Editing SSC/Virial/PolytropesSummary (section)

==Statement of Concern==
Throughout our discussion of embedded (pressure-truncated) polytropes &#8212; both on this "summary" page and in an [[User:Tohline/SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying chapter]] where critical background derivations are presented &#8212;  we have used expressions for the structural form factors that include an overall leading factor of <math>~(5-n)^{-1}</math>.  For clarity, the form factors that we have used [[User:Tohline/SSC/Virial/Polytropes#Summary|for ''isolated'' polytropes]] is reprinted on the lefthand side of the following table while the ones that we have used [[User:Tohline/SSC/Virial/Polytropes#PTtable|for ''pressure-truncated'' polytropes]] is reprinted on the righthand side of the table.

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
  </th>
  <th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
  </th>
</tr>
<tr>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1}  
</math>
  </td>
</tr>
</table>
  </td>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi}  + \tilde\Theta^{n+1}
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>
</div>
This factor seemed destined to become a nuisance in the specific case of <math>~n=5</math> polytropic structures.  But we did not let its appearance in these expressions deter us from using a free-energy analysis to study the equilibrium and stability of spherical polytropes because, after all, the factor of <math>~(5-n)^{-1}</math> appears in [[User:Tohline/Appendix/References|Chandrasekhar's [C67]]] expression for the gravitational potential energy of ''isolated'' polytropes &#8212; see his Equation (90), p. 101. In retrospect, its appearance in the structural form factors for ''isolated'' polytropes did not prove to be a problem because, via a free-energy and virial theorem analysis, the [[User:Tohline/SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|derived expression for the configuration's equilibrium radius]] depends on the ratio of <math>~f_W</math> to <math>~f_A</math>, so the awkward factor of <math>~(5-n)^{-1}</math> cancels out.

However, in our discussion of ''pressure-truncated'' <math>~n=5</math> polytropic structures, the factor of <math>~(5-n)^{-1}</math> did not conveniently cancel out at the appropriate time and we were forced to carry out some logical contortions [[User:Tohline/SSC/Virial/PolytropesSummary#Plotting_the_Virial_Theorem_Relation|as we tried to compare the mass-radius relation obtained from the virial theorem]] to Stahler's mass-radius relation, which was derived from detailed force-balance arguments.  This leads us to seriously question whether our, rather casually derived, expressions for the structural form factors in ''pressure-truncated'' polytropes are correct.