Editing Appendix/Ramblings/PatrickMotl (section)

==May 5 (following a phone conversation with Patrick)==

===Tying Expressions into H_Book Context===

In our wiki-based chapter titled, "[[PGE/FirstLawOfThermodynamics#First_Law_of_Thermodynamics|First Law of Thermodynamics]]," we have introduced the concept of an ''entropy tracer,'' <math>~\tau</math>.  In the subsubsection of this chapter that is titled, "[[PGE/FirstLawOfThermodynamics#Substantiation|Substantiation]]," we show that an expression for the specific entropy of a fluid element is,
<div align="center">
<math>~s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant}  \, .</math>
</div>
In addition, from our wiki-based chapter titled, "[[SR/IdealGas#Ideal_Gas_Equation_of_State|Ideal Gas Equation of State]]," we find the relations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c_P - c_V </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Re}{\bar\mu} </math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;
  <td align="right">
<math>~\gamma_g </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{c_P}{c_V} </math>
  </td>
  <td align="center">&nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp;
  <td align="right">
<math>~c_P </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_g}{(\gamma_g-1)} \biggl( \frac{\Re}{\bar\mu} \biggr) \, .</math>
  </td>
</tr>
</table>
Hence this expression for the entropy may be rewritten as,
<div align="center">
<math>~s = \frac{\gamma_g}{\gamma_g-1} \biggl( \frac{\Re}{\bar\mu} \biggr) \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant}  \, .</math>
</div>
Aside from the factor of <math>~({\bar\mu})^{-1}</math> that appears on the RHS &#8212; more on this [[#Pick_a_Different_Molecular-Weight_Ratio|below]] &#8212; these are the expressions that Patrick has used to generate the <font color="red">'''s.ps'''</font> plot, where the (unlabeled) ordinate is the normalized specific entropy, <math>~s/\Re</math>.

At the end of another subsubsection titled, "[[PGE/FirstLawOfThermodynamics#Initial_Recognition|Initial Recognition]]," we also find a relevant expression, namely,
<div align="center">
<math>~\tau \equiv (\rho\epsilon)^{1/\gamma_g} = \biggl[ \frac{P}{(\gamma_g - 1)} \biggr]^{1/\gamma_g} \, .</math>
</div>
Hence, ignoring the additive constant, in general we may write,
<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{\tau}{\rho}\biggr)^{\gamma_g} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>

===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.

===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>

===Eigenvectors of Marginally Unstable Models===

In preparation for our examination of the relative stability of bipolytropic structures having <math>~(n_c, n_e) = (5, 1)</math> &#8212; via numerical integration of the Linear-Adiabatic Wave Equation (LAWE) &#8212; we have demonstrated that we understand technically how to solve this type of eigenvalue problem by quantitatively reproducing related analyses that have been previously published by other groups.

