Editing SSC/Structure/OtherAnalyticRamblings (section)

===Polytropic Configurations===
Let's compare this presentation of the LAWE to the form of the LAWE that has been derived [[SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation|specifically for polytropic equilibrium configurations]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + 
\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta} - 
\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot \frac{(n+1)V(x)}{\xi^2} \biggr]  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~V(\xi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{\xi}{(\theta/\theta_c)} \frac{d (\theta/\theta_c)}{d\xi} = \frac{g_0}{a_n}\biggl(\frac{a_n^2\rho_0}{P_0}\biggr)\frac{\xi}{(n+1)} \, .</math>
  </td>
</tr>
</table>
</div>
[Note that <math>~\theta_c = 1</math> and, therefore for all practical purposes, it can be dropped.  This notation was introduced in our [[SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation|separate discussion of the polytropic LAWE]] in order to make it clear how our derivations have overlapped earlier published work.]  Regrouping terms, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} - 
\biggl[ \frac{\alpha (n+1)V(x)}{\xi^2} \biggr]  x</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\xi^4}\frac{d}{d\xi}\biggl(\xi^4 \frac{dx}{d\xi}\biggr) -\biggl[\frac{(n+1)V(\xi)}{\xi} \biggr] 
\biggl[\frac{1}{\xi^\alpha}\frac{d}{d\xi} \biggl(\xi^\alpha x\biggr)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\xi^4}\frac{d}{d\xi}\biggl(\xi^4 \frac{dx}{d\xi}\biggr) +\biggl[(n+1)\frac{d\ln(\theta/\theta_c)}{d\xi} \biggr] 
\biggl[\frac{1}{\xi^\alpha}\frac{d}{d\xi} \biggl(\xi^\alpha x\biggr)\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Next we note that, written in terms of the traditional polytropic radial coordinate, <math>~\xi</math>, the fractional radius,
<div align="center">
<math>~\chi_0 \equiv \frac{r_0}{R} = \frac{\xi}{\xi_1} = \frac{a_n \xi}{R} \, .</math>
</div>
Hence, multiplying the polytropic LAWE through by the quantity, <math>~(R/a_n)^2</math>, gives,

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

<tr>
  <td align="right">
<math>~\frac{1}{\chi_0^4}\frac{d}{d\chi_0}\biggl(\chi_0^4 \frac{dx}{d\chi_0}\biggr) +\biggl[(n+1)\frac{d\ln(\theta/\theta_c)}{d\chi_0} \biggr] 
\biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0} \biggl(\chi_0^\alpha x\biggr)\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[\omega^2 \biggl(\frac{R^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta}\biggr]x</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl(\frac{\theta_c}{\theta}\biggr)\sigma^2 x \, .</math>
  </td>
</tr>
</table>
</div>

Finally, noting that, for polytropic configurations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\theta}{\theta_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{P_0}{P_c} \biggr)\biggl( \frac{\rho_0}{\rho_c} \biggr)^{-1} 
= \biggl( \frac{P_0}{P_c} \biggr)^{1/(n+1)} \, ,
</math>
  </td>
</tr>
</table>
</div>
we can rewrite the polytropic LAWE in the form,

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

<tr>
  <td align="right">
<math>~\frac{1}{\chi_0^4}\frac{d}{d\chi_0}\biggl(\chi_0^4 \frac{dx}{d\chi_0}\biggr) +\biggl[\frac{d\ln(P_0/P_c)}{d\chi_0} \biggr] 
\biggl[\frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0} \biggl(\chi_0^\alpha x\biggr)\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( \frac{P_0}{P_c} \biggr)^{-1}\biggl( \frac{\rho_0}{\rho_c} \biggr)\sigma^2 x \, ,</math>
  </td>
</tr>
</table>
</div>
which precisely matches the general expression for the LAWE presented at the end of our [[SSC/Structure/OtherAnalyticModels#Generic_Setup|generic setup, directly above]].  

This seems to be a particularly insightful way to write the LAWE, as the only structural functions that appear explicitly are <math>~P_0(\chi_0)</math> and <math>~\rho_0(\chi_0)</math>.  It appears as though the eigenfunctions that describe ''adiabatic'' radial pulsations do not explicitly depend ''a priori'' on the radial dependence of the equilibrium gravitational acceleration.