Editing SSC/Perturbations (section)

===Review Article by Cox (1974)===
In an excellent review of "Pulsating Stars", [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974, Reports on Progress in Physics, 37, 563)] presents a full derivation of, what he refers to as, the ''adiabatic wave equation''.  It appears as equation (5.34) in the form displayed in the following boxed-in image:

<div align="center">
<table border="1" cellpadding="5" width="75%">
<tr>
  <th align="center">
''Adiabatic Wave Equation'' extracted<sup>&dagger;</sup> from [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974)]<p></p>
"''Pulsating Stars''"<p></p>
Reports on Progress in Physics, vol. 37, pp. 563 - 698 &copy; [http://iopscience.iop.org/ IOP Publishing]
  </th>
<tr>
  <td>
[[File:Christy1966Eq5.png|600px|center|Cox (1974)]]
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Equation displayed here, as a single digital image, exactly as it appears in the original publication.</td></tr>
</table>
</div>

Recognizing the correspondence with our derived expression requires, first, switching the independent variable from <math>~m</math> to <math>~r_0</math> via the relation [[#Consistent_Lagrangian_Formulation|identified, above]], namely,

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

<tr>
  <td align="right">
<math>~
\frac{d}{dm}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{4\pi \rho_0 r_0^2} \cdot \frac{d}{dr_0}
\, ,
</math>
  </td>
</tr>
</table>
</div>
to obtain,

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

<tr>
  <td align="right">
<math>~
\sigma^2 \xi 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{(4\pi \rho_0 r_0^4)}\frac{d}{dr_0}\biggl[ (4\pi \Gamma_1 P_0 r_0^4) \frac{d\xi}{dr_0} \biggr] - 
\frac{1}{\rho_0 r_0} \biggl\{ \frac{d}{dr_0} [ (3\Gamma_1-4)P_0] \biggr\} \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\Gamma_1 P_0 }{\rho_0 }\frac{d^2 \xi}{dr_0^2} 
- \frac{\Gamma_1 }{(\rho_0 r_0^4)} \biggl[4P_0 r_0^3  + r_0^4\frac{dP_0}{dr_0} \biggr]  \frac{d\xi}{dr_0} 
+ \frac{(4-3\Gamma_1)}{\rho_0 r_0} \biggl[ \frac{dP_0}{dr_0} \biggr] \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\Gamma_1 P_0 }{\rho_0 }\biggl\{ \frac{d^2 \xi}{dr_0^2} 
+ \biggl[\frac{4}{r_0} + \frac{1}{P_0}\frac{dP_0}{dr_0} \biggr]  \frac{d\xi}{dr_0} \biggr\}
- \biggl\{ \sigma^2 - \frac{(4-3\Gamma_1)}{\rho_0 r_0} \biggl[ \frac{dP_0}{dr_0} \biggr] \biggr\} \xi \, .
</math> 
  </td>
</tr>
</table>
</div>

Then, using the definition of <math>~g_0</math>, also [[#Euler_.2B_Poisson_Equations|as provided above]], to facilitate the substitution,
<div align="center">
<math>~\frac{dP_0}{dr_0} \rightarrow - g_0 \rho_0 \, ,</math>
</div>
gives,

<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^2 \xi}{dr_0^2} 
+ \biggl[\frac{4}{r_0} - \frac{g_0 \rho_0}{P_0}\biggr]  \frac{d\xi}{dr_0}
+ \frac{\rho_0 }{\Gamma_1 P_0 }\biggl\{ \sigma^2 + \frac{(4-3\Gamma_1)g_0}{r_0} \biggr\} \xi \, ,
</math> 
  </td>
</tr>
</table>
</div>

which, apart from the adoption of different variable names, exactly matches our derived expression.