Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3B (section)

=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>


* 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>
==Focus on Zero-Zero Free-Energy Expression==
Here, we will draw heavily from the following accompanying chapters:
*  [[SSC/Structure/BiPolytropes/Analytic00#Step_7:__Surface_Boundary_Condition|Analytic Detailed Force Balance Models]]
*  [[SSC/Structure/BiPolytropes/FreeEnergy00#Free_Energy_of_BiPolytrope_with|Free-Energy Analysis]]


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

====Equilibrium Radius====
=====First View=====

In an [[SSC/Structure/BiPolytropes/FreeEnergy00#Virial_Theorem|accompanying chapter]] we find,

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

<tr>
  <td align="right">
<math>~
\frac{P_0 R_\mathrm{eq}^4}{G M_\mathrm{tot}^2 } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2
\biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr]
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<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>
1+
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)  \biggl(\frac{1}{q^2}-1 \biggr) +\biggl( \frac{\rho_e}{\rho_c}  
\biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr]  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{F} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[  (-2q^2 + 3q^3 - q^5) +
\frac{3}{5} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-1 +5q^2 - 5q^3 + q^5) \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\rho_e}{\rho_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{q^3(1-\nu)}{\nu(1-q^3)} \, .
</math>
  </td>
</tr>
</table>
</div>

Here, we prefer to normalize the equilibrium radius to <math>~R_\mathrm{norm}</math>.  So, let's replace the central pressure with its expression in terms of <math>~K_c</math>.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K_c \rho_c^{\gamma_c} = K_c \biggl[ \frac{3M_\mathrm{core}}{4\pi R_i^3} \biggr]^{\gamma_c}
= K_c \biggl[ \frac{3\nu M_\mathrm{tot}}{4\pi q^3 R_\mathrm{eq}^3} \biggr]^{(n_c+1)/n_c} 
~~~\Rightarrow~~~ \frac{P_0}{P_\mathrm{norm}} = \biggl[ \frac{3}{4\pi}\biggl(\frac{\nu}{q^3}\biggr) \frac{1}{\chi_\mathrm{eq}^3}\biggr]^{(n_c+1)/n_c} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~K_c \biggl[ \frac{3\nu M_\mathrm{tot}}{4\pi q^3 R_\mathrm{eq}^3} \biggr]^{(n_c+1)/n_c}
\frac{R_\mathrm{eq}^4}{G M_\mathrm{tot}^2 } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2
\biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~R_\mathrm{eq}^{(n_c-3)/n_c}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{G}{K_c}\biggr) M_\mathrm{tot}^{(n_c-1)/n_c} \biggl[ \frac{3\nu }{4\pi q^3 } \biggr]^{-(n_c+1)/n_c}
\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2
\biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\chi_\mathrm{eq}^{(n_c-3)/n_c} \equiv \biggl[\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr]^{(n_c-3)/n_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl(\frac{4\pi}{3} \biggr)^{1/n_c}
\biggl( \frac{\nu}{q^3}\biggr)^{(n_c-1)/n_c}
\biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Or, in terms of <math>~\gamma_c</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl(\frac{3}{4\pi} \biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3}\biggr)^{2-\gamma_c}
\biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

=====Second View=====

Alternatively, from our derivation and discussion of [[SSC/Structure/BiPolytropes/Analytic00#CentralPressure|analytic detailed force-balance models]],

<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>
\biggl[ \frac{R^4}{GM_\mathrm{tot}^2} \biggr] P_0</math>
  </td>
  <td align="center">
&nbsp; <math>~=</math>&nbsp; 
  </td>
  <td align="left">
<math>\biggl( \frac{3}{2^3\pi} \biggr) \frac{\nu^2 g^2}{q^4} \, ,</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~[g(\nu,q)]^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
1  + \biggl(\frac{\rho_e}{\rho_0}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + 
\frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

In order to show that this expression is the same as the other one, [[#First_View_2|above]], we need to show that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="left">
<math>~\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2
\biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3}{2^3\pi} \biggr) \frac{\nu^2 g^2}{q^4} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
