Editing SSC/FreeEnergy/Powerpoint (section)

====Case P====
Set <math>~K</math> and <math>~P_e</math> constant and examine how the free-energy behaves as a function of the coordinates, <math>~(R,M_\mathrm{tot})</math>.  In this case (see, for example, [[SSC/FreeEnergy/PolytropesEmbedded#Case_P|here]]),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n\biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}}  
\biggl( \frac{n+1}{n} \biggr)^{3/(n-3)}
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{3(n-1)/[n(n-3)]} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl( \frac{n+1}{n} \biggr)^{3/(n-3)} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(5-n)/(n-3)}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~R_0 = 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/(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}{n} \biggr)^{3/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, .</math>
  </td>
</tr>
</table>
</div>

where the structural form factors for pressure-truncated polytropes are precisely defined [[SSCpt1/Virial/FormFactors#PTtable|here]]. If we set all three structural form-factors to unity, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{a}{3c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{b}{nc}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]^{(n+1)/n} 
</math>
  </td>
</tr>
</table>
</div>

=====Virial Equilibrium=====
So, the statement of virial equilibrium becomes,

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

<tr>
  <td align="right">
<math>~\biggl[ \biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]^{(n+1)/n} x^{(n-3)/n }_\mathrm{eq} 
-  \frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} -  x^{4}_\mathrm{eq}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  0 \, .</math>
  </td>
</tr>
</table>
</div>


<div align="center">
<math>~\xi_1</math>
</div>


<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="4">
Known Analytic Lane-Emden Functions
  </td>
</tr>
<tr>
  <td align="center">
<math>~n</math>
  </td>
  <td align="center">
<math>~\theta_n(\xi)</math>
  </td>
  <td align="center">
<math>~-\theta^'_n</math>
  </td>
  <td align="center">
<math>~\xi_1</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~0</math>
  </td>
  <td align="center">
<math>~1-\frac{\xi^2}{6}</math>
  </td>
  <td align="center">
<math>~\frac{\xi}{3}</math>
  </td>
  <td align="center">
<math>~\sqrt{6}</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~1</math>
  </td>
  <td align="center">
<math>~\frac{\sin\xi}{\xi}</math>
  </td>
  <td align="center">
<math>~\frac{\sin\xi}{\xi^2} - \frac{\cos\xi}{\xi}</math>
  </td>
  <td align="center">
<math>~\pi</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~5</math>
  </td>
  <td align="center">
<math>~\biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-1/ 2}</math>
  </td>
  <td align="center">
<math>~\frac{\xi}{3}\biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-3/ 2} </math>
  </td>
  <td align="center">
<math>~\infty</math>
  </td>
</tr>
</table>
</div>

=====Dynamical Instability=====
Along the "Case P" equilibrium sequence, the transition from stable to unstable configurations occurs at,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[x_\mathrm{eq}]^4_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{3(n+1)}  \biggl[ \frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{20\pi n}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} 
= \frac{2^2\pi (n-3)}{3^2\cdot 5 n}  \biggl[\frac{3}{4\pi}\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]^{2} 
\, ,
</math>
  </td>
</tr>
</table>
</div>
which, in combination with the "Case P" virial equilibrium expression gives,

<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\{ \biggl[ \frac{3^2\cdot 5 n}{2^2\pi (n-3)} \biggr]^{1/2} x_\mathrm{crit}^2 \biggr\}^{(n+1)/n} x^{(n-3)/n }_\mathrm{crit} 
-  \frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl\{ \biggl[\frac{20\pi n}{(n-3)}\biggr] x^4_\mathrm{crit}  \biggr\} -  x^{4}_\mathrm{crit}  </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 n}{2^2\pi (n-3)} \biggr]^{(n+1)/(2n)} x^{(3n-1)/n }_\mathrm{crit} 
- x^4_\mathrm{crit}  \biggl\{ \biggl[\frac{3(n+1)}{(n-3)}\biggr] +1 \biggr\} </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 n}{2^2\pi (n-3)} \biggr]^{(n+1)/(2n)} x^{(3n-1)/n }_\mathrm{crit} 
- \biggl[\frac{4n}{(n-3)}\biggr] x^4_\mathrm{crit} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ x^{(n+1)/n}_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{(n-3)}{4n}\biggr] \biggl[ \frac{3^2\cdot 5 n}{2^2\pi (n-3)} \biggr]^{(n+1)/(2n)} 
</math>
  </td>
</tr>
<!--
<tr>
  <td align="right">
<math>~\Rightarrow~~~ [x_\mathrm{eq}]_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\cdot 2^{-(3n+1)/(n+1)} \biggl( \frac{5}{\pi} \biggr)^{1/2} \biggl[\frac{(n-3)}{n}\biggr]^{(n-1)/[2(n+1)]} \, .
</math>
  </td>
</tr>
-->
</table>
</div>

=====Turning Points=====
Let's simplify the notation, defining,
<div align="center">
<math>~m \equiv \frac{3}{4\pi}\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr) \, .</math>
</div>
The statement of virial equilibrium becomes,

