Editing SSC/Stability/UniformDensity (section)

===The Approach Taken by Sterne (1937)===
{{ Sterne37 }} begins his analysis by deriving the

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>
in a manner explicitly designed to reproduce [[SSC/Perturbations#Eddington_.281926.29|Eddington's ''pulsation equation'']] &#8212; it appears as equation (1.8) in {{ Sterne37hereafter }} &#8212; and, along with it, the [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations|surface boundary condition]],

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

<tr>
  <td align="right" >
<math>r_0 \frac{d\ln x}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, ,</math>
  </td>
</tr>
</table>
</div>
which appears in {{ Sterne37hereafter }} as equation (1.9).  Then, as shown in the following paragraph extracted directly from his paper, {{ Sterne37hereafter }} rewrites both of these expressions in, what he considers to be, "more convenient forms."


<div align="center">
<table border="2" cellpadding="10" width="80%">
<tr>
  <td align="center" colspan="1">
Reprint of the last paragraph of &sect;1 (p. 585) from &hellip;<br />
{{ Sterne37figure }}
  </td>
<tr>
  <td colspan="1">
<!-- [[File:Sterne1937B.png|600px|center|Sterne (1937)]] -->
<font color="darkgreen">
"Before proceeding further we write equations (1.8) and (1.9) in more convenient forms.  Let <math>\xi_0 = Rx</math>.  Then (1.8) becomes
<table border="0" cellpadding="5" align="center" width="100%">

<tr>
  <td align="center" >
<math>\xi_1^{''} + \frac{4-\mu}{x} \xi_1^' 
+ \frac{R\rho_0}{P_0}\biggl( \frac{n^2 R}{\gamma} - \frac{\alpha g_0}{x} \biggr)\xi_1 = 0 \, ;</math>
  </td>
  <td align="right" width="10%">
(1.91)
  </td>
</tr>
</table>
and the boundary condition (1.9) becomes, from (1.5),
<table border="0" cellpadding="5" align="center" width="100%">

<tr>
  <td align="center" >
<math>\biggl( \frac{n^2 R}{\gamma g_0} - \alpha \biggr) \xi_1 = \xi_1^'</math>
  </td>
  <td align="right" width="10%">
(1.92)
  </td>
</tr>
</table>
at <math>x = 1</math>.  Here dashes denote differentiation with respect to <math>x</math>, a convention henceforth to be followed."
</font>
  </td>
</tr>
<tr><td align="center">
 
  <table border="0" width="75%" cellpadding="4">
  <tr>
  <td align="center" colspan="3">
'''Notation:'''
  </td>
  </tr>
  <tr>
<th align="center" width="40%">Sterne's<p></p>
----</th>
<td width="20%">&nbsp;</td>
<th align="center">Ours<p></p>
----</th>
  </tr>
  <tr>
<td align="center" width="40%"><math>~\xi_0 = Rx</math></td>
<td>&nbsp;</td>
<td align="center"><math>~r_0</math></td>
  </tr>
  <tr>
<td align="center" width="40%"><math>~\xi_1</math></td>
<td>&nbsp;</td>
<td align="center"><math>~x</math></td>
  </tr>
  <tr>
<td align="center" width="40%"><math>~n^2</math></td>
<td>&nbsp;</td>
<td align="center"><math>~\omega^2</math></td>
  </tr>
  <tr>
<td align="center" width="40%"><math>~\alpha</math></td>
<td>&nbsp;</td>
<td align="center"><math>~3-4/\gamma_g</math></td>
  </tr>
  <tr>
<td align="center" width="40%"><math>~\mu</math></td>
<td>&nbsp;</td>
<td align="center"><math>~g_0 \rho_0 r_0/P_0</math></td>
  </tr>
  <tr>
<td align="center" width="40%"><math>~g_0 R^2</math></td>
<td>&nbsp;</td>
<td align="center"><math>~GM_\mathrm{tot}</math></td>
  </tr>
  </table>

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