Editing SSC/FreeEnergy/PolytropesEmbedded/Pt2

=Free Energy of Embedded Polytropes=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt1|Part I: &nbsp; Synopsis]]<br />&nbsp;</td>
  <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt2|Part II:&nbsp; Truncated Polytropes]]<br />&nbsp;</td>
  <td align="center" rowspan="1" colspan="3"bgcolor="lightblue"><br />Part III: &nbsp;Free-Energy of Bipolytropes<br />&nbsp;</td>
</tr>
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="17%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A|IIIA:&nbsp; Focus on (5, 1) Bipolytropes]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="17%"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt3B|IIIB:&nbsp; Focus on (0, 0) Bipolytropes]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/FreeEnergy/PolytropesEmbedded/Pt3C|IIIC:&nbsp; Overview]]<br />&nbsp;</td>
</tr>
</table>


=Free-Energy of Truncated Polytropes=
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  <td align="right">
<math>~\frac{\partial\mathfrak{G}|_{K,M,P_e}}{\partial R} = 0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\biggl[ \frac{\partial^2\mathfrak{G}|_{K,M,P_e}}{\partial R^2} \biggr]_\mathrm{eq} = 0</math>
  </td>
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In this case, the Gibbs-like free energy is given by the sum of three separate energies,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_eV</math>
  </td>
</tr>
<!-- HIDE INTERMEDIATE EXPRESSION ...
<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] \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 
</math>
  </td>
</tr>
END EXPRESSION HIDING -->
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 3\mathcal{A} \biggl[\frac{GM^2}{R} \biggr] + n\mathcal{B} \biggl[ \frac{KM^{(n+1)/n}}{R^{3/n}} \biggr] + \frac{4\pi}{3} \cdot P_e R^3 \, ,</math>
  </td>
</tr>
</table>
</div>

where the constants, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A} \equiv \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>\mathcal{B} \equiv \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, ,</math>
  </td>
</tr>
</table>
</div>

and, as [[SSCpt1/Virial/FormFactors#PTtable|derived elsewhere]],

<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 <math>~(n \ne 5)</math>
</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\tilde\theta^'}{\tilde\xi} \biggr) </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\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
</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{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} +  (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
  </td>
</tr>
</table>

</td>
</tr>
<tr>
  <td align="left" colspan="1">
As [[SSCpt1/Virial/FormFactors#Summary_.28n.3D5.29|we have shown separately]], for the singular case of <math>~n = 5</math>, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
( 1 + \ell^2 )^{-3/2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
\frac{5}{2^4} \cdot \ell^{-5}  
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{2^3} \ell^{-3}  [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]  
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\ell \equiv \tilde\xi/\sqrt{3} </math>
  </td>
</tr>
</table>
</div>


In general, then, the warped free-energy surface drapes across a four-dimensional parameter "plane" such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{G}(R, K, M, P_e) \, .</math>
  </td>
</tr>
</table>
</div>

In order to effectively visualize the structure of this free-energy surface, we will reduce the parameter space from four to two, in two separate ways:  First, we will hold constant the parameter pair, <math>~(K,M)</math>; giving a nod to [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura's (1981b)] nomenclature, we will refer to the resulting function, <math>~\mathfrak{G}_{K,M}(R,P_e)</math>, as a "Case M" free-energy surface because the mass is being held constant.   Second, we will hold constant the parameter pair, <math>~(K,P_e)</math>, and examine the resulting "Case P" free-energy surface, <math>~\mathfrak{G}_{K,P_e}(R,M)</math>.

==Virial Equilibrium and Dynamical Stability==
The first (partial) derivative of <math>~\mathfrak{G}</math> with respect to <math>~R</math> is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \mathfrak{G}}{\partial R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{R} \biggl[
3\mathcal{A} GM^2 R^{-1} - 3\mathcal{B}KM^{(n+1)/n} R^{-3/n} + 4\pi P_e R^3
\biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and the second (partial) derivative is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{R^2} \biggl[
-6\mathcal{A} GM^2 R^{-1} + \biggl(\frac{n+3}{n}\biggr) 3\mathcal{B}KM^{(n+1)/n} R^{-3/n} + 8\pi P_e R^3
\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
The virial equilibrium radius is identified by setting the first derivative to zero. This means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~3\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3\mathcal{A} GM^2 R_\mathrm{eq}^{-1}  + 4\pi P_e R_\mathrm{eq}^3 \, .
</math>
  </td>
</tr>
</table>
</div>

