Editing Appendix/Ramblings/PatrickMotl (section)

===Understanding the Step Function at the Core-Envelope Interface===

Now, turning to the [[SSC/Structure/BiPolytropes/Analytic51#Profile|accompanying tabular summary of the ''Radial Profile of Various Physical Variables'']], we are able to determine how the specific entropy behaves throughout the core and, separately, throughout the envelope in <math>~(n_c, n_e) = (5, 1)</math> bipolytropes. 

<table border="0" align="center" width="80%" cellpadding="10"><tr><td align="left">
<font color="red"><b>CORE:</b></font> &nbsp; Throughout the core, we see that,
<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>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{6}{5} \, .</math>
  </td>
</tr>
</table>
Hence, independent of the radial location, <math>~\xi</math>, throughout the core,
<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>~
5\ln (5) \, . 
</math>
  </td>
</tr>
</table>

<font color="red"><b>ENVELOPE:</b></font> &nbsp; Throughout the envelope, we see that,
<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>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \, .</math>
  </td>
</tr>
</table>
Hence, independent of the radial location, <math>~\eta</math>, throughout the envelope,
<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{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{-4} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr] \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

It is therefore clear that the core is a uniform specific-entropy sphere and the envelope is a uniform specific-entropy spherical shell, but in general the specific entropy of material in the envelope is different from the specific entropy of material in the core.  

According to our [[SSC/Stability/BiPolytropes#Free-Energy_Stability_Evaluation|free-energy based evaluation of the stability of bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>]], the marginally unstable model along the <math>~\mu_e/\mu_c = 1</math> sequence has the following properties: <math>~(\xi_i, q, \nu, \rho_c/\bar\rho) = (2.41610822, 0.59520261, 0.68306067, 16.3788)</math>; the red arrow in the following diagram points to this model's position in the <math>~q-\nu</math> parameter plane.  In an effort to test whether or not this model does identify the transition from stable to unstable configurations along the <math>~\mu_e/\mu_c = 1</math> sequence, Patrick picked a pair of models (highlighted in green in the following table) that straddle the location of the marginally unstable model, and followed the dynamical evolution of both models using a fully 3-D hydrodynamics code.  In particular, for the pair of models that Patrick evolved, we find:

<table border="0" align="center"><tr><td align="center">
<table border="1" align="right" cellpadding="8">
<tr><td align="center">'''Figure 2'''</td></tr>
<tr>
<td align="center"><math>~\gamma_c = 6/5</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>~\gamma_e = 2</math><br />&nbsp;<br />[[File:Entropy01Annotated.png|350px|Entropy distribution]]
</td></tr></table>
<table border="1" align="left" cellpadding="8" width="50%">
<tr><td align="center" colspan="4">'''Figure 1'''</td></tr>
<tr>
  <td align="center" colspan="4" width="100%"><b>Initial Model Parameters<br />for<br />Patrick's Pair of Simulations</b><br /><font size="-1>(green background)</font></td>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" width="25%"><math>~\xi_i</math></td>
  <td align="center" width="25%"><math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math></td>
  <td align="center" width="25%"><math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center" bgcolor="lightgreen">2.39184</td>
  <td align="center">8.04719</td>
  <td align="center">2.13422</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">2.41610822</td>
  <td align="center">8.04719</td>
  <td align="center">2.16080</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center" bgcolor="lightgreen">2.44016</td>
  <td align="center">8.04719</td>
  <td align="center">2.18706</td>
</tr>
<tr>
  <td align="center" colspan="4" width="100%">[[File:MotlVirialDetermination02.png|350px|Free-Energy determination of marginally unstable model]]</td>
</tr>
<tr>
  <td align="left" colspan="4" width="100%">For more details, see the accompanying discussion titled, ''[[SSC/Stability/BiPolytropes#Free-Energy_Stability_Evaluation|Free-Energy Stability Evaluation]]''</td>
</tr>
</table>
</td></tr></table>
These tabulated values of the normalized specific entropy in the ''core'' and, separately, in the ''envelope'' &#8212; also see the plot shown here on the right &#8212; appear to be consistent with Patrick's <font color="red">s.ps</font> plot of specific entropy.  In particular, this confirms that a step function should appear at the core-envelope interface and that the specific entropy of the envelope material should be ''lower'' than the specific entropy of the core material.  Therefore, the [[2DStructure/AxisymmetricInstabilities#Modeling_Implications_and_Advice|Schwarzschild criterion]] is violated at the interface and we should not have been surprised to see convective motions develop &#8212; initially, only at the interface.