC__


=Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations=
{| class="2ndOrderTVE" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Ellipsoidal_.26_Ellipsoidal-Like|<b>Steady-State<br />2<sup>nd</sup>-Order<br />Tensor Virial<br />Equations</b>]]</font>
|}
Drawing from our [[VE#Virial_Equations_.28Rotating_Frame.29|accompanying discussion of virial equations as viewed from a rotating frame of reference]], here we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx \, ,
</math>
</td>
</tr>
</table>
to determine the equilibrium conditions of uniform-density <math>~(\rho)</math> ellipsoids that have semi-axes, <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c),</math> and an internal velocity field, <math>~\vec{u}</math> (as [[#Adopted_.28Internal.29_Velocity_Field|prescribed below]]), that preserves this specified ellipsoidal shape, as viewed from a frame of reference that is rotating with angular velocity, <math>~\vec\Omega</math>. Because each of the indices, <math>~i</math> and <math>~j</math>, run from 1 to 3, inclusive, this TVE appears to provide nine equilibrium constraints; and once the values of the density and the three semi-axes are specified, there appear to be seven unknowns: <math>~\Pi</math> and the three pairs of velocity-field components <math>~(\Omega_1, \zeta_1)</math>, <math>~(\Omega_2, \zeta_2)</math>, <math>~(\Omega_3, \zeta_3).</math> In practice, however, only five constraints are relevant/independent because, as is encapsulated in …
<table border="0" width="60%" align="center" cellpadding="10"><tr><td align="left">
<div align="center"><font color="maroon">'''Riemann's Fundamental Theorem'''</font></div>
<font color="darkgreen">… non-trivial solutions are obtained only if no more than two of the three pairs of velocity-field components are different from zero.</font>
</td></tr></table>
<span id="SummaryTable">Following EFE</span>, we will set <math>~\Omega_1 = \zeta_1 = 0</math>, in which case the only applicable TVE constraint relations are the five identified in the following table of equations.

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">Each Associated 2<sup>nd</sup>-Order TVE Expression</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
+ \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2
+ \biggl\{
\Omega_2^2 + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
- (2\pi G \rho)A_3
\biggr\}c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

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

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

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

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>
</table>

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

==General Coefficient Expressions==

In the context of our discussion of configurations that are triaxial ellipsoids, we begin by adopting the <math>~(\ell, m, s)</math> subscript notation to identify which semi-axis length is the (largest, medium-length, smallest). As has been detailed in an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|accompanying chapter]], the gravitational potential anywhere inside or on the surface of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~\frac{A_\ell}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~\frac{2}{a_\ell^3}
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{A_s}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{2}{a_\ell^3} \biggl[ \frac{(a_m/a_s) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{A_m}{a_\ell a_m a_s} = \frac{2 - (A_\ell + A_s)}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{ 2}{a_\ell^3 }
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_s/a_m)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_s}{a_\ell} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_m/a_\ell)^2}{1 - (a_s/a_\ell)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>2</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_1 > a_2 > a_3 > 0</math> — such as Jacobi, Dedekind, or ''most'' Riemann S-Type ellipsoids — we must adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_m, a_m)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_1
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[ \frac{(a_2/a_1) \sin\theta - (a_3/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 - (A_1+A_3) \, ,</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §17, Eq. (32)</font> ]</td></tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===

When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in [[Appendix/References#EFE|EFE]] as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_1
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 \biggl( \frac{a_2}{a_1} \biggr)\biggl( \frac{a_3}{a_1} \biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2 \biggl( \frac{a_3}{a_1} \biggr) \biggl[ \frac{(a_3/a_1) \sin\theta - (a_2/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3 = 2 - (A_1 + A_2)
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{2a_2 a_3}{a_1^2}
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_2/a_3)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals of the first and second kind are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_2}{a_1} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_3/a_1)^2}{1 - (a_2/a_1)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48d, footnote to Table VII (p. 143)</font> ]</td></tr>
</table>
</div>
NOTE: All ''irrotational'' ellipsoids belong to this category of configurations.

===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_2 > a_1 > a_3 > 0</math> — for example, ''most'' Riemann ellipsoids of Types I, II, & III — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>~(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl( \frac{a_1}{a_2}\biggr) \biggl[ \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_1 = 2 - (A_2 + A_3)
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{ 2a_1 a_3}{a_2^2 }
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_2} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_1/a_2)^2}{1 - (a_3/a_2)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

===Oblate Spheroids [a<sub>2</sub> = a<sub>1</sub> > a<sub>3</sub>]===

Starting with the case of <math>~a_2 > a_1 > a_3 > 0</math> and setting <math>~a_2 = a_1</math>, we recognize, first, that <math>~k = 0</math>. Hence, we have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl[ \frac{ \sin\theta - (a_3/a_1)E(\theta,0)}{\sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>
</table>

==Adopted (Internal) Velocity Field==

EFE (p. 130) states that the … <font color="#007700">kinematical requirement, that the motion <math>~(\vec{u})</math>, associated with <math>~\vec{\zeta}</math>, preserves the ellipsoidal boundary, leads to the following expressions for its components:</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~u_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~u_2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~u_3</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (1)</font> ]</td></tr>
</table>

==Equilibrium Expressions==
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \delta_{ij}\Pi \, .</math>
</td>
</tr>
</table>
</div>
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2<sup>nd</sup>-order TVE takes on the more general form:
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 2, §12, Eq. (64)</font> ]</td></tr>
</table>

EFE (p. 57) also shows that … <font color="#007700">The potential energy tensor … for a homogeneous ellipsoid is given by</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2A_i I_{ij} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (128)</font> ]</td></tr>
</table>
<font color="#007700">where</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~I_{ij}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (129)</font> ]</td></tr>
</table>

<font color="#007700">is the moment of inertia tensor.</font> Expressions for all nine components of the kinetic energy tensor, <math>~\mathfrak{T}_{ij}</math> are derived in [[#Appendix_E:_.C2.A0_Kinetic_Energy_Components|Appendix E]], below; and expressions for each of the six Coriolis components can be found in [[#Appendix_B:_.C2.A0Coriolis_Component_u1x2|Appendices B, C, & D]].

===The Three Diagonal Elements===
For <math>~i = j = 1</math>, we have,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi
+ \Omega^2 I_{11} - \Omega_1\Omega_k I_{k1} + 2\epsilon_{1lm}\Omega_m \int_V \rho u_lx_1 d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11}
- \Omega_1^2I_{11}
+ 2 \Omega_3 \int_V \rho u_2x_1 ~d^3x
- 2\Omega_2 \int_V \rho u_3x_1 ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi
+( \Omega_2^2 + \Omega_3^2) I_{11}
+ 2 \Omega_3\rho \int_V u_2x ~d^3x
- 2\Omega_2\rho \int_V u_3 x~ d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}
~-~(2\pi G\rho) A_1 I_{11} + \Pi
+( \Omega_2^2 + \Omega_3^2) I_{11}
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 I_{11}
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 I_{11}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+ \biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} I_{11}
+
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2 \, .
</math>
</td>
</tr>
</table>

Once we choose the values of the (semi) axis lengths <math>~(a, b, c)</math> of an ellipsoid — from which the value of <math>~A_1</math> can be immediately determined — along with a specification of <math>~\rho</math>, this equation has the following five unknowns: <math>~\Pi, \Omega_2, \Omega_3, \zeta_2, \zeta_3</math>. Similarly, for <math>~i = j = 2</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi
+ \Omega^2 I_{22} - \Omega_2\Omega_k I_{k2} + 2\epsilon_{2lm}\Omega_m \int_V \rho u_lx_2 d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi
+ (\Omega_1^2 + \Omega_3^2) I_{22} + 2\Omega_1 \rho \int_V u_3 y ~d^3x
- 2\Omega_3 \rho \int_V u_1 y ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33}
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
~-~( 2\pi G \rho) A_2 {I}_{22}
+ \Pi
+ (\Omega_1^2 + \Omega_3^2) I_{22}
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1 I_{22}
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 I_{22}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}{I}_{22}
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~-\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 c^2 \, .
</math>
</td>
</tr>
</table>

This gives us a second equation, but an additional pair of (for a total of seven) unknowns: <math>~\Omega_1, \zeta_1</math>. For the third diagonal element — that is, for <math>~i=j=3</math> — we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi
+ \Omega^2 I_{33} - \Omega_3\Omega_k I_{k3} + 2\epsilon_{3lm}\Omega_m \int_V \rho u_lx_3 ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi
+ (\Omega_1^2 + \Omega_2^2) I_{33} + 2\Omega_2 \rho \int_V u_1 z ~d^3x
- 2\Omega_1 \rho \int_V u_2 z ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
- (2\pi G \rho)A_3 I_{33} + \Pi
+ (\Omega_1^2 + \Omega_2^2) I_{33} + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 I_{33}
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
+ \biggl\{
(\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1
- (2\pi G \rho)A_3
\biggr\}I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 b^2
+ \biggl\{
(\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1
- (2\pi G \rho)A_3
\biggr\}c^2 \, .
</math>
</td>
</tr>
</table>
This gives us three equations ''vs.'' seven unknowns.

