Editing SSC/Virial/PolytropesSummary (section)

====Second Table====

Here, we have decided to look for a stable equilibrium state that is bounded by the same external pressure as the ''unstable'' state that has been identified in the above figure and table.  Rather than going straight to the free-energy expression in search of the desired stable configuration, we cheated a bit.  Using the properties of an <math>~n=4</math> polytrope, as tabulated on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)], in conjunction with the algebraic expression found in the next-to-last row of the above table, namely,
<div align="center">
<math>
\frac{P_e}{P_\mathrm{norm}} = \biggl[  \biggl( \frac{5^3}{4\pi} \biggr) \tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5} \, ,
</math>
</div>
we examined how <math>~P_e</math> varies with <math>~\tilde\xi</math>.  We found that <math>~P_e/P_\mathrm{norm} = 1.71\times 10^4</math> at <math>~\tilde\xi = 2.6</math>, which is almost identical to the value of the normalized external pressure that we determined was  associated with the unstable equilibrium state (at <math>~\tilde\xi = 4.81</math>) above.  As is illustrated by the figure and table that follows, we determined that the stable equilibrium state associated with this normalized external pressure is the minimum that occurs on the free energy curve having parameters, <math>~(n, \Pi_\mathrm{ad}) = (4, 0.02369)</math>.
 

<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="4">
[[File:TryN4Pi0.0237.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, 0.02369)</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>~1.274</math> </td>
  <td align="center"> <math>~1.317</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>~0.0624</math></td>
  <td align="center"> <math>~0.0713</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>~0.02369</math></td>
  <td align="center"> <math>~0.02369 </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"> ---- </td>
  <td align="center"><math>~2.6</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\tilde\theta</math> (approx.) </td>
  <td align="center">---- </td>
  <td align="center"><math>~0.5048</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
  <td align="center"> ---- </td>
  <td align="center"> <math>~0.175</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"> ---- </td>
  <td align="center"> <math>~0.0714</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"> ---- </td>
  <td align="center"><math>~0.2019</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"> ---- </td>
  <td align="center"><math>~0.2039</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"> ---- </td>
  <td align="center"><math>~0.4922</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"> ---- </td>
  <td align="center"><math>~2.542</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"> ---- </td>
  <td align="center"><math>~1.71 \times 10^4</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"> ---- </td>
  <td align="center"><math>~0.0316</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"> ---- </td>
  <td align="center"><math>~1.71 \times 10^4</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"> ---- </td>
  <td align="center"><math>~0.0316</math></td>
</tr>
</table>