Editing SSC/FreeEnergy/Powerpoint (section)

====Detailed Force-Balance Models====
=====Structural Form Factors=====

The following table of structural form factors has been drawn from [[SSCpt1/Virial/FormFactors#PTtable|here]],

<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">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td 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>~ \biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </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(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1} 
</math>
  </td>
</tr>
</table>

</td>

<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>
</table>
</div>

and [[SSCpt1/Virial/FormFactors#Summary_.28n.3D5.29|here]],

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors for <font color="red">Pressure-Truncated</font> n = 5 Polytropes
</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>~  
( 1 + \ell^2 )^{-3/2}  
</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{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>~\tilde\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>
<tr>
<td align="left" colspan="3">
where, &nbsp;  &nbsp;  &nbsp; <math>~\ell \equiv \frac{\tilde\xi}{\sqrt{3}}</math>
</td>
</tr>
</table>

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

=====Case M Equilibrium Conditions=====
Employing the [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|renormalization factors]],

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

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n}\biggr]^{1/(n-3)} \, ,</math>
  </td>
</tr>

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

we find from detailed force-balance analyses that the [[SSC/Structure/PolytropesEmbedded#General_Properties|equilibrium radius and corresponding external pressure]] for "Case M" configurations are,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \cdot \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n}\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \, ,
</math>
  </td>
</tr>
</table>
</div>
which matches the expression derived in an [[SSC/Structure/Polytropes#Lane-Emden_Equation|ASIDE box found with our introduction of the Lane-Emden equation]], and
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{P_\mathrm{e}}{P_\mathrm{norm}} = \frac{P_\mathrm{e}}{P_\mathrm{Horedt}} \cdot \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, .
</math>
  </td>
</tr>
<!-- Next two equations are in a form suitable for the powerpoint presentation
<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n}\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
</math>
  </td>
</tr>

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

There are two turning points:  One associated with a maximum in <math>~P_e</math> and one associated with a maximum in <math>~R_\mathrm{eq}</math>.  According to [[SSC/Structure/PolytropesEmbedded#Turning_Points|Kimura's discussion]], the first of these occurs in the configuration for which,

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

<tr>
  <td align="right">
<math>~\frac{\tilde\theta^{n+1}}{(\tilde\theta^')^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{2} 
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
For n = 5, this occurs when <math>~\tilde\xi = 3</math>
  </td>
</tr>
</table>
</div>
This point along the equilibrium sequence is identified by the dark green circular dot in Figure 3, below.  The second occurs in the configuration for which,

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

<tr>
  <td align="right">
<math>~\frac{\tilde\xi \tilde\theta^{n}}{(-\tilde\theta^')} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{(n-1)} 
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
For n = 5, this occurs when <math>~\tilde\xi = \sqrt{15} \approx 3.87298</math>
  </td>
</tr>
</table>
</div>

This point along the equilibrium sequence is identified by the yellow circular dot in Figure 3, below.  In addition, we have identified the point of dynamical instability.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(n-3)}{3^2(n+1)} \cdot \frac{a}{c} = \frac{1}{3^3} \cdot \frac{3}{2^2\pi} \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr)^{-1} \frac{3}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^2\cdot 3 \cdot 5\pi} \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr)^{-1} \frac{5}{2^4} 
(1+\ell^2)^3 \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>~\Rightarrow ~~~ \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4</math> 
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^6\cdot 3 \cdot \pi \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>
</table>
</div>
But the equilibrium condition for n = 5 configurations is,
<div align="center">
<table border="0" cellpadding="3">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2 \cdot 3^3}{\pi}\biggr]^{3}
\tilde\theta_n^{6}\tilde\xi^{12}( - \tilde\theta' )^{6} 
\biggl\{ \frac{1}{6}\biggl[ \frac{\pi}{2\cdot 3^3}\biggr]^{1/2}
\tilde\xi^{-3} ( - \tilde\theta' )^{-2} \biggr\}^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^4\cdot 3^4}
\biggl[ \frac{2 \cdot 3^3}{\pi}\biggr]
\tilde\theta_n^{6} ( - \tilde\theta' )^{-2} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^3\pi} \cdot \ell^{-2}
</math>
  </td>
</tr>

