Editing SSC/Stability/n1PolytropeLAWE (section)

==Setup==
From our derived [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|structure of an n = 1 polytrope]], in terms of the configuration's radius <math>R</math> and mass <math>M</math>, the central pressure and density are, respectively,
<div align="center">
<math>P_c = \frac{\pi G}{8}\biggl( \frac{M^2}{R^4} \biggr) </math> , 
</div>
and
<div align="center">
<math>\rho_c = \frac{\pi M}{4 R^3} </math> .
</div>
Hence the characteristic time and acceleration are, respectively,
<div align="center">
<math> 
\tau_\mathrm{SSC} = \biggl[ \frac{R^2 \rho_c}{P_c} \biggr]^{1/2} = 
\biggl[ \frac{2R^3 }{GM} \biggr]^{1/2} = 
\biggl[ \frac{\pi}{2 G\rho_c} \biggr]^{1/2},
</math><br />
</div>
and,
<div align="center">
<math> 
g_\mathrm{SSC} = \frac{P_c}{R \rho_c} = \biggl( \frac{GM}{2R^2} \biggr) .
</math><br />
</div>

The required functions are,
* <font color="red">Density</font>: 
<div align="center">
<math>\frac{\rho_0(\chi_0)}{\rho_c} = \frac{\sin(\pi\chi_0)}{\pi\chi_0} </math> ;
</div>

* <font color="red">Pressure</font>: 
<div align="center">
<math>\frac{P_0(\chi_0)}{P_c} = \biggl[ \frac{\sin(\pi\chi_0)}{\pi\chi_0} \biggr]^2 </math> ;
</div>

* <font color="red">Gravitational acceleration</font>: 
<div align="center">
<math> 
\frac{g_0(r_0)}{g_\mathrm{SSC}} = \frac{2}{\chi_0^2} \biggl[ \frac{M_r(\chi_0)}{M}\biggr]  =
\frac{2}{\pi \chi_0^2} \biggl[ \sin (\pi\chi_0 ) - \pi\chi_0 \cos (\pi\chi_0 ) \biggr].
</math><br />
</div>

So our desired Eigenvalues and Eigenvectors will be solutions to the following ODE:

<div align="center">
<math>
\frac{d^2x}{d\chi_0^2} + \frac{2}{\chi_0} \biggl[ 1 +  \pi\chi_0 \cot (\pi\chi_0 ) \biggr]  \frac{dx}{d\chi_0} + \frac{1}{\gamma_\mathrm{g}} \biggl\{ \frac{\pi \chi_0}{\sin(\pi\chi_0)} \biggl[ \frac{\pi \omega^2}{2G\rho_c} \biggr] + \frac{2}{\chi_0^2 } (4 - 3\gamma_\mathrm{g}) \biggl[ 1 - \pi\chi_0 \cot (\pi\chi_0 ) \biggr] \biggr\}  x = 0 ,
</math><br />
</div>
<br />
or, replacing <math>\chi_0</math> with <math>\xi \equiv \pi\chi_0</math> and dividing the entire expression by <math>\pi^2</math>, we have,

<div align="center">
<math>
\frac{d^2x}{d\xi^2} + \frac{2}{\xi} \biggl[ 1 +  \xi \cot \xi \biggr]  \frac{dx}{d\xi} + \frac{1}{\gamma_\mathrm{g}} \biggl\{ \frac{\xi}{\sin \xi} \biggl[ \frac{\omega^2}{2\pi G\rho_c} \biggr] + \frac{2}{\xi^2 } (4 - 3\gamma_\mathrm{g}) \biggl[ 1 - \xi \cot \xi \biggr] \biggr\}  x = 0 .
</math><br />
</div>
<br />

This is identical to the formulation of the wave equation that is relevant to  the (n = 1) core of the composite polytrope studied by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy &amp; R. Fiedler (1985b)]; for comparison, their expression is displayed, in the following boxed-in image.

<div align="center">
<table border="2" cellpadding="10" id="MurphyFiedler1985b">
<tr>
  <td align="center">
n = 1 Polytropic Formulation of the LAWE as Presented by &hellip;

{{ MF85bfigure }}
<!--[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] -->
  </td>
<tr>
  <td>
<!-- [[File:MurphyFiedlerN1formulation.png|700px|center|Murphy &amp; Fiedler (1985b)]] -->
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center">
<math>
\frac{d^2\eta}{d\zeta^2} 
+ \frac{1}{\zeta}\biggl[ 4 + \frac{2(\zeta \cos\zeta - \sin\zeta)}{\sin\zeta}\biggr]\frac{d\eta}{d\zeta}
+ \biggl[ \frac{\omega_k^2 \zeta}{\sin\zeta} + \frac{2\alpha^* (\zeta \cos\zeta - \sin\zeta)}{\zeta^2\sin\zeta} \biggr]\eta 
= 0
</math>
  </td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

{{ SGFworkInProgress }}

<span id="WorkInProgress">&nbsp;</span>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
From an [[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|accompanying discussion]], we find the,
<div align="center">
<font color="maroon"><b>Polytropic LAWE (linear adiabatic wave equation)</b></font><br />
{{ Math/EQ_RadialPulsation02 }}
</div>
For an isolated n = 1 <math>(\gamma_g = 2, \alpha = 1)</math> polytrope, we know that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\theta</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>\frac{\sin\xi}{\xi}</math>
  </td>
  <td align="right">
&nbsp; &nbsp; <math>\Rightarrow ~~~ Q(\xi) \equiv - \frac{d \ln \theta}{d\ln \xi}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>\biggl[1 - \xi\cot\xi\biggr] \, .</math>
  </td>
</tr>
</table>
Hence, the relevant LAWE is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{d^2 x}{d\xi^2} 
+ 
\biggl[4 - 2(1 - \xi\cot\xi)\biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}
+
2 \biggl[\biggl( \frac{\sigma_c^2}{12}\biggr)\frac{\xi^3}{\sin\xi} - (1 - \xi\cot\xi)  \biggr]\frac{x}{\xi^2}
</math>
  </td>
</tr>
</table>

<div align="center">
<font color="maroon"><b>LAWE for n = 1 Polytrope</b></font><br />
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\frac{d^2 x}{d\xi^2} 
+ 
\frac{2}{\xi} \biggl[1 +  \xi\cot\xi\biggr] \frac{dx}{d\xi}
+
\frac{1}{2} \biggl[\biggl( \frac{\sigma_c^2}{3}\biggr)\frac{\xi}{\sin\xi} - \frac{4}{\xi^2} \biggl(1 - \xi\cot\xi \biggr)  \biggr]x
</math>
  </td>
</tr>
</table>
</div>
This matches precisely the expression derived immediately above.

[[SSC/Stability/Polytropes#Boundary_Conditions|Surface boundary condition]]:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>- \frac{d\ln x}{d\ln \xi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\biggl( \frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)}
\biggl[ \frac{\xi}{\theta'}\biggr]_\mathrm{surf}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ - \frac{d\ln x}{d\ln \xi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
1 + \frac{\sigma_c^2}{12}
\biggl[ \frac{\xi^3}{(\xi \cos\xi - \sin\xi)}\biggr]_{\xi=\pi}
=
1 - \frac{\pi^2 \sigma_c^2}{12}
</math>
  </td>
</tr>
</table>

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