This expression can be usefully rewritten in the following forms:
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <td colspan="2" align="center">Virial Equilibrium Condition</td>
</tr>
<tr>
  <td align="center">Case 1:</td>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~3(n+3)\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3(n+3)\mathcal{A} GM^2 R_\mathrm{eq}^{-1}  + 4\pi (n+3) P_e R_\mathrm{eq}^3 
</math>
  </td>
</tr>
</table>
  </td>
</tr>

<tr>
  <td align="center">Case 2:</td>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
-6n\mathcal{A} GM^2 R_\mathrm{eq}^{-1}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 8\pi nP_e R_\mathrm{eq}^3 - 6n\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n}  </math>
  </td>
</tr>
</table>
  </td>
</tr>

<tr>
  <td align="center">Case 3:</td>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
8\pi nP_e R_\mathrm{eq}^3 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~6n\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n} - 6n\mathcal{A} GM^2 R_\mathrm{eq}^{-1} </math>
  </td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

Dynamical stability is determined by the sign of the second derivative expression ''evaluated at the equilibrium radius''; setting the second derivative to zero identifies the transition from stable to unstable configurations.  The criterion is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
-6n\mathcal{A} GM^2 R^{-1} + 3(n+3) \mathcal{B}KM^{(n+1)/n} R^{-3/n} + 8\pi nP_e R^3\biggr]_{R_\mathrm{eq}}
</math>
  </td>
</tr>
</table>
</div>

===Case 1 Stability Criterion===
Using the "Case 1" virial expression to define the equilibrium radius means that the stability criterion is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-6n\mathcal{A} GM^2 R_\mathrm{eq}^{-1} + 3(n+3)\mathcal{A} GM^2 R_\mathrm{eq}^{-1}  + 4\pi (n+3) P_e R_\mathrm{eq}^3 + 8\pi nP_e R_\mathrm{eq}^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathcal{A} GM^2 R_\mathrm{eq}^{-1} [3(n+3)- 6n ] + 4\pi P_e R_\mathrm{eq}^3 [(n+3)   + 2n]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
4\pi P_e R_\mathrm{eq}^3 [3(n+1) ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathcal{A} GM^2 R_\mathrm{eq}^{-1} [3(n-3)] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
4\pi P_e R_\mathrm{eq}^4 (n+1) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathcal{A} GM^2 (n-3)
</math>
  </td>
</tr>
</table>
</div>

===Case 2 Stability Criterion===
Using the "Case 2" virial expression to define the equilibrium radius means that the stability criterion is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 8\pi nP_e R_\mathrm{eq}^3 - 6n\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n}  + 3(n+3) \mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n} + 8\pi nP_e R_\mathrm{eq}^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
16\pi nP_e R_\mathrm{eq}^3 - [3(n-3)]\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
16\pi nP_e R_\mathrm{eq}^3  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[3(n-3)]\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
16\pi nP_e R_\mathrm{eq}^{3(n+1)/n}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[3(n-3)]\mathcal{B}KM^{(n+1)/n}  
</math>
  </td>
</tr>
</table>
</div>

===Case 3 Stability Criterion===
Using the "Case 3" virial expression to define the equilibrium radius means that the stability criterion is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-6n\mathcal{A} GM^2 R_\mathrm{eq}^{-1} + 3(n+3) \mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n} + 6n\mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n} - 6n\mathcal{A} GM^2 R_\mathrm{eq}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-12n\mathcal{A} GM^2 R_\mathrm{eq}^{-1}  
+ [6n +3(n+3)] \mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
9(n+1) \mathcal{B}KM^{(n+1)/n} R_\mathrm{eq}^{-3/n} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
12n\mathcal{A} GM^2 R_\mathrm{eq}^{-1}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
R_\mathrm{eq}^{n-3} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4n\mathcal{A}}{3(n+1) \mathcal{B}} \biggr]^n \biggl(\frac{G}{K}\biggr)^n M^{n-1} 
</math>
  </td>
</tr>
</table>
</div>

===Case M===
Now, in our discussion of "Case M" sequence analyses, the configuration's radius is normalized to,
<div align="center">
<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>~[ G^n K^{-n} M^{n-1} ]^{1/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>
Our "Case 3" stability criterion directly relates.  We conclude that the transition from stability to dynamical instability occurs when,