====Good Comparisons With Previously Published Studies====

<ul>
  <li>
<table border="1" align="right" cellpadding="3"><tr><td align="center" bgcolor="black"><font size="-1" color="white">Schwarzschild (1941)</font><br />[[File:Schwarzschild1941movie.gif|150px|Eigenfunctions for Standard Model]]</td></tr><tr><td align="center" bgcolor="black"><font size="-1" color="white">Taff &amp; Van Horn (1974)</font><br />[[File:TaffVanHorn1974Fundamental.gif|150px|Fundamental mode animation]]</td></tr><tr><td align="center" bgcolor="black"><font size="-1" color="white">Murphy &amp; Fiedler (1985b)</font><br />[[File:MF85Figure3.png|150px|Figure 3 (Model 17) from Murphy &amp; Fiedler (1985b)]]</td></tr></table>
[http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)] numerically integrated the LAWE for ''isolated'' <math>~n=3</math> polytropic spheres to find eigenvectors (''i.e.,'' the spatially discrete eigenfunction and corresponding eigenfrequency) for five separate oscillation modes (the fundamental mode, plus the 1<sup>st</sup>, 2<sup>nd</sup>, 3<sup>rd</sup>, and 4<sup>th</sup> overtones) for models having four different adopted adiabatic indexes <math>~\gamma_g = \tfrac{4}{3}, \tfrac{10}{7}, \tfrac{20}{13}, \tfrac{5}{3})</math>.  In an [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|accompanying chapter of this H_Book]], we demonstrate that we have been able to reproduce in detail Schwarzschild's results for the specific case of <math>~\gamma_g = \tfrac{20}{13}</math>.
  </li>
  <li>
[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff &amp; Van Horn (1974)] examined radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal during the oscillations. For models having nine different truncation radii &#8212; chosen to straddle the position along the equilibrium sequence where the marginally unstable model was expected to arise &#8212; they determined and published the fundamental-mode eigenvalues.  For three of these models they also determined and published eigenvalues for the first harmonic mode of oscillation; the radial eigenfunctions associated with both the fundamental mode and the first harmonic mode of these three models also has been displayed in their Figure 1.  In a separate [[SSC/Stability/Isothermal#From_the_Analysis_of_Taff_and_Van_Horn_.281974.29|accompanying discussion]], we demonstrate that we have been able to reproduce in detail the subset of eigenfunctions and associated eigenvalues that have been previously published by [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff &amp; Van Horn (1974)].
  </li>
  <li>
In their published study of bipolytropes having <math>~(n_c, n_e) = (1,5)</math> with <math>~\mu_e/\mu_c = 1</math>, [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] integrated a coupled pair of LAWEs &#8212; one for the core and another for the envelope &#8212; to determine the eigenfunctions and corresponding eigenvalues of various radial modes of oscillation in more than a dozen different equilibrium models, assuming that during the oscillations, <math>~\gamma_g = 5/3</math> throughout both the core and the envelope.  In an accompanying chapter of this H_Book titled, ''[[SSC/Stability/MurphyFiedler85#Review_of_the_BiPolytrope_Stability_Analysis_by_Murphy_.26_Fiedler_.281985b.29|Review of the BiPolytrope Stability Analysis by Murphy &amp; Fiedler (1985b)]],'' we show that we have been able to duplicate in quantitative detail the eigenvectors associated with their equilibrium Models 10 and 17.
  </li>
</ul>
<span id="Fig6">Building upon this set of successful</span> comparisons with stability analyses published by other groups, we have carried out numerical integrations of the relevant LAWE to identify the eigenvectors associated with the fundamental-mode of radial oscillation in pressure-truncated, <math>~n = 5</math> polytropic configurations.  Details of this analysis are provided in yet [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|another chapter of this H_Book]].  The following animation sequence illustrates the results of this analysis.  As far as we have been able to determine, an analysis of this type has not previously been conducted for pressure-truncated, <math>~n = 5</math> polytropes.

<div align="center">
'''Figure 6'''<br />
[[File:N5Truncated2.gif|500px|Fundamental-mode eigenvectors for pressure-truncated n = 5 polytropes]]
</div>

In a [[SSC/Stability/n5PolytropeLAWE#Search_for_Analytic_Solutions_to_the_LAWE|subsection of this separate chapter]], we have also shown that, at the maximum-mass turning point along the pressure-truncated <math>~n=5</math> equilibrium sequence &#8212; identified by the green circular marker in the left-hand panel of this animation &#8212; the fundamental-mode eigenfrequency is precisely zero and the associated eigenfunction is described exactly by the formula for a parabola.

====Our Numerical Analysis of Bipolytropes Having (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)====
In an [[SSC/Stability/BiPolytropes#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|accompanying discussion]] we have shown that we can integrate the linear adiabatic wave equation (LAWE) &#8212; that is, we effectively have been able to solve the eigenvalue problem &#8212; to obtain the eigenvector associated with marginally unstable models along equilibrium sequences for bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>.  The marginally unstable models that have been identified via this more rigorous approach &#8212; see the orange triangles in the following figure &#8212; fall at different points along each equilibrium sequence than the points that were identified via our free-energy analysis &#8212; see the red-dashed demarcation curve.  Assuming that the algorithm that we developed to integrate the LAWE was basically error-free, we trust the model identifications generated via this more rigorous technique.  

Our analysis indicates that, along the <math>~\mu_e/\mu_c = 1</math> equilibrium sequence, the core-envelope interface of the marginally unstable model is located at <math>~\xi_i = 1.6686460157</math>; in the following figure, the red arrow points to this location along that sequence.  In the brief table that accompanies the figure, we have listed values for the dimensionless specific entropy in the core and, separately, in the envelope, along with the interface location, <math>~\xi_i</math>.


