Editing SSC/VirialStability (section)

==Summary Expressions (New)==
In the above derivations, we have adopted the notation, 
<div align="center">
<math>
\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R_0^3} \, .
</math>
</div>
Now, guided by the [[SSCpt1/Virial#Nonrotating_Configuration_Embedded_in_an_External_Medium|earlier discussion of pressure-bounded isothermal spheres]], we choose the following normalization energy and radius:
<div align="center">
<math>
E_0 = 3M_\mathrm{tot} c_s^2 
</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>
R_0 = \frac{GM_\mathrm{tot}}{5c_s^2} \, .
</math>
</div>
Also, by analogy, it is useful to define the dimensionless parameter,
<div align="center">
<math>
\Pi_I \equiv \frac{K_e \rho_\mathrm{norm}^{1/n_e}}{c_s^2}
= \frac{K_e}{c_s^2} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr]^{1/n_e}
= \biggl( \frac{3\cdot 5^3}{4\pi} \biggr)^{1/n_e} \frac{K_e}{c_s^2} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr]^{1/n_e} \, .
</math>
</div>
(It is worth noting that if we set <math>n_e = -1</math>, the dimensionless parameter <math>\Pi_I</math> becomes identical to the parameter <math>\Pi</math> as defined [[SSCpt1/Virial#P-V_Diagram|in the context of our discussion of the Bonnor-Ebert sphere]].  But in order to complete the analogy with the Bonnor-Ebert sphere discussion, we would also need to change the sign  on the last term in the above expression for the free energy because in the earlier discussion the external pressure was an external, confining condition whereas here it is included as an internal energy of the system.)


<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="3">
Relevant Expressions for Isothermal Core
  </th>
</tr>

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

<tr>
  <td align="center">
<math>
\chi \equiv \frac{R}{R_0} 
</math>
  </td>
  <td align="center">
<math>
\Pi_I^{n_e/3}  [\nu^{-n_e}\ (1-\nu)^{n_e+1} ]^{1/3} q^{n_e} (1-q^3)^{-(n_e+1)/3} 
</math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\chi^3 = \biggl(\frac{R}{R_0} \biggr)^{3}
</math>
  </td>
  <td align="center">
<math>
\Pi_I^{n_e}  \biggl( \frac{\rho_e}{\rho_c}\biggr)^{n_e + 1} \biggl[ q^3 + \biggl( \frac{\rho_e}{\rho_c} \biggr)(1-q^3)  \biggr]^{-1} 
</math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{A}{E_0} 
</math>
  </td>
  <td align="center">
<math>
\nu^2 \xi_s \biggl\{ 1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
  </td>
  <td align="left">
<math>= \biggl( \frac{\nu^2}{q} \biggr) \biggl\{ f_A(\nu,q) \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{B_e}{E_0} 
</math>
  </td>
  <td align="center">
<math>
\Pi_I \biggl( \frac{n_e}{3} \biggr) (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e}  (\xi_s^3 - 1)^{-1/n_e}
</math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{B_I}{E_0} 
</math>
  </td>
  <td align="center">
<math>\nu </math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{\mathfrak{G}}{E_0} 
</math>
  </td>
  <td align="center">
<math>
- \frac{A}{E_0} \chi^{-1}  - \frac{B_I}{E_0} \ln\chi + \frac{B_e}{E_0} \chi^{-3/n_e} 
</math>
  </td>
  <td align="left">
<math>
= \nu \biggl[ - \biggl( \frac{\nu}{q} \biggr) \frac{f_A(\nu,q)}{\chi} - \ln\chi + \frac{n_e}{3} \biggl(\frac{1}{q^3}-1\biggr)\biggr]
</math>
  </td>
</tr>
</table>

[On <font color="red">8 November 2013</font>, J. E. Tohline wrote: I just confirmed that the simpler expression for the normalized total free energy, <math>\mathfrak{G}/E_0</math>, matches the more complicated version.  I don't like the result because the third term in the free energy -- the one contributed by the internal energy of the envelope -- is independent of the radius of the configuration, <math>\chi</math>; it works out this way because my expression for <math>\Pi_I</math> has a radial dependence that exactly cancels out the explicit radial dependence that appears in the more complicated expression.  But maybe it's okay after all because this expression is intended to show how the free energy varies across the <math>(q,\nu)</math> plane, and the effect of <math>\Pi_I</math> appears implicitly through the specification of <math>\chi</math>, or visa versa.]