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

<tr>
  <td align="right">
<math>~m^{(n+1)/n} x^{(n-3)/n } 
-  c_0m^2 -  x^{4}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  0 \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~c_0 \equiv \biggl[ \frac{4\pi(n+1)}{15n} \biggr]  \, .</math>
</div>
Differentiating gives,

<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(\frac{n+1}{n}\biggr)m^{1/n} x^{(n-3)/n } dm 
+ \biggl(\frac{n-3}{n}\biggr)m^{(n+1)/n} x^{-3/n } dx
-  2c_0m dm -  4x^{3}  dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ (n+1)m^{1/n} x^{(n-3)/n } -  2c_0 n m \biggr] dm 
+ \biggl[ (n-3) m^{(n+1)/n} x^{-3/n }  -  4nx^{3} \biggr] dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{dm}{dx}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4nx^{3} - (n-3) m^{(n+1)/n} x^{-3/n }  }{(n+1)m^{1/n} x^{(n-3)/n } -  2c_0 n m} \, .
</math>
  </td>
</tr>
</table>
</div>

----

One turning point occurs where the numerator is zero, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4nx^{3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(n-3) m^{(n+1)/n} x^{-3/n } </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 4nx^{3(n+1)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(n-3) m^{(n+1)/n} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{m}{x^3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{4n}{(n-3)}\biggr]^{n/(n+1)} \, .</math>
  </td>
</tr>
</table>
</div>

Plugging this into the virial equilibrium expression gives,

<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\{  \biggl[\frac{4n}{(n-3)}\biggr]^{n/(n+1)} x^3 \biggr\}^{(n+1)/n} x^{(n-3)/n } 
-  c_0\biggl\{ \biggl[\frac{4n}{(n-3)}\biggr]^{n/(n+1)} x^3 \biggr\}^2 -  x^{4}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{4n}{(n-3)}\biggr] x^4 
-  c_0 \biggl[\frac{4n}{(n-3)}\biggr]^{2n/(n+1)} x^6  -  x^{4}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~  c_0 \biggl[\frac{4n}{(n-3)}\biggr]^{2n/(n+1)} x^2  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{4n}{(n-3)}\biggr]  - 1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~  x^2  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{15n}{4\pi(n+1)} \biggr]\biggl[\frac{3(n+1)}{(n-3)}\biggr] \biggl[\frac{(n-3)}{4n}\biggr]^{2n/(n+1)}  </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 n}{4\pi(n-3)} \biggr]\biggl[\frac{(n-3)}{4n}\biggr]^{2n/(n+1)}  \, .</math>
  </td>
</tr>
</table>
</div>

The associated mass is,

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

<tr>
  <td align="right">
<math>\biggl( \frac{M_\mathrm{max}}{M_\mathrm{SWS}}\biggr) = \biggl(\frac{4\pi}{3} \biggr)m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4\pi}{3} \biggl[\frac{n-3}{4n}\biggr]^{-n/(n+1)}
\biggl[ \frac{3^2\cdot 5 n}{4\pi(n-3)} \biggr]^{3/2}\biggl[\frac{(n-3)}{4n}\biggr]^{3n/(n+1)}  
</math>
  </td>
</tr>

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

Notice that, for <math>~n=3</math>,

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

<tr>
  <td align="right">
<math>\biggl( \frac{M_\mathrm{max}}{M_\mathrm{SWS}}\biggr)_{n=3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{3^4\cdot 5^3 }{2^{8} \pi} \biggr]^{1/2}
\, .</math>
  </td>
</tr>
</table>
</div>

----

Another turning point occurs where the denominator is zero, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(n+1)m^{1/n} x^{(n-3)/n }  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2c_0 n m</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (n+1)x^{(n-3)/n }  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2c_0 n m^{(n-1)/n}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{x^{n-3}}{ m^{n-1}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2c_0 n}{(n+1)} \biggr]^n</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{8\pi}{15} \biggr]^n</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ x^{(n-3)/(n-1)} \biggl(  \frac{15}{8\pi}  \biggr)^{n/(n-1)}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Plugging this into the virial equilibrium expression gives,

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

<tr>
  <td align="right">
<math>~  x^{4}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ x^{(n-3)/(n-1)} \biggl(  \frac{15}{8\pi}  \biggr)^{n/(n-1)}\biggr]^{(n+1)/n} x^{(n-3)/n } 
-  c_0\biggl[ x^{(n-3)/(n-1)} \biggl(  \frac{15}{8\pi}  \biggr)^{n/(n-1)}\biggr]^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(  \frac{15}{8\pi}  \biggr)^{(n+1)/(n-1)} x^{(n-3)(n+1)/[n(n-1)]} \cdot x^{(n-3)/n } 
-  c_0 \biggl(  \frac{15}{8\pi}  \biggr)^{2n/(n-1)} x^{2(n-3)/(n-1)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(  \frac{15}{8\pi}  \biggr)^{(n+1)/(n-1)} x^{2(n-3)/(n-1)}  
-  \biggl[ \frac{(n+1)}{2n} \biggr]\biggl(  \frac{15}{8\pi}  \biggr)^{(n+1)/(n-1)} x^{2(n-3)/(n-1)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(  \frac{15}{8\pi}  \biggr)^{(n+1)/(n-1)} x^{2(n-3)/(n-1)}  \biggl[1
-  \frac{(n+1)}{2n} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ x^{2(n+1)/(n-1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \frac{(n-1)}{2n} \biggl(  \frac{15}{8\pi}  \biggr)^{(n+1)/(n-1)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{R_\mathrm{max}}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \biggl[\frac{(n-1)}{2n} \biggr]^{(n-1)/[2(n+1)]} \biggl(  \frac{15}{8\pi}  \biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>