===Off-Diagonal Elements===

Notice that the off-diagonal components of both <math>~I_{ij}</math> and <math>~\mathfrak{W}_{ij}</math> are zero. Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2<sup>nd</sup>-order TVE is,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j d^3x
\, .
</math>
</td>
</tr>
</table>

For example — as is explicitly illustrated on p. 130 of EFE — for <math>~i=2</math> and <math>~j=3</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 d^3x}
- 2\Omega_3 \int_V \rho u_1x_3 d^3x \, ,
</math>
</td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (3)</font> ]</td></tr>
</table>
whereas for <math>~i=3</math> and <math>~j=2</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 d^3x
- 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 d^3x}
\, .
</math>
</td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (4)</font> ]</td></tr>
</table>

<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_V \rho u_i x_j d^3x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>
</td>
<td align="right">      if    <math>~i = j \, .</math>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (5)</font> ]</td></tr>
</table>
This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{T}_{32} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{23}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} \int_V \rho v_2 v_3 d^3x \, .</math>
</td>
</tr>
</table>
</td></tr></table>

Our first off-diagonal element is, then,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \rho \int_V u_1 z d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- ~
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_2\Omega_3 c^2
- 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_3 \zeta_2 c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
\Omega_2\Omega_3
+ \biggl[ \frac{\zeta_2 a^2}{a^2 + c^2 }\biggr] \biggl[ 2\Omega_3 + \frac{\zeta_3 b^2}{b^2+a^2}\biggr]
\biggr\} c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2 \, .
</math>
</td>
</tr>
</table>

The second is,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \rho \int_V u_1 y d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- ~
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_3 \Omega_2 b^2
- 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\Omega_2 \zeta_3 b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
\Omega_2 \Omega_3
+ \biggl[ \frac{\zeta_3 a^2}{a^2+b^2}\biggr] \biggl[2\Omega_2 + \frac{\zeta_2 c^2}{c^2 + a^2}\biggr]
\biggr\} b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2 \, .
</math>
</td>
</tr>
</table>

===How Solution is Obtained ===
Adding this pair of governing expressions we obtain,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
+
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4 \mathfrak{T}_{23} - \Omega_2\Omega_3(I_{22}+ I_{33} )
+
2 \int_V \rho u_1 (\Omega_2 x_2 - \Omega_3 x_3) dx \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (6)</font> ]</td></tr>
</table>
and subtracting the pair gives,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
-
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2\Omega_3 (I_{22} - I_{33} )
- 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (7)</font> ]</td></tr>
</table>

=Various Degrees of Simplification=

==Riemann Ellipsoids of Types I, II, & III==
In this, most general, case, the two vectors <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>~y-z</math>-plane — that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>. For a given specified density <math>~(\rho)</math> and choice of the three semi-axes <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in our above [[#SummaryTable|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns: <math>~\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>. Here we show how these five unknowns can be derived from the five constraint equations, closely following the analysis that is presented in §47 (pp. 129 - 132) of [ [[User:Tohline/Appendix/References#EFE|EFE]] ].

===Constraints Due to Off-Diagonal Elements===
We begin by subtracting the constraint equation provided by the first off-diagonal element <math>~(i, j) = (2, 3)</math> from the constraint equation provided by the second off-diagonal element <math>~(i, j) = (3, 2) </math>. This gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 1 + \frac{\zeta_3}{2 \Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \biggl[1 + \frac{\zeta_2}{2\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
+ \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ \frac{b^2}{b^2+a^2} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]
+ \frac{\zeta_3}{\Omega_3} \cdot \frac{\zeta_2}{\Omega_2} \biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (11)</font> ]</td></tr>
</table>

Adding the two instead gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
+
\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr]
+ \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 c^2}{(a^2 + c^2)( b^2+a^2 ) }\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (10)</font> ]</td></tr>
</table>

The first of these relations cleanly gives an expression for the frequency ratio, <math>~\zeta_3/\Omega_3</math>, in terms of the ''other'' frequency ratio, <math>~\zeta_2/\Omega_2</math>. This allows us to rewrite the second relation in terms of the ratio, <math>~\zeta_2/\Omega_2</math>, alone. We obtain,

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

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \biggl\{ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) } \biggr] \cdot \biggl\{ \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 }{( b^2+a^2 ) }\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \biggl\{ c^2 - b^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) } \biggr] \cdot \biggl\{ c^2 - b^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2c^2
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] (4a^2 + c^2 - b^2 )
+ \biggl\{ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr] \biggr\}^22a^2
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
ASIDE:   Alternatively, given that,

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

<tr>
<td align="right">
<math>~
\frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2a^2}\biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
</math>
</td>
</tr>
</table>
the quadratic equation that governs the value of the frequency ratio, <math>~\zeta_3/\Omega_3</math> is …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] (4a^2 + c^2 - b^2 )
+ \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] (4a^2 + c^2 - b^2 )
+ ( b^2 - c^2) (4a^2 + c^2 - b^2 )
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (b^2 - c^2)^2 + 2(b^2 - c^2) \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ ( b^2 - c^2) (4a^2 + c^2 - b^2 )
+ (b^2 - c^2)^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
\biggl[ (4a^2 + c^2 - b^2 )+ 2(b^2 - c^2) \biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
(4a^2 + b^2 - c^2 )
+ 4 a^2 b^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ 0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \biggl(\frac{\zeta_3}{\Omega_3} \biggr)^2 \frac{a^2 b^2}{(a^2+b^2)^2}\biggr]
+ \frac{1}{2}\biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{1}{a^2+b^2}\biggr) \biggr]
(4a^2 + b^2 - c^2 )
+ 1 \, .
</math>
</td>
</tr>
</table>

Now, in our discussion of Riemann S-Type ellipsoids, there is also a quadratic equation that governs the equilibrium frequency ratio, <math>~f \equiv \zeta_3/\Omega_3</math>. It is, specifically,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2 b^2}{(a^2 + b^2)^2} \biggr] f^2
+ \biggl[ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]\frac{f}{a^2 + b^2} + 1 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§48, Eq. (35)</font> ]</td></tr>
</table>
Notice that the first and third terms of this quadratic equation exactly match the first and third terms of the quadratic equation, which we have just derived, that governs the same frequency ratio in Riemann ellipsoids of Types I, II & III. Does the second term match? That is, is the coefficient of the linear term the same in both quadratic relations? Well, …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3 + a^2 b^2 \biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]^{-1} \biggl[A_2 + a^2\biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3(a^2 - b^2) + a^2 b^2 (A_1 - A_2) \biggr]^{-1} \biggl[a^2 A_1 - b^2 A_2\biggr] \, .
</math>
</td>
</tr>
</table>

Even appreciating that we can make the substitution, <math>~A_3 = (2 - A_1 - A_2)</math>, I don't see any way that this coefficient expression can be manipulated to match the associated coefficient in the other expression, namely, <math>~(4a^2 + b^2 - c^2)/2</math>.

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

This is a quadratic equation whose solution gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~4a^2 \cdot \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + c^2 - b^2 ) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \, .
</math>
</td>
</tr>
</table>

For the other frequency ratio we therefore find,

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

<tr>
<td align="right">
<math>~
2\biggl\{ b^2 -c^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \cdot \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + c^2 - b^2 ) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~
4a^2 \cdot \frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + b^2 -c^2) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
</td>
</tr>
</table>
<span id="OffDiagonal"> </span>
<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">
<div align="center">'''SUMMARY:   Riemann Ellipsoids of Types I, II, & III'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\beta \equiv~ -~\frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2 -b^2 + c^2 ) \mp \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (16)</font> ]</td></tr>

<tr>
<td align="right">
<math>~
\gamma \equiv~-~\frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2 + b^2 -c^2) \mp \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (17)</font> ]</td></tr>
</table>

As is emphasized in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, §47, p. 131) "<font color="darkgreen">… the signs in front of the radicals, in the two expressions, go together.</font> Furthermore, "<font color="darkgreen">the two roots … correspond to the fact that, consistent with Dedekind's theorem, two states of internal motions are compatible with the same external figure.</font>"

----

As has also been pointed out in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, §51, p. 158), from the steps that have led to the development and solution of the above pair of quadratic equations we can demonstrate that the following relations also hold:

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

<tr>
<td align="right">
<math>~\beta^2 - 2\beta + \frac{c^2}{a^2} = \biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta \, ,</math>
</td>
<td align="center">
       
</td>
<td align="left">
<math>~\gamma^2 -2\gamma + \frac{b^2}{a^2} = \biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma</math>
</td>
</tr>

<tr>
<td align="right">
<math>~1 - 2\beta + \biggl(\frac{a^2}{c^2}\biggr)\beta^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4a^2 - b^2 - 3c^2}{2c^2}\biggr]\beta \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~1 - 2\gamma + \biggl(\frac{a^2}{b^2}\biggr)\gamma^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eqs. (161) - (163)</font> ]</td></tr>
</table>

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

===Constraints Due to Diagonal Elements===

Next, to simplify manipulations, let's replace the frequency ratios by these newly defined — and ''known'' — parameters, <math>~\beta</math> and <math>~\gamma</math>, in the three diagonal-element expressions that are written out in our above [[#SummaryTable|Summary Table]].

