Editing SSC/SynopsisStyleSheet (section)

===Bipolytropes===

{| class="Synopsis1F" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|- 
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: &nbsp; Applicable to Bipolytropic Configurations</b></font>
|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2467;</font></b>&nbsp; &nbsp;<b>Variational Principle</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">&#x2469;</font></b>&nbsp; &nbsp;<b>Free-Energy Analysis of Stability</b>

|- 
! style="vertical-align:top; text-align:left;" |
<div align="center">
<font color="#770000">'''Governing Variational Relation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2  dM_r^* </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^*  \, .
</math>
  </td>
</tr>
</table>
</div>
! style="vertical-align:top; text-align:left;" rowspan="3"|
As we have detailed in an [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives|accompanying discussion]], the first derivative of the relevant free-energy expression is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R ~\frac{\partial \mathfrak{G}}{\partial R}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2S_\mathrm{tot} + W_\mathrm{tot} 
\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math>
  </td>
</tr>
</table>
and the second derivative of that free-energy function is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\biggl[
W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env} 
\biggr] \, .
</math>
  </td>
</tr>
</table>

----


This stability criterion may be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
2[(3\gamma_c -4) S_\mathrm{core}
+ (3\gamma_e -4) S_\mathrm{env} ] \, .
</math>
  </td>
</tr>
</table>
Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio,
<table border="0" cellpadding="5" align="center">

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

See the [[SSC/Stability/BiPolytropes#What_to_Expect_for_Equilibrium_Configurations|accompanying discussion]].

----

If &#8212; based for example on <b><font color="maroon" size="+1">&#x2466;</font></b> &#8212; we make the reasonable assumption that, in equilibrium, the statements,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>
  </td>
</tr>
</table>
hold separately, then we satisfy the virial equilibrium condition, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math>
  </td>
</tr>
</table>
and the second derivative of the relevant free-energy function can be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(W_\mathrm{core} + W_\mathrm{env})
+ (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core}) 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env})
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e) 
+ (4-3\gamma_c ) W_\mathrm{core}
+ (4-3\gamma_e)W_\mathrm{env} \, .
</math>
  </td>
</tr>
</table>

Note the similarity with <b><font color="maroon" size="+1">&#x2468;</font></b> &#8212; temporarily, see [[SSC/Stability/BiPolytropes#Revised_Free-Energy_Analysis|this discussion]].

|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2468;</font></b>&nbsp; &nbsp;<b>Approximation: &nbsp; Homologous Expansion/Contraction</b>

|- 
! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R  r^2  dM_r</math> 
  </td>
  <td align="center">
<math>\leq</math>
  </td>
  <td align="left">
<math>
(4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e)   P_i V_\mathrm{core} \, .
</math>
  </td>
</tr>
</table>
</div>

|}