Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1 (section)

===Structural Form Factors===

In our [[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying detailed analysis]], we demonstrate that the expressions given above for the free-energy function and the virial theorem are correct in sufficiently strict detail that they can be used to precisely match &#8212; and assist in understanding &#8212; the equilibrium of embedded polytropes whose structures have been determined from the set of detailed force-balance equations.  In order to draw this association, it is only necessary to realize that, very broadly, the constant coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, in the above [[SSC/Virial/PolytropesSummary#FreeEnergyExpression|algebraic free-energy expression]] are expressible in terms of three [[SSCpt1/Virial#Structural_Form_Factors|structural form factors]], <math>\tilde\mathfrak{f}_M</math>, <math>\tilde\mathfrak{f}_W</math>, and <math>\tilde\mathfrak{f}_A</math>, as follows:
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  
\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A 
= \frac{4\pi}{3} 
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3(n+1)/n} \biggr]_\mathrm{eq} 
\cdot \tilde\mathfrak{f}_A \, ;
</math>
  </td>
</tr>
</table>
</div>
and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write &hellip;

<div align="center" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr><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>~\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>~
~\frac{\tilde\mathfrak{f}_A - \tilde\theta^{n+1} }{\tilde\mathfrak{f}_W}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 
</math>
  </td>
</tr>
</table>

</td>
</tr>
</table>
</div>


After plugging these nontrivial expressions for <math>\mathcal{A}</math> and <math>\mathcal{B}</math> into the righthand sides of the above equations for <math>\Pi_\mathrm{ad}</math> and <math>\Chi_\mathrm{ad}</math> and, simultaneously, using the {{ Horedt70 }} detailed force-balanced expressions for <math>r_a</math> and <math>p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>P_e/P_\mathrm{norm}</math> in these same equations &#8212; see [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Summary_2|our accompanying discussion]] &#8212; we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
  </td>
</tr>
</table>
</div>
where the newly identified, key physical parameter, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta_\mathrm{ad} \equiv \frac{b_\mathrm{ad}}{a_\mathrm{ad}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3\cdot 5~ \tilde\theta^{n+1}}{(n+1) \tilde\xi^2 \mathfrak{f}_W} \, .</math>
  </td>
</tr>
</table>
</div>

It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[#ConciseVirial|virial theorem presented above]].