Editing Appendix/Ramblings/PatrickMotl (section)

====Use the Same Ratio of Specific Heats Throughout====

Let's examine the initial model's entropy profile under the assumption that the system is evolved with <math>~\gamma_g = 5/3</math> throughout the bipolytrope.  From the above analysis, in this case the relevant general expression for the specific entropy profile should be,

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

<tr>
  <td align="right">
<math>~\frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{2} \ln \biggl[ \frac{3P}{2\rho^{5/3}} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \biggl(\frac{3P}{2}\biggr)^{3 / 2} \rho^{-5/2} \biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="0" align="center" width="80%" cellpadding="10"><tr><td align="left">
<font color="red"><b>CORE:</b></font> &nbsp; Given that, throughout the core, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-3}</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-5/2} \, ,</math>
  </td>
</tr>
</table>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl\{ 
\biggl(\frac{3}{2}\biggr)^{3 / 2} \biggl[ \biggl(1+ \frac{\xi^2}{3}\biggr)^{-9/2} \biggr] 
\biggl[ \biggl(1+ \frac{\xi^2}{3}\biggr)^{25/4} \biggr] 
\biggr\}
=
\frac{1}{4}\cdot\ln \biggl[ 
\biggl(\frac{3}{2}\biggr)^{6} \biggl(1+ \frac{\xi^2}{3}\biggr)^{7} \biggr] \, .
</math>
  </td>
</tr>
</table>

<font color="red"><b>ENVELOPE:</b></font> &nbsp; Given that, throughout the envelope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\theta_i^6 [\phi(\eta)]^2</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)] \, ,</math>
  </td>
</tr>
</table>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl\{ \biggl(\frac{3}{2}\biggr)^{3 / 2} 
\biggl[ \theta_i^6 [\phi(\eta)]^2  \biggr]^{3/2} 
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)] \biggr]^{-5/2} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \cdot \ln \biggl\{ \biggl(\frac{3}{2}\biggr)^{3 } \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5}
\theta_i^{-7} [\phi(\eta)]  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{4} \cdot \ln \biggl\{ \biggl(\frac{3}{2}\biggr)^{6 } 
\biggl(1+ \frac{\xi_i^2}{3}\biggr)^{7} 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10}  [\phi(\eta)]^2
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

<table border="1" align="center" cellpadding="8">
<tr><td align="center">'''Figure 5'''</td></tr>
<tr><td align="center">[[File:Entropy05Annotated.png|450px|Entropy distribution]]</td></tr></table>
Notice that, because <math>~[\gamma_c = 5/3] > [(n_c + 1)/n_c = 6/5]</math>, the specific entropy increases with radius throughout the core, so according to the [[2DStructure/AxisymmetricInstabilities#Modeling_Implications_and_Advice|Schwarzschild criterion]] the core is stable against convection.  However, because <math>~[\gamma_e = 5/3] < [(n_e + 1)/n_e = 2]</math>, the specific entropy decreases with radius throughout the envelope, so according to the [[2DStructure/AxisymmetricInstabilities#Modeling_Implications_and_Advice|Schwarzschild criterion]] the envelope must be unstable toward convection.


Note:  {{ MF85bfull }} have examined radial oscillation modes in bipolytropic configurations that have a ''flipped'' set of indexes &#8212; that is, they studied equilibrium structures having <math>~(n_c, n_e) = (1, 5)</math> &#8212; assuming, as we have examined here, that oscillations in both the core and the envelope are governed by <math>~\gamma_g = 5/3</math>.  The chapter of this H_Book in which we discuss the detailed analysis presented by {{ MF85b }}, we have inserted a short subsection titled, ''[[SSC/Stability/MurphyFiedler85#Aside_Regarding_Convectively_Unstable_Core|Aside Regarding Convectively Unstable Core]],'' where we point out that the ''cores'' of the Murphy &amp; Fiedler models should be convectively unstable whereas their ''envelopes'' should be stable against convection.