Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3A

=Free Energy of Embedded Polytropes=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt1|Part I: &nbsp; Synopsis]]<br />&nbsp;</td>
  <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt2|Part II:&nbsp; Truncated Polytropes]]<br />&nbsp;</td>
  <td align="center" rowspan="1" colspan="3"bgcolor="lightblue"><br />Part III: &nbsp;Free-Energy of Bipolytropes<br />&nbsp;</td>
</tr>
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="17%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A|IIIA:&nbsp; Focus on (5, 1) Bipolytropes]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="17%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt3B|IIIB:&nbsp; Focus on (0, 0) Bipolytropes]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt3C|IIIC:&nbsp; Overview]]<br />&nbsp;</td>
</tr>
</table>


=Free-Energy of Bipolytropes=

In this case, the Gibbs-like free energy is given by the sum of four separate energies,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{core} + \biggl[W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{env} \, .
</math>
  </td>
</tr>

</table>
</div>

In addition to specifying (generally) separate polytropic indexes for the core, <math>~n_c</math>, and envelope, <math>~n_e</math>, and an envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>, we will assume that the system is fully defined via specification of the following five physical parameters:
* Total mass, <math>~M_\mathrm{tot}</math>;
* Total radius, <math>~R</math>; 
* Interface radius, <math>~R_i</math>, and associated dimensionless interface marker, <math>~q \equiv R_i/R</math>;
* Core mass, <math>~M_c</math>, and associated dimensionless mass fraction, <math>~\nu \equiv M_c/M_\mathrm{tot}</math>;
* Polytropic constant in the core, <math>~K_c</math>.

In general, the warped free-energy surface drapes across a five-dimensional parameter "plane" such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{G}(R, K_c, M_\mathrm{tot}, q, \nu) \, .</math>
  </td>
</tr>
</table>
</div>
==Order of Magnitude Derivation==
Let's begin by providing very rough, approximate expressions for each of these four terms, assuming that <math>~n_c = 5</math> and <math>~n_e = 1</math>.  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W_\mathrm{grav}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- \mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot} M_c}{(R_i/2)} \biggr] 
= - 2\mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{\nu}{q}\biggr) \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~W_\mathrm{grav}\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- \mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot} M_e}{(R_i+R)/2} \biggr] 
= - 2\mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{core} = U_\mathrm{int}\biggr|_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\mathfrak{b}_c \cdot n_cK_c M_c ({\bar\rho}_c)^{1/n_c}
= 5\mathfrak{b}_c \cdot K_c M_\mathrm{tot}\nu \biggl[ \frac{3M_c}{4\pi R_i^3} \biggr]^{1/5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} K_c (M_\mathrm{tot}\nu)^{6/5} (Rq)^{-3/5}  
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{env} = U_\mathrm{int}\biggr|_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\mathfrak{b}_e \cdot n_eK_e M_\mathrm{env} ({\bar\rho}_e)^{1/n_e}
= \mathfrak{b}_e \cdot K_e M_\mathrm{tot}(1-\nu) \biggl[ \frac{3M_\mathrm{env}}{4\pi (R^3-R_i^3)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr) K_e [M_\mathrm{tot}(1-\nu)]^2 [R^3(1-q^3)]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
In writing this last expression, it has been necessary to (temporarily) introduce a sixth physical parameter, namely, the polytropic constant that characterizes the envelope material, <math>~K_e</math>.  But this constant can be expressed in terms of <math>~K_c</math> via a relation that ensures continuity of pressure across the interface while taking into account the drop in mean molecular weight across the interface, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K_e ({\bar\rho}_e)^{(n_e+1)/n_e}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~K_c ({\bar\rho}_c)^{(n_c+1)/n_c}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ K_e \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr) {\bar\rho}_c\biggr]^{2}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~K_c ({\bar\rho}_c)^{6/5}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{K_e}{K_c} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3M_\mathrm{tot}\nu}{4\pi (Rq)^3} \biggr]^{-4/5} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the fourth energy term may be rewritten in the form,

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

<tr>
  <td align="right">
<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{env} = U_\mathrm{int}\biggr|_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr) \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
K_c\biggl[ \frac{3M_\mathrm{tot}\nu}{4\pi (Rq)^3} \biggr]^{-4/5}  [M_\mathrm{tot}(1-\nu)]^2 [R^3(1-q^3)]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
K_c M_\mathrm{tot}^{6/5}R^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5}  \frac{(1-\nu)^2}{(1-q^3)} \, .
</math>
  </td>
</tr>
</table>
</div>