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

<tr>
  <td align="right">
<math>~
\biggl[\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr]_\mathrm{crit}^{n-3} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4n\mathcal{A}}{3(n+1) \mathcal{B}} \biggr]^n 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr]_\mathrm{crit}^{(n-3)/n} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4n}{15(n+1) }  \biggl(\frac{4\pi}{3}\biggr)^{1/n}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_A \tilde{\mathfrak{f}}_M^{(n-1)/n}} 
</math>
  </td>
</tr>
</table>
</div>
Also in the "Case M" discussions, the external pressure is normalized to,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[ G^{-3(n+1)} K^{4n} M^{-2(n+1)} ]^{1/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>

If we raise the "Case 1" stability criterion expression to the <math>~(n-3)</math> power, then divide it by the "Case 3" stability criterion expression raised to the fourth power, we find,

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
[P_e]_\mathrm{crit}^{n-3} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{\mathcal{A} GM^2 (n-3)}{4\pi (n+1)}\biggr]^{n-3}\biggl\{ \biggl[\frac{4n\mathcal{A}}{3(n+1) \mathcal{B}} \biggr]^n \biggl(\frac{G}{K}\biggr)^n M^{n-1}  \biggr\}^{-4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{\mathcal{A}  (n-3)}{4\pi (n+1)}\biggr]^{n-3} G^{n-3} M^{2(n-3)} \biggl[\frac{3(n+1) \mathcal{B}}{4n\mathcal{A}} \biggr]^{4n} \biggl(\frac{K}{G}\biggr)^{4n} M^{4(1-n)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{  (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} K^{4n} M^{-2(n+1)}  G^{-3(n+1)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\biggl[\frac{P_e}{P_\mathrm{norm}} \biggr]_\mathrm{crit}^{n-3} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{  (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} 
</math>
  </td>
</tr>

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

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

===Case P===
Flipping around this expression for <math>~[P_e]_\mathrm{crit}</math>, we also can write,

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

<tr>
  <td align="right">
<math>~
[M]_\mathrm{crit}^{2(n+1)}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{  (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} K^{4n}   G^{-3(n+1)} P_e^{3-n} \, .
</math>
  </td>
</tr>
</table>
</div>

Now, in our "Case P" discussions we normalized the mass to
<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}{n}\biggr)^{3/2} G^{-3/2} K^{2n/(n+1)} P_e^{(3-n)/[2(n+1)]} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, we have,

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

<tr>
  <td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{SWS}} \biggr]_\mathrm{crit}^{2(n+1)}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{  (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} \biggl(\frac{n+1}{n}\biggr)^{-3(n+1)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{  (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{3 }{4} \biggr)^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} \, ,
</math>
  </td>
</tr>
</table>
</div>

where the constants, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A} \equiv \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>\mathcal{B} \equiv \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, .</math>
  </td>
</tr>
</table>
</div>

So we can furthermore conclude that,

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

<tr>
  <td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{SWS}} \biggr]_\mathrm{crit}^{2(n+1)}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{  (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{3 }{4} \biggr)^{4n} \biggl\{ \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr\}^{-3(n+1)} 
\biggl\{ \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \biggr\}^{4n} 
</math>
  </td>
</tr>

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


Our expression for <math>~[M]_\mathrm{crit}^{2(n+1)}</math> can also be combined with the "Case 2 stability criterion" to eliminate the mass entirely, giving,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl\{ 16\pi nP_e R_\mathrm{eq}^{3(n+1)/n}  \biggr\}^{2n}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ [3(n-3)]\mathcal{B}K\biggr\}^{2n} 
\biggl[\frac{  (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{4n} K^{4n}   G^{-3(n+1)} P_e^{3-n}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
R_\mathrm{eq}^{6(n+1)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{ 3(n-3)}{16\pi n} \biggr]^{2n} 
\biggl[\frac{  (n-3)}{4\pi (n+1)}\biggr]^{n-3} \biggl[\frac{3(n+1) }{4n} \biggr]^{4n} \mathcal{A}^{-3(n+1)} \mathcal{B}^{6n} K^{6n}   G^{-3(n+1)} P_e^{3(1-n)}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
R_\mathrm{eq}^{2(n+1)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ \frac{ (n-3)}{4\pi n} \biggr]^{2n} \biggl[\frac{  (n-3)}{4\pi n}\biggr]^{n-3} \biggl[\frac{(n+1) }{n} \biggr]^{4n+(3-n)} \biggl(\frac{3 }{4} \biggr)^{6n} \biggr\}^{1/3}
\mathcal{A}^{-(n+1)} \mathcal{B}^{2n} K^{2n}   G^{-(n+1)} P_e^{(1-n)}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{ (n-3)}{4\pi n} \biggr]^{(n-1)} \biggl[\frac{(n+1) }{n} \biggr]^{(n+1)} \biggl(\frac{3 }{4} \biggr)^{2n} 
\mathcal{A}^{-(n+1)} \mathcal{B}^{2n} K^{2n}   G^{-(n+1)} P_e^{(1-n)}   \, .
</math>
  </td>
</tr>
</table>
</div>