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">Rewritten Diagonal-Element Expressions</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]
~-~(2\pi G\rho) A_1
\biggr\} a^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 b^2
- \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2 c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
- \Omega_2^2 - \Omega_3^2
+ 2 \Omega_3^2 \gamma
+ 2 \Omega_2^2 \beta
~+~(2\pi G\rho) A_1
\biggr\} a^2
- \biggl( \frac{a^4}{b^2}\biggr) \Omega_3^2\gamma^2
- \biggl( \frac{a^4}{c^2}\biggr) \Omega_2^2 \beta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{ \Omega_2^2 \biggl[2 \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr) \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2 \gamma - 1 - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2 \biggr]
~+~(2\pi G\rho) A_1
\biggr\}a^2
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (158)</font> ]</td></tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 a^2
- \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma^2 a^2
- \Omega_3^2 b^2
+ 2 a^2 \Omega_3^2 \gamma
~+~( 2\pi G \rho) b^2A_2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~a^2 \Omega_3^2 \biggl[\gamma^2 - 2\gamma
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]
~+~( 2\pi G \rho) b^2A_2
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (159)</font> ]</td></tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2 a^2
- \biggl\{
\Omega_2^2 + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]
- (2\pi G \rho)A_3
\biggr\}c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \Omega_2^2 \beta^2 a^2
-\Omega_2^2c^2 + 2 a^2\Omega_2^2 \beta
+ (2\pi G \rho)c^2 A_3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+ (2\pi G \rho)c^2 A_3
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (160)</font> ]</td></tr>
</table>

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

Using the <math>~(i, j) = (3, 3)</math> element to preplace <math>~\Pi</math> in the other two expressions, we obtain,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+
\Omega_2^2 \biggl[2 \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr) \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2 \gamma - 1 - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2 \biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
</td>
</tr>
</table>

and,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2 \biggl[\gamma^2 - 2\gamma
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]
~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 - \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr] \, .
</math>
</td>
</tr>
</table>

Inserting the [[#OffDiagonal|various relations highlighted above]], these two expressions may be rewritten as,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta
~-~\Omega_2^2 \biggl[ \frac{4a^2 - b^2 - 3c^2}{2c^2}\biggr]\beta
~-~\Omega_3^2 \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~-~\Omega_3^2 \gamma \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
</td>
</tr>
</table>

<span id="Temporary">and,</span>

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2
\biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 - \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \Omega_2^2 \beta
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~-~\Omega_3^2 \gamma
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ -~\Omega_3^2 \gamma
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr] \, .
</math>
</td>
</tr>
</table>
Together, then,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr]
~+~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\biggl\{
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
~+~\frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~2\pi G \rho \biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr]
\biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2[ c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)] + (4a^2 - c^2 - 3b^2)a^2 c^2}{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho \biggl\{
\biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~\biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr]
\biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2 c^2(c^2 - b^2) + a^2 b^2(c^2 - b^2 ) + ( c^2 - b^2)a^2 c^2 + a^2 (4a^2 -2b^2 - 2c^2 )(c^2 - b^2 ) }{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho
\biggl[ \frac{b^2(a^2A_1 - c^2 A_3) + a^2(3b^2-4a^2 + c^2)B_{23} }{a^2b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{a^2b^2} \biggr] \, ,
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (170)</font> ]</td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B_{23}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §21, Eqs. (105) & (107)</font> ]</td></tr>
</table>

Similarly, given that ([[#Temporary|see just above]]),

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

<tr>
<td align="right">
<math>~\Omega_2^2 \beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 - b^2)B_{23} }{c^2 } \biggr] \, ,
</math>
</td>
</tr>
</table>
we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 - b^2)B_{23} }{c^2 } \biggr]
\biggr\}
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{a^2b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho \biggl\{
\biggl[ \frac{ [4a^4 - a^2 (b^2 + c^2) + b^2 c^2 ](c^2 - b^2)B_{23} }{a^2 b^2 c^2 }
\biggr]
~+~
\biggl[ \frac{a^2c^2 (3b^2-4a^2 + c^2)B_{23} + b^2c^2(a^2A_1 - c^2 A_3) }{a^2b^2 c^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho
\biggl[ \frac{ a^2( b^2 + 3c^2 - 4a^2 ) B_{23} +c^2(a^2A_1 - b^2 A_2) }{a^2 c^2 }
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (171)</font> ]</td></tr>
</table>

Finally, looking back at the <math>~(i, j) = (3, 3)</math> constraint and recognizing that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~-~
\Omega_2^2\beta (c^2 - b^2)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr] \, ,
</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~2a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+ (4\pi G \rho)c^2 A_3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4\pi G \rho)c^2 A_3
-~
(c^2 - b^2 )\Omega_2^2 \beta
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4\pi G \rho)c^2 A_3
+~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi G \rho c^2 \biggl\{ A_3
+~
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

==Riemann S-Type Ellipsoids==
In this case, we assume that <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are aligned with each other and, as well, are aligned with the <math>~z</math>-axis; that is to say, in addition to setting <math>~(\Omega_1, \zeta_1) = (0, 0)</math> we also set <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. So, there are only three unknowns — <math>~\Pi, (\Omega_3, \zeta_3)</math> — and they can be determined by ignoring off-axis expressions and simultaneously solving the ''diagonal element'' expressions displayed in our above [[#SummaryTable|''Summary Table'']]. Furthermore, two of the three diagonal-element expressions can be simplified because we are setting <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. The three relevant equilibrium constraints are:

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">2<sup>nd</sup>-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
- (2\pi G \rho)A_3 c^2
</math>
</td>
</tr>
</table>

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

The <math>~(i, j) = (3, 3)</math> component expression immediately identifies the value of one of the unknowns, namely,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^3\pi^2}{3\cdot 5} \biggr) G \rho^2A_3 a b c^3 \, .
</math>
</td>
</tr>
</table>

From the remaining pair of diagonal-element expressions, we therefore have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^2 \Omega_3^2
+ 2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \, ,
</math>
</td>
</tr>
</table>

and,
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+
b^2 \Omega_3^2
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \, .
</math>
</td>
</tr>
</table>
Multiplying the first of these two expressions through by <math>~b^2</math> and the second through by <math>~a^2</math>, then subtracting the second from the first gives,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2\biggl\{
2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \biggr\}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-~
a^2\biggl\{
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
2 \biggl[ \frac{b^4 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )b^2 \biggr\}
~-~
\biggl\{
2 \biggl[ \frac{a^4 b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) a^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_3 c^2 - A_2 b^2) a^2 ~-~(A_3 c^2 - A_1 a^2 )b^2}{ b^2 - a^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (30)</font> ]</td></tr>
</table>
Note that — as EFE has done and as we have recorded in a [[ThreeDimensionalConfigurations/JacobiEllipsoids#Equilibrium_Conditions_for_Jacobi_Ellipsoids|related discussion]] — the first term on the right-hand-side of this last expression can be expressed more compactly in terms of the coefficient, <math>~A_{12}</math>.

Alternatively, dividing the first expression through by <math>~a^2</math> and the second by <math>~b^2</math>, then adding the pair of expressions gives,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )\frac{1}{a^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~+~
\biggl[ \frac{a^2 b^2}{(b^2+a^2)^2}\biggr] \zeta_3^2
+
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \frac{1}{b^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_3^2 + 2 \Omega_3 \zeta_3
+ 2\biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
~+~2\pi G\rho \biggl[ \frac{A_3 c^2 - A_1 a^2 }{a^2} + \frac{A_3c^2 - A_2 b^2}{b^2}\biggr] \, .
</math>
</td>
</tr>
</table>

If we divide through by 2, then replace the product, <math>~\Omega_3\zeta_3</math>, in this expression by the relation derived immediately above, we have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
\Omega_3^2
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~-~\pi G\rho \biggl[ \frac{b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 ) }{a^2b^2} \biggr]
~-~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2)}{ b^2 - a^2} \biggr]\biggl[ \frac{b^2+a^2}{b^2 a^2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) }
\biggl\{ [ b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 )](b^2-a^2)
~+~
[ (A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2) ](b^2+a^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) }
\biggl\{ [ - A_1 a^2 b^2 - A_2 a^2 b^2 ](b^2-a^2)
~+~
(A_1 - A_2)a^2b^2 (b^2+a^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) }
\biggl[
A_1 a^2
- A_2 b^2
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (29)</font> ]</td></tr>
</table>

