Editing SSC/Structure/OtherAnalyticRamblings (section)

====Dabbling with Equilibrium Condition====
In the meantime, I've found it instructive to play with the first of these two expressions to see how it might be restructured in order to most directly confirm that it is satisfied by the expressions presented in [[SSC/Structure/OtherAnalyticModels#Examples|Table 1]].  Adopting the shorthand notation,
<div align="center">
<math>~\Gamma \equiv 4\pi G\rho_c \tau_\mathrm{SSC}^2</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\varpi \equiv \frac{\rho_0}{\rho_c} \, ,</math>
</div>
and multiplying the "equilibrium" relation through by <math>~(-\varpi p)</math>, we have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~\Gamma \varpi^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \varpi p^'\biggl\{
\frac{1}{\chi_0^2 p^'} \frac{d}{d\chi_0} (\chi_0^2 p^') - \frac{1}{\varpi}\frac{d\varpi}{d\chi_0}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p^' \varpi^' - \varpi \frac{dp^'}{d\chi_0}  -\frac{2\varpi p^'}{\chi_0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p^' }{\chi_0}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \varpi \frac{dp^'}{d\chi_0}  \, ;
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\Gamma \varpi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p^' }{\chi_0 \varpi}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \frac{dp^'}{d\chi_0}  \, .
</math>
  </td>
</tr>
</table>
</div>

=====Specific Cases=====

<font color="red">'''Case 1''' (Parabolic)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = 1 -\chi_0^2</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = -2\chi_0 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^' = -5\chi_0 + 8\chi_0^3 - 3\chi_0^5 = \chi_0(1-\chi_0^2)(-5+3\chi_0^2)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{p^'}{\chi_0 \varpi} = -5 +3\chi_0^2  \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
Also, note: 
  </td>
  <td align="left">
<math>~\frac{d(p^')}{d\chi_0} = -5 +24\chi_0^2 -15\chi_0^4 \, .</math>
  </td>
</tr>
</table>
</div>

For the parabolic case, therefore, the RHS of the "equilibrium" expression is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<font size="+1">RHS</font>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-5+3\chi_0^2)\biggl[ -2\chi_0^2 - 2(1-\chi_0^2) \biggr] - (-5 +24\chi_0^2 -15\chi_0^4)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(10 - 6\chi_0^2) + (5 -24\chi_0^2 +15\chi_0^4)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~15(1-2\chi_0^2+\chi_0^4) \, ,
</math>
  </td>
</tr>
</table>
</div>
which, indeed, matches the LHS of the "equilibrium" relation, if,
<div align="center">
<math>~\Gamma = 15</math> 
&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
<math>~\tau_\mathrm{SSC}^2 = \frac{15}{4\pi G \rho_c} \, .</math>
</div>
This has all worked satisfactorily because, [[SSC/Structure/OtherAnalyticModels#Stabililty_2|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the parabolic density distribution.


<font color="red">'''Case 2''' (Linear)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = 1 -\chi_0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = -1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^' = \tfrac{12}{5}[- 4\chi_0 + 7\chi_0^2 - 3\chi_0^3]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{p^'}{\chi_0 \varpi} = \tfrac{12}{5}(-4 +3\chi_0)  \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
Also, note: 
  </td>
  <td align="left">
<math>~\frac{d(p^')}{d\chi_0} = \tfrac{12}{5}[- 4 + 14\chi_0 - 9\chi_0^2] \, .</math>
  </td>
</tr>
</table>
</div>

For the linear case, therefore, the RHS of the "equilibrium" expression is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<font size="+1">RHS</font>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{12}{5}(-4 +3\chi_0)\biggl[ -\chi_0  - 2(1-\chi_0) \biggr] - \tfrac{12}{5}(- 4 + 14\chi_0 - 9\chi_0^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{12}{5}\biggl[ (4 -3\chi_0)( 2-\chi_0 ) + (4 - 14\chi_0 + 9\chi_0^2) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{12}{5}(12-24\chi_0 + 12\chi_0^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{2^4\cdot 3^2}{5}(1-2\chi_0 + \chi_0^2) \, ,
</math>
  </td>
</tr>
</table>
</div>
which, indeed, matches the LHS of the "equilibrium" relation, if,
<div align="center">
<math>~\Gamma = \frac{2^4\cdot 3^2}{5}</math> 
&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
<math>~\tau_\mathrm{SSC}^2 = \frac{2^2\cdot 3^2}{5\pi G \rho_c} \, .</math>
</div>
This has all worked satisfactorily because, [[SSC/Structure/OtherAnalyticModels#Lagrangian_Approach|as presented above]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of the linear density distribution.