And the associated mass is,

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

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr|_\mathrm{turn}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3}\biggl(  \frac{15}{8\pi}  \biggr)^{n/(n-1)} 
\biggl\{ \biggl[\frac{(n-1)}{2n} \biggr]^{(n-1)/[2(n+1)]} \biggl(  \frac{15}{8\pi}  \biggr)^{1/2}  \biggr\}^{(n-3)/(n-1)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3}\biggl(  \frac{15}{8\pi}  \biggr)^{3/2} \biggl[\frac{(n-1)}{2n} \biggr]^{(n-3)/[2(n+1)]}  \, .</math>
  </td>
</tr>
</table>
</div>

=====Case P Summary=====
<div align="center">
<table border="1" align="center" cellpadding="8">
<tr><th align="center" colspan="2">
<font size="+1">Case P</font>
</th></tr>
<tr><td align="center" colspan="2">
Order-of-Magnitude Analysis:  Assume <math>~{\tilde\mathfrak{f}}_M = {\tilde\mathfrak{f}}_W = {\tilde\mathfrak{f}}_A = 1</math>
</td></tr>

<tr>
  <td align="center">
Virial Equilibrium:
  </td>
  <td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \biggl(\frac{3}{4\pi} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]^{(n+1)/n} 
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n }
-  \frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} 
-  \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>
  </td>
</tr>

<tr>
  <td align="center">
<p>Dynamical Instability:</p>
<math>~(n > 3)</math>
  </td>
  <td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="center">
<p>Turning Point <math>~(M_\mathrm{max} )</math>:</p>
<math>~(n>3)</math>
  </td>
  <td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_\mathrm{max}}{M_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4\pi}{3} \biggl[\frac{n-3}{4n}\biggr]^{2n/(n+1)}
\biggl[ \frac{3^2\cdot 5 n}{4\pi(n-3)} \biggr]^{3/2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr|_\mathrm{turn}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(n-3)}{4n}\biggr]^{n/(n+1)} \biggl[ \frac{3^2\cdot 5 n}{2^2\pi(n-3)} \biggr]^{1/2} 
</math>
  </td>
</tr>
</table>
</div>
  </td>
</tr>


<tr>
  <td align="center">
<p>Turning Point <math>~(R_\mathrm{max} )</math>:</p>
<math>~(n>1)</math>
  </td>
  <td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{R_\mathrm{max}}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \biggl[\frac{(n-1)}{2n} \biggr]^{(n-1)/[2(n+1)]} \biggl(  \frac{15}{8\pi}  \biggr)^{1/2} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr|_\mathrm{turn}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3}\biggl(  \frac{15}{8\pi}  \biggr)^{3/2} \biggl[\frac{(n-1)}{2n} \biggr]^{(n-3)/[2(n+1)]} </math>
  </td>
</tr>
</table>
</div>
  </td>
</tr>

</table>
</div>


<div align="left">
<table border="2" cellpadding="5" align="center">
<tr><th align="center" colspan="2">Figure 2</th></tr>
<tr>
<td align="center">{{ Stahler83figure }}</td>
<td align="center" rowspan="2">
[[File:MimicStahlerPlot3.png|center|350px|Case P equilibrium sequences with key configurations highlighted]]
</td>
</tr>
<tr>
<td align="center">
[[File:Stahler_MRdiagram1.png|350px|center|Stahler (1983) Figure 17 (edited)]]
</td>
</tr>
</table>
</div>

The right-hand panel of Figure 2 presents substantial segments of ''Case P'' virial equilibrium sequences for a range of polytropic indexes (n = 1, 2, 2.8, 3, 3.5, 4, 5).  For each sequence, the location of the <math>~R_\mathrm{max}</math> and <math>~M_\mathrm{max}</math> turning points &#8212; if they exist &#8212; are denoted by a yellow or red circular dot, respectively.  The point along each <math>~(n \geq 3)</math> sequence at which the transition from dynamically stable to dynamically unstable structures occurs coincides with the location of <math>~M_\mathrm{max}</math> (''i.e.,'' with the red circular dot).

For display purposes, all normalized masses <math>~(M_\mathrm{tot}/M_\mathrm{SWS})</math> have been further normalized to the maximum mass on the n = 3 sequence.
</div>