<span id="fDefined">It has become customary to characterize each Riemann S-Type ellipsoid by the value of its equilibrium frequency ratio, </span>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{\zeta_3}{\Omega_3} \, ,</math>
</td>
</tr>
</table>
in which case the relevant pair of constraint equations becomes,

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] f \Omega_3^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\Omega_3^2 \biggl\{1
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] f^2 \biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) }
\biggl[
A_1 a^2
- A_2 b^2
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (33)</font> ]</td></tr>
</table>

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

These two equations can straightforwardly be combined to generate a quadratic equation for the frequency ratio, <math>~f</math>. Then, once the value of <math>~f</math> has been determined, either expression can be used to determine the corresponding equilibrium value for <math>~\Omega_3</math> in the unit of <math>~(\pi G \rho)^{1 / 2}</math>. The fact that the value of <math>~f</math> is determined from the solution of a quadratic equation underscores the realization that, for a given specification of the ellipsoidal geometry <math>~(a, b, c)</math>, if an equilibrium exists — ''i.e.,'' if the solution for <math>~f</math> is real rather than imaginary — then two equally valid, and usually different (''i.e.,'' non-degenerate), values of <math>~f</math> will be realized. This means that two different underlying flows — one ''direct'' and the other ''adjoint'' — will sustain the shape of the ellipsoidal configuration, as viewed from a frame that is rotating about the <math>~z</math>-axis with frequency, <math>~\Omega_3</math>.

==Jacobi and Dedekind Ellipsoids==
Describe …

==Maclaurin Spheroids==

Describe …

=Appendices:  Various Integrals Over Ellipsoid Volume=
Throughout this set of appendices, we work with a uniform-density ellipsoid whose surface is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \, .
</math>
</td>
</tr>
</table>

==Appendix A:  Volume==

Here we seek to find the volume of the ellipsoid via the ''Cartesian'' integral expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint dx ~dy ~dz \, .
</math>
</td>
</tr>
</table>

===Preliminaries===
First, we will integrate over <math>~x</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x_\ell</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
a\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \, ;
</math>
</td>
</tr>
</table>

second, we will integrate over <math>~z</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~z_\ell</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
c\biggl[ 1 - \frac{y^2}{b^2} \biggr]^{1 / 2} \, ;
</math>
</td>
</tr>
</table>
third, we will integrate over <math>~y</math> and set the limits of integration as <math>~\pm b</math>.

===Carry Out the Integration===
Following thestrategy that has just been outlined, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iint dy ~dz \int_{-x_\ell}^{+x_\ell} dx
=
\iint dy ~dz \biggl[ x \biggr]_{-x_\ell}^{+x_\ell}
=
2\int dy \int x_\ell ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2a\int dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
=
\frac{2a}{c} \int dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{c} \int \frac{dy}{2} \biggl[ z\sqrt{ z_\ell^2- z^2 } + z_\ell^2 \sin^{-1} \biggl( \frac{z}{|z_\ell |} \biggr) \biggr]_{-z_\ell}^{+z_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{c} \int \biggl[ z_\ell \cancelto{0}{\sqrt{ z_\ell^2- z_\ell^2 }} + z_\ell^2 \sin^{-1} \biggl(1\biggr) \biggr] dy
=
\frac{2a}{c} \int \biggl[ \frac{\pi}{2} z_\ell^2 \biggr] dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi a c \int_{-b}^{+b} \biggl( 1 - \frac{y^2}{b^2} \biggr) dy
=
\pi a c \biggl[ y - \frac{y^3}{3b^2} \biggr]_{-b}^{+b}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\pi}{3} \cdot a b c\, .
</math>
</td>
</tr>
</table>

==Appendix B:  Coriolis Component u<sub>1</sub>x<sub>2</sub>==
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_1 y] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} y ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \iiint y^2 ~dx ~dy ~dz
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint yz ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int dz \int_{-x_\ell}^{+x_\ell} dx
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z ~dz \int_{-x_\ell}^{+x_\ell} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{2a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int x_\ell dz
+
\biggl[ \frac{2a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~x_\ell ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
+
\biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz
~+~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int_{-z_\ell}^{+z_\ell} z~\biggl[ z_\ell^2 - z^2 \biggr]^{1 / 2} ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z \sqrt{z_\ell^2 - z^2} + z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}
~-~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \cdot \frac{1}{3} \biggl\{ \biggl[ z_\ell^2 - z^2 \biggr]^{3 / 2} \biggr\}_{-z_\ell}^{+z_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}
=
- \pi a~c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int_{-b}^b y^2 \biggl[1 - \frac{y^2}{b^2} \biggr] dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \pi ac\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{y^3}{3} - \frac{y^5}{5b^2} \biggr]_{-b}^{+b}
=
- 2\pi a b^3 c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{2}{15} \biggr]
=
- \frac{4\pi abc}{3} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{b^2}{5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{I_{22}}{\rho} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (9a)</font> ]</td></tr>
</table>

==Appendix C:  Coriolis Component u<sub>1</sub>x<sub>3</sub>==

Here we will additionally make use of the integration limits,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~y_\ell^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~b^2 \biggl(1 - \frac{z^2}{c^2}\biggr) \, .</math>
</td>
</tr>
</table>
Integration over the relevant Coriolis component gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_1 z] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} z ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3\iiint \cancelto{0}{y z ~dx ~dy ~dz}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint z^2 ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \int_{-x_\ell}^{+x_\ell} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2a\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \biggl\{ \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int_{-y_\ell}^{+y_\ell} \biggl[ y_\ell^2 - y^2 \biggr]^{1 / 2} dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \cdot \frac{1}{2}\biggl\{ y \sqrt{y_\ell^2 - y^2} + y_\ell^2 \sin^{-1}\biggr( \frac{y}{|y_\ell |} \biggr)\biggr\}_{-y_\ell}^{+y_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ \frac{\pi}{2} y_\ell^2 \biggr\} dz
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ 1 - \frac{z^2}{c^2} \biggr\} dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{z^3}{3} - \frac{z^5}{5c^2} \biggr\}_{-c}^{+c}
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{1}{3} - \frac{1}{5} \biggr\}2c^3
=
\frac{4 \pi a b c}{3}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{c^2}{5} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~+
~\frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (9b)</font> ]</td></tr>
</table>

==Appendix D:   The Other Four Coriolis Components ==

It follows that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_2 x] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z} + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x \biggr\} x ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+~\frac{I_{11}}{\rho}\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_2 z] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z + \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x} \biggr\} z ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{I_{33}}{\rho} \biggl[\frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_3 x] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x + \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y} \biggr\} x ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{I_{11}}{\rho} \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_3 y] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x} + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y \biggr\} y ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+~\frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 \, .
</math>
</td>
</tr>
</table>

==Appendix E:   Kinetic Energy Components ==

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

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{11} = \int_V u_1 u_1 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_1^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 y^2
- 2\cancelto{0}{\biggl[ \frac{a^2}{a^2 + b^2}\biggr]
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \zeta_3} yz
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 z^2
\biggr\} ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \iiint y^2 ~dx ~dy ~dz
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2\iiint z^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{22}}{\rho} \biggr]
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{33}}{\rho} \biggr] \, .</math>
</td>
</tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{22} = \int_V u_2 u_2 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{33}}{\rho} \biggr]
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{11}}{\rho} \biggr] \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{33} = \int_V u_3 u_3 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{11}}{\rho} \biggr]
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{22}}{\rho} \biggr] \, .</math>
</td>
</tr>
</table>

===Off-Diagonal Elements===
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{23} = \int_V u_2 u_3 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2 u_3] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z \biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~+ \iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~\iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 \biggr\}
x^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- ~\frac{I_{11}}{\rho}
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (8)</font> ]</td></tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{12} = \int_V u_1 u_2 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_1 u_2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~
\biggl[ \frac{a^2}{a^2+c^2}\biggr]
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2
\iiint z^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ \frac{I_{33}}{\rho}
\biggl[ \frac{a^2}{a^2+c^2}\biggr]
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{31} = \int_V u_3 u_1 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_3 u_1] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -~
\biggl[ \frac{c^2}{c^2+b^2}\biggr]
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3
\iiint y^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -~ \frac{I_{22}}{\rho}
\biggl[ \frac{c^2}{c^2+b^2}\biggr]
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3 \, .
</math>
</td>
</tr>
</table>

And, finally,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{T}_{32}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{23} \, ;</math>
</td>
<td align="center">     </td>
<td align="right">
<math>~\mathfrak{T}_{21}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{12} \, ;</math>
</td>
<td align="center">      and,     </td>
<td align="right">
<math>~\mathfrak{T}_{13}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{31} \, .</math>
</td>
</tr>
</table>

=See Also=
* [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Properties of Maclaurin Spheroids]]
* [[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]

{{SGFfooter}}

VE/RiemannEllipsoids

2024-07-02T22:00:41Z

70.180.56.157: Created page with "__FORCETOC__  =Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations= {| class="2ndOrderTVE" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#9390DB;" | <font size="-1"><b>Steady-State<br />2<sup>nd</sup>-Order<br />Tensor Virial<br />Equations</b></font> |} Drawing fro..."