Putting all the terms together gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
- 2\mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{\nu}{q}\biggr) \biggr] 
- 2\mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] 
+ \mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} K_c (M_\mathrm{tot}\nu)^{6/5} (Rq)^{-3/5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
K_c M_\mathrm{tot}^{6/5}R^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5}  \frac{(1-\nu)^2}{(1-q^3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggr] 
+ \mathcal{B}_\mathrm{biP} K_c \biggl[\frac{(\nu M_\mathrm{tot})^{2}}{ qR} \biggr]^{3/5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggr] \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2}
+ \mathcal{B}_\mathrm{biP} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} K_c \biggl[\frac{M_\mathrm{tot}^{2}}{ R} \biggr]^{3/5}\biggl(\frac{G^3}{K_c^5}\biggr)^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{R_\mathrm{norm}}{R} \biggr] 
+ \mathcal{B}_\mathrm{biP} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} \biggl[\frac{R_\mathrm{norm}}{ R} \biggr]^{3/5} \, ,
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{biP}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \mathfrak{a}_c\biggl(\frac{\nu}{q}\biggr)  + \mathfrak{a}_e  \biggl(\frac{1-\nu}{1+q}\biggr)  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{biP}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3}{2^2\pi} \biggr)^{1/5} \biggl[5\mathfrak{b}_c  
+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
==Equilibrium Radius==
===Order of Magnitude Estimate===
This means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial\mathfrak{G}}{\partial R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ 2 \mathcal{A}_\mathrm{biP}\biggl[ \frac{GM_\mathrm{tot}^2 }{R^2} \biggr] 
- \frac{3}{5} \mathcal{B}_\mathrm{biP} K_c \biggl[\frac{\nu^{2}}{ q} \biggr]^{3/5} M_\mathrm{tot}^{6/5} R^{-8/5}
\, .
</math>
  </td>
</tr>
</table>
</div>

Hence, because equilibrium radii are identified by setting <math>~\partial\mathfrak{G}/\partial R = 0</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\cdot 5}{3}\biggr)^{5/2} \biggl[\frac{\mathcal{A}_\mathrm{biP} }{\mathcal{B}_\mathrm{biP}}\biggr]^{5/2} \biggl(\frac{ q} {\nu^{2}}\biggr)^{3/2} \, .
</math>
  </td>
</tr>
</table>
</div>

===Reconcile With Known Analytic Expression===
From our [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_2|earlier derivations]], it appears as though,

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

<tr>
  <td align="right">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3^8}{2^5\pi} \biggr)^{-1/2}  
\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2\cdot 5}{3}\biggr)^{5/2} \biggl(\frac{q}{\nu^2} \biggr)^{3/2} 
\biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2}  \biggl(\frac{\nu^2}{q} \biggr)^{5/2} 
\frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr] \, .
</math>
  </td>
</tr>
</table>

This implies that,

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

<tr>
  <td align="right">
<math>\frac{\mathcal{A}_\mathrm{biP} }{\mathcal{B}_\mathrm{biP}}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2}  \biggl(\frac{\nu^2}{q} \biggr)^{5/2} 
\frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr]^{2/5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{\nu^2}{q} \biggr)
\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/5}   \frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c\biggl(\frac{\nu}{q}\biggr)  + \mathfrak{a}_e  \biggl(\frac{1-\nu}{1+q}\biggr)  \biggr] </math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{1}{2^2\cdot 5}\biggl(\frac{\nu^2}{q} \biggr)
\frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c  
+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c  + \mathfrak{a}_e \cdot  \frac{q(1-\nu)}{\nu(1+q)}  \biggr] </math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{\nu}{2^2\cdot 5}
\frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c  
+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr] 
</math>
  </td>
</tr>
</table>
==Focus on Five-One Free-Energy Expression==