 f - 1-\mathfrak{F} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2q^3} \biggl[g^2-1\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2q^3}  \biggl(\frac{\rho_e}{\rho_0}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + 
\frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2q^5} \biggl(\frac{\rho_e}{\rho_0}\biggr)  \biggl\{ 2 ( q^2 - q^3 ) 
+ \frac{\rho_e}{\rho_0}\biggl[  1 - 3q^2+ 2q^3 \biggr] \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>

Let's see &hellip;

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

<tr>
  <td align="right">
<math>~
 f - 1-\mathfrak{F} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)  \biggl(\frac{1}{q^2}-1 \biggr) +\biggl( \frac{\rho_e}{\rho_c}  
\biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr] -
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[  (-2q^2 + 3q^3 - q^5) +
\frac{3}{5} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-1 +5q^2 - 5q^3 + q^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)  \biggl(\frac{1}{q^2}-1 \biggr) 
- \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[  (-2q^2 + 3q^3 - q^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ \frac{3}{5} \biggl( \frac{\rho_e}{\rho_c}\biggr) (-1 +5q^2 - 5q^3 + q^5) \biggr]
+\biggl( \frac{\rho_e}{\rho_c}  \biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl\{ (q^3- q^5 ) 
+ (2q^2 - 3q^3 + q^5) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \frac{1}{q^5} \biggl[ 3 (1 -5q^2 + 5q^3 - q^5) \biggr]
+\frac{1}{2} \biggl( \frac{\rho_e}{\rho_c}  \biggr)^2 \frac{1}{q^5} \biggl[ 2 - 2q^5 + 5\biggl( q^5-q^3\biggr)\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (q^3- q^5 ) + (2q^2 - 3q^3 + q^5) \biggr]
+ \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \frac{1}{q^5} \biggl[  3 (1 -5q^2 + 5q^3 - q^5)+2 - 2q^5 + 5( q^5-q^3)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ 2q^2 - 2q^3  \biggr]
+ \frac{5}{2q^5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[  1 - 3q^2 + 2q^3  \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Q.E.D.

Hence, the equilibrium radius can also be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl(\frac{3}{4\pi} \biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3}\biggr)^{2-\gamma_c} q^2 g^2 \, ;
</math>
  </td>
</tr>
</table>
</div>
or, in terms of the polytropic index,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{n_c-3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} \, .
</math>
  </td>
</tr>
</table>
</div>

====Gravitational Potential Energy====

Also from our [[SSC/Structure/BiPolytropes/FreeEnergy00#Gravitational_Potential_Energy|accompanying discussion]], we have,
<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>
- \Chi^{-1} \biggl( \frac{3}{5}\biggr) \biggl(\frac{\nu}{q^3} \biggr)^2 q^5
\biggl[ \frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} \biggr]^{-1/(n_c-3)}  f(\nu,q) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \Chi^{-1} \biggl( \frac{6}{5}\biggr) q^5 f 
\biggl[ 2^{n_c-(n_c-3)} \biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{(1-n_c)+2(n_c-3)} b_\xi^{n_c} \biggr]^{1/(n_c-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \Chi^{-1} \biggl( \frac{6}{5}\biggr) q^5 f 
\biggl[ \biggl(\frac{6}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^{n_c} \biggr]^{1/(n_c-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

====Internal Energy Components====

=====First View=====

Before writing out the expressions for the internal energy of the core and of the envelope, we [[SSC/Structure/BiPolytropes/FreeEnergy00#Virial_Theorem|note from our separate detailed derivation]] that, in either case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{P_i \chi^{3\gamma}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\biggl(\frac{P_i }{P_0} \biggr) \biggl(\frac{P_0 }{P_\mathrm{norm}} \biggr)\chi^{3}\biggr]_\mathrm{eq} \biggl[\frac{\chi}{\chi_\mathrm{eq}}\biggr]^{3-3\gamma}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma} \, ,</math>
  </td>
</tr>
</table>
</div>
where, in equilibrium,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{P_i }{P_0} \biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - b_\xi q^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_\xi q^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\frac{2}{5}q^3 f + \biggl[1 - \frac{2}{5} q^3( 1+\mathfrak{F} ) \biggr]\biggr\}^{-1} </math>
  </td>
</tr>

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

