Editing SSC/Virial/PolytropesSummary (section)

====First Table====

<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="4">
[[File:TryN4Pi0.01.png|450px|Dimensionless Free-Energy Curve]]
  </th>
</tr>
<tr>
  <th align="center" colspan="4">
Determined from Plot of Renormalized Free-Energy 

with <math>~(n, \Pi_\mathrm{ad}) = (4, 0.01)</math>
  </th>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <th align="center">&nbsp;</th>
  <th align="center" width="25%">Maximum</th>
  <th align="center" width="25%">Minimum</th>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\Chi</math> </td>
  <td align="center"> <math>~1.0494</math> </td>
  <td align="center"> <math>~2.13905</math> </td>
</tr>
<tr>
  <th align="center" colspan="4">
Immediate Implications from Virial Theorem 
  </th>
</tr>
<tr>
  <th align="center"><math>~\Chi^{1/4} - 1</math></th>
  <td align="center"> <math>~\eta_\mathrm{ad}</math> </td>
  <td align="center"> <math>~0.012128</math></td>
  <td align="center"> <math>~0.20936</math> </td>
</tr>
<tr>
  <th align="center"><math>~(\Chi^{1/4} - 1)\cdot \Chi^{-4}</math></th>
  <td align="center"> <math>~\Pi_\mathrm{ad}</math> </td>
  <td align="center"> <math>~1.000024 \times 10^{-2}</math></td>
  <td align="center"> <math>~1.000018 \times 10^{-2}</math> </td>
</tr>
<tr>
  <th align="center" colspan="4">
Associated Detailed Force-Balanced Model Parameters 

obtained via interpolation of tabulated numbers on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]
  </th>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\tilde\xi</math> (approx.) </td>
  <td align="center"> <math>~4.81</math></td>
  <td align="center"><math>~1.624</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\tilde\theta</math> (approx.) </td>
  <td align="center"> <math>~0.251</math></td>
  <td align="center"><math>~0.709</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
  <td align="center"><math>~0.0727</math></td>
  <td align="center"> <math>~0.239</math></td>
</tr>
<tr>
  <th align="center"> <math>~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2}</math></th>
  <td align="center"> <math>~\eta</math> (check) </td>
  <td align="center"><math>~0.0126</math></td>
  <td align="center"> <math>~0.2091</math></td>
</tr>
<tr>
  <th align="center" colspan="4">
and, hence, Implied Structural Form Factors &amp; Coefficients <math>~\mathcal{B}</math> &amp; <math>~\mathcal{A}</math>
  </th>
</tr>
<tr>
  <th align="center"> <math>~3(-\tilde\theta^')/\tilde\xi</math></th>
  <td align="center"> <math>~\mathfrak{f}_M</math></td>
  <td align="center"> <math>~0.0453</math></td>
  <td align="center"><math>~0.4415</math></td>
</tr>
<tr>
  <th align="center"> <math>~5[3(-\tilde\theta^')/\tilde\xi]^2</math></th>
  <td align="center"> <math>~\mathfrak{f}_W</math></td>
  <td align="center"> <math>~0.01028</math></td>
  <td align="center"><math>~0.975</math></td>
</tr>
<tr>
  <th align="center"> <math>~15(-\tilde\theta^')^2 + \tilde\theta^5</math></th>
  <td align="center"> <math>~\mathfrak{f}_A</math></td>
  <td align="center"> <math>~0.08028</math></td>
  <td align="center"><math>~1.036</math></td>
</tr>
<tr>
  <th align="center"> <math>~\biggl(\frac{3}{4\pi} \biggr)^{1/4} \mathfrak{f}_M^{-5/4} \cdot \mathfrak{f}_A</math></th>
  <td align="center"> <math>~\mathcal{B}</math></td>
  <td align="center"> <math>~2.682</math></td>
  <td align="center"><math>~2.0122</math></td>
</tr>
<tr>
  <th align="center"> <math>~\frac{\tilde\mathfrak{f}_W}{5 \tilde\mathfrak{f}_M^2} </math></th>
  <td align="center"> <math>~\mathcal{A}</math></td>
  <td align="center"> <math>~1</math></td>
  <td align="center"><math>~1</math></td>
</tr>
<tr>
  <th align="center" colspan="4">
Given <math>~\Pi_\mathrm{ad}</math>, <math>~\Chi</math>, and <math>~\mathcal{B}</math>, we obtain
  </th>
</tr>
<tr>
  <th align="center"> <math>~\frac{3}{4\pi}\mathcal{D} = \frac{3}{4\pi} \Pi_\mathrm{ad} \mathcal{B}^{16} </math></th>
  <td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
  <td align="center"> <math>~1.71 \times 10^4</math></td>
  <td align="center"><math>~1.72 \times 10^2</math></td>
</tr>
<tr>
  <th align="center"> <math>~\Chi \mathcal{B}^{-4}</math></th>
  <td align="center"> <math>~\chi_\mathrm{eq}</math></td>
  <td align="center"> <math>~0.0203</math></td>
  <td align="center"><math>~0.1305</math></td>
</tr>
<tr>
  <th align="center" colspan="4">
Compare with Horedt's Equilibrium Parameters obtained from DFB Models
  </th>
</tr>
<tr>
  <th align="center"><math>\biggl[  \biggl( \frac{5^3}{4\pi} \biggr) 
\tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5} 
</math>
</th>
  <td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
  <td align="center"> <math>~1.76 \times 10^4</math></td>
  <td align="center"><math>~1.73 \times 10^2</math></td>
</tr>
<tr>
  <th align="center"><math>
\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}  
</math>
</th>
  <td align="center"> <math>~\chi_\mathrm{eq}</math></td>
  <td align="center"> <math>~0.0203</math></td>
  <td align="center"><math>~0.130</math></td>
</tr>
</table>

Now, we are convinced that both extrema identify perfectly valid equilibrium configurations.  However, in the context of astrophysics, the two identified equilibria are not connected to one another in any meaningful way.  In particular, two of the free-energy coefficients, <math>~\mathcal{B}</math> and <math>~\mathcal{D}</math>, have different values in the two cases; and, by inference, the normalized external pressure, <math>~P_e/P_\mathrm{norm}</math>, is different in the two cases.  So the plotted free-energy curve does not represent a "constant pressure" evolutionary trajectory.  How do we identify two equilibria that are associated with the same normalized external pressure?  And how do we identify the free-energy "evolutionary trajectory" that connects the two states?