===Approximate Expressions===

Let's plug this equilibrium radius back into each term of the free-energy expression.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- 2\mathfrak{a}_c \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} \biggl[ \frac{GM_\mathrm{tot}^2 }{R_\mathrm{eq}} \biggl(\frac{\nu}{q}\biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2\mathfrak{a}_c \biggl(\frac{\nu}{q}\biggr) \biggl[ \frac{R_\mathrm{norm} }{R_\mathrm{eq}} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- 2\mathfrak{a}_e \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} \biggl[ \frac{GM_\mathrm{tot}^2 }{R_\mathrm{eq}} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2\mathfrak{a}_e \biggl(\frac{1-\nu}{1+q}\biggr) \biggl[ \frac{R_\mathrm{norm} }{R_\mathrm{eq}} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{S_\mathrm{core}}{E_\mathrm{norm}} = \biggl[\frac{3(\gamma_c-1)}{2}\biggr] \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr|_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{3}{2\cdot 5}\biggr]\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} 
K_c (M_\mathrm{tot}\nu)^{6/5} (R_\mathrm{eq}q)^{-3/5}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{3}{2\cdot 5}\biggr]\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5}  
\biggl(\frac{\nu^2}{q}\biggr)^{3/5} \biggl(\frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)^{3/5}
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{S_\mathrm{env}}{E_\mathrm{norm}} = \biggl[\frac{3(\gamma_e-1)}{2}\biggr] \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr|_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{3}{2}\biggr]
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl(\frac{G^3}{K_c^5}\biggr)^{1/2} K_c M_\mathrm{tot}^{6/5}R_\mathrm{eq}^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5}  \frac{(1-\nu)^2}{(1-q^3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{3}{2}\biggr]
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
\biggl[ \frac{q^3}{\nu} \biggr]^{4/5}  \frac{(1-\nu)^2}{(1-q^3)} 
\biggl(\frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)^{3/5} \, .
</math>
  </td>
</tr>

</table>
</div>

===From Detailed Force-Balance Models===

In the following derivations, we will use the expression,

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

<tr>
  <td align="right">
<math>\chi_\mathrm{eq} \equiv \frac{ R_\mathrm{eq}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^3 \biggl( \frac{\pi}{2^3} \biggr)^{1/2} \frac{1}{A^2\eta_s}
= \biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5} \, .</math>
  </td>
</tr>
</table>

Keep in mind, as well &#8212; as derived in an [[SSC/Structure/BiPolytropes/Analytic51#Background|accompanying discussion]] &#8212; that,

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

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

where,
<div align="center">
<math>m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
</div>

From the [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|accompanying Table 1 parameter values]], we also can write,

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

<tr>
  <td align="right">
<math>q</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\eta_i}{\eta_s}
= \eta_i \biggl\{\frac{\pi}{2} + \eta_i + \tan^{-1}\biggl[ \frac{1}{\eta_i} - \ell_i \biggr]  \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i \biggl\{\eta_i + \cot^{-1}\biggl[ \ell_i - \frac{1}{\eta_i} \biggr]  \biggr\}^{-1} \, ,
</math>
  </td>
</tr>
</table>

where,

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

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

Let's also define the following shorthand notation:

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

<tr>
  <td align="right">
<math>\mathfrak{L}_i</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{(\ell_i^4-1)}{\ell_i^2} + \frac{(1+\ell_i^2)^3}{\ell_i^3} \cdot \tan^{-1}\ell_i \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{K}_i</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr] + \frac{\Lambda_i}{\eta_i} \, .</math>
  </td>
</tr>
</table>


====Gravitational Potential Energy of the Core====

Pulling from [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|our detailed derivations]],
<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\biggl[  \frac{W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}
\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3}  + \tan^{-1}(\ell_i) \biggr] \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -\chi_\mathrm{eq} \biggl[  \frac{W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}
\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3}  + \tan^{-1}(\ell_i) \biggr] 
\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^5}
\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr)   + (1 + \ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr] 
</math>
  </td>
</tr>
</table>
</div>

Out of equilibrium, then, we should expect,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{core}}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \chi^{-1}
\biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^5}
\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr)   + (1 + \ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~
- \chi^{-1}
\biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^2}
\biggl[ \mathfrak{L}_i - \frac{8}{3} \biggr]  
\, ,
</math>
  </td>
</tr>
</table>
</div>
which, in comparison with our above approximate expression, implies,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~\mathfrak{a}_c  </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3}{2^5} \biggr) \frac{\nu}{\ell_i^5}
\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr)   + (1 + \ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

====Thermal Energy of the Core====

Again, pulling from [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|our detailed derivations]],