So, copying from our [[SSC/Structure/BiPolytropes/FreeEnergy00#InternalEnergies|accompanying detailed derivation]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi/3 }{({\gamma_c}-1)}  
\biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma_c} 
\biggl\{ \biggl( \frac{P_0}{P_{ic}} \biggr) \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1 }{({\gamma_c}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{3-3\gamma_c} 
q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi/3 }{({\gamma_e}-1)}  
\biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma_e} 
\biggl\{ (1-q^3) +  b_\xi \biggl(\frac{P_0}{P_{ie} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F}
\biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_e}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]  
\Chi^{3-3\gamma_e} \biggl(\frac{P_i }{P_0} \biggr)
\biggl\{ (1-q^3) +  b_\xi \biggl(\frac{P_0}{P_{ie} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F}
\biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_e}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]  
\Chi^{3-3\gamma_e} 
\biggl\{ (1-b_\xi q^2)(1-q^3) +  b_\xi  \biggl[\frac{2}{5} q^5 \mathfrak{F} \biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_e}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]  
\Chi^{3-3\gamma_e} (1-q^3)
\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Furthermore,

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

<tr>
  <td align="right">
<math>~\biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{4\pi}\biggr)^{\gamma_c - 1}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}
\biggl\{\chi_\mathrm{eq}^{4-3\gamma_c}\biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{4\pi}\biggr)^{(\gamma_c - 1)/(4-3\gamma_c)}
\biggl( \frac{\nu}{q^3} \biggr)^{(6-5\gamma_c)(4-3\gamma_c)}
\biggl\{\frac{q^2}{2}
\biggl[ 1 + \frac{2}{5} q^3( f - 1-\mathfrak{F} )\biggr] \biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)}
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
n_c  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{-3/n_c} 
q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
n_c  
\biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} 
\Chi^{-3/n_c} 
q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
n_e 
\biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} 
\Chi^{-3/n_e} (1-q^3)
\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>


=====Second View=====
In our [[SSC/Structure/BiPolytropes/Analytic00#PiDefinition|accompanying discussion of energies associated with detailed force balance models]], we used the notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{2^3\pi}\biggr) \frac{GM_\mathrm{tot}^2}{R^4} \biggl(\frac{\nu}{q^3}\biggr)^2
= P_\mathrm{norm} \chi^{-4}\biggl(\frac{3}{2^3\pi}\biggr) \biggl(\frac{\nu}{q^3}\biggr)^2 \, ,
</math>
  </td>
</tr>
</table>
</div>
which allows us to rewrite the [[#Second_View|above quoted relationship]] between the central pressure and the radius of the bipolytrope as,
<div align="center">
<math>~P_0 = \Pi (qg)^2 \, .</math>
</div>
We [[SSC/Structure/BiPolytropes/Analytic00#Virial_Equilibrium|also showed]] that, in equilibrium, the relationship between the central pressure and the interface pressure is,
<div align="center">
<math>~P_0 =P_i + \Pi_\mathrm{eq} q^2 \, .</math>
</div>
This means that, in equilibrium, the ratio of the interface pressure to the central pressure is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{P_i}{P_0}\biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\Pi_\mathrm{eq} q^2}{P_0}
= 1- \frac{1}{g^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
or given that (see [[#Second_View|above]]),

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

<tr>
  <td align="right">
<math>~
 \frac{5}{2q^3} \biggl[g^2-1\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f - 1-\mathfrak{F} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
g^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 1+\frac{2}{5} q^3 ( f - 1-\mathfrak{F} ) \, ,
</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{P_i}{P_0}\biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\Pi_\mathrm{eq} q^2}{P_0}
= 1- \biggl[ 1+\frac{2}{5} q^3 ( f - 1-\mathfrak{F} )  \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
This is exactly the pressure-ratio expression presented in our "first view" and unveils the notation association,
<div align="center">
<table border="0">

<tr>
  <td align="right">
<math>~b_\xi q^2</math>
  </td>
  <td align="center">
<math>~\leftrightarrow~</math>
  </td>
  <td align="left">
<math>
\frac{1}{g^2} \, .
