Editing SSC/BipolytropeGeneralization/Pt2

=Bipolytrope Generalization (Pt 2)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization|Part I:&nbsp; Bipolytrope Generalization]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization/Pt2|Part II:&nbsp; Derivations]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization/Pt3|Part III:&nbsp; Examples]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSC/BipolytropeGeneralization/Pt4|Part IV:&nbsp; Best of the Best]]
&nbsp;
  </td>
</tr>
</table>


{{ SGFworkInProgress }}

==Detailed Derivations==
Dividing the free-energy expression through by <math>~E_\mathrm{norm}</math> generates,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{E_\mathrm{norm}} \biggr) \biggl( \frac{1}{R} \biggr) \cdot \mathfrak{f}_{WM}
+ \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr] \biggl[ \frac{M_\mathrm{tot} K_c \rho_{ic}^{\gamma_c-1} }{E_\mathrm{norm}} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math> 
~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl[ \frac{M_\mathrm{tot} K_e \rho_{ie}^{\gamma_e-1} }{E_\mathrm{norm}}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl[ \frac{K_c G^{(3\gamma_c -4)}M_\mathrm{tot}^{2(3\gamma_c -4)}}{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}} \biggr]^{1/(3\gamma_c -4)} \biggl( \frac{1}{R} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr] \biggl[ \frac{K_c M_\mathrm{tot}^{3\gamma_c -4} K_c^{3\gamma_c -4}  }{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}} 
\biggr]^{1/(3\gamma_c -4)}  \biggl( \frac{\rho_{ic}}{\bar\rho} \biggr)^{\gamma_c-1} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R^3} \biggr]^{\gamma_c-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math> 
~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl[ \frac{K_c M_\mathrm{tot}^{3\gamma_c -4} K_e^{3\gamma_c -4}  }{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6}} 
\biggr]^{1/(3\gamma_c -4)}  \biggl( \frac{\rho_{ie}}{\bar\rho} \biggr)^{\gamma_e-1} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R^3} \biggr]^{\gamma_e-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl( \frac{R_\mathrm{norm}}{R} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr] \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr)^{\gamma_c-1} 
\biggl[ \frac{ K_c^{3\gamma_c -3}  }{G^{3\gamma_c-3} M_\mathrm{tot}^{2\gamma_c-2}} 
\biggr]^{1/(3\gamma_c -4)}  \biggl[ \frac{R_\mathrm{norm}}{R} \biggr]^{3(\gamma_c-1)} 
\biggl[ \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} \biggr]^{-3(\gamma_c-1)/(3\gamma_c -4)}
\biggl( \frac{\rho_{ic}}{\bar\rho} \biggr)^{\gamma_c-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math> 
~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr)^{\gamma_e-1}
\biggl[ \frac{K_c^{3\gamma_c - 3} (K_e/K_c)^{3\gamma_c -4}  }{G^{3\gamma_c-3} M_\mathrm{tot}^{2\gamma_c-2}} \biggr]^{1/(3\gamma_c -4)}  
\biggl[ \frac{R_\mathrm{norm}}{R} \biggr]^{3(\gamma_e-1)}
\biggl[ \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} \biggr]^{-3(\gamma_e-1)/(3\gamma_c -4)} \biggl( \frac{\rho_{ie}}{\bar\rho} \biggr)^{\gamma_e-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl( \frac{R_\mathrm{norm}}{R} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ \biggl[ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggr]  \biggl( \frac{3}{4\pi} \biggr)^{\gamma_c-1} \biggl[ M_\mathrm{tot}^{3\gamma_c-4} \biggr]^{(\gamma_c-1)/(3\gamma_c-4)}
\biggl[ M_\mathrm{tot}^{-2} \biggr]^{(\gamma_c-1)/(3\gamma_c -4)}  
\biggl[ M_\mathrm{tot}^{-3\gamma_c+6} \biggr]^{(\gamma_c-1)/(3\gamma_c -4)}
\biggl[ \frac{R_\mathrm{norm}}{R} \biggr]^{3(\gamma_c-1)} 
\biggl( \frac{\rho_{ic}}{\bar\rho} \biggr)^{\gamma_c-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math> 
~+ \biggl[ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggr] \biggl( \frac{3}{4\pi} \biggr)^{\gamma_e-1} \biggl( \frac{K_e}{K_c} \biggr)
\biggl[ M_\mathrm{tot}^{2(\gamma_e-1)-2(\gamma_c-1)}\biggr]^{1/(3\gamma_c-4)}
\biggl[ \frac{R_\mathrm{norm}}{R} \biggr]^{3(\gamma_e-1)}
\biggl[ \biggl( \frac{K_c}{G} \biggr)^{(\gamma_c-1)-(\gamma_e-1)} \biggr]^{3/(3\gamma_c -4)} \biggl( \frac{\rho_{ie}}{\bar\rho} \biggr)^{\gamma_e-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} 
+ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} \biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} 
\biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{-3(\gamma_c-1)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math> 
~+ \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggl( \frac{K_e}{K_c} \biggr)
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2}  \biggr]^{(\gamma_c-\gamma_e)/(3\gamma_c -4)} 
\biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ie}}{\bar\rho} \biggr]^{\gamma_e-1} \biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{-3(\gamma_e-1)} \, .
</math>
  </td>
</tr>
</table>
</div>

