Editing SSC/BipolytropeGeneralization (section)

==Simplest Bipolytrope==

===Familiar Setup===
As has been shown in [[#.280.2C_0.29_Bipolytropes|an accompanying presentation]], for an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_{WM}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\nu^2}{q} \cdot f \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~s_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
1 + \Lambda_\mathrm{eq} \, ,
</math>
  </td>
</tr>

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

and where (see, for example, [[SSC/Structure/BiPolytropes/Analytic00#Expression_for_Free_Energy|in the context of its original definition]], or another,  [[SSC/Structure/BiPolytropes/Analytic00#LambdaDeff|separate derivation]]),
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

and where (see the [[SSC/VirialStability#Expressions_for_Mass|associated discussion of relevant mass integrals]]),
<div align="center">
<math>
\frac{\rho_c}{\bar\rho} = \frac{\nu}{q^3} \, ; ~~~~~ \frac{\rho_e}{\bar\rho} = \frac{1-\nu}{1-q^3} \, ; ~~~~~ 
\frac{\rho_e}{\rho_c} = \frac{q^3(1-\nu)}{\nu (1-q^3)} ~~\Rightarrow ~~~ \frac{q^3}{\nu} = \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q^3) + q^3 \, .
</math>
</div>

===Cleaner Virial Presentation===

In an effort to show the similarity in structure among the several energy terms, we have also found it useful to write their expressions in the following forms:

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

<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \frac{\nu^2}{q} \cdot f 
= - 4\pi P_i R^3 \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} \cdot \lambda_i f \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~S_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi P_{ic} R^3 \biggl[ q^3 + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} \cdot \lambda_{ic} \biggr]   \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~S_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi P_{ie} R^3 \biggl[ (1-q^3) + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} \cdot \lambda_{ie} \mathfrak{F} \biggr]   \, ,</math>
  </td>
</tr>
</table>
</div>
where (see an [[SSC/Structure/BiPolytropes/Analytic00#Gravitational_Potential_Energy|associated discussion]] or the [[SSC/VirialStability#Energy_Expressions|original derivation]]),
<div align="center">
<math>
f\biggl(q, \frac{\rho_e}{\rho_c}\biggr) = 1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (q^3 - q^5 )
+ \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{2}{5} - q^3 + \frac{3}{5}q^5 \biggr) \biggr] \, ,
</math>
</div>

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

<tr>
  <td align="right">
<math>~\lambda_i</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{GM_\mathrm{tot}^2}{R^4 P_i} \, ,</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">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\lambda_{ie}} \biggl( \frac{2^2 \cdot 5\pi}{3} \biggr) \frac{q(1-q^3)}{\nu^2} (s_\mathrm{env} -1)
 \, .</math>
  </td>
</tr>
</table>
</div>

<div align="center">
<table border="1" cellpadding="10" width="80%">
<tr><td align="left">
This also means that the three key terms used as shorthand notation in the above expressions for the three energy terms have the following definitions:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_{WM}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\nu^2}{q} \cdot f \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~s_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
1 + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q^4} \cdot \lambda_{ic} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~s_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
1 + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q(1-q^3)} \cdot \lambda_{ie} \mathfrak{F} 
\, ,</math>
  </td>
</tr>
</table>
</div>

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

Hence, if all the interface pressures are equal &#8212; that is, if <math>~P_i = P_{ic} = P_{ie}</math> and, hence also, <math>~\lambda_{i} = \lambda_{ic} = \lambda_{ie}</math> &#8212; then the total thermal energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~S_\mathrm{tot} = S_\mathrm{core} + S_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
2\pi P_{i} R^3 \biggl[ 1 + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} \cdot \lambda_{i} (1+\mathfrak{F}) \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and the virial is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~2S_\mathrm{tot} + W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
4\pi P_{i} R^3 \biggl[ 1 + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} \cdot \lambda_{i} (1+\mathfrak{F} - f ) \biggr] 
\, .
</math>
  </td>
</tr>
</table>
</div>