</math>
  </td>
</tr>
</table>
</div>

From [[SSC/Structure/BiPolytropes/Analytic00#Thermal_Energy_Content|our separate derivation]], we have, in equilibrium,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~\mathfrak{G}_\mathrm{core} = \biggl(\frac{2n_c}{3}\biggr) S_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2n_c}{3}\biggr) \biggl( \frac{4\pi}{5} \biggr) R_\mathrm{eq}^3 q^5 \biggl (\frac{5P_i}{2q^2} + \Pi \biggr)_\mathrm{eq} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl( \frac{ q^5n_c}{5} \biggr) R_\mathrm{eq}^3  \biggl( \frac{2^3\pi}{3} \biggr) \Pi_\mathrm{eq} \biggl[\frac{5}{2q^2} \biggl( \frac{P_i}{\Pi} \biggr)_\mathrm{eq} + 1 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl( \frac{ n_c}{5} \biggr) \biggl[ R_\mathrm{norm}^3  
P_\mathrm{norm} \biggr] \chi_\mathrm{eq}^{-1} \biggl(\frac{\nu^2}{q}\biggr)  
\biggl[\frac{5}{2q^2} \biggl( \frac{P_i}{P_0} \biggr)_\mathrm{eq}\biggl( \frac{P_0}{\Pi} \biggr)_\mathrm{eq} + 1 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{\mathfrak{G}_\mathrm{core} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{ n_c}{5} \biggr) \biggl(\frac{\nu^2}{q}\biggr)  
\biggl[\frac{5}{2q^2} \biggl( 1-\frac{1}{g^2} \biggr)\biggl( q^2g^2\biggr) + 1 \biggr] \chi_\mathrm{eq}^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{ n_c}{2} \biggr) \biggl(\frac{\nu^2}{q}\biggr)  
\biggl[ g^2-\frac{3}{5}  \biggr] 
\biggl\{\frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} 
\biggr\}^{-1/(n_c-3)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2}  \biggr]
\biggl( \frac{ 1}{2} \biggr) \biggl(\frac{\nu^2}{q}\biggr)  g^2 
\biggl\{2^{n_c}\biggl(\frac{3}{4\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{1-n_c} (q g)^{-2n_c} 
\biggr\}^{1/(n_c-3)}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2}  \biggr]
\biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{3n_c-15}  g^{-6} 
\biggr\}^{1/(n_c-3)} \, .</math>
  </td>
</tr>
</table>
</div>
Finally, switching from the <math>~g</math> notation to the <math>~b_\xi</math> notation gives,

<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~\biggl[ \frac{\mathfrak{G}_\mathrm{core} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) b_\xi q^2  \biggr]
\biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{3n_c-15}  b_\xi^3 q^{6} 
\biggr\}^{1/(n_c-3)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c q^3 \biggl[ 1- \biggl(\frac{3}{5}\biggr) b_\xi q^2  \biggr]
\biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^3 \biggr\}^{1/(n_c-3)} \, ,</math>
  </td>
</tr>
</table>
</div>
which, after setting <math>~\Chi = 1</math>, precisely matches the above, "first view" expression.  Also from our [[SSC/Structure/BiPolytropes/Analytic00#Thermal_Energy_Content|previous derivation]], we can write,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~\mathfrak{G}_\mathrm{env} = \biggl(\frac{2n_e}{3}\biggr) S_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 2\pi\biggl(\frac{2n_e}{3}\biggr)
R_\mathrm{eq}^3 \Pi_\mathrm{eq} \biggl\{ (1-q^3) \biggl(\frac{P_i  }{\Pi}\biggr)_\mathrm{eq}  
+ \biggl( \frac{\rho_e}{\rho_0} \biggr)\biggl[ (-2q^2 + 3q^3 - q^5 )
  + \frac{3}{5} \biggl( \frac{\rho_e}{\rho_0} \biggr) ( -1 + 5q^2 -5q^3 +  q^5 )\biggr]\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 2\pi\biggl(\frac{2n_e}{3}\biggr)
R_\mathrm{eq}^3 \biggl[  P_\mathrm{norm} \chi^{-4}\biggl(\frac{3}{2^3\pi}\biggr) \biggl(\frac{\nu}{q^3}\biggr)^2\biggr]_\mathrm{eq}