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3}   + \tan^{-1}(\ell_i) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \chi_\mathrm{eq}^{3} \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]^5_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{2^5} \biggl( \frac{3^8}{2^5\pi} \biggr)^{5/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3}   + \tan^{-1}(\ell_i) \biggr]^5 
\biggl[\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5}\biggr]^{3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\pi}\biggl(\frac{3}{2^{2}}\biggr)^{11} 
\biggl(\frac{\nu^2}{q}\biggr)^{3} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3}   + \tan^{-1}(\ell_i) \biggr]^5 
\biggl[\frac{(1+\ell_i^2)^9}{\ell_i^{15}}\biggr] \, .
</math>
  </td>
</tr>

</table>
</div>

Out of equilibrium, we should then expect,

<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~\frac{S_\mathrm{core}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl(\frac{3}{2^2\pi} \biggr)^{1/5}\biggl[ \chi^{-1} 
\biggl(\frac{\nu^2}{q}\biggr) \frac{1}{(1+\ell_i^2)^{2}} \biggr]^{3/5} \biggl(\frac{3}{2^{2}}\biggr)^{2}\mathfrak{L}_i 
\, . 
</math>
  </td>
</tr>

</table>
</div>

In comparison with our above approximate expression, we therefore have,

<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~
\biggl[ \biggl(\frac{3}{2\cdot 5}\biggr)\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5}  
\biggl(\frac{\nu^2}{q}\biggr)^{3/5} \biggr]^5</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\pi}\biggl(\frac{3}{2^{2}}\biggr)^{11} 
\biggl(\frac{\nu^2}{q}\biggr)^{3} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3}   + \tan^{-1}(\ell_i) \biggr]^5 
\biggl[\frac{(1+\ell_i^2)^9}{\ell_i^{15}}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\mathfrak{b}_c  
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\frac{ 3 }{2^3\ell_i^{3}(1+\ell_i^2)^{6/5}}
\biggl[ \ell_i (\ell_i^4 - 1 )   + (1+\ell_i^2)^{3}\tan^{-1}(\ell_i) \biggr] \, .
</math>
  </td>
</tr>

</table>
</div>


====Gravitational Potential Energy of the Envelope====

Again, pulling from [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|our detailed derivations]] and appreciating, in particular, that (see, for example, [[SSC/Structure/BiPolytropes/Analytic51#Equilibrium_Condition|our notes on equilibrium conditions]]),
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\frac{\eta_i}{\sin(\eta_i - B)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(\eta_s - B)</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\pi \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\eta_i - B</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\frac{\pi}{2} - \tan^{-1}(\Lambda_i)\, ,</math>
  </td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \sin(\eta_i -B) = (1+\Lambda_i^2)^{-1/2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>~\sin[2(\eta_i-B)] = 2\Lambda_i(1 + \Lambda_i^2)^{-1} \ ,</math>
  </td>
</tr>
</table>
</div>
we have,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -\biggl( \frac{1}{2^3\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2
\biggl\{ \biggl[6(\eta_s-B) - 3\sin[2(\eta_s - B)] -4\eta_s\sin^2(\eta_s-B) + 4B\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ - \biggl[6(\eta_i-B) - 3\sin[2(\eta_i - B)] -4\eta_i\sin^2(\eta_i-B) + 4B \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -\biggl( \frac{1}{2^3\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl[\frac{\eta_i}{\sin(\eta_i - B)} \biggr]^2
\biggl\{ 6\pi - \biggl[6(\eta_i-B) - 3\sin[2(\eta_i - B)] -4\eta_i\sin^2(\eta_i-B) \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -\biggl( \frac{1}{2^3\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2(1+\Lambda_i^2)
\biggl\{ 6\pi - 6\biggl[\frac{\pi}{2} - \tan^{-1}(\Lambda_i)\biggr] + 6\biggl[ \frac{\Lambda_i}{(1 + \Lambda_i^2)}  \biggr] 
+ 4\eta_i \biggl[ \frac{1}{(1+\Lambda_i^2)} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -\biggl( \frac{3^2}{2\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2
\biggl\{ (1+\Lambda_i^2)\biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \Lambda_i + \frac{2}{3} \cdot \eta_i \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