The virial should sum to zero in equilibrium, which means,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} \cdot (f - 1- \mathfrak{F} ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl[ \biggl( \frac{2^2\cdot 5\pi}{3} \biggr) \frac{q}{\nu^2} \biggr] \frac{R_\mathrm{eq}^4 P_i}{GM_\mathrm{tot}^2} </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 ~~~~ \biggl( \frac{\rho_e}{\rho_c} \biggr)^{-1} \biggl[ \biggl( \frac{2^3\pi}{3} \biggr) \frac{q^6}{\nu^2} \biggr] \frac{R_\mathrm{eq}^4 P_i}{GM_\mathrm{tot}^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ (q^3 - q^5 )+ \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{2}{5} - q^3 + \frac{3}{5}q^5 \biggr) \biggr] -
\biggl[  (-2q^2 + 3q^3 - q^5) + \biggl( \frac{\rho_e}{\rho_c}\biggr) (-\frac{3}{5}  +3q^2 - 3q^3 + \frac{3}{5} q^5) \biggr]
</math>
  </td>
</tr>

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

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

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

===Shift to Central Pressure Normalization===
Let's rework the definition of <math>~\lambda_i</math> in two ways:  (1) Normalize <math>~R_\mathrm{eq}</math> to <math>~R_\mathrm{norm}</math> and normalize the pressure to <math>~P_\mathrm{norm}</math>; (2) shift the referenced pressure from the pressure at the interface <math>~(P_i)</math> to the central pressure <math>~(P_0)</math>, because it is <math>~P_0</math> that is directly related to <math>~K_c</math> and <math>~\rho_c</math>; specifically, <math>P_0 = K_c \rho_c^{\gamma_c}</math>.  Appreciating that, in equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0 - q^2 \Pi_\mathrm{eq}
= K_c \rho_c^{\gamma_c} - \frac{3}{2^3 \pi} 
\biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \biggl( \frac{\nu^2}{q^6} \biggr) q^2 \, ,</math>
  </td>
</tr>
</table>
</div>

the left-hand-side of the last expression, above, can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}^4 P_i}{GM_\mathrm{tot}^2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} 
\biggl[ P_0 - \frac{3}{2^3 \pi} 
\biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \biggl( \frac{\nu^2}{q^6} \biggr) q^2\biggr] </math>
  </td>
</tr>

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

Hence, the virial equilibrium condition gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{R_\mathrm{eq}^4 P_0}{GM_\mathrm{tot}^2} - \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr)
\frac{\nu}{q^3} \biggr]^2 q^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^2 q^2 (g^2-1)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~ \frac{R_\mathrm{eq}^4 P_0}{GM_\mathrm{tot}^2}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{2} q^2 g^2  \, .
</math>
  </td>
</tr>
</table>
</div>

This result precisely matches [[SSC/Structure/BiPolytropes/Analytic00#CentralPressure|the result obtained via the detailed force-balanced conditions]] imposed through hydrostatic equilibrium.

Adopting our new variable normalizations and realizing, in particular, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_\mathrm{norm}^4 P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~GM_\mathrm{tot}^2 \, ,</math>
  </td>
</tr>
</table>
</div>
the expression alternatively can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}^4 P_i}{GM_\mathrm{tot}^2} 
= \chi_\mathrm{eq}^4 \biggl( \frac{P_i}{P_\mathrm{norm}} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\chi_\mathrm{eq}^4 \biggl\{  \frac{K_c \rho_c^{\gamma_c}}{P_\mathrm{norm}} - \frac{2\pi}{3} 
\biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2 \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4P_\mathrm{norm}} \biggr) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\chi_\mathrm{eq}^4 \biggl\{  \frac{K_c }{P_\mathrm{norm}} \biggl[ \frac{\rho_c}{\bar\rho} \biggl( \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3} \biggr) \chi_\mathrm{eq}^{-3} \biggr]^{\gamma_c} 
- \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2 \chi_\mathrm{eq}^{-4} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\chi_\mathrm{eq}^{4-3\gamma_c} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \frac{K_c }{P_\mathrm{norm}} \biggl( \frac{M_\mathrm{tot}^{\gamma_c}}{R_\mathrm{norm}^{3\gamma_c}} \biggr) 
- \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2
</math>
  </td>
</tr>

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

Normalized in this manner, the virial equilibrium (as well as the hydrostatic balance) condition gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