__FORCETOC__


=Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations=
{| class="2ndOrderTVE" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Ellipsoidal_.26_Ellipsoidal-Like|<b>Steady-State<br />2<sup>nd</sup>-Order<br />Tensor Virial<br />Equations</b>]]</font>
|}
Drawing from our [[VE#Virial_Equations_.28Rotating_Frame.29|accompanying discussion of virial equations as viewed from a rotating frame of reference]], here we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx \, ,
</math>
</td>
</tr>
</table>
to determine the equilibrium conditions of uniform-density <math>~(\rho)</math> ellipsoids that have semi-axes, <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c),</math> and an internal velocity field, <math>~\vec{u}</math> (as [[#Adopted_.28Internal.29_Velocity_Field|prescribed below]]), that preserves this specified ellipsoidal shape, as viewed from a frame of reference that is rotating with angular velocity, <math>~\vec\Omega</math>. Because each of the indices, <math>~i</math> and <math>~j</math>, run from 1 to 3, inclusive, this TVE appears to provide nine equilibrium constraints; and once the values of the density and the three semi-axes are specified, there appear to be seven unknowns: <math>~\Pi</math> and the three pairs of velocity-field components <math>~(\Omega_1, \zeta_1)</math>, <math>~(\Omega_2, \zeta_2)</math>, <math>~(\Omega_3, \zeta_3).</math> In practice, however, only five constraints are relevant/independent because, as is encapsulated in …
<table border="0" width="60%" align="center" cellpadding="10"><tr><td align="left">
<div align="center"><font color="maroon">'''Riemann's Fundamental Theorem'''</font></div>
<font color="darkgreen">… non-trivial solutions are obtained only if no more than two of the three pairs of velocity-field components are different from zero.</font>
</td></tr></table>
<span id="SummaryTable">Following EFE</span>, we will set <math>~\Omega_1 = \zeta_1 = 0</math>, in which case the only applicable TVE constraint relations are the five identified in the following table of equations.

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">Each Associated 2<sup>nd</sup>-Order TVE Expression</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
+ \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2
+ \biggl\{
\Omega_2^2 + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
- (2\pi G \rho)A_3
\biggr\}c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

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

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

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

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>
</table>

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

==General Coefficient Expressions==

In the context of our discussion of configurations that are triaxial ellipsoids, we begin by adopting the <math>~(\ell, m, s)</math> subscript notation to identify which semi-axis length is the (largest, medium-length, smallest). As has been detailed in an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|accompanying chapter]], the gravitational potential anywhere inside or on the surface of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~\frac{A_\ell}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~\frac{2}{a_\ell^3}
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{A_s}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{2}{a_\ell^3} \biggl[ \frac{(a_m/a_s) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{A_m}{a_\ell a_m a_s} = \frac{2 - (A_\ell + A_s)}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{ 2}{a_\ell^3 }
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_s/a_m)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_s}{a_\ell} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_m/a_\ell)^2}{1 - (a_s/a_\ell)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>2</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_1 > a_2 > a_3 > 0</math> — such as Jacobi, Dedekind, or ''most'' Riemann S-Type ellipsoids — we must adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_m, a_m)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_1
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[ \frac{(a_2/a_1) \sin\theta - (a_3/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 - (A_1+A_3) \, ,</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §17, Eq. (32)</font> ]</td></tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===

When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in [[Appendix/References#EFE|EFE]] as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_1
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 \biggl( \frac{a_2}{a_1} \biggr)\biggl( \frac{a_3}{a_1} \biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2 \biggl( \frac{a_3}{a_1} \biggr) \biggl[ \frac{(a_3/a_1) \sin\theta - (a_2/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3 = 2 - (A_1 + A_2)
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{2a_2 a_3}{a_1^2}
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_2/a_3)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals of the first and second kind are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_2}{a_1} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_3/a_1)^2}{1 - (a_2/a_1)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48d, footnote to Table VII (p. 143)</font> ]</td></tr>
</table>
</div>
NOTE: All ''irrotational'' ellipsoids belong to this category of configurations.

===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_2 > a_1 > a_3 > 0</math> — for example, ''most'' Riemann ellipsoids of Types I, II, & III — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>~(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl( \frac{a_1}{a_2}\biggr) \biggl[ \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_1 = 2 - (A_2 + A_3)
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{ 2a_1 a_3}{a_2^2 }
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_2} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_1/a_2)^2}{1 - (a_3/a_2)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

===Oblate Spheroids [a<sub>2</sub> = a<sub>1</sub> > a<sub>3</sub>]===

Starting with the case of <math>~a_2 > a_1 > a_3 > 0</math> and setting <math>~a_2 = a_1</math>, we recognize, first, that <math>~k = 0</math>. Hence, we have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl[ \frac{ \sin\theta - (a_3/a_1)E(\theta,0)}{\sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>
</table>

==Adopted (Internal) Velocity Field==

EFE (p. 130) states that the … <font color="#007700">kinematical requirement, that the motion <math>~(\vec{u})</math>, associated with <math>~\vec{\zeta}</math>, preserves the ellipsoidal boundary, leads to the following expressions for its components:</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~u_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~u_2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~u_3</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (1)</font> ]</td></tr>
</table>

==Equilibrium Expressions==
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \delta_{ij}\Pi \, .</math>
</td>
</tr>
</table>
</div>
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2<sup>nd</sup>-order TVE takes on the more general form:
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 2, §12, Eq. (64)</font> ]</td></tr>
</table>

EFE (p. 57) also shows that … <font color="#007700">The potential energy tensor … for a homogeneous ellipsoid is given by</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2A_i I_{ij} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (128)</font> ]</td></tr>
</table>
<font color="#007700">where</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~I_{ij}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (129)</font> ]</td></tr>
</table>

<font color="#007700">is the moment of inertia tensor.</font> Expressions for all nine components of the kinetic energy tensor, <math>~\mathfrak{T}_{ij}</math> are derived in [[#Appendix_E:_.C2.A0_Kinetic_Energy_Components|Appendix E]], below; and expressions for each of the six Coriolis components can be found in [[#Appendix_B:_.C2.A0Coriolis_Component_u1x2|Appendices B, C, & D]].

===The Three Diagonal Elements===
For <math>~i = j = 1</math>, we have,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi
+ \Omega^2 I_{11} - \Omega_1\Omega_k I_{k1} + 2\epsilon_{1lm}\Omega_m \int_V \rho u_lx_1 d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11}
- \Omega_1^2I_{11}
+ 2 \Omega_3 \int_V \rho u_2x_1 ~d^3x
- 2\Omega_2 \int_V \rho u_3x_1 ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi
+( \Omega_2^2 + \Omega_3^2) I_{11}
+ 2 \Omega_3\rho \int_V u_2x ~d^3x
- 2\Omega_2\rho \int_V u_3 x~ d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}
~-~(2\pi G\rho) A_1 I_{11} + \Pi
+( \Omega_2^2 + \Omega_3^2) I_{11}
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 I_{11}
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 I_{11}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+ \biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} I_{11}
+
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2 \, .
</math>
</td>
</tr>
</table>

Once we choose the values of the (semi) axis lengths <math>~(a, b, c)</math> of an ellipsoid — from which the value of <math>~A_1</math> can be immediately determined — along with a specification of <math>~\rho</math>, this equation has the following five unknowns: <math>~\Pi, \Omega_2, \Omega_3, \zeta_2, \zeta_3</math>. Similarly, for <math>~i = j = 2</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi
+ \Omega^2 I_{22} - \Omega_2\Omega_k I_{k2} + 2\epsilon_{2lm}\Omega_m \int_V \rho u_lx_2 d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi
+ (\Omega_1^2 + \Omega_3^2) I_{22} + 2\Omega_1 \rho \int_V u_3 y ~d^3x
- 2\Omega_3 \rho \int_V u_1 y ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33}
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
~-~( 2\pi G \rho) A_2 {I}_{22}
+ \Pi
+ (\Omega_1^2 + \Omega_3^2) I_{22}
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1 I_{22}
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 I_{22}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}{I}_{22}
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~-\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 c^2 \, .
</math>
</td>
</tr>
</table>

This gives us a second equation, but an additional pair of (for a total of seven) unknowns: <math>~\Omega_1, \zeta_1</math>. For the third diagonal element — that is, for <math>~i=j=3</math> — we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi
+ \Omega^2 I_{33} - \Omega_3\Omega_k I_{k3} + 2\epsilon_{3lm}\Omega_m \int_V \rho u_lx_3 ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi
+ (\Omega_1^2 + \Omega_2^2) I_{33} + 2\Omega_2 \rho \int_V u_1 z ~d^3x
- 2\Omega_1 \rho \int_V u_2 z ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
- (2\pi G \rho)A_3 I_{33} + \Pi
+ (\Omega_1^2 + \Omega_2^2) I_{33} + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 I_{33}
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
+ \biggl\{
(\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1
- (2\pi G \rho)A_3
\biggr\}I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 b^2
+ \biggl\{
(\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1
- (2\pi G \rho)A_3
\biggr\}c^2 \, .
</math>
</td>
</tr>
</table>
This gives us three equations ''vs.'' seven unknowns.

