Editing SSC/FreeEnergy/PolytropesEmbedded/Pt2 (section)

==Case P Free-Energy Surface==

Again, it is useful to rewrite the free-energy function in terms of dimensionless parameters.  But here we need to pick normalizations for energy, radius, and mass that are expressed in terms of the gravitational constant, <math>~G</math>, and the two fixed parameters, <math>~K</math> and <math>~P_e</math>.  As is [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|detailed in an accompanying discussion]], we have chosen to use the normalizations defined by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)], namely,

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<table border="0" cellpadding="3">

<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+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,</math>
  </td>
</tr>

<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^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}  \, .</math>
  </td>
</tr>

</table>
</div>

The self-consistent energy normalization is,

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<table border="0" cellpadding="3">

<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>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .</math>
  </td>
</tr>
</table>
</div>

After implementing these normalizations &#8212; see our [[SSC/Virial/PolytropesEmbeddedOutline#Our_Case_M_Analysis|accompanying analysis]] for details &#8212; the expression that describes the "Case P" free-energy surface is,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}_{K,P_e}^* \equiv \frac{\mathfrak{G}_{K,P_e}}{E_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
<!-- HIDE LONG RE-DERIVATION ...
  <td align="left">
<math>~
\biggl\{\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{- \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R} 
+ \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
+ \frac{4\pi}{3} \cdot P_e R^3 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R} 
- \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
- \frac{4\pi}{3} \cdot P_e R^3 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \frac{R_\mathrm{SWS}}{R} \biggl[G M_\mathrm{SWS}^2 R_\mathrm{SWS}^{-1}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl[K M_\mathrm{SWS}^{(n+1)/n} R_\mathrm{SWS}^{-3/n}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl[ P_e R_\mathrm{SWS}^3 \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 
\biggl(\frac{R_\mathrm{SWS}}{R}\biggr) G \biggl[\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}  \biggr]^2 \biggl[\biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}  \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] 
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}  \biggr]^{(n+1)/n} \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}  \biggr]^{-3/n}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 P_e \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^3
\biggr\}
</math>
  </td>
</tr>
</table>
</div>


<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}_{K,P_e}^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 
\biggl(\frac{R_\mathrm{SWS}}{R}\biggr) G \biggl[\biggl( \frac{n+1}{nG} \biggr)^{3} K^{4n/(n+1)} P_\mathrm{e}^{(3-n)/[(n+1)]}  \biggr] 
\biggl[\biggl( \frac{n+1}{nG} \biggr)^{-1/2} K^{-n/(n+1)} P_\mathrm{e}^{(n-1)/[2(n+1)]}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] 
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3(n+1)/(2n)} K^{2} P_\mathrm{e}^{(3-n)/(2n)}  \biggr]
\biggl[ \biggl( \frac{n+1}{nG} \biggr)^{-3/2n} K^{-3/(n+1)} P_\mathrm{e}^{3(n-1)/[2n(n+1)]}  \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 P_e \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{3n/(n+1)} P_\mathrm{e}^{3(1-n)/[2(n+1)]} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 
\biggl(\frac{R_\mathrm{SWS}}{R}\biggr) \biggl[\biggl( \frac{n+1}{n} \biggr)^{5/2} G^{-3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] 
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}  \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 
\biggl(\frac{R_\mathrm{SWS}}{R}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] 
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
+ \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)\biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] 
\biggl(\frac{M}{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>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
END SUPPRESSION OF LONG DERIVATION -->
  <td align="left">
<math>~- 3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
+ n\mathcal{B} \biggl(\frac{M}{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>

Given the polytropic index, <math>~n</math>, we expect to obtain a different "Case P" free-energy surface for each choice of the dimensionless truncation radius, <math>~\tilde\xi</math>; this choice will imply corresponding values for <math>~\tilde\theta</math> and <math>~\tilde\theta^'</math> and, hence also, corresponding (constant) values of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.