===Free-Energy Coefficients===
Therefore, for an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, the coefficients in the normalized free-energy function are,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3\gamma_c} 
=\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3\gamma_c-4} 
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \nu \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c-1} - 
\chi_\mathrm{eq}^{3\gamma_c-4} \biggl( \frac{3^2}{2^3\cdot 5\pi} \biggr) \frac{\nu^2}{q^4} \biggl( \frac{4\pi }{3} \biggr) q^3 \biggr\}
=
\nu \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c-1} - 
\chi_\mathrm{eq}^{3\gamma_c-4} \biggl( \frac{3}{10} \biggr) \frac{\nu^2}{q} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{C}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\biggl( \frac{4\pi }{3} \biggr) (1-q^3) s_\mathrm{env} 
\biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \, .</math>
  </td>
</tr>
</table>
</div>

Note that, because <math>~P_{ie} = P_{ic}</math> in equilibrium, the ratio of coefficients,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathcal{C}}{\mathcal{B}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_\mathrm{eq}^{3(\gamma_e - \gamma_c)}\biggl\{\frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \chi_\mathrm{eq}^{3(\gamma_c - \gamma_e)} \biggl( \frac{\mathcal{C}}{\mathcal{B}} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{  20\pi q(1-q^3) \lambda_i^{-1} + 3\nu^2 \mathfrak{F}   }{ 20\pi q^4 \lambda_i^{-1} + 3\nu^2   }  \, .
</math>
  </td>
</tr>
</table>
</div>


The equilibrium condition is,
<div align="center">
<math>\frac{\mathcal{A}}{\mathcal{B} + \mathcal{C}^'} = \chi_\mathrm{eq}^{4-3\gamma_c}  
\, ,</math>
</div>
where,
<div align="center">
<math>
\mathcal{C}^' \equiv \mathcal{C} \chi_\mathrm{eq}^{3(\gamma_c-\gamma_e)}  \, .
</math>
</div>

===More General Derivation of Free-Energy Coefficients B and C===
Keep in mind that, generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>GM_\mathrm{tot}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_\mathrm{norm}^4 P_\mathrm{norm} = E_\mathrm{norm} R_\mathrm{norm}  \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{R^4 P_i}{GM_\mathrm{tot}^2} = \biggl( \frac{P_i}{P_\mathrm{norm}} \biggr) \chi^4 
</math>
&nbsp;&nbsp;&hellip; and, note that &hellip; &nbsp;&nbsp;
<math>
\frac{1}{\Lambda} = \biggl( \frac{3\cdot 5}{2^2\pi} \biggr) \frac{1}{\lambda_i} \cdot \frac{1}{q^2 \sigma^2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\Pi}{P_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3}{2^3\pi} \biggl( \frac{GM_\mathrm{tot}^2}{P_\mathrm{norm} R^4} \biggr) \frac{\nu^2}{q^6} 
=\biggl( \frac{2\pi}{3} \biggr) \sigma^2 \chi^{-4} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{K_c \rho_c^{\gamma_c} }{P_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{K_c}{P_\mathrm{norm}} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{\gamma_c} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R^3} \biggr]^{\gamma_c}
= \frac{K_c M_\mathrm{tot}^{\gamma_c} }{R_\mathrm{norm}^{3\gamma_c} P_\mathrm{norm}} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \chi^{-3\gamma_c} = \sigma^{\gamma_c} \chi^{-3\gamma_c} \, ,
</math>
  </td>
</tr>
</table>
</div>
where we have introduced the notation, 
<div align="center">
<math>
\sigma \equiv \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \, .
</math>
</div>

So, the free-energy coefficient,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3\gamma_c} 
=\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3\gamma_c-4} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 \biggl[ \chi_\mathrm{eq}^4 \biggl( \frac{P_{ic}}{P_\mathrm{norm}} \biggr)_\mathrm{eq}  +
 \biggl( \frac{4\pi}{3\cdot 5} \biggr) q^2 \sigma^2 \biggr] \chi_\mathrm{eq}^{3\gamma_c-4} \, .
</math>
  </td>
</tr>
</table>
</div>


And the free-energy coefficient,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{C}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) (1-q^3) s_\mathrm{env} 
\biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} 
= \biggl( \frac{4\pi }{3} \biggr) (1-q^3) s_\mathrm{env} 
\biggl[ \frac{1 }{\lambda_{ie}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3\gamma_e-4}
</math>
  </td>
</tr>

