Editing SSC/Stability/Isothermal (section)

===Playing Around===
Our own analysis of this problem decidedly validates the results published by earlier authors.  In particular, it appears as though the {{ Yabushita688hereafter }} conjecture is correct; specifically, a careful LAWE analysis seems to indicate that &#8212; if we consider only isothermal fluctuations, that is, <math>\gamma_\mathrm{g}=1</math> &#8212; the dynamical instability sets in at the point along the equilibrium sequence where the pressure maximum arises.  Let's see if we can put this conjecture on an even firmer foundation by manipulating the LAWE analytically.

When <math>\gamma_\mathrm{g} = 1</math>, the LAWE for pressure-truncated isothermal spheres 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>
\frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+  \biggl[\frac{\sigma_c^2}{6} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr]  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{d\ln x}{d\ln\xi} 
+  \xi^2\biggl[\frac{\sigma_c^2}{6} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{d\ln \xi}\biggl[ \frac{d\ln x}{d\ln\xi}\biggr] + \biggl[\frac{d\ln x}{d\ln\xi}-1 \biggr] \frac{d\ln x}{d\ln\xi}
+ \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{d\ln x}{d\ln\xi} 
+  \xi^2\biggl[\frac{\sigma_c^2}{6} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{dy_\xi}{d\ln \xi} + \biggl[y_\xi-1 \biggr] y_\xi
+ \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] y_\xi 
+  \xi^2\biggl[\frac{\sigma_c^2}{6} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, we have introduced the shorthand notation,
<div align="center">
<math>y_\xi \equiv \frac{d\ln x}{d\ln\xi} \, .</math>
</div>

Now, precisely ''at'' the onset of dynamical instability, we should find that, <math>\sigma_c^2 = 0</math>.  We also know, from the work of {{ Bonnor56 }}, that the equilibrium configuration at the pressure maximum is identified by the model whose Lane-Emden function ''at its surface'' exhibits the property,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{d\psi}{d\xi}\biggr)_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ 2^{1 / 2} e^{-\psi/2} \biggr]_\mathrm{surf} \, .</math>
  </td>
</tr>
</table>
</div>

Therefore, ''at the surface'' the condition expected from the LAWE 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\{
\xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi-1 ) y_\xi
+ \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] y_\xi 
+  \xi \biggl( \frac{d\psi}{d\xi} \biggr) 
\biggr\}_\mathrm{surf}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi + 3 ) y_\xi 
+  \xi (1-y_\xi)\biggl( \frac{d\psi}{d\xi} \biggr) 
\biggr\}_\mathrm{surf}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi + 3 ) y_\xi 
+  2^{1 / 2}\xi (1-y_\xi)e^{-\psi/2} 
\biggr\}_\mathrm{surf} \, .
</math>
  </td>
</tr>
</table>
</div>

Let's see what this implies if, at the surface of the configuration, the logarithmic derivative of the fundamental-mode eigenfunction has the value,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>[y_\xi]_\mathrm{surf} = \frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-3 \, .</math>
  </td>
</tr>
</table>
</div>

In this case, we have,

<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{dy_\xi}{d\xi}  
+  2^{5 / 2}e^{-\psi/2} 
\biggr\}_\mathrm{surf} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\frac{dy_\xi}{d\xi}  
+  4 \cdot \frac{d\psi}{d\xi} 
\biggr\}_\mathrm{surf} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\frac{d}{d\xi}  \biggl[y_\xi + 4\psi \biggr]
\biggr\}_\mathrm{surf} \, ,
</math>
  </td>
</tr>
</table>
</div>
<!--
Not sure what to make of this!
-->

which means that, at the surface,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>y_\xi + 4\psi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>b_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d\ln x}{d\ln\xi} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>b_0 - 4\psi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ b_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\psi_\mathrm{surf} - 3 \, .</math>
  </td>
</tr>
</table>
</div>
[NOTE:  We have deduced empirically that, <math>b_0 \approx 7.568796</math>.]  

<!--  INCORRECT TO ASSUME THAT THE "SURFACE CONDITION" RELATION CAN BE DIFFERENTIATED OR INTEGRATED ...
Let's go back to Bonnor's condition at the surface and see if we can integrate that relation once.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{\psi/2}d\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1 / 2} d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{2} e^{\psi/2}d\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2} d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ e^{\psi/2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2} \xi + a_0</math>
  </td>
</tr>
</table>
</div>
[NOTE:  We have deduced empirically that, <math>~a_0 \approx -0.8140441</math>.]  

Returning to the [[SSC/Structure/IsothermalSphere#Chandrasekhar|isothermal Lane-Emden equation]], we know as well that,
<div align="center">
{{ Math/EQ_SSLaneEmden02 }}
</div>
So, when combined with the desired condition,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{d\psi}{d\xi}\biggr)_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 2^{1 / 2} e^{-\psi/2} \biggr]_\mathrm{surf} \, ,</math>
  </td>
</tr>
</table>
</div>
we expect that, at the surface,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi^2 e^{-\psi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1 / 2} \cdot \frac{d}{d\xi} \biggl[ \xi^2 e^{-\psi/2} \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1 / 2} \biggl[ 2\xi e^{-\psi/2} - \frac{\xi^2}{2} e^{-\psi/2} \frac{d\psi}{d\xi}\biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 2^{3 / 2}\xi e^{-\psi/2} - \xi^2 e^{-\psi} \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 2 e^{-\psi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2^{3 / 2}}{\xi} e^{-\psi/2} \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ e^{-\psi/2} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2^{1 / 2}}{\xi} \biggr]_\mathrm{surf} \, .</math>
  </td>
</tr>
</table>
</div>
Given this relation, we can derive expressions for both of the above integration constants &#8212; <math>~a_0</math> and <math>~b_0</math> &#8212; in terms of <math>~\xi_\mathrm{surf}</math>.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[e^{\psi/2} - \frac{\xi}{\sqrt{2}}  \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\sqrt{2}\biggl(\frac{\xi}{4-\xi}\biggr) - \frac{\xi}{\sqrt{2}}  \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2}\biggl[ \frac{2\xi - \xi(4-\xi)}{4-\xi}  \biggr]_\mathrm{surf}</math>
  </td>
</tr>
</table>
</div>
END DELETION DUE TO IMPROPER DIFFERENTIATION AND INTEGRATION -->