So, in equilibrium we can write,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~-\chi_\mathrm{eq}\biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{3^2}{2\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2
\biggl\{ (1+\Lambda_i^2)\biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \Lambda_i + \frac{2}{3} \cdot \eta_i \biggr\} 
\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{3}{2^2} \biggl(\frac{\eta_i}{m_3}\biggr)^3
\biggl\{ \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \frac{\Lambda_i}{\eta_i} + \frac{2}{3} \biggr\} 
\frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{\ell_i^5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{3}{2^2} \biggl(\frac{\nu^2}{q} \biggr) \frac{1}{\ell_i^2} 
\biggl\{ \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \frac{\Lambda_i}{\eta_i} + \frac{2}{3} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And out of equilibrium,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\frac{W_\mathrm{env}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~
-\chi^{-1}\cdot \frac{3}{2^2} \biggl(\frac{\nu^2}{q} \biggr) \frac{1}{\ell_i^2} 
\biggl[\mathfrak{K}_i+ \frac{2}{3} \biggr] 
\, .
</math>
  </td>
</tr>
</table>
</div>

This, in turn, implies that both in and out of equilibrium,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\mathfrak{a}_e </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{3}{2^3} \biggl[\frac{\nu^2(1+q)}{q(1-\nu)} \biggr] \frac{1}{\ell_i^2} 
\biggl\{ \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \frac{\Lambda_i}{\eta_i} + \frac{2}{3} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

====Thermal Energy of the Envelope====

Again, pulling from [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|our detailed derivations]],

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~ \biggl( \frac{1}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 
\biggl\{  \biggl[6(\eta_s - B) - 3\sin[2(\eta_s-B)] \biggr] - \biggl[6(\eta_i - B) - 3\sin[2(\eta_i-B)] \biggr] \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~ \biggl( \frac{1}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \biggl[\frac{\eta_i}{\sin(\eta_i - B)} \biggr]^2 
\biggl\{ 6\pi  - 6(\eta_i - B) + 3\sin[2(\eta_i-B)]  \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~ \biggl( \frac{1}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2 (1 + \Lambda_i^2)  
\biggl\{ 6\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + 6\biggl[\Lambda_i(1 + \Lambda_i^2)^{-1} \biggr] \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~\frac{1}{2} \biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2  
\biggl\{  (1 + \Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \Lambda_i \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>

So, in equilibrium we can write,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{3}\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~\frac{1}{2} \biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2  
\biggl\{  (1 + \Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \Lambda_i \biggr\} 
\biggl[\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5}\biggr]^{3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{3^2\pi^2}{2^{12}} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^3  
\biggl\{  \frac{(1 + \Lambda_i^2)}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \frac{\Lambda_i}{\eta_i} \biggr\} 
\biggl[\frac{(1+\ell_i^2)^9}{3^9\ell_i^{15}}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{\pi}{2^{6}\cdot 3^5} \biggr)   \biggl[\frac{(1+\ell_i^2)^6}{\ell_i^{12}}\biggr]
\biggl\{  \frac{(1 + \Lambda_i^2)}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \frac{\Lambda_i}{\eta_i} \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>

And, out of equilibrium,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~
\chi^{-3}\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{\pi}{2^{6}\cdot 3^5} \biggr)   
\biggl[\frac{(1+\ell_i^2)^6}{\ell_i^{12}}\biggr]\mathfrak{K} 
\, .
</math>
  </td>
</tr>

</table>
</div>

====Combined in Equilibrium====

Notice that, in combination,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{2S_\mathrm{env} + W_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{2}{3}\biggl( \frac{3^2}{2\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^3 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2}  
\biggl[\frac{\ell_i^3}{( 1 + \ell_i^2)^3}\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Also, from above,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\biggl[  \frac{2S_\mathrm{core}+W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}
\biggl[ \ell_i \biggl(- \frac{8}{3} \ell_i^2 \biggr) (1 + \ell_i^2)^{-3}   \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ + \biggl( \frac{2\cdot 3^6}{\pi } \biggr)^{1/2}
\biggl[ \frac{\ell_i^3}{(1 + \ell_i^2)^{3}}   \biggr] \, .</math>
  </td>
</tr>

</table>
</div>

So, in equilibrium, these terms from the core and envelope sum to zero, as they should.