===Off-Diagonal Elements===

Notice that the off-diagonal components of both <math>~I_{ij}</math> and <math>~\mathfrak{W}_{ij}</math> are zero. Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2<sup>nd</sup>-order TVE is,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j d^3x
\, .
</math>
</td>
</tr>
</table>

For example — as is explicitly illustrated on p. 130 of EFE — for <math>~i=2</math> and <math>~j=3</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 d^3x}
- 2\Omega_3 \int_V \rho u_1x_3 d^3x \, ,
</math>
</td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (3)</font> ]</td></tr>
</table>
whereas for <math>~i=3</math> and <math>~j=2</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 d^3x
- 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 d^3x}
\, .
</math>
</td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (4)</font> ]</td></tr>
</table>

<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_V \rho u_i x_j d^3x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>
</td>
<td align="right">      if    <math>~i = j \, .</math>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (5)</font> ]</td></tr>
</table>
This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{T}_{32} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{23}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} \int_V \rho v_2 v_3 d^3x \, .</math>
</td>
</tr>
</table>
</td></tr></table>

Our first off-diagonal element is, then,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \rho \int_V u_1 z d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- ~
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_2\Omega_3 c^2
- 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_3 \zeta_2 c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
\Omega_2\Omega_3
+ \biggl[ \frac{\zeta_2 a^2}{a^2 + c^2 }\biggr] \biggl[ 2\Omega_3 + \frac{\zeta_3 b^2}{b^2+a^2}\biggr]
\biggr\} c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2 \, .
</math>
</td>
</tr>
</table>

The second is,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \rho \int_V u_1 y d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- ~
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_3 \Omega_2 b^2
- 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\Omega_2 \zeta_3 b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
\Omega_2 \Omega_3
+ \biggl[ \frac{\zeta_3 a^2}{a^2+b^2}\biggr] \biggl[2\Omega_2 + \frac{\zeta_2 c^2}{c^2 + a^2}\biggr]
\biggr\} b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2 \, .
</math>
</td>
</tr>
</table>

===How Solution is Obtained ===
Adding this pair of governing expressions we obtain,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
+
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4 \mathfrak{T}_{23} - \Omega_2\Omega_3(I_{22}+ I_{33} )
+
2 \int_V \rho u_1 (\Omega_2 x_2 - \Omega_3 x_3) dx \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (6)</font> ]</td></tr>
</table>
and subtracting the pair gives,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
-
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2\Omega_3 (I_{22} - I_{33} )
- 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (7)</font> ]</td></tr>
</table>

=Various Degrees of Simplification=

==Riemann Ellipsoids of Types I, II, & III==
In this, most general, case, the two vectors <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>~y-z</math>-plane — that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>. For a given specified density <math>~(\rho)</math> and choice of the three semi-axes <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in our above [[#SummaryTable|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns: <math>~\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>. Here we show how these five unknowns can be derived from the five constraint equations, closely following the analysis that is presented in §47 (pp. 129 - 132) of [ [[User:Tohline/Appendix/References#EFE|EFE]] ].

===Constraints Due to Off-Diagonal Elements===
We begin by subtracting the constraint equation provided by the first off-diagonal element <math>~(i, j) = (2, 3)</math> from the constraint equation provided by the second off-diagonal element <math>~(i, j) = (3, 2) </math>. This gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 1 + \frac{\zeta_3}{2 \Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \biggl[1 + \frac{\zeta_2}{2\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
+ \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ \frac{b^2}{b^2+a^2} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]
+ \frac{\zeta_3}{\Omega_3} \cdot \frac{\zeta_2}{\Omega_2} \biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (11)</font> ]</td></tr>
</table>

Adding the two instead gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
+
\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr]
+ \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 c^2}{(a^2 + c^2)( b^2+a^2 ) }\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (10)</font> ]</td></tr>
</table>

The first of these relations cleanly gives an expression for the frequency ratio, <math>~\zeta_3/\Omega_3</math>, in terms of the ''other'' frequency ratio, <math>~\zeta_2/\Omega_2</math>. This allows us to rewrite the second relation in terms of the ratio, <math>~\zeta_2/\Omega_2</math>, alone. We obtain,

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

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \biggl\{ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) } \biggr] \cdot \biggl\{ \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 }{( b^2+a^2 ) }\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \biggl\{ c^2 - b^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) } \biggr] \cdot \biggl\{ c^2 - b^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2c^2
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] (4a^2 + c^2 - b^2 )
+ \biggl\{ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr] \biggr\}^22a^2
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
ASIDE:   Alternatively, given that,

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

<tr>
<td align="right">
<math>~
\frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2a^2}\biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
</math>
</td>
</tr>
</table>
the quadratic equation that governs the value of the frequency ratio, <math>~\zeta_3/\Omega_3</math> is …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] (4a^2 + c^2 - b^2 )
+ \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] (4a^2 + c^2 - b^2 )
+ ( b^2 - c^2) (4a^2 + c^2 - b^2 )
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (b^2 - c^2)^2 + 2(b^2 - c^2) \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ ( b^2 - c^2) (4a^2 + c^2 - b^2 )
+ (b^2 - c^2)^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
\biggl[ (4a^2 + c^2 - b^2 )+ 2(b^2 - c^2) \biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
(4a^2 + b^2 - c^2 )
+ 4 a^2 b^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ 0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \biggl(\frac{\zeta_3}{\Omega_3} \biggr)^2 \frac{a^2 b^2}{(a^2+b^2)^2}\biggr]
+ \frac{1}{2}\biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{1}{a^2+b^2}\biggr) \biggr]
(4a^2 + b^2 - c^2 )
+ 1 \, .
</math>
</td>
</tr>
</table>

Now, in our discussion of Riemann S-Type ellipsoids, there is also a quadratic equation that governs the equilibrium frequency ratio, <math>~f \equiv \zeta_3/\Omega_3</math>. It is, specifically,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2 b^2}{(a^2 + b^2)^2} \biggr] f^2
+ \biggl[ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]\frac{f}{a^2 + b^2} + 1 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§48, Eq. (35)</font> ]</td></tr>
</table>
Notice that the first and third terms of this quadratic equation exactly match the first and third terms of the quadratic equation, which we have just derived, that governs the same frequency ratio in Riemann ellipsoids of Types I, II & III. Does the second term match? That is, is the coefficient of the linear term the same in both quadratic relations? Well, …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3 + a^2 b^2 \biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]^{-1} \biggl[A_2 + a^2\biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3(a^2 - b^2) + a^2 b^2 (A_1 - A_2) \biggr]^{-1} \biggl[a^2 A_1 - b^2 A_2\biggr] \, .
</math>
</td>
</tr>
</table>

Even appreciating that we can make the substitution, <math>~A_3 = (2 - A_1 - A_2)</math>, I don't see any way that this coefficient expression can be manipulated to match the associated coefficient in the other expression, namely, <math>~(4a^2 + b^2 - c^2)/2</math>.

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

This is a quadratic equation whose solution gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~4a^2 \cdot \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + c^2 - b^2 ) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \, .
</math>
</td>
</tr>
</table>

For the other frequency ratio we therefore find,

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

<tr>
<td align="right">
<math>~
2\biggl\{ b^2 -c^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \cdot \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + c^2 - b^2 ) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~
4a^2 \cdot \frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + b^2 -c^2) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
</td>
</tr>
</table>
<span id="OffDiagonal"> </span>
<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">
<div align="center">'''SUMMARY:   Riemann Ellipsoids of Types I, II, & III'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\beta \equiv~ -~\frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2 -b^2 + c^2 ) \mp \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (16)</font> ]</td></tr>

<tr>
<td align="right">
<math>~
\gamma \equiv~-~\frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2 + b^2 -c^2) \mp \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (17)</font> ]</td></tr>
</table>

As is emphasized in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, §47, p. 131) "<font color="darkgreen">… the signs in front of the radicals, in the two expressions, go together.</font> Furthermore, "<font color="darkgreen">the two roots … correspond to the fact that, consistent with Dedekind's theorem, two states of internal motions are compatible with the same external figure.</font>"

----

As has also been pointed out in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, §51, p. 158), from the steps that have led to the development and solution of the above pair of quadratic equations we can demonstrate that the following relations also hold:

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

<tr>
<td align="right">
<math>~\beta^2 - 2\beta + \frac{c^2}{a^2} = \biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta \, ,</math>
</td>
<td align="center">
       
</td>
<td align="left">
<math>~\gamma^2 -2\gamma + \frac{b^2}{a^2} = \biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma</math>
</td>
</tr>

<tr>
<td align="right">
<math>~1 - 2\beta + \biggl(\frac{a^2}{c^2}\biggr)\beta^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4a^2 - b^2 - 3c^2}{2c^2}\biggr]\beta \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~1 - 2\gamma + \biggl(\frac{a^2}{b^2}\biggr)\gamma^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eqs. (161) - (163)</font> ]</td></tr>
</table>

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

===Constraints Due to Diagonal Elements===

Next, to simplify manipulations, let's replace the frequency ratios by these newly defined — and ''known'' — parameters, <math>~\beta</math> and <math>~\gamma</math>, in the three diagonal-element expressions that are written out in our above [[#SummaryTable|Summary Table]].

