Editing SSCpt1/Virial/FormFactors (section)

==Expectation in Context of Pressure-Truncated Polytropes==
For pressure-truncated polytropic configurations, the normalized virial theorem states that,
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  <td align="right">
<math>2 \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr) + \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} \, .</math>
  </td>
</tr>
</table>
</div>
This provides one mechanism by which the correctness of our form-factor expressions can be checked.  Specifically, having determined <math>S_\mathrm{therm}</math> and <math>W_\mathrm{grav}</math> from the derived form factors, we can see whether the sum of these energies as specified on the lefthand-side of this virial theorem expression indeed match the normalized energy term involving the external pressure, as specified on the righthand side.  In order to facilitate this "reality check" at the end of each example, below, we will use [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's detailed force-balanced solution of the equilibrium structure of embedded polytropes]] to provide an expression for the term on the righthand side of the virial theorem expression.

We begin by plugging our [[SSCpt1/Virial#Normalizations|general expression for <math>E_\mathrm{norm}</math>]] into this righthand-side term and grouping factors to facilitate insertion of Stahler's expressions.
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<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{tot}^{(5-n)} \biggr]^{1/(n-3)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)/(n-3)} P_e R_\mathrm{eq}^3 
\biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)} \, . </math>
  </td>
</tr>
</table>
</div>
From [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's equilibrium solution]], we have,
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<table border="0" cellpadding="3">

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<math>
R_\mathrm{eq}
</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\tilde\xi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}
</math>
  </td>
</tr>

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  <td align="right">
<math>\Rightarrow~~~~
~P_e R_\mathrm{eq}^3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{1 + 3(1-n)/[2(n+1)]}
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
M_\mathrm{limit}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2 
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\tilde\xi}
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} 
</math>
  </td>
</tr>

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  <td align="right">
<math>\Rightarrow~~~~
K^{-n} G^3 M_\mathrm{limit}^{(5-n)}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3-3(5-n)/2} K_n^{-n +2n(5-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3(n-3)/2} K_n^{3n(3-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} \, ;
</math>
  </td>
</tr>

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  <td align="right">
<math>\Rightarrow~~~~
P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} \biggr\}^{(5-n)/(n-3)} G^{3/2} K_n^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} 
</math>
  </td>
</tr>

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&nbsp;
  </td>
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&nbsp;
  </td>
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<math>\times ~\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(5-n)/2}  
\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^{3(n-3)}_{\tilde\xi} \biggl( \frac{n+1}{4\pi} \biggr)^{3(n-3)/2} \biggr\}^{1/(n-3)}
</math>
  </td>
</tr>

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&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
(n+1)^{3[(5-n)+(n-3)]/2} (4\pi)^{[(n-5)+(9-3n)]/2}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{[(n-3)(5-n) + 3(n-1)(n-3)]/2} \tilde\xi^{[2(5-n) + 3(n-3)]}
\biggr\}^{1/(n-3)}
</math>
  </td>
</tr>

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&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>
\biggl\{ (n+1)^{3} (4\pi)^{(2-n)}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{(n+1)(n-3)} \tilde\xi^{(n+1)}
\biggr\}^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the expectation based on Stahler's equilibrium models is that,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)} 
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

As a cross-check, multiplying this expression through by <math>[(R_\mathrm{eq}/R_\mathrm{norm})(M_\mathrm{norm}/M_\mathrm{limit})^2]</math> &#8212; where the expression for <math>R_\mathrm{eq}/R_\mathrm{norm}</math> can be obtained from our [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|discussions of detailed force-balanced models]] &#8212; gives a related result that can be obtained directly from [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|Horedt's expressions]], namely,

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  <td align="right">
<math>\biggl[ \frac{4\pi P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(n+1)( -\tilde\theta' )^{2}}   \, .
</math>
  </td>
</tr>
</table>
</div>