Editing SSC/VirialStability (section)

==Examples==
===Isothermal Core with <math>n_e=3/2</math>===
Consider the case examined by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)], that is, the case of an isothermal core and an envelope with <math>n_e = 3/2</math>.  The equilibrium radius is given by the expression,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\chi_E^{-1}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\alpha - \beta_I \chi_E \,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ \beta_I \chi_E^{2} -\alpha \chi_E + 1
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
0 \,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ \chi_E
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2\beta_I} \biggl[\alpha \pm \biggr(\alpha^2-4\beta_I \biggr)^{1/2}\biggr] \,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\alpha}{2\beta_I} \biggl[1 \pm \biggr(1-\frac{4\beta_I}{\alpha^2} \biggr)^{1/2}\biggr] \, .
</math>
  </td>
</tr>

</table>
</div>
And the system is stable when,
<div align="center">
<math>
\chi_E < \chi_0 \equiv  \frac{\alpha }{2\beta_I} \, .
</math>
</div>

A couple of physical attributes are now clear:
*Physical configurations only exist for <math>(4\beta_I/\alpha^2) \le 1</math>.
*For each value of <math>(4\beta_I/\alpha^2) < 1 \,</math>, there are two equilibrium configurations, given by the <math>\pm</math> roots of the quadratic equation for <math>\chi_E</math>; the "negative" branch is stable but the "positive" branch is unstable.

Note that,
<div align="center">
<math>
\chi_0 \equiv \frac{\alpha }{2\beta_I}  = \biggl( \frac{GM_\mathrm{tot}}{10R_0 c_s^2} \biggr) \frac{\nu f(\nu,q)}{q} ~~~\Rightarrow ~~~ 
\biggl( \frac{10R_0 c_s^2}{GM_\mathrm{tot}} \biggr) = \frac{\nu f(\nu,q)}{q \chi_0}  \, ,
</math>
</div>
and,
<div align="center">
<math>
\frac{4\beta_I}{\alpha^2 }  = \biggl( \frac{10 R_0 c_s^2 }{GM_\mathrm{tot}} \biggr)^2 \biggl[ \frac{K_e (\rho_e|_0)^{1/n_e}}{c_s^2} \biggr] \frac{q^2(1-\nu)}{\nu^3 f^2(\nu,q)}  
= \biggl[ \frac{K_e (\rho_e|_0)^{1/n_e}}{c_s^2} \biggr] \frac{q^2(1-\nu)}{\nu^3 f^2(\nu,q)} \biggl[\frac{\nu f(\nu,q)}{q \chi_0} \biggr]^2 
= \biggl[ \frac{\mu_c}{\mu_e} \biggr] \biggl( \frac{1}{\nu} - 1 \biggr) \, .
</math>
</div>
But, this last expression must be less than or equal to unity, which implies,
<div align="center">
<math>
\frac{1}{\nu} \le 1 + \frac{\mu_e}{\mu_c} ~~~\Rightarrow ~~~ \nu \ge \biggl(1 + \frac{\mu_e}{\mu_c}  \biggr)^{-1}
</math>
This doesn't seem to have the correct behavior, for example, the smaller values of <math>\nu</math> should be the stable ones, so there must be a mistake in the derivation.
</div>

===Adiabatic Core with <math>n_c = 5</math> and <math>n_e=1</math>===
Consider the case with an analytical structure derived by  [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eagleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831), that is, the case of an adiabatic core having <math>n_c=5</math> and an envelope with <math>n_e = 1</math>.  The equilibrium radius is,