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">Rewritten Diagonal-Element Expressions</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]
~-~(2\pi G\rho) A_1
\biggr\} a^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 b^2
- \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2 c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
- \Omega_2^2 - \Omega_3^2
+ 2 \Omega_3^2 \gamma
+ 2 \Omega_2^2 \beta
~+~(2\pi G\rho) A_1
\biggr\} a^2
- \biggl( \frac{a^4}{b^2}\biggr) \Omega_3^2\gamma^2
- \biggl( \frac{a^4}{c^2}\biggr) \Omega_2^2 \beta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{ \Omega_2^2 \biggl[2 \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr) \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2 \gamma - 1 - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2 \biggr]
~+~(2\pi G\rho) A_1
\biggr\}a^2
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (158)</font> ]</td></tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 a^2
- \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma^2 a^2
- \Omega_3^2 b^2
+ 2 a^2 \Omega_3^2 \gamma
~+~( 2\pi G \rho) b^2A_2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~a^2 \Omega_3^2 \biggl[\gamma^2 - 2\gamma
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]
~+~( 2\pi G \rho) b^2A_2
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (159)</font> ]</td></tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2 a^2
- \biggl\{
\Omega_2^2 + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]
- (2\pi G \rho)A_3
\biggr\}c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \Omega_2^2 \beta^2 a^2
-\Omega_2^2c^2 + 2 a^2\Omega_2^2 \beta
+ (2\pi G \rho)c^2 A_3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+ (2\pi G \rho)c^2 A_3
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (160)</font> ]</td></tr>
</table>

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

Using the <math>~(i, j) = (3, 3)</math> element to preplace <math>~\Pi</math> in the other two expressions, we obtain,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+
\Omega_2^2 \biggl[2 \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr) \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2 \gamma - 1 - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2 \biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
</td>
</tr>
</table>

and,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2 \biggl[\gamma^2 - 2\gamma
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]
~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 - \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr] \, .
</math>
</td>
</tr>
</table>

Inserting the [[#OffDiagonal|various relations highlighted above]], these two expressions may be rewritten as,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta
~-~\Omega_2^2 \biggl[ \frac{4a^2 - b^2 - 3c^2}{2c^2}\biggr]\beta
~-~\Omega_3^2 \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~-~\Omega_3^2 \gamma \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
</td>
</tr>
</table>

<span id="Temporary">and,</span>

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2
\biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 - \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \Omega_2^2 \beta
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~-~\Omega_3^2 \gamma
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ -~\Omega_3^2 \gamma
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr] \, .
</math>
</td>
</tr>
</table>
Together, then,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr]
~+~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\biggl\{
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
~+~\frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~2\pi G \rho \biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr]
\biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2[ c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)] + (4a^2 - c^2 - 3b^2)a^2 c^2}{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho \biggl\{
\biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~\biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr]
\biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2 c^2(c^2 - b^2) + a^2 b^2(c^2 - b^2 ) + ( c^2 - b^2)a^2 c^2 + a^2 (4a^2 -2b^2 - 2c^2 )(c^2 - b^2 ) }{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho
\biggl[ \frac{b^2(a^2A_1 - c^2 A_3) + a^2(3b^2-4a^2 + c^2)B_{23} }{a^2b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{a^2b^2} \biggr] \, ,
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (170)</font> ]</td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B_{23}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §21, Eqs. (105) & (107)</font> ]</td></tr>
</table>

Similarly, given that ([[#Temporary|see just above]]),

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

<tr>
<td align="right">
<math>~\Omega_2^2 \beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 - b^2)B_{23} }{c^2 } \biggr] \, ,
</math>
</td>
</tr>
</table>
we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 - b^2)B_{23} }{c^2 } \biggr]
\biggr\}
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{a^2b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho \biggl\{
\biggl[ \frac{ [4a^4 - a^2 (b^2 + c^2) + b^2 c^2 ](c^2 - b^2)B_{23} }{a^2 b^2 c^2 }
\biggr]
~+~
\biggl[ \frac{a^2c^2 (3b^2-4a^2 + c^2)B_{23} + b^2c^2(a^2A_1 - c^2 A_3) }{a^2b^2 c^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho
\biggl[ \frac{ a^2( b^2 + 3c^2 - 4a^2 ) B_{23} +c^2(a^2A_1 - b^2 A_2) }{a^2 c^2 }
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (171)</font> ]</td></tr>
</table>

Finally, looking back at the <math>~(i, j) = (3, 3)</math> constraint and recognizing that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~-~
\Omega_2^2\beta (c^2 - b^2)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr] \, ,
</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~2a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+ (4\pi G \rho)c^2 A_3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4\pi G \rho)c^2 A_3
-~
(c^2 - b^2 )\Omega_2^2 \beta
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4\pi G \rho)c^2 A_3
+~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi G \rho c^2 \biggl\{ A_3
+~
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

==Riemann S-Type Ellipsoids==
In this case, we assume that <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are aligned with each other and, as well, are aligned with the <math>~z</math>-axis; that is to say, in addition to setting <math>~(\Omega_1, \zeta_1) = (0, 0)</math> we also set <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. So, there are only three unknowns — <math>~\Pi, (\Omega_3, \zeta_3)</math> — and they can be determined by ignoring off-axis expressions and simultaneously solving the ''diagonal element'' expressions displayed in our above [[#SummaryTable|''Summary Table'']]. Furthermore, two of the three diagonal-element expressions can be simplified because we are setting <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. The three relevant equilibrium constraints are:

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">2<sup>nd</sup>-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
- (2\pi G \rho)A_3 c^2
</math>
</td>
</tr>
</table>

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

The <math>~(i, j) = (3, 3)</math> component expression immediately identifies the value of one of the unknowns, namely,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^3\pi^2}{3\cdot 5} \biggr) G \rho^2A_3 a b c^3 \, .
</math>
</td>
</tr>
</table>

From the remaining pair of diagonal-element expressions, we therefore have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^2 \Omega_3^2
+ 2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \, ,
</math>
</td>
</tr>
</table>

and,
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+
b^2 \Omega_3^2
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \, .
</math>
</td>
</tr>
</table>
Multiplying the first of these two expressions through by <math>~b^2</math> and the second through by <math>~a^2</math>, then subtracting the second from the first gives,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2\biggl\{
2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \biggr\}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-~
a^2\biggl\{
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
2 \biggl[ \frac{b^4 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )b^2 \biggr\}
~-~
\biggl\{
2 \biggl[ \frac{a^4 b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) a^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_3 c^2 - A_2 b^2) a^2 ~-~(A_3 c^2 - A_1 a^2 )b^2}{ b^2 - a^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (30)</font> ]</td></tr>
</table>
Note that — as EFE has done and as we have recorded in a [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Equilibrium_Conditions_for_Jacobi_Ellipsoids|related discussion]] — the first term on the right-hand-side of this last expression can be expressed more compactly in terms of the coefficient, <math>~A_{12}</math>.