====Out of Equilibrium====
And now, in combination ''out'' of equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathfrak{G}}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-1} \biggl\{ \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}  
+ \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}\biggr\}
+\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3/5} \biggl(\frac{2n_c}{3}\biggr) \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
+\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3} \biggl(\frac{2n_e}{3}\biggr)\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
\, .
</math>
  </td>
</tr>
</table>
</div>

Hence, quite generally ''out'' of equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial \chi} \biggl[ \frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-1} \biggl\{ \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}  
+ \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}\biggr\}
-\frac{3}{5}\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3/5} \biggl(\frac{10}{3}\biggr) \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
-3\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3} \biggl(\frac{2}{3}\biggr)\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} \, .
</math>
  </td>
</tr>
</table>
</div>
Let's see what the value of this derivative is if the dimensionless radius, <math>~\chi</math>, is set to the value that has been determined, via a detailed force-balanced analysis, to be the equilibrium radius, namely, <math>~\chi = \chi_\mathrm{eq}</math>.  In this case, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl\{\frac{\partial}{\partial \chi} \biggl[ \frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr] \biggr\}_\mathrm{\chi \rightarrow \chi_\mathrm{eq}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\chi_\mathrm{eq}^{-1}\biggl\{
\biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}  
+ \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
+2\biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
+2\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
But, according to the virial theorem &#8212; and, as we have just demonstrated &#8212; the four terms inside the curly braces sum to zero.  So this demonstrates that the derivative of our out-of-equilibrium free-energy expression does go to zero at the equilibrium radius, as it should!

===Summary51===
In summary, the desired ''out'' of equilibrium free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathfrak{G}}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{W_\mathrm{core}}{E_\mathrm{norm}}  + \frac{W_\mathrm{env}}{E_\mathrm{norm}}
+\biggl(\frac{2n_c}{3}\biggr)\frac{S_\mathrm{core}}{E_\mathrm{norm}}
+\biggl(\frac{2n_e}{3}\biggr)\frac{S_\mathrm{env}}{E_\mathrm{norm}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
- \chi^{-1}
\biggl( \frac{3}{2^4} \biggr) \frac{\nu^2}{q} \cdot \frac{1}{\ell_i^2}
\biggl[ \mathfrak{L}_i - \frac{8}{3} \biggr]  
-\chi^{-1}\cdot \frac{3}{2^2} \biggl(\frac{\nu^2}{q} \biggr) \frac{1}{\ell_i^2} 
\biggl[\mathfrak{K}_i+ \frac{2}{3} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+
\biggl(\frac{2\cdot 5}{3}\biggr) \biggl(\frac{3}{2^2\pi} \biggr)^{1/5}\biggl[ \chi^{-1} 
\biggl(\frac{\nu^2}{q}\biggr) \frac{1}{(1+\ell_i^2)^{2}} \biggr]^{3/5} \biggl(\frac{3}{2^{2}}\biggr)^{2}\mathfrak{L}_i 
+\biggl(\frac{2}{3}\biggr)
\chi^{-3}\biggl(\frac{\nu^2}{q} \biggr)^3 \biggl( \frac{\pi}{2^{6}\cdot 3^5} \biggr)   
\biggl[\frac{(1+\ell_i^2)^6}{\ell_i^{12}}\biggr]\mathfrak{K} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
- 
\biggl( \frac{3}{2^4} \biggr) \biggl[\chi^{-1}\frac{\nu^2}{q} \cdot \frac{1}{\ell_i^2}\biggr]
\biggl[ \mathfrak{L}_i + 4\mathfrak{K}_i \biggr]  
+
\biggl(\frac{3}{2^2\pi} \biggr)^{1/5}\biggl(\frac{3\cdot 5}{2^3}\biggr) \biggl[ \chi^{-1} 
\biggl(\frac{\nu^2}{q}\biggr) \frac{1}{(1+\ell_i^2)^{2}} \biggr]^{3/5} 
\mathfrak{L}_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl( \frac{\pi}{2^{5}\cdot 3^6} \biggr)
\biggl[\chi^{-1}\biggl(\frac{\nu^2}{q} \biggr)   
\frac{(1+\ell_i^2)^2}{\ell_i^{4}}\biggr]^3\mathfrak{K} \, .
</math>
  </td>
</tr>
</table>
</div>