Finally, recognizing that in our "Case P" discussions we normalized the radius to
<div align="center">
<table border="0" cellpadding="5" align="center">

<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}{n}\biggr)^{1/2} G^{-1/2} K^{n/(n+1)} P_e^{(1-n)/[2(n+1)]} \, ,
</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
[R_\mathrm{eq}]_\mathrm{crit}^{2(n+1)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{ (n-3)}{4\pi n} \biggr]^{(n-1)} \biggl(\frac{n+1 }{n} \biggr)^{(n+1)} \biggl(\frac{3 }{4} \biggr)^{2n} 
\mathcal{A}^{-(n+1)} \mathcal{B}^{2n} \biggl\{ R_\mathrm{SWS}\biggl(\frac{n+1 }{n} \biggr)^{-1/2} \biggr\}^{2(n+1)}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr]_\mathrm{crit}^{2(n+1)} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{ (n-3)}{4\pi n} \biggr]^{(n-1)} \biggl(\frac{3 }{4} \biggr)^{2n} 
\mathcal{A}^{-(n+1)} \mathcal{B}^{2n}    
</math>
  </td>
</tr>

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

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

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

==Case M Free-Energy Surface==

It is useful to rewrite the free-energy function in terms of dimensionless parameters.  Here we need to pick normalizations for energy, radius, and pressure that are expressed in terms of the gravitational constant, <math>~G</math>, and the two fixed parameters, <math>~K</math> and <math>~M</math>.  We have chosen to use,
<!-- Equation for use in PowerPoint presentation
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R}{R_0}</math>
  </td>
</tr>
</table>
</div>
-->

<div align="center">
<table border="0" cellpadding="3">

<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)^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^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  \, ,</math>
  </td>
</tr>

</table>
</div>

which, as is detailed in an [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|accompanying discussion]], are similar but not identical to the normalizations used by [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and by [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)].  The self-consistent energy normalization is,

<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~P_\mathrm{norm} R^3_\mathrm{norm} \, .</math>
  </td>
</tr>

</table>
</div>

As we have [[SSCpt1/Virial#Gathering_it_all_Together|demonstrated elsewhere]], after implementing these normalizations, the expression that describes the "Case M" free-energy surface is,
<div align="center">
<math>
\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} = 
-3\mathcal{A} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} +~ n\mathcal{B} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n} 
+~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, ,
</math>
</div>

Given the polytropic index, <math>~n</math>, we expect to obtain a different "Case M" 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>.

<!-- Supports PowerPoint summary
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_e}{P_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{20\pi x_\mathrm{eq}^4} \biggl[ 15\biggl(\frac{3}{4\pi}\biggr)^{1/n} x_\mathrm{eq}^{(n-3)/n} - 3\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~[x_\mathrm{eq}]_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4n}{15(n+1)} \biggl( \frac{4\pi}{3} \biggr)^{1/n} \biggr]^{n/(n-3)}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~[x_\mathrm{eq}]_\mathrm{turn}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4n}{15(n+1)} \biggl( \frac{4\pi}{3} \biggr)^{1/n} \biggr]^{n/(n-3)}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{P_\mathrm{max}}{P_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{20\pi} \biggl( \frac{n-3}{n+1} \biggr) \biggl[ \frac{15(n+1)}{4n}\biggl(\frac{3}{4\pi}\biggr)^{1/n} \biggr]^{4n/(n-3)}</math>
  </td>
</tr>
</table>
</div>
-->

==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,

<div align="center">
<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,

<div align="center">
<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,
<div align="center">
<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>.

