Editing SSC/FreeEnergy/PolytropesEmbedded/Pt2 (section)

==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.