Editing SSC/VirialStability (section)

==Rescaled Energy Expressions==
===Isothermal Core===
In the case of an isothermal core, the expression for <math>W</math> shown above needs only a slight modification to put it in the appropriate form, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>W \,</math> 
  </td>
  <td align="left">
<math>
= - \frac{3}{5}  M_\mathrm{tot} c_s^2 \chi^{-1} \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \, ,
</math>
  </td>
</tr>

</table>

In the case of an isothermal core, the expression for the total internal energy may be rewritten as,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{U}{M_\mathrm{tot}} \,
</math>
  </td>
  <td align="center">
<math>= \,</math>
  </td>
  <td align="left">
<math>
\nu c_s^2 \ln(\rho_c/\rho_0)  + (1-\nu) n_e K_e \rho_e^{1/n_e}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\nu c_s^2 [\ln(\rho_c/\bar{\rho}) + \ln(\bar\rho/\rho_0)]   + (1-\nu) n_e K_e \rho_0^{1/n_e} \biggl[\biggl( \frac{\rho_e}{\bar\rho} \biggr) \biggl( \frac{\bar\rho}{\rho_0} \biggr) \biggr]^{1/n_e} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\nu c_s^2 [\ln(\nu/q^3) -3 \ln\chi]   + (1-\nu) n_e K_e \rho_0^{1/n_e} \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} \chi^{-3/n_e} \, .
</math>
  </td>
</tr>
</table>
Hence (in the case of an isothermal core),

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{M_\mathrm{tot} c_s^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
 - \frac{3}{5}  \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \chi^{-1} 
+ \nu [\ln(\nu/q^3) -3 \ln\chi]   + (1-\nu) n_e \biggl[ \frac{K_e \rho_0^{1/n_e}}{c_s^2} \biggr] \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} \chi^{-3/n_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathfrak{G}_0^* - A \chi^{-1} - C \ln\chi  + D \chi^{-3/n_e} \, ,
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>\mathfrak{G}_0^* \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
\nu \ln(\nu/q^3)   \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
\frac{3}{5} \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>C \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
3 \nu  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>D \, </math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
(1-\nu) n_e \biggl[ \frac{K_e \rho_0^{1/n_e}}{c_s^2} \biggr] \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} 
= (1-\nu) n_e \biggl( \frac{K_e}{c_s^2} \biggr) \biggl[  \frac{3}{4\pi} \biggl(\frac{c_s^6}{G^3M^2_\mathrm{tot}}\biggr)   \biggr]^{1/n_e} \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} \, .
</math>
  </td>
</tr>
</table>

===Polytropic Core===
In the case of a polytropic core,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{M_\mathrm{tot} K_c \rho_0^{1/n_c}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- A \chi^{-1}  + B_c \chi^{-3/n_c} + B_e \chi^{-3/n_e} \, ,
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>A \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
\frac{3}{5} \biggl[ \frac{GM_\mathrm{tot}}{R_0 K_c \rho_0^{1/n_c}} \biggr] \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) =
\frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/n_c} \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_c \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
n_c \nu \biggl( \frac{\nu}{q^3}\biggr)^{1/n_c} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_e \, </math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
n_e (1-\nu) \biggl[ \biggl( \frac{K_e }{K_c}\biggr)\rho_0^{1/n_e - 1/n_c} \biggr] \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} 
\, .
</math>
  </td>
</tr>
</table>