Editing SSC/Structure/StahlerMassRadius (section)

==Review==
In [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|an accompanying chapter]] that discusses detailed force-balanced models of embedded (and pressure-truncated) polytropes, we have summarized the derivation by {{ Stahler83full }} of the pair of parametric [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|relations for the equilibrium mass and equilibrium radius]] for such systems, namely,
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<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
M
</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\}_{\xi_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<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\}_{\xi_e}
</math>
  </td>
</tr>
</table>
</div>
where,
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<math>M_\mathrm{SWS} = 
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
</div>
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<math>
R_\mathrm{SWS} = \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .
</math>
</div>

As [[SSC/Structure/PolytropesEmbedded/n5#Overlap_with_Stahler's_Presentation|we also have reviewed]], {{ Stahler83 }} &#8212; hereafter, {{ Stahler83hereafter }} &#8212; also explicitly states (see his equation B13) that [[SSC/Structure/PolytropesEmbedded#Overlap_with_Stahler.27s_Presentation_2|the relevant mass-radius relationship]] for <math>~n = 5</math> embedded polytropes is,
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<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{20\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~0 \, .
</math>
  </td>
</tr>
</table>
</div>

In what was intended to be a complementary discussion, our [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|free-energy analysis of embedded polytropes]] produced a virial equilibrium expression of the general form,
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<math>
\mathcal{A} - \mathcal{B}\chi_\mathrm{eq}^{4 -3\gamma_g}  +~ \mathcal{D}\chi_\mathrm{eq}^4 = 0 \, ,
</math>
</div>
where,
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<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{norm}} 
\, ,</math>
  </td>
</tr>

<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}^{\gamma}
\cdot \tilde\mathfrak{f}_A \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, ,
</math>
  </td>
</tr>
</table>
</div>
and,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma_g} \biggr]^{1/(4-3\gamma_g)} 
=\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{(n-1)} \biggr]^{1/(n-3)}
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{K^4}{G^{3\gamma_g} M_\mathrm{tot}^{2\gamma_g}} \biggr]^{n/(n-3)}  
= \biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} 
\, ,</math>
  </td>
</tr>
</table>
and,

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<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">

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  <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>
When we went back to compare the mass-radius relationship that results from our very general virial equilibrium expression to the one published by Stahler for pressure-truncated <math>~n = 5</math> polytropes, they did not appear to agree.  In what follows, we methodically plow through this comparison in considerable detail to uncover whatever discrepancies might exist.