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

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

<div align="center" id="DerivationTable">
<table border="1" align="center" cellpadding="10">
<tr>
  <td align="center">
OLD DERIVATION
<div align="center">
<math>P_{ic} = K_c \rho_c^{\gamma_c}</math>
</div>
  </td>
  <td align="center">
NEW DERIVATION
<div align="center">
<math>P_{ic} = P_0 - q^2\Pi = K_c \rho_c^{\gamma_c} - q^2\Pi</math>
</div>
  </td>
</tr>


<tr>
  <td align="center" colspan="2">
&hellip; therefore &hellip;
  </td>
</tr>


<tr>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{OLD}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 \biggl[ \sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} +
 \biggl( \frac{4\pi}{3\cdot 5} \biggr) q^2 \sigma^2 \biggr] \chi_\mathrm{eq}^{3\gamma_c-4} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 \biggl[ \sigma^{\gamma_c}  +
 \biggl( \frac{4\pi}{3\cdot 5} \biggr) q^2 \sigma^2 \chi_\mathrm{eq}^{3\gamma_c-4} \biggr]  
</math>
  </td>
</tr>
</table>
  </td>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{NEW}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 \biggl[ \sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} 
-  \biggl( \frac{2\pi}{3} \biggr) q^2 \sigma^2
+ \biggl( \frac{4\pi}{3\cdot 5} \biggr) q^2 \sigma^2 \biggr] \chi_\mathrm{eq}^{3\gamma_c-4} 
</math>
  </td>
</tr>

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

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



<tr>
  <td align="center" colspan="2">
&hellip; and, enforcing in equilibrium <math>~P_{ie} = P_{ic}</math> &hellip;
  </td>
</tr>


<tr>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{C}_\mathrm{OLD}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) 
\biggl\{ (1-q^3) \biggl[ \sigma^{\gamma_c} \chi_\mathrm{eq}^{4-3\gamma_c} \biggr] + \biggl( \frac{2\pi}{3} \biggr) \sigma^2 
\biggl[ \frac{2}{5} q^5 \mathfrak{F} \biggr] \biggr\}_\mathrm{eq} \chi_\mathrm{eq}^{3\gamma_e-4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) 
\biggl[ (1-q^3) \sigma^{\gamma_c}  + \biggl( \frac{2\pi}{3} \biggr) \sigma^2 
\biggl( \frac{2}{5} q^5 \mathfrak{F} \biggr) \chi_\mathrm{eq}^{3\gamma_e-4}\biggr] 
</math>
  </td>
</tr>
</table>
  </td>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{C}_\mathrm{NEW}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) 
\biggl\{ (1-q^3) \biggl[ \sigma^{\gamma_c} \chi_\mathrm{eq}^{4-3\gamma_c} - \biggl( \frac{2\pi}{3} \biggr) q^2  \sigma^2 \biggr] + \biggl( \frac{2\pi}{3} \biggr) \sigma^2 
\biggl[ \frac{2}{5} q^5 \mathfrak{F} \biggr] \biggr\}_\mathrm{eq} \chi_\mathrm{eq}^{3\gamma_e-4} 
</math>
  </td>
</tr>

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

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



<tr>
  <td align="center" colspan="2">
&hellip; and, also &hellip;
  </td>
</tr>


<tr>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{1}{\Lambda_\mathrm{eq}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{q^2 } \biggl( \frac{3\cdot 5}{2^2\pi} \biggr) \sigma^{\gamma_c - 2} \chi_\mathrm{eq}^{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{5q}{\nu} \biggr) \sigma^{\gamma_c - 1} \chi_\mathrm{eq}^{4 - 3\gamma_c}  
</math>
  </td>
</tr>
</table>
  </td>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} -  \biggl( \frac{2\pi}{3} \biggr) q^2 \sigma^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{1}{\Lambda_\mathrm{eq}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{q^2 \sigma^2} \biggl( \frac{3\cdot 5}{2^2\pi} \biggr) \biggl[
\sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} -  \biggl( \frac{2\pi}{3} \biggr) q^2 \sigma^2 \biggr] 
</math>
  </td>
</tr>

