Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3B (section)

====Summary00====
In summary, the desired ''out'' of equilibrium free-energy expression is,
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  <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,

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  <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>

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  <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,
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<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,

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  <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.
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  <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
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<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>

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  <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>

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

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  <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>

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  <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:
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  <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">

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  <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,

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  <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,

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<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,
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  <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>