<font color="red">'''Case 3''' (n = 1 polytrope)</font>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = \frac{\sin(\pi\chi_0)}{\pi\chi_0}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi^' = \frac{\cos(\pi\chi_0)}{\chi_0} -  \frac{\sin(\pi\chi_0)}{\pi\chi_0^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^' = \frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{p^'}{\chi_0 \varpi} = \frac{2}{(\pi\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]  \, .</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
Also, note: &nbsp; &nbsp; &nbsp;<math>~\frac{d(p^')}{d\chi_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2\pi \cdot \cos(\pi\chi_0)}{(\pi^2\chi_0^3)} - \frac{6\sin(\pi\chi_0)}{(\pi^2\chi_0^4)}  \biggr] \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr] 
+\frac{2\sin(\pi\chi_0)}{(\pi^2\chi_0^3)} \biggl[ \pi\cos(\pi\chi_0) - \pi^2\chi_0 \sin(\pi\chi_0) - \pi \cos(\pi\chi_0) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\pi^2 \chi_0^4} \biggl\{
\biggl[ 2\pi \chi_0\cdot \cos(\pi\chi_0) - 6\sin(\pi\chi_0)  \biggr] \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr] 
+2 \chi_0 \sin(\pi\chi_0) \biggl[ - \pi^2\chi_0 \sin(\pi\chi_0) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\pi^2 \chi_0^4} \biggl\{
2\pi^2\chi_0^2\cos^2(\pi\chi_0) -8\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+6\sin^2(\pi\chi_0)
- 2 \pi^2 \chi_0^2 \sin^2(\pi\chi_0)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{2}{\pi^2 \chi_0^4} \biggl\{3\sin^2(\pi\chi_0)
-4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+\pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

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

For the case of an n = 1 polytropic configuration, therefore, the equilibrium requirement is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~\Gamma \varpi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{p^' }{\chi_0 \varpi}\biggl[ \chi_0 \varpi^' - 2\varpi \biggr] - \frac{dp^'}{d\chi_0}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi\chi_0^3)} \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]\biggl[ \cos(\pi\chi_0) -  \frac{\sin(\pi\chi_0)}{\pi\chi_0} - \frac{2\sin(\pi\chi_0)}{\pi\chi_0} \biggr]   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{2}{\pi^2 \chi_0^4} \biggl\{3\sin^2(\pi\chi_0)
-4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0)+\pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi^2\chi_0^4)} \biggl\{ \biggl[ \pi\chi_0 \cos(\pi\chi_0) - \sin(\pi\chi_0) \biggr]\biggl[\pi\chi_0 \cos(\pi\chi_0) -  3\sin(\pi\chi_0) \biggr]   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-3\sin^2(\pi\chi_0) + 4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0) - \pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi^2\chi_0^4)} \biggl\{3\sin^2(\pi\chi_0) - 4\pi\chi_0\sin(\pi\chi_0)\cos(\pi\chi_0) + (\pi\chi_0)^2 \biggl[1-\sin^2(\pi\chi_0) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-3\sin^2(\pi\chi_0) + 4\pi\chi_0 \sin(\pi\chi_0) \cos(\pi\chi_0) - \pi^2\chi_0^2\biggl[1  - 2\sin^2(\pi\chi_0)  \biggr]
\biggr\}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\pi^2\chi_0^4)} \biggl\{ (\pi\chi_0)^2 \sin^2(\pi\chi_0)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 \biggl[\frac{\sin(\pi\chi_0)}{\pi\chi_0}  \biggr]^2
</math>
  </td>
</tr>
</table>
</div>
So, the equilibrium condition is satisfied if,
<div align="center">
<math>~\Gamma = 2\pi^2</math> 
&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
<math>~\tau_\mathrm{SSC}^2 = \frac{2\pi^2}{4\pi G \rho_c} = \frac{\pi}{2G \rho_c}  \, .</math>
</div>
This has all worked satisfactorily because, [[SSC/Stability/Polytropes#Setup|as presented in a separate chapter discussion]], this is the correct value of <math>~\tau_\mathrm{SSC}^2</math> in the case of an n = 1 polytropic configuration.