Editing SSC/Stability/n1PolytropeLAWE (section)

====Rationale====
From our [[SSC/Structure/Polytropes#.3D_1_Polytrope|review of the properties of <math>~n=1</math> polytropic spheres]], we know that the equilibrium density distribution is given by the sinc function, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sin\xi}{\xi} \, ,</math>
  </td>
</tr>

</table>
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where,
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<math>~\xi \equiv \pi \biggl(\frac{r_0}{R_0} \biggr) \, .</math>
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The total mass is,
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<math>~M_\mathrm{tot} =  \frac{4}{\pi} \cdot \rho_c R_0^3 \, ,</math>
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and the fractional mass enclosed within a given radius, <math>~r</math>, is,

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

<tr>
  <td align="right">
<math>~\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi} [\sin\xi - \xi \cos\xi] \, .</math>
  </td>
</tr>

</table>
</div>
Let's guess that, during the fundamental mode of radial oscillation, the sinc-function profile is preserved as the system's total radius varies.  In particular, we will assume that the system's time-varying radius is,
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<math>R = R_0 \biggl( 1 + \frac{\delta R}{R_0} \biggr) = R_0 ( 1 + \epsilon_R) \, ,</math>
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and seek to determine how the displacement vector, <math>~\epsilon \equiv \delta r/r_0</math>, varies with <math>~r_0</math> in order to preserve the overall sinc-function profile.  As is usual, we will only examine small perturbations away from equilibrium, that is, we will assume that everywhere throughout the configuration, <math>~|\epsilon| \ll 1 </math>.


Let's begin by defining a new dimensionless coordinate,
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<math>~\eta \equiv \pi \biggl(\frac{r}{R} \biggr) = \pi \biggl[\frac{r_0(1+\epsilon)}{R_0(1+\epsilon_R)} \biggr] 
\approx \xi (1 + \epsilon) \, ,</math>
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and recognize that, in the new perturbed state, the fractional mass enclosed within a given radius, <math>~r</math>, is,

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

<tr>
  <td align="right">
<math>~\frac{M_r(\eta)}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi} [\sin\eta - \eta \cos\eta] \, .</math>
  </td>
</tr>

</table>
</div>
In order to associate each mass shell in the perturbed configuration with its corresponding mass shell in the unperturbed, equilibrium state, we need to set the two <math>~M_r</math> functions equal to one another, that is, demand that,

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

<tr>
  <td align="right">
<math>~\sin\xi - \xi \cos\xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin\eta - \eta \cos\eta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\sin[\xi(1+\epsilon)] - \xi(1+\epsilon) \cos[\xi(1+\epsilon)]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \sin\xi \cos(\xi\epsilon) + \cos\xi \sin (\xi\epsilon) \biggr] 
- \xi(1+\epsilon) \biggl[ \cos\xi \cos(\xi\epsilon) - \sin\xi \sin (\xi\epsilon) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\sin\xi \biggl[1 - \frac{1}{2}(\xi\epsilon)^2 \biggr] + (\xi\epsilon)\cos\xi 
- \xi(1+\epsilon) \cos\xi \biggl[1 - \frac{1}{2}(\xi\epsilon)^2 \biggr]  + \xi^2 \epsilon(1+\epsilon) \sin\xi </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>
~\sin\xi  -\xi\cos\xi - \frac{1}{2}(\xi\epsilon)^2 \sin\xi  + (\xi\epsilon)\cos\xi 
- (\xi \epsilon) \cos\xi  + \frac{1}{2} \xi^3 \epsilon^2\cos\xi
+ \xi^2 \epsilon \sin\xi + (\xi \epsilon)^2\sin\xi  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~- \xi^2 \epsilon \sin\xi </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{(\xi \epsilon)^2}{2} \biggl[\xi \cos\xi+ \sin\xi  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{1}{\epsilon}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{2} \biggl[\xi \cdot \frac{\cos\xi}{\sin\xi} + 1 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\epsilon</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~-
2\biggl[1 + \xi \cdot \frac{\cos\xi}{\sin\xi} \biggr]^{-1}
= - 2\sin\xi \biggl[\sin\xi + \xi \cos\xi \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>