Alternatively, dividing the first expression through by <math>~a^2</math> and the second by <math>~b^2</math>, then adding the pair of expressions gives,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )\frac{1}{a^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~+~
\biggl[ \frac{a^2 b^2}{(b^2+a^2)^2}\biggr] \zeta_3^2
+
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \frac{1}{b^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_3^2 + 2 \Omega_3 \zeta_3
+ 2\biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
~+~2\pi G\rho \biggl[ \frac{A_3 c^2 - A_1 a^2 }{a^2} + \frac{A_3c^2 - A_2 b^2}{b^2}\biggr] \, .
</math>
</td>
</tr>
</table>

If we divide through by 2, then replace the product, <math>~\Omega_3\zeta_3</math>, in this expression by the relation derived immediately above, we have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
\Omega_3^2
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~-~\pi G\rho \biggl[ \frac{b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 ) }{a^2b^2} \biggr]
~-~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2)}{ b^2 - a^2} \biggr]\biggl[ \frac{b^2+a^2}{b^2 a^2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) }
\biggl\{ [ b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 )](b^2-a^2)
~+~
[ (A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2) ](b^2+a^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) }
\biggl\{ [ - A_1 a^2 b^2 - A_2 a^2 b^2 ](b^2-a^2)
~+~
(A_1 - A_2)a^2b^2 (b^2+a^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) }
\biggl[
A_1 a^2
- A_2 b^2
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (29)</font> ]</td></tr>
</table>

<span id="fDefined">It has become customary to characterize each Riemann S-Type ellipsoid by the value of its equilibrium frequency ratio, </span>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{\zeta_3}{\Omega_3} \, ,</math>
</td>
</tr>
</table>
in which case the relevant pair of constraint equations becomes,

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] f \Omega_3^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\Omega_3^2 \biggl\{1
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] f^2 \biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) }
\biggl[
A_1 a^2
- A_2 b^2
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (33)</font> ]</td></tr>
</table>

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

These two equations can straightforwardly be combined to generate a quadratic equation for the frequency ratio, <math>~f</math>. Then, once the value of <math>~f</math> has been determined, either expression can be used to determine the corresponding equilibrium value for <math>~\Omega_3</math> in the unit of <math>~(\pi G \rho)^{1 / 2}</math>. The fact that the value of <math>~f</math> is determined from the solution of a quadratic equation underscores the realization that, for a given specification of the ellipsoidal geometry <math>~(a, b, c)</math>, if an equilibrium exists — ''i.e.,'' if the solution for <math>~f</math> is real rather than imaginary — then two equally valid, and usually different (''i.e.,'' non-degenerate), values of <math>~f</math> will be realized. This means that two different underlying flows — one ''direct'' and the other ''adjoint'' — will sustain the shape of the ellipsoidal configuration, as viewed from a frame that is rotating about the <math>~z</math>-axis with frequency, <math>~\Omega_3</math>.

==Jacobi and Dedekind Ellipsoids==
Describe …

==Maclaurin Spheroids==

Describe …

=Appendices:  Various Integrals Over Ellipsoid Volume=
Throughout this set of appendices, we work with a uniform-density ellipsoid whose surface is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \, .
</math>
</td>
</tr>
</table>

==Appendix A:  Volume==

Here we seek to find the volume of the ellipsoid via the ''Cartesian'' integral expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint dx ~dy ~dz \, .
</math>
</td>
</tr>
</table>

===Preliminaries===
First, we will integrate over <math>~x</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x_\ell</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
a\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \, ;
</math>
</td>
</tr>
</table>

second, we will integrate over <math>~z</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~z_\ell</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
c\biggl[ 1 - \frac{y^2}{b^2} \biggr]^{1 / 2} \, ;
</math>
</td>
</tr>
</table>
third, we will integrate over <math>~y</math> and set the limits of integration as <math>~\pm b</math>.

===Carry Out the Integration===
Following thestrategy that has just been outlined, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iint dy ~dz \int_{-x_\ell}^{+x_\ell} dx
=
\iint dy ~dz \biggl[ x \biggr]_{-x_\ell}^{+x_\ell}
=
2\int dy \int x_\ell ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2a\int dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
=
\frac{2a}{c} \int dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{c} \int \frac{dy}{2} \biggl[ z\sqrt{ z_\ell^2- z^2 } + z_\ell^2 \sin^{-1} \biggl( \frac{z}{|z_\ell |} \biggr) \biggr]_{-z_\ell}^{+z_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{c} \int \biggl[ z_\ell \cancelto{0}{\sqrt{ z_\ell^2- z_\ell^2 }} + z_\ell^2 \sin^{-1} \biggl(1\biggr) \biggr] dy
=
\frac{2a}{c} \int \biggl[ \frac{\pi}{2} z_\ell^2 \biggr] dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi a c \int_{-b}^{+b} \biggl( 1 - \frac{y^2}{b^2} \biggr) dy
=
\pi a c \biggl[ y - \frac{y^3}{3b^2} \biggr]_{-b}^{+b}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\pi}{3} \cdot a b c\, .
</math>
</td>
</tr>
</table>

==Appendix B:  Coriolis Component u<sub>1</sub>x<sub>2</sub>==
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_1 y] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} y ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \iiint y^2 ~dx ~dy ~dz
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint yz ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int dz \int_{-x_\ell}^{+x_\ell} dx
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z ~dz \int_{-x_\ell}^{+x_\ell} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{2a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int x_\ell dz
+
\biggl[ \frac{2a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~x_\ell ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
+
\biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz
~+~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int_{-z_\ell}^{+z_\ell} z~\biggl[ z_\ell^2 - z^2 \biggr]^{1 / 2} ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z \sqrt{z_\ell^2 - z^2} + z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}
~-~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \cdot \frac{1}{3} \biggl\{ \biggl[ z_\ell^2 - z^2 \biggr]^{3 / 2} \biggr\}_{-z_\ell}^{+z_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}
=
- \pi a~c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int_{-b}^b y^2 \biggl[1 - \frac{y^2}{b^2} \biggr] dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \pi ac\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{y^3}{3} - \frac{y^5}{5b^2} \biggr]_{-b}^{+b}
=
- 2\pi a b^3 c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{2}{15} \biggr]
=
- \frac{4\pi abc}{3} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{b^2}{5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{I_{22}}{\rho} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (9a)</font> ]</td></tr>
</table>

==Appendix C:  Coriolis Component u<sub>1</sub>x<sub>3</sub>==

Here we will additionally make use of the integration limits,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~y_\ell^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~b^2 \biggl(1 - \frac{z^2}{c^2}\biggr) \, .</math>
</td>
</tr>
</table>
Integration over the relevant Coriolis component gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_1 z] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} z ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3\iiint \cancelto{0}{y z ~dx ~dy ~dz}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint z^2 ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \int_{-x_\ell}^{+x_\ell} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2a\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \biggl\{ \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int_{-y_\ell}^{+y_\ell} \biggl[ y_\ell^2 - y^2 \biggr]^{1 / 2} dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \cdot \frac{1}{2}\biggl\{ y \sqrt{y_\ell^2 - y^2} + y_\ell^2 \sin^{-1}\biggr( \frac{y}{|y_\ell |} \biggr)\biggr\}_{-y_\ell}^{+y_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ \frac{\pi}{2} y_\ell^2 \biggr\} dz
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ 1 - \frac{z^2}{c^2} \biggr\} dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{z^3}{3} - \frac{z^5}{5c^2} \biggr\}_{-c}^{+c}
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{1}{3} - \frac{1}{5} \biggr\}2c^3
=
\frac{4 \pi a b c}{3}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{c^2}{5} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~+
~\frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (9b)</font> ]</td></tr>
</table>

==Appendix D:   The Other Four Coriolis Components ==

It follows that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_2 x] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z} + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x \biggr\} x ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+~\frac{I_{11}}{\rho}\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_2 z] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z + \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x} \biggr\} z ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{I_{33}}{\rho} \biggl[\frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_3 x] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x + \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y} \biggr\} x ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{I_{11}}{\rho} \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_3 y] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x} + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y \biggr\} y ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+~\frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 \, .
</math>
</td>
</tr>
</table>

==Appendix E:   Kinetic Energy Components ==

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

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{11} = \int_V u_1 u_1 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_1^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 y^2
- 2\cancelto{0}{\biggl[ \frac{a^2}{a^2 + b^2}\biggr]
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \zeta_3} yz
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 z^2
\biggr\} ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \iiint y^2 ~dx ~dy ~dz
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2\iiint z^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{22}}{\rho} \biggr]
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{33}}{\rho} \biggr] \, .</math>
</td>
</tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{22} = \int_V u_2 u_2 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{33}}{\rho} \biggr]
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{11}}{\rho} \biggr] \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{33} = \int_V u_3 u_3 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{11}}{\rho} \biggr]
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{22}}{\rho} \biggr] \, .</math>
</td>
</tr>
</table>

===Off-Diagonal Elements===
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{23} = \int_V u_2 u_3 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2 u_3] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z \biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~+ \iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~\iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 \biggr\}
x^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- ~\frac{I_{11}}{\rho}
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (8)</font> ]</td></tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{12} = \int_V u_1 u_2 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_1 u_2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~
\biggl[ \frac{a^2}{a^2+c^2}\biggr]
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2
\iiint z^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ \frac{I_{33}}{\rho}
\biggl[ \frac{a^2}{a^2+c^2}\biggr]
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{31} = \int_V u_3 u_1 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_3 u_1] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -~
\biggl[ \frac{c^2}{c^2+b^2}\biggr]
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3
\iiint y^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -~ \frac{I_{22}}{\rho}
\biggl[ \frac{c^2}{c^2+b^2}\biggr]
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3 \, .
</math>
</td>
</tr>
</table>

And, finally,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{T}_{32}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{23} \, ;</math>
</td>
<td align="center">     </td>
<td align="right">
<math>~\mathfrak{T}_{21}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{12} \, ;</math>
</td>
<td align="center">      and,     </td>
<td align="right">
<math>~\mathfrak{T}_{13}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{31} \, .</math>
</td>
</tr>
</table>

=See Also=
* [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Properties of Maclaurin Spheroids]]
* [[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]

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