Editing Appendix/Ramblings/PatrickMotl (section)

===Can the Step Function be Flipped or Erased===

====Assume a Mean-Molecular-Weight Ratio of Unity====
If we continue to examine equilibrium models that have <math>~\mu_e/\mu_c = 1</math>, is there a value of the interface radius, <math>~\xi_i</math>, for which the entropy step-function disappears and above which the step function flips?  The answer appears to be, "Yes."  It occurs along the equilibrium sequence where the normalized specific entropy has the same value in the core and as in the envelope.  That is, it occurs when,
<table border="0" align="center" width="80%"><tr><td align="center">
<table border="1" align="right" cellpadding="8">
<tr><td align="center">'''Figure 3'''</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:Entropy02Annotated.png|250px|Entropy distribution]]</td></tr></table>
<table border="0" cellpadding="5" align="left">

<tr>
  <td align="right">
<math>~
\ln \biggl[ \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr]_\mathrm{smooth} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5\ln(5)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~ \biggl( 1 + \frac{\xi_i^2}{3}\biggr)_\mathrm{smooth} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5^{5 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~ [\xi_i ]_\mathrm{smooth}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[3 \biggl(5^{5 / 2} - 1 \biggr)\biggr]^{1 / 2} \approx 12.83375</math> &hellip;
  </td>
</tr>
</table>
</td></tr></table>
<font color="red">'''Prediction:'''</font> &nbsp; Any initial model with <math>~\mu_e/\mu_c = 1</math> and <math>~\xi_i > [\xi_i]_\mathrm{smooth}</math> will be stable against convection, but will be globally dynamically unstable.

====Pick a Different Molecular-Weight Ratio====

<font color="red">'''Caution:'''</font> &nbsp; This subsection contains a derivation that is based on an interpretation of the specific entropy normalization at the interface that may not be physically correct.  It needs more thought &hellip; After more thought:&nbsp; In order to determine whether the entropy of the envelope is greater than the entropy of the core it would be wise to always plot the specific entropy relative to <math>~\mu_c</math>; therefore, when the ratio <math>~(\tau/\rho)^{\gamma_g}</math> is used to calculate the envelope entropy, <math>~[s/(\Re/\bar\mu)]_\mathrm{env}</math>, as [[#Tying_Expressions_into_H_Book_Context|above]], the calculated answer should then be ''divided'' by <math>~(\mu_e/\mu_c)</math> in order to obtain the proper normalization.


In principle, we can determine in a similar fashion the values of <math>~[\xi_i]_\mathrm{smooth}</math> that are relevant to equilibrium model sequences having  <math>~\mu_e/\mu_c < 1</math>.  But in doing this, we must take into account that in most of our above derivations the mean-molecular-weight appears in the denominator of the (LHS) expression for the normalized specific entropy.  More generally, the prescription for <math>~[\xi_i]_\mathrm{smooth}</math> should come from the demand that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{s_\mathrm{env}}{\Re} - \frac{s_\mathrm{core}}{\Re}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\mu_c}
\biggl\{\biggl( \frac{\mu_c}{\mu_e} \biggr) \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr]_\mathrm{smooth} - 5  \ln(5) \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr]_\mathrm{smooth} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl[ 5 \biggr]^{5(\mu_e/\mu_c)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl( 1 + \frac{\xi_i^2}{3}\biggr)_\mathrm{smooth} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5^{2.5(\mu_e/\mu_c)} \biggl(\frac{\mu_e}{\mu_c}\biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ [\xi_i]_\mathrm{smooth} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3^{1 / 2}\biggl[ 5^{2.5(\mu_e/\mu_c)} \biggl(\frac{\mu_e}{\mu_c}\biggr) -1\biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Here are a few examples:

