Editing SSC/Stability/NeutralMode (section)

==Equilibrium Structure==

In an article titled, "Radial Oscillations of a Stellar Model," [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P C. Prasad (1949, MNRAS, 109, 103)] investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,
<div align="center">
<math>\rho_0 = \rho_c\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr] \, ,</math>
</div>
where, <math>\rho_c</math> is the central density and, <math>R</math> is the radius of the star.  Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{r_0} 4\pi r_0^2 \rho_0 dr_0</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi\rho_c r_0^3}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
in which case we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_0 \equiv \frac{G M_r }{r_0^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi G \rho_c r_0}{3}  
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2\biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Delta \equiv \frac{M_r }{4\pi r_0^3\rho_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1 }{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1}
\, .</math>
  </td>
</tr>
</table>

Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P Prasad (1949)] determines that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi G\rho_c^2 R^2}{15} 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\, ,</math>
  </td>
</tr>
</table>
</div>
where, it can readily be deduced, as well, that the central pressure is,
<div align="center">
<math>P_c = \frac{4\pi}{15} G\rho_c^2 R^2 \, .</math>
</div>

<table border="1" width="90%" cellpadding="8" align="center"><tr><td align="left">
<div align="center">'''Specific Entropy Distribution'''</div>


For purposes of later discussion, we find from [[Appendix/Ramblings/PatrickMotl#Tying_Expressions_into_H_Book_Context|a separate examination of specific entropy distributions]], <math>s_0(r_0)</math>, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s_0}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{(\gamma_g-1)}\ln \biggl(\frac{\tau_0}{\rho_0}\biggr)^{\gamma_g} 
=
\frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P_0}{(\gamma_g-1)\rho_0^{\gamma_g}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]s_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \frac{P_c}{(\gamma_g-1)\rho_c^{\gamma_g}} \biggr] 
+
\ln \biggl\{ \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
 \biggr\} 
+
\ln \biggl\{ \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-\gamma_g}\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \frac{P_c}{(\gamma_g-1)\rho_c^{\gamma_g}} \biggr] 
+
\ln \biggl\{ 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
 \biggr\} 
+ (2- \gamma_g)
\ln \biggl\{ \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]\biggr\} \, .
</math>
  </td>
</tr>
</table>

Notice that, independent of the value of <math>\gamma_g</math>, the specific entropy varies with <math>r_0</math> throughout the structure.  According to the [[2DStructure/AxisymmetricInstabilities#Schwarzschild_Criterion|Schwarzschild criterion]], spherically symmetric equilibrium configurations will be stable against convection if the specific entropy increases outward, and unstable toward convection if the specific entropy decreases outward.  Let's examine the slope, <math>ds_0/dr_0</math>, throughout configurations that have a parabolic density distribution.    

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

<tr>
  <td align="right">
<math>\biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]\frac{ds_0}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{r_0}{R^2} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
- \frac{2(2- \gamma_g)r_0}{R^2}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{R^2}{r_0}
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]
\biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]\frac{ds_0}{dr_0}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
- 2(2- \gamma_g)
\biggl[ 1 - \frac{1}{2}\biggl(\frac{r_0}{R} \biggr)^2 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2\gamma_g- 5)
+ (3 - \gamma_g)
\biggl[\biggl(\frac{r_0}{R} \biggr)^2 \biggr] 
</math>
  </td>
</tr>
</table>

[[File:EntropyDistribution245.png|right|400px]]
The slope is zero when,

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

<tr>
  <td align="right">
<math>
\biggl(\frac{r_0}{R} \biggr)^2 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5 - 2\gamma_g}{3 - \gamma_g} \, .
</math>
  </td>
</tr>
</table>
Moving from the center of the configuration to its surface, <math>0 < (r_0/R)^2 < 1</math>, the slope will go to zero &#8212; hence, the slope of the entropy will change sign

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