<span id="pressureMatch">We also want to ensure that envelope pressure matches the core pressure at the interface.</span>  This means,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K_e \rho_{ie}^{\gamma_e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K_c \rho_{ic}^{\gamma_c}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\frac{K_e}{K_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_{ic}^{\gamma_c} \rho_{ie}^{-\gamma_e} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} 
\rho_\mathrm{norm}^{\gamma_c - \gamma_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} \biggl[ \frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} 
\biggl\{ \frac{3}{4\pi} \biggl[ \frac{G^3 M_\mathrm{tot}^2}{K_c^3} \biggr]^{1/(3\gamma_c -4)}  \biggr\}^{\gamma_c - \gamma_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\frac{K_e}{K_c} \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2} \biggr]^{(\gamma_c - \gamma_e)/(3\gamma_c -4)}  
\biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ie}}{\bar\rho} \biggr]^{\gamma_e-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\rho_\mathrm{norm}} \biggr]^{\gamma_c} 
\biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ie}}{\rho_\mathrm{norm}} \biggr]^{-\gamma_e} 
\biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ie}}{\bar\rho} \biggr]^{\gamma_e-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} 
\biggl( \frac{\rho_{ic}}{\rho_{ie}} \biggr) \biggl( \frac{\rho_\mathrm{norm}}{ \bar\rho } \biggr)^{\gamma_e - \gamma_c}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} 
\biggl( \frac{\rho_{ic}}{\rho_{ie}} \biggr) \biggl( \frac{ R}{R_\mathrm{norm}} \biggr)^{3(\gamma_e - \gamma_c)}
</math>
  </td>
</tr>
</table>
</div>
Hence, we can write the normalized (and dimensionless) free energy as,

<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>
- \frac{3\cdot \mathfrak{f}_{WM}}{5} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} 
+ \biggl\{ \frac{\nu s_\mathrm{core} }{(\gamma_c - 1)} + \frac{(1-\nu) s_\mathrm{env} }{(\gamma_e - 1)} \biggl( \frac{ \rho_{ic} }{ \rho_{ie} } \biggr)\biggr\} 
\biggl[ \biggl( \frac{3}{4\pi} \biggr)\frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c-1} 
\biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{-3(\gamma_c-1)} \, .
</math>
  </td>
</tr>
</table>
</div>

Keep in mind that, if the envelope and core both have uniform (but different) densities, then <math>~\rho_{ic} = \rho_c</math>, <math>~\rho_{ie} = \rho_e</math>, and
<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)} \, .
</math>
</div>

===Free Energy and Its Derivatives===

Now, the free energy can be written as,
<div align="center">
<table border="0">

<tr>
  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~U_\mathrm{tot} + W</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env} + W</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] C_\mathrm{core} R^{3-3\gamma_c} 
+ \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] C_\mathrm{env} R^{3-3\gamma_e} - A R^{-1} \, .</math>
  </td>
</tr>
</table>
</div>

The first derivative of the free energy with respect to radius is, then,

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

<tr>
  <td align="right">
<math>~\frac{d\mathfrak{G}}{dR}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -2 C_\mathrm{core} R^{2-3\gamma_c} -2 C_\mathrm{env} R^{2-3\gamma_e} + A R^{-2} \, .</math>
  </td>
</tr>
</table>
</div>

And the second derivative is,

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

<tr>
  <td align="right">
<math>~\frac{d^2\mathfrak{G}}{dR^2}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ -2 (2-3\gamma_c) C_\mathrm{core} R^{1-3\gamma_c} -2 (2-3\gamma_e) C_\mathrm{env} R^{1-3\gamma_e} - 2A R^{-3} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{2}{R^2} \biggl[(3\gamma_c-2) C_\mathrm{core} R^{3-3\gamma_c} +
(3\gamma_e-2) C_\mathrm{env} R^{3-3\gamma_e} - A R^{-1} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{2}{R^2} \biggl[(3\gamma_c-2) S_\mathrm{core} +
(3\gamma_e-2) S_\mathrm{env}  +W \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