</table>
</div>

Putting the two expressions  together gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{2^3\pi} \cdot \ell^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^6\cdot 3 \cdot \pi \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>~\Rightarrow ~~~2^3\cdot 3 \cdot \ell^{3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\ell^5 - \frac{8}{3}\ell^3 - \ell   + (1+\ell^2)^3 \tan^{-1}\ell 
</math>
  </td>
</tr>

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

<!-- Next equation in format suitable for powerpoint presentation
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{(1+\ell^2)^3}{\ell}\biggr] \tan^{-1}\ell   </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
1 +  \biggl[ \frac{2^4\cdot 5}{3} \biggr] \ell^{2} - \ell^4
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
For n = 5, this occurs when <math>~\tilde\xi \approx 3.85065</math>
  </td>
</tr>
</table>
</div>
-->
This agrees with the [[SSC/FreeEnergy/PolytropesEmbedded#Case_M|expression derived in a separate <font color="red">ASIDE</font>]]; as was pointed out in that context, the root of this equation is:  <math>~\ell \approx 2.223175</math>, that is, <math>~\tilde\xi \approx 3.85065</math>.  This point along the equilibrium sequence is identified by the red circular dot in Figure 3, below.  It is almost &#8212; but definitely not &#8212; coincident with the configuration along the sequence (marked by the yellow circular dot) that is associated with the minimum-radius turning point.

<div align="center" id="Figure3">
<table border="0" align="center"><tr><th align="center">
Figure 3: &nbsp; Case M Equilibrium Sequence 
</th></tr>
<tr><td align="center">
[[File:CaseMcompareTogether.png|700px|center|Case M pressure-truncated n=5 polytrope]]
</td></tr>
</table>
</div>


Now for a movie!
<div align="center">
<table border="1" align="center">
<tr>
  <td align="center">
[[File:CaseMfreeEnergy50.gif|750px|Case M movie of free-energy structures]]
  </td>
</tr>
</table>
</div>

=====Case P Equilibrium Conditions=====

The [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|equilibrium radius and corresponding configuration mass]] from a "Case P" analysis are,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \tilde\xi {\tilde\theta}_n^{(n-1)/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{M_\mathrm{tot}}{M_\mathrm{SWS} }
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n^3}{4\pi} \biggr)^{1/2}  {\tilde\theta}_n^{(n-3)/2} (-\tilde\xi^2 \tilde\theta^') \, .
</math>
  </td>
</tr>

</table>
</div>

According to our [[SSC/Structure/PolytropesEmbedded#Other_Limits|review of, especially, Kimura's work]], the turning point associated with <math>~M_\mathrm{max}</math> occurs where,

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

<tr>
  <td align="right">
<math>~\frac{\tilde\theta^{n+1}}{(\tilde\theta^')^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{2} 
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
For n = 5, this occurs when <math>~\tilde\xi = 3 \, .</math>
  </td>
</tr>
</table>
</div>
And a turning point associated with <math>~R_\mathrm{max}</math> occurs where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\tilde\xi (-\tilde\theta^')}{\tilde\theta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{(n-1)} \, .</math>
  </td>
</tr>
<!--  Next expression for PowerPoint presentation
<tr>
  <td align="right">
<math>~\frac{\tilde\xi (-\tilde\theta^')}{\tilde\theta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{(n-1)} </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
For n = 5, this occurs when <math>~\tilde\xi = \sqrt{3} </math>
  </td>
</tr>
-->
</table>
</div>

For <math>~n=5</math> configurations, this means,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tilde\xi \biggl( 1 + \frac{\tilde\xi^2}{3}\biggr)^{1/ 2} \frac{\tilde\xi}{3} \biggl( 1 + \frac{\tilde\xi^2}{3}\biggr)^{-3/ 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\ell^2}{(1+\ell^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (1+\ell^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\ell^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \ell^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \tilde\xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{3} \, .
</math>
  </td>
</tr>
</table>
</div>