\biggl\{ (1-q^3) q^2(g^2-1)  + \biggl(\frac{2}{5}\biggr) q^5 \mathfrak{F} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \biggl[  P_\mathrm{norm} R_\mathrm{norm}^3 \biggr] \frac{n_e}{2}
\biggl(\frac{\nu^2}{q^4}\biggr)(1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} 
\chi^{-1}_\mathrm{eq} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\biggl[ \frac{\mathfrak{G}_\mathrm{env} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} \frac{q^2}{2}\biggl(\frac{\nu}{q^3}\biggr)^2
\biggl[\frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c}\biggr]^{-1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} 
\biggl[2^{[n_c-(n_c-3)]} \biggl(\frac{3}{4\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{(1-n_c)+2(n_c-3)} q^{2(n_c-3)-2n_c} g^{-2n_c} \biggr]^{1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} 
\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6} g^{-2n_c} \biggr]^{1/(n_c-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
And, finally, switching from the <math>~g</math> notation to the <math>~b_\xi</math> notation gives,

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{\mathfrak{G}_\mathrm{env} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3) (b_\xi q^2)^{-1}
\biggl\{  1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} 
\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6} (b_\xi q^2)^{n_c} \biggr]^{1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3) 
\biggl\{  1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} 
\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6-2(n_c-3)+2n_c}  b_\xi^{3-n_c+n_c} \biggr]^{1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^{3} \biggr]^{1/(n_c-3)} (1-q^3) 
\biggl\{  1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
which, after setting <math>~\Chi = 1</math>, precisely matches the above, "first view" expression.


====Summary00====
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>~ 
A_0\Chi^{-3/n_c} + B_0\Chi^{-3/n_e} - C_0\Chi^{-1}
</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~A_0 \equiv \biggl( \frac{\mathfrak{S}_\mathrm{core}}{E_\mathrm{norm}} \biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{n_c}{b_\xi}  
\biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^{n_c}\biggr]^{1/(n_c-3)} 
q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B_0 \equiv \biggl( \frac{\mathfrak{S}_\mathrm{env}}{E_\mathrm{norm}} \biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{n_e}{b_\xi} 
\biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^{n_c} \biggr]^{1/(n_c-3)} (1-q^3)
\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C_0 \equiv \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
\biggl( \frac{6}{5}\biggr) q^5 f 
\biggl[ \biggl(\frac{2\cdot 3}{\pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^{n_c} \biggr]^{1/(n_c-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

Or, in a more compact form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^* \equiv \biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^{n_c}\biggr]^{-1/(n_c-3)} 
\biggl[\frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
n_c A_1\Chi^{-3/n_c} + n_e B_1\Chi^{-3/n_e} - 3C_1\Chi^{-1}
</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~A_1 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{1}{b_\xi}  (q^3) \biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B_1 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{1}{b_\xi} (1-q^3)\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C_1 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
 <td align="left">
<math>
\biggl( \frac{2}{5}\biggr) q^5 f  \, .
