Editing SSC/Stability/n3PolytropeLAWE (section)

==Specific case of n = 3 Polytropes==
===Homologous Collapse===

If we examine only an <math>~n=3</math> polytropic configuration, then the last term disappears.  This means that ''in this very special case'', a perfectly valid solution to the LAWE is <math>~x = \mathrm{constant}</math>.  This is presumably the eigenfunction that Schwarzschild deduced; the fundamental-mode "oscillations" are perfectly homologous. Given that the model is marginally unstable, an ensuing dynamical collapse will presumably begin in a perfectly homologous fashion.  This is precisely the type of "free-fall" collapse that was discussed and modeled by [[Apps/GoldreichWeber80#Homologous_Solution|Goldreich &amp; Weber (1980)]].

===Another Potential Option===
We have wondered whether, in this very special case, one or more additional fundamental-mode eigenfunction(s) might satisfy the governing LAWE.  Here is a relevant line of arguments, beginning with the LAWE for the n = 3 polytropic sphere.
<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} + 4( 1 +  Q ) \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{dy}{d\xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 4( 1 +  Q ) \frac{y}{\xi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{4}\cdot \frac{d\ln y}{d\ln\xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- ( 1 +  Q ) \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~y \equiv \frac{dx}{d\xi} \, .</math>
</div>

But, by definition, the function <math>~Q(\xi)</math> is a logarithmic derivative of the Lane-Emden function.  Hence, we also can write,

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

<tr>
  <td align="right">
<math>~-1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{4}\cdot \frac{d\ln y}{d\ln\xi} + \frac{d\ln\theta}{d\ln\xi}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln (\theta y^{1/4})}{d\ln\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~- d\ln\xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ d\ln (\theta y^{1/4}) \, . </math>
  </td>
</tr>
</table>
</div>
Integrating this equation once gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln(\theta y^{1/4}) + \ln\xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln (c_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi \theta y^{1/4} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{dx}{d\xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_0}{\xi\theta}\biggr)^4 \, .</math>
  </td>
</tr>
</table>
</div>

Referring to the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicLaneEmden|power-series expansion of the polytropic Lane-Emden function]], <math>~\theta(\xi)</math>, about the configuration's center, we see that the product, <math>~\xi\theta</math>, goes to zero as the first power of <math>~\xi</math>.  This means that the right-hand side of this last differential equation blows up at the center.  This, therefore, does not appear to provide a physically viable avenue by which to identify an alternative fundamental-mode eigenfunction.