<table border="0" align="center" width="80%"><tr><td align="center">
<table border="1" align="right" cellpadding="8">
<tr><td align="center">'''Figure 8'''</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:Entropy04Annotated.png|350px|Entropy distribution]]</td></tr></table>
<table border="1" align="left" cellpadding="8" width="40%">
<tr><td align="center" colspan="4">'''Figure 7'''</td></tr>
<tr>
  <td align="center" colspan="4" width="80%"><b>Initial Model Parameters<br />for<br />LAWE-Determined<br />Marginally Unstable Model</b></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">1.6686460157</td>
  <td align="center">8.04719</td>
  <td align="center">1.31310</td>
</tr>
<tr>
  <td align="center" colspan="4" width="100%">[[File:MotlLAWEdetermination02.png|350px|LAWE 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#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|Eigenvectors from Solution of LAWE]]''</td>
</tr>
</table>
</td></tr></table>

===Summary===

For three separate equilibrium sequences &#8212; specifically, for the cases of <math>~\mu_e/\mu_c = 1, \tfrac{1}{2}, \tfrac{1}{3}</math> &#8212; the table on the left, below, identifies the location <math>~(\xi_i)</math> along the sequence where the model is marginally unstable toward a global, dynamical instability as ''estimated'' by the our free-energy analysis (column 2) and as determined by our numerical integration of the coupled pair of governing LAWEs (column 6).  For each identified model, the corresponding values of <math>~q, \nu</math> and <math>~\rho_c/\bar\rho</math> are also provided.  Additional details can be found in our accompanying discussion of, respectively, our [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|free-energy analysis]] and our numerical solution of the relevant [[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models|eigenvalue problem]].

<table border="0" align="center" width="90%" cellpadding="8"><tr><td align="left">
<table border="1" align="left" cellpadding="9">
<tr>
  <td align="center" colspan="10">Marginally Unstable (dynamically)<br />assuming <math>~\gamma_c = \tfrac{6}{5}</math> and <math>~\gamma_e = 2</math></td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" colspan="4">Determined via<br />[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Free-Energy Analysis]]</td>
  <td align="center" rowspan="5" bgcolor="lightgrey">&nbsp;</td>
  <td align="center" colspan="4">Determined via<br />[[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models|Eigenvector Analysis]]</td>
</tr>
<tr>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~q</math></td>
  <td align="center"><math>~\nu</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~q</math></td>
  <td align="center"><math>~\nu</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">2.416</td>
  <td align="center">0.595</td>
  <td align="center">0.683</td>
  <td align="center">16.4</td>
  <td align="center">1.66865</td>
  <td align="center">0.539</td>
  <td align="center">0.498</td>
  <td align="center">8.42</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="center">4.186</td>
  <td align="center">0.328</td>
  <td align="center">0.701</td>
  <td align="center">354.</td>
  <td align="center">2.27926</td>
  <td align="center">0.185</td>
  <td align="center">0.234</td>
  <td align="center">63.3</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="center">8.548</td>
  <td align="center">0.099</td>
  <td align="center">0.479</td>
  <td align="center">63,000.</td>
  <td align="center">2.58201</td>
  <td align="center">0.176</td>
  <td align="center">0.218</td>
  <td align="center">230.</td>
</tr>
</table>

<table border="1" align="right" cellpadding="5">
<tr><td align="center">'''Figure 9'''</td></tr>
<tr><td align="center"><math>~\gamma_c = 6/5</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>~\gamma_e = 2</math><br />
[[File:SmoothAndUnstable01.png|375px|Convectively stable while dynamically unstable]]
</td></tr></table>
</td></tr></table>

In the figure shown here on the right, the orange-dashed curve shows how <math>~\xi_i</math> (table column 6) varies with <math>~\mu_e/\mu_c</math> (table column 1) for marginally unstable models as determined by our numerical eigenvector analysis.  The square of the fundamental-mode eigenfrequency is negative for all models above this orange-dashed curve, indicating that they should be ''globally'' dynamically unstable.  The solid blue curve shows how <math>~[\xi_i]_\mathrm{smooth}</math> varies with <math>~\mu_e/\mu_c</math>, as [[#Pick_a_Different_Molecular-Weight_Ratio|derived above]].  Models above and to the left of this blue curve should be stable against convection at the core-envelope interface.