===Equilibrium===
The radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration(s) is determined by setting the first derivative of the free energy to zero.  Hence,

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

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 2 C_\mathrm{core} R_\mathrm{eq}^{2-3\gamma_c} + 2 C_\mathrm{env} R_\mathrm{eq}^{2-3\gamma_e} - A R_\mathrm{eq}^{-2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ R_\mathrm{eq}^{-1} \biggl[ 2 C_\mathrm{core} R_\mathrm{eq}^{3-3\gamma_c} + 
2 C_\mathrm{env} R_\mathrm{eq}^{3-3\gamma_e} - A R_\mathrm{eq}^{-1} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ R_\mathrm{eq}^{-1} \biggl[ 2 S_\mathrm{core} + 
2 S_\mathrm{env}  +W \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ 2 S_\mathrm{tot} + W </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
This is the familiar statement of virial equilibrium.  From it we should always be able to derive the radius of equilibrium configurations.

===Stability===

To assess the relative stability of an equilibrium configuration, we need to determine the sign of the second derivative of the free energy, evaluated at the equilibrium radius.  If the sign of the second derivative is positive, the system is dynamically stable; if the sign is negative, he system is dynamically unstable.  Using the above statement of virial equilibrium, that is, setting,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~2 S_\mathrm{tot} + W</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ S_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~- S_\mathrm{core} - \frac{W}{2} \, ,</math>
  </td>
</tr>
</table>
</div>
we obtain,

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

<tr>
  <td align="right">
<math>~\frac{d^2\mathfrak{G}}{dR^2}\biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{2}{R_\mathrm{eq}^2} \biggl[ (3\gamma_c-2) S_\mathrm{core} 
+W - (3\gamma_e-2)\biggl( S_\mathrm{core} + \frac{W}{2}\biggr) \biggr]_\mathrm{eq} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{2}{R_\mathrm{eq}^2} \biggl[ 3(\gamma_c-\gamma_e) S_\mathrm{core} 
+ \biggl(2 - \frac{3}{2}\gamma_e\biggr)W \biggr]_\mathrm{eq} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{6}{R_\mathrm{eq}^2} \biggl[ (\gamma_c-\gamma_e) S_\mathrm{core} 
+ \frac{1}{2}\biggl(\frac{4}{3} - \gamma_e\biggr)W \biggr]_\mathrm{eq} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{6}{R_\mathrm{eq}^2} \biggl[ -\frac{W}{2}\biggl( \gamma_e - \frac{4}{3}\biggr) - 
(\gamma_e-\gamma_c) S_\mathrm{core} \biggr]_\mathrm{eq} \, .</math>
  </td>
</tr>
</table>
</div>
So, if when evaluated at the equilibrium state, the expression inside of the square brackets of this last expression is negative, the equilibrium configuration will be dynamically unstable.  We have chosen to write the expression in this particular final form because we generally will be interested in bipolytropes for which the adiabatic exponent of the envelope is greater than <math>~4/3</math> and the adiabatic exponent of the core is less than or equal to <math>~4/3</math> &#8212; that is, <math>~\gamma_e > 4/3 \ge \gamma_c</math>.  Hence, because the gravitational potential energy, <math>~W</math>, is intrinsically negative, the system will be dynamically unstable only if the second term (involving <math>~S_\mathrm{core}</math>) is greater in magnitude than the first term (involving <math>~W</math>).

It is worth noting that, instead of drawing upon <math>~S_\mathrm{core}</math> and <math>~W</math> to define the stability condition, we could have used an appropriate combination of <math>~S_\mathrm{env}</math> and <math>~W</math>, or the <math>~S_\mathrm{core}</math> and <math>~S_\mathrm{env}</math> pair.  Also, for example, because the virial equilibrium condition is <math>~S_\mathrm{tot} = -W/2</math>, it is easy to see that the following inequality also equivalently defines stability:
<div align="center">
<table border="0">

<tr>
  <td align="right">
<math>~ S_\mathrm{tot}\biggl( \gamma_e - \frac{4}{3}\biggr) - 
(\gamma_e-\gamma_c) S_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~>~</math>
  </td>
  <td align="left">
<math>~ 0 \, .</math>
  </td>
</tr>
</table>
</div>

=Related Discussions=
* [[SSC/BipolytropeGeneralizationVersion2|Newer, Version2 of ''Bipolytrope Generalization'' derivations]].
* [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|Analytic solution]] with <math>n_c = 0</math> and <math>n_e=0</math>.
* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic solution]] with <math>n_c = 5</math> and <math>n_e=1</math>.

{{ SGFfooter }}