Editing SSCpt1/Virial/PolytropesEmbeddedOutline (section)

===Our Case M Analysis===

====Coefficient Definitions (M)====
As has been detailed in [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseFreeEnergyExpression|an accompanying discussion]], our <font color="red">'''Case M'''</font> analysis has demonstrated that the free-energy expression governing the equilibrium structure and stability of pressure-truncated polytropic configurations can also be written as,
<div align="center" id="CaseMFreeEnergyExpression">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathfrak{G}^*_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} 
+~ n\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} 
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3}  \, . 
</math>
  </td>
</tr>
</table>
</div>
Written in this form, the expression highlights the functional dependence on the configuration's mass while assuming that the external pressure is held fixed.  Still, the expression is identical to the [[#Overview|free-energy function given above]], but viewed in this manner the appropriate coefficient and variable substitutions are: 
<div align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
Our <font color="red">Case M</font> Analysis
</td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>b</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>n\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}   </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>\frac{4\pi}{3} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>x</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>\frac{R}{R_\mathrm{SWS}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{G}</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>\mathfrak{G}^{*}_\mathrm{SWS} \equiv \frac{\mathfrak{G}}{E_\mathrm{SWS}} </math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>

where, drawing from [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Steven W. Stahler's (1983)] work &#8212; see also our [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>R_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>E_\mathrm{SWS} \equiv \biggl( \frac{n}{n+1}\biggr)\frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}} </math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}  \, ,</math>
  </td>
</tr>
</table>
</div>

and, in terms of the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Structural_Form_Factors|structural form factors]], <math>\tilde\mathfrak{f}_M</math>, <math>\tilde\mathfrak{f}_A</math>, and <math>\tilde\mathfrak{f}_W</math>,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\mathcal{A}_{M_\ell}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}  =
\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{B}_{M_\ell}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{-(n+1)/n} =
\biggl( \frac{3}{4\pi}\biggr)^{1/n} 
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, .
</math>
  </td>
</tr>
</table>
</div>

====Virial Equilibrium (M)====
Plugging these coefficient assignments into the [[#Equilibrium_Configurations|above mathematical prescription of the virial theorem]] gives the mass-radius relationship for pressure-truncated, polytropic equilibrium configurations, namely,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} 
+ \frac{4\pi}{3} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4 \, ,
</math>
  </td>
</tr>

</table>
</div>
which matches the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseFreeEnergyExpression|statement of virial equilibrium presented in our accompanying, more detailed analysis]].

====Stability (M)====
Similarly, according to the [[#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}  </math>
  </td>
  <td align="center">
<math>></math>
  </td>
  <td align="left">
<math>\biggl[ \frac{\mathcal{A}_{M_\ell}(n-3)}{4\pi n}\biggr]^{1/4} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{1/2} 
\, .
</math>
  </td>
</tr>

</table>
</div>
Or, flipped around, the criterion for stability may be written as,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} </math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n-3)}\biggr]^{1/2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^2
\, .
</math>
  </td>
</tr>

</table>
</div>


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