</math>
  </td>
</tr>
</table>
</div>

Let's examine the behavior of the first radial derivative.
<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>~\frac{3}{\Chi} \biggl[ - A_1\Chi^{-3/n_c} - B_1\Chi^{-3/n_e} + C_1\Chi^{-1} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Let's see whether the sum of terms inside the square brackets is zero at the derived equilibrium radius, that is, when <math>~\Chi = 1</math> and, hence, when
<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{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} \biggr]^{1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} b_\xi^{-n_c} \biggr]^{1/(n_c-3)} 
\, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~
C_1 - A_1 - B_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2}{5}\biggr) q^5 f 
- \frac{1}{b_\xi}  (q^3) \biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr]
- \frac{1}{b_\xi} (1-q^3)\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2}{5}\biggr) q^5 f 
- \frac{1}{b_\xi} \biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\}
+ \frac{q^3}{b_\xi} \biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\}
- \frac{q^3}{b_\xi}  \biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2}{5}\biggr) q^5 f - \frac{1}{b_\xi} 
+ \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]q^2  
+ \frac{q^3}{b_\xi} 
- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]q^5  
- \frac{q^3}{b_\xi} + \biggl( \frac{3}{5} \biggr) q^5 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2\biggl\{
\biggl( \frac{2}{5}\biggr) q^3 f - \frac{1}{b_\xi q^2} 
+ \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]  (1-q^3)
+ \biggl( \frac{3}{5} \biggr) q^3
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2\biggl\{0\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Q.E.D.

Even slightly better:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{q^2}\biggl[ \biggl(\frac{\pi}{2\cdot 3}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(5-n_c)} b_\xi^{-n_c}\biggr]^{1/(n_c-3)} 
\biggl[\frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
n_c A_2\Chi^{-3/n_c} + n_e B_2\Chi^{-3/n_e} - 3C_2\Chi^{-1} \, ,
</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 ''Structural'' </font> <math>~(n_c, n_e) = (0, 0)</math>
</th>
</tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~2\biggl(\frac{q^2}{\nu}\biggr)^2 \chi_\mathrm{eq} 
\biggl[\frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
n_c A_2\Chi^{-3/n_c} + n_e B_2\Chi^{-3/n_e} - 3C_2\Chi^{-1}
</math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>
where, keeping in mind that,

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

<tr>
  <td align="right">
<math>~\frac{1}{(b_\xi q^2)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 + \frac{2}{5}q^3 (f - 1-\mathfrak{F} ) \biggr] \, , </math>
  </td>
</tr>
</table>
</div>
we have,

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

<tr>
  <td align="right">
<math>~A_2 \equiv \frac{A_1}{q^2} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{q^3}{(b_\xi q^2)}  \biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{5}q^3  \biggl[1 + q^3 (f - 1-\mathfrak{F} ) \biggr]   \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B_2 \equiv \frac{B_1}{q^2} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{1}{(b_\xi q^2)} (1-q^3)\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
(1-q^3)\biggl\{ \frac{1}{(b_\xi q^2)} -1  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{5} q^3 \biggl\{ (1-q^3) (f - 1-\mathfrak{F} )  + \mathfrak{F} \biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{5} q^3  f - A_2 \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C_2 \equiv \frac{C_1}{q^2} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
 <td align="left">
<math>
\frac{2}{5} q^3 f  \, .
</math>
  </td>
</tr>
</table>
</div>

As before, the equilibrium system is dynamically unstable if <math>~\partial^2 \mathfrak{G}/\partial \Chi^2 < 0</math>.  We have deduced that the system is unstable if,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{n_e}{3}\biggl[ \frac{3-n_e}{n_c-n_e} \biggr] </math>
  </td>
  <td align="center">
<math>~< </math>
  </td>
  <td align="left">
<math>~
\frac{A_2}{C_2}
= \frac{1}{f} \biggl[1 + q^3 (f - 1-\mathfrak{F} ) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>