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

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

</table>
</div>

===Extrema===
Extrema in the free energy occur when,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c} + \mathcal{C} \chi_\mathrm{eq}^{4-3\gamma_e} \, .</math>
  </td>
</tr>
</table>
</div>

Also, as stated above, because <math>~P_{ie} = P_{ic}</math> in equilibrium, the ratio of coefficients,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathcal{C}}{\mathcal{B}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_\mathrm{eq}^{3(\gamma_e - \gamma_c)}\biggl[ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
When put together, these two relations imply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c} + \chi_\mathrm{eq}^{4-3\gamma_c} \mathcal{B} 
\biggl[ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c} \biggl[ 1+ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

But the definition of <math>~\mathcal{B}</math> gives,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, extrema occur when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq}  \biggl[ 1+ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl( \frac{3}{2^2 \cdot 5\pi } \biggr) \frac{\nu^2}{q} \cdot f</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>  
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq}   \biggl[ q^3 s_\mathrm{core}  +  (1-q^3) s_\mathrm{env} \biggr] </math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>  
\biggl( \frac{3}{2^2 \cdot 5\pi } \biggr) \frac{\nu^2}{q} \cdot (f - 1 - \mathfrak{F})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>  
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2} q^2 (g^2 - 1 ) \, .
</math>
  </td>
</tr>
</table>
</div>

In what follows, keep in mind that,
<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>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4-3\gamma_c} 
= R_\mathrm{eq}^{4-3\gamma_c} \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~K_c \rho_c^{\gamma_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
K_c \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{\gamma_c} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R^3} \biggr]^{\gamma_c}
= K_c \sigma^{\gamma_c} M_\mathrm{tot}^{\gamma_c} R^{-3\gamma_c} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3}{2^3 \pi} \biggl(  \frac{GM_\mathrm{tot}^2}{R^4} \biggr) \frac{\nu^2}{q^6} 
= \frac{2\pi}{3} \biggl(  \frac{GM_\mathrm{tot}^2}{R^4} \biggr) \sigma^2 \, .
</math>
  </td>
</tr>
</table>
</div>



<div align="center">
<table border="1" align="center" cellpadding="10">
<tr>
  <td align="center">
OLD DERIVATION
<div align="center">
<math>P_{i} = K_c \rho_c^{\gamma_c}</math>

<math>\Rightarrow ~~~~ K_c  = P_{i} \sigma^{-\gamma_c} M_\mathrm{tot}^{-\gamma_c} R^{+3\gamma_c}  </math></div>
  </td>
  <td align="center">
NEW DERIVATION
<div align="center">
<math>P_0 = K_c \rho_c^{\gamma_c} </math>

<math>\Rightarrow ~~~~ K_c = P_0 \sigma^{-\gamma_c} M_\mathrm{tot}^{-\gamma_c} R^{+3\gamma_c}  </math>
</div>
  </td>
</tr>


<tr>
  <td align="center" colspan="2">
&hellip; hence, as derived in the above table &hellip;
  </td>
</tr>


<tr>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} 
</math>
  </td>
</tr>
</table>
  </td>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} -  \biggl( \frac{2\pi}{3} \biggr) q^2 \sigma^2 
</math>
  </td>
</tr>

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

<tr>
  <td align="center" colspan="2">
&hellip; which, when combined with the condition that identifies extrema, gives &hellip;
  </td>
</tr>


<tr>
  <td align="left">
<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>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 (g^2 - 1 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ R_\mathrm{eq}^{4-3\gamma_c} \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 (g^2 - 1 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{ R_\mathrm{eq}^{4}  P_i }{GM_\mathrm{tot}^{2} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2} q^2 (g^2-1)
</math>
  </td>
</tr>

</table>
  </td>
  <td align="left">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \chi_\mathrm{eq}^{4 - 3\gamma_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 g^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ R_\mathrm{eq}^{4-3\gamma_c} \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 g^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{ R_\mathrm{eq}^{4}  P_0 }{GM_\mathrm{tot}^{2} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2} q^2 g^2
</math>
  </td>
</tr>

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

<tr>
  <td align="center" colspan="2">
These are consistent results because they result in the detailed force-balance relation, <math>P_0 - P_i = q^2 \Pi_\mathrm{eq} \, .</math>
  </td>
</tr>


</table>
</div>