<table border="0" align="center" width="60%"><tr><td align="center">
<table border="1" align="right" cellpadding="8">
<tr><td align="center">'''Figure 4'''</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:Entropy03Annotated.png|250px|Entropy distribution]]</td></tr></table>
<table border="1" align="left" cellpadding="8">
<tr>
  <td align="center" width="50%"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~[\xi_i]_\mathrm{smooth}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right">12.83375</td>
</tr>
<tr>
  <td align="center">1/2</td>
  <td align="right">2.86620</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="right">1.07128</td>
</tr>
<tr>
  <td align="center">1/3</td>
  <td align="right">0.90754</td>
</tr>
<tr>
  <td align="center">0.31</td>
  <td align="right">0.4871479</td>
</tr>
<tr>
  <td align="center">0.299577998</td>
  <td align="right">0.00000000</td>
</tr>
</table>
</td></tr></table>
Note: &nbsp;  <math>[\xi_i]_\mathrm{smooth} ~\rightarrow~ 0</math> when the argument inside the square brackets goes to zero, which occurs when <math>(\mu_e/\mu_c)\ln(\mu_e/\mu_c) ~\rightarrow ~ - 2.5\ln(5)</math>, that is, when <math>(\mu_e/\mu_c) \approx 0.299577998</math>.  The implication is that ''all'' of the models along a sequence are stable against convection when <math>(\mu_e/\mu_c) < 0.299577998</math>.

<div align="center">[[File:ConvectiveBoundaryLabeled.png|400px|Convective Boundary]]</div>

====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.

====Behavior At Interface During Radial Oscillation====


From [[#Tying_Expressions_into_H_Book_Context|above]], we have,
<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{1}{(\gamma_g-1)}\ln \biggl[ \frac{P}{(\gamma_g-1)\rho^{\gamma_g}} \biggr] \, .
</math>
  </td>
</tr>
</table>
From the perspective of the core, at the interface we should find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
5\ln \biggl[ \frac{5(P_i+\delta P)}{(\rho_i+\delta\rho_c)^{6/5}} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
5\ln \biggl[ \frac{5P_i}{\rho_i^{6/5}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
(P_i+\delta P)(\rho_i+\delta\rho_c)^{-6/5}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{P_i}{\rho_i^{6/5}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl[1 + \frac{\delta P}{P_i}\biggr] \biggl[ 1 + \frac{\delta\rho_c}{\rho_i}\biggr]^{-6/5}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\delta P}{P_i} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 + \frac{\delta\rho_c}{\rho_i}\biggr]^{6/5} - 1
\, .
</math>
  </td>
</tr>
</table>
Whereas, from the perspective of the envelope, at the interface we should find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\ln \biggl\{ (P_i +\delta P) \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)\rho_i + \delta\rho_e \biggr]^{-2} \biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\ln \biggl\{ P_i \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)\rho_i\biggr]^{-2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
P_i \biggl[ 1 + \frac{\delta P}{P_i} \biggr] \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)\rho_i\biggr]^{-2}
\biggl[1 + \frac{\delta\rho_e }{ (\mu_e/\mu_c)\rho_i} \biggr]^{-2} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
P_i \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)\rho_i\biggr]^{-2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\delta P}{P_i} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 + \frac{\delta\rho_e }{ (\mu_e/\mu_c)\rho_i} \biggr]^{2} -1
\, .
</math>
  </td>
</tr>
</table>
In order to ensure that <math>\delta P/P_i</math> is the same at the core/envelope interface, this means that,

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

<tr>
  <td align="right">
<math>
\biggl[ 1 + \frac{\delta\rho_c}{\rho_i}\biggr]^{6/5}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 + \frac{\delta\rho_e }{ (\mu_e/\mu_c)\rho_i} \biggr]^{2}
\, ,
</math>
  </td>
</tr>
</table>
or, more generally,

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

<tr>
  <td align="right">
<math>
\biggl[ 1 + \frac{\delta\rho_c}{\rho_i}\biggr]^{\gamma_c}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 + \frac{\delta\rho_e }{ (\mu_e/\mu_c)\rho_i} \biggr]^{\gamma_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \gamma_c\biggl(\frac{\delta\rho_c}{\rho_i}\biggr)
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\gamma_e\biggl[ \frac{\delta\rho_e }{ (\mu_e/\mu_c)\rho_i} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ 
\frac{\delta\rho_e }{ \rho_i}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\gamma_c}{\gamma_e} \cdot \frac{\mu_e}{\mu_c}\biggr)\frac{\delta\rho_c}{\rho_i}
\, .
</math>
  </td>
</tr>
</table>