==Summary==

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center">&nbsp;</th>
  <th align="center">DFB Equilibrium</th>
  <th align="center">Onset of Dynamical Instability</th>
</tr>
<tr>
  <th align="center" rowspan="2"><font size="+1">Case M:</font></th>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr]^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{4\pi}{(n+1)^n} \biggr] {\tilde\xi}^{(n-3)} (-{\tilde\xi}^2 \tilde{\theta^'})^{(1-n)}
</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr]_\mathrm{crit}^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{4n}{15(n+1) }\biggr]^n  \biggl(\frac{4\pi}{3}\biggr)  \frac{\tilde{\mathfrak{f}}_W^n}{\tilde{\mathfrak{f}}_A^n \tilde{\mathfrak{f}}_M^{(n-1)}} 
</math>
  </td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(n+1)} {\tilde\theta}^{(n+1)(n-3)} (-{\tilde\xi}^2 \tilde{\theta^'})^{2(n+1)}
</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]_\mathrm{crit}^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3}{4\pi}\biggr)^4 \biggl[\frac{  (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{n+1}{n} \biggr)^{3(n+1)} 
\biggl[ \frac{ 5^3 \tilde{\mathfrak{f}}_M^2}{ \tilde{\mathfrak{f}}_W^3 } \biggr]^{n+1} \biggl( \frac{3\tilde{\mathfrak{f}}_A}{4}\biggr)^{4n} 
</math>
  </td>
</tr>
</table>

  </td>
</tr>
<tr>
  <th align="center" rowspan="2"><font size="+1">Case P:</font></th>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{n}{4\pi}\biggr) {\tilde\xi}^2 {\tilde\theta}^{n-1}
</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr]_\mathrm{crit}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{3^2\cdot 5}{4\pi \tilde{\mathfrak{f}}_W}\biggr] 
\biggl[ \frac{n-3}{n} \biggr]^{(n-1)/(n+1)} 
\biggl[\frac{ \tilde{\mathfrak{f}}_A}{4} \biggr]^{2n/(n+1)}   
</math>
  </td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr]^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{n^3}{4\pi}\biggr) {\tilde\theta}^{n-3} (-{\tilde\xi}^2 {\tilde{\theta^'}})^2
</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr]_\mathrm{crit}^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{5^3\tilde{\mathfrak{f}}_M^2}{\tilde{\mathfrak{f}}_W^3} \biggr]\biggl(\frac{3}{4\pi} \biggr)^{4/(n+1)}
\biggl[\frac{  (n-3)}{4\pi n}\biggr]^{(n-3)/(n+1)} \biggl(\frac{3\tilde{\mathfrak{f}}_A }{4} \biggr)^{4n/(n+1)} 
</math>
  </td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="left" colspan="3">
In all four cases, the expression on right intersects (is equal to) the expression on the left when the following condition applies:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="left">
For <math>~n \ne 5</math>: &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">
<math>~
2(9-2n){\tilde\theta}^{n+1}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3(n-3) \biggl[ (-{\tilde\theta}^')^2 -  \biggl( -\frac{\tilde\theta {\tilde\theta}^'}{\tilde\xi} \biggr)\biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="left">
For <math>~n = 5</math>: &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">
<math>~
\biggl[\frac{2^4\cdot 5}{3}\biggr] \ell^3
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1+\ell^2)^3 \tan^{-1}\ell + \ell(\ell^4-1) \, .
</math>
  </td>
</tr>
</table>

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


If (for <math>n\ne 5</math>) we adopt the shorthand notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Upsilon</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[3 (-{\tilde\theta}^')^2 - {\tilde\mathfrak{f}}_M \tilde\theta] =
3\biggl[ (-{\tilde\theta}^')^2 -  \biggl( -\frac{\tilde\theta {\tilde\theta}^'}{\tilde\xi} \biggr)\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
and
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tau</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~{\tilde\theta}^{n+1} \, ,
</math>
  </td>
</tr>
</table>
</div>

then the critical condition becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(n-3)\Upsilon</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2(9-2n)\tau \, ,</math>
  </td>
</tr>
</table>
</div>
and ''at'' the critical state, the expressions for the structural form-factors become,