Or, in terms of the ratio, 
<div align="center">
<math>\Chi \equiv \frac{\chi}{\chi_\mathrm{eq}} \, ,</math>
</div>
and pulling from the above expressions,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~\biggl[  \frac{W_\mathrm{core}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}
\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3}  + \tan^{-1}(\ell_i) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3}
\biggl[ \mathfrak{L}_i - \frac{8}{3}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -\biggl( \frac{3^2}{2\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2
\biggl\{ (1+\Lambda_i^2)\biggl[\frac{\pi}{2}+\tan^{-1}(\Lambda_i)\biggr] + \Lambda_i + \frac{2}{3} \cdot \eta_i \biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -\biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2}  \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3}
\biggl[4\mathfrak{K}_i + \frac{8}{3} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3}   + \tan^{-1}(\ell_i) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3}\mathfrak{L}_i  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}} \biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~\frac{1}{2} \biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^2  
\biggl\{  (1 + \Lambda_i^2)\biggl[\frac{\pi}{2} + \tan^{-1}(\Lambda_i) \biggr] + \Lambda_i \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ ~\frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]^{3}  (4\mathfrak{K}_i) \, ,
</math>
  </td>
</tr>
</table>
</div>

we have the streamlined,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{2^5\pi}{3^6} \biggr)^{1/2} \biggl[ \frac{(1+\ell_i^2)}{\ell_i} \biggr]^{3}  \biggl[\frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
+\Chi^{-3/5} (5 \mathfrak{L}_i)
+\Chi^{-3} (4\mathfrak{K}_i)
-\Chi^{-1} (3\mathfrak{L}_i  +12\mathfrak{K}_i )
</math>
  </td>
</tr>
</table>
</div>

or, better yet,

<div align="center" id="BiPolytropeFreeEnergy">
<table border="1" cellpadding="5" align="center">
<tr>
<th align="center">
<font size="+1">Out-of-Equilibrium, Free-Energy Expression for BiPolytropes with</font> <math>~(n_c, n_e) = (5, 1)</math>
</th>
</tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~2^4\biggl( \frac{q\ell_i^2}{\nu^2}\biggr) \chi_\mathrm{eq} \biggl[\frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\Chi^{-3/5} (5 \mathfrak{L}_i)
+\Chi^{-3} (4\mathfrak{K}_i)
-\Chi^{-1} (3\mathfrak{L}_i  +12\mathfrak{K}_i )
</math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>


where,

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

<tr>
  <td align="right">
<math>~\mathfrak{L}_i</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{(\ell_i^4-1)}{\ell_i^2} + \frac{(1+\ell_i^2)^3}{\ell_i^3} \cdot \tan^{-1}\ell_i \, ,</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Lambda_i</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{\eta_i} - \ell_i \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\eta_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[\frac{\ell_i }{(1+\ell_i^2)}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

From the [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|accompanying Table 1 parameter values]], we also can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{q}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\eta_s}{\eta_i}
= 1 + \frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nu</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\ell_i q}{(1+\Lambda_i^2)^{1/2}} \, .
</math>
  </td>
</tr>
</table>
</div>

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">Radial Derivatives</td></tr>
<tr><td 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>~
-\Chi^{-8/5} (3 \mathfrak{L}_i)
-\Chi^{-4} (12\mathfrak{K}_i)
+\Chi^{-2} (3\mathfrak{L}_i  +12\mathfrak{K}_i )
</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>~\frac{3}{5}\biggl[ 
\Chi^{-13/5} (8\mathfrak{L}_i)
+\Chi^{-5} (80\mathfrak{K}_i)
-\Chi^{-1} (10\mathfrak{L}_i  +40\mathfrak{K}_i )\biggr]
</math>
  </td>
</tr>
</table>

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

Consistent with our [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|the stability of bipolytropes having]] <math>~(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>~\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>~\Chi = 1</math>; that is, the equilibrium condition is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi = \chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5} \, .</math>
  </td>
</tr>
</table>
</div>

Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, it is unstable whenever,

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

<tr>
  <td align="right">
<math>~\frac{ \mathfrak{L}_i}{\mathfrak{K}_i}</math>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>~20 \, .</math>
  </td>
</tr>
</table>
</div>
[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Table 1 of an accompanying chapter]] &#8212; and the red-dashed curve in the figure adjacent to that table &#8212; identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the mean-molecular weight ratio, <math>~\mu_e/\mu_c</math>.

=See Also=
In <font color="red">October 2023</font>, this very long chapter was subdivided in order to more effectively accommodate edits.  Here is a list of the resulting set of shorter chapters:
<ol>
  <li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt1|Free-Energy Synopsis]]</li>
  <li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt2|Free-Energy of Truncated Polytropes]]</li>
  <li>Free-Energy of BiPolytropes
    <ul>
<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A|Focus on Five-One Free-Energy Expression]]</li>
<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3B|Focus on Zero-Zero Free-Energy Expression]]</li>
<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3C|Overview]]</li>
    </ul>
  </li>
</ol>


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