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

<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(5-n)} \biggl[6\tau + (n+1)\Upsilon  \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(5-n)} \biggl\{ 6 + (n+1)\biggl[ \frac{2(9-2n)}{n-3} \biggr]  \biggr\}\tau </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(5-n)} \biggl[ \frac{6(n-3) + 2(9-2n)(n+1)}{n-3} \biggr]  \tau </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(5-n)} \biggl[ \frac{4n(5-n)}{n-3} \biggr]  \tau </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4n\tau}{(n-3)} \, ;</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~{\tilde\mathfrak{f}}_W</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3\cdot 5}{(5-n) {\tilde\xi}^2} \biggl[\tau + \Upsilon  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3\cdot 5}{(5-n) {\tilde\xi}^2} \biggl\{1 + \biggl[ \frac{2(9-2n)}{n-3} \biggr]  \biggr\}\tau</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3\cdot 5}{(5-n) {\tilde\xi}^2} \biggl[ \frac{3(5-n)}{n-3} \biggr]  \tau</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5 \tau}{(n-3) {\tilde\xi}^2}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{5^3\tilde{\mathfrak{f}}_M^2}{\tilde{\mathfrak{f}}_W^3}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(n-3) {\tilde\xi}^2}{3^2\tau} \biggr]^{3} \biggl(-\frac{3 {\tilde\theta}^'}{\tilde\xi} \biggr)^2
= 3^2\biggl[\frac{(n-3) {\tilde\xi}^2}{3^2\tau} \biggr]^{3} \biggl(-\frac{{\tilde\xi}^2 {\tilde\theta}^'}{ {\tilde\xi}^3 } \biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(n-3)^3}{3^4\tau^3} \biggr] (-{\tilde\xi}^2 {\tilde\theta}^' )^2 \, .
</math>
  </td>
</tr>
</table>
</div>

Hence (1),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr]_\mathrm{crit}^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n }\biggr]
\biggl[ \frac{4n}{15}\biggr]^n  \biggl(\frac{1}{3}\biggr)   
\biggl[ \frac{3^2\cdot 5 }{4n {\tilde\xi}^2}   \biggr]^n 
\tilde{\mathfrak{f}}_M^{1-n}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n }\biggr]   
{\tilde\xi}^{n-3}
(-{\tilde\xi}^2{\tilde\theta}^')^{1-n}
</math>
  </td>
</tr>
</table>
</div>
Q.E.D.


And (2),
<div align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]_\mathrm{crit}^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3}{4\pi}\biggr)^4 \biggl[\frac{  (n-3)}{4\pi n}\biggr]^{n-3} \biggl(\frac{n+1}{n} \biggr)^{3(n+1)} 
\biggl[ \frac{ 5^3 \tilde{\mathfrak{f}}_M^2}{ \tilde{\mathfrak{f}}_W^3 } \biggr]^{n+1} \biggl( \frac{3\tilde{\mathfrak{f}}_A}{4}\biggr)^{4n} 
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]_\mathrm{crit}^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{n+1}
(-{\tilde\xi}^2 {\tilde\theta}^' )^{2(n+1)}  {\tilde\theta}^{(n+1)(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
Q.E.D.


And (3),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr]^{2(n+1)}_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{3^2\cdot 5}{4\pi \tilde{\mathfrak{f}}_W}\biggr]^{n+1}
\biggl[ \frac{n-3}{n} \biggr]^{(n-1) } 
\biggl[\frac{ \tilde{\mathfrak{f}}_A}{4} \biggr]^{2n }   
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr]^{2}_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{n}{4\pi }\biggr) {\tilde\xi}^2 {\tilde\theta}^{n-1}
</math>
  </td>
</tr>
</table>
</div>
Q.E.D.

And (4),
<div align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr]_\mathrm{crit}^{2(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{5^3\tilde{\mathfrak{f}}_M^2}{\tilde{\mathfrak{f}}_W^3} \biggr]^{n+1} \biggl(\frac{3}{4\pi} \biggr)^{4}
\biggl[\frac{  (n-3)}{4\pi n}\biggr]^{(n-3)} \biggl(\frac{3\tilde{\mathfrak{f}}_A }{4} \biggr)^{4n} 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr]_\mathrm{crit}^{2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{n^3}{4\pi}\biggr](-{\tilde\xi}^2 {\tilde\theta}^' )^{2} 
{\tilde\theta}^{n-3}
</math>
  </td>
</tr>
</table>
</div>
Q.E.D.


=See Also=


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