Editing Appendix/Ramblings/PatrickMotl (section)

==May 11 (from Joel)==

<table border="1" cellpadding="10" width="60%" align="center"><tr><td align="left">
Patrick,

Following our phone conversation earlier this week, I have added a number of paragraphs, derivations, and tables/figures to my VisTrails wiki page &#8212; see the [[#May_5_.28following_a_phone_conversation_with_Patrick.29|above subsection]] dated, ''May 5.''

Here is a quick summary, assuming evolutions are conducted with <math>~(\gamma_c, \gamma_e) = (6/5, 2)</math>:
<ol type="1">
<li>
I completely agree with your expression for the specific entropy, except I believe that the gas constant needs to be divided by the mean-molecular weight <math>~(\Re/\bar\mu)</math>.
</li>
<li>
I concur with your <font color="red">s.ps</font> plot.  In particular, for the bipolytropic model having <math>~\mu_e/\mu_c = 1</math> and <math>~\xi_i = 2.4161</math> (this is the model that my free-energy-based analysis identifies as the marginally unstable model), I get <math>~[s/(\Re/\bar\mu)]_\mathrm{core} = 8.04719</math> and <math>~[s/(\Re/\bar\mu)]_\mathrm{env} = 2.16080</math>.  See Figure 2 and the table accompanying my Figure 1.
</li>
<li>
The size of the entropy jump **does** depend on the location of the interface.  If we stay on the <math>~\mu_e/\mu_c = 1</math> sequence, the entropy of the envelope will equal the entropy of the core if <math>~\xi_i = 12.83375</math>.  See Figure 3.  (I  now label this critical value, <math>~[\xi_i]_\mathrm{smooth}</math>.)  At values of <math>~\xi_i > [\xi_i]_\mathrm{smooth}</math>, the envelope's entropy will be larger than the core's entropy.
</li>
<li>
If you choose a sequence for which <math>~\mu_e/\mu_c = 0.5</math>, then my analysis says that <math>~[\xi_i]_\mathrm{smooth} = 2.86620</math>.  See Figure 4.
</li>
</ol>

If you set <math>~\mu_e/\mu_c = 1</math> but assume evolutions are conducted with <math>~\gamma_c = \gamma_e = 5/3</math>:
<ol type="1" start="5">
<li>
The entropy discontinuity disappears at the interface, but the entropy is no longer constant throughout the core or the envelope.  See Figure 5.  Moving radially outward from the center, the entropy steadily increases through the core (convectively stable) then it steadily drops through the envelope (convectively unstable).
</li>
</ol>

Finally, we should consider the following before embarking on more 3D simulations:
<ul>
<li>
Along any given equilibrium sequence, a free-energy analysis only provides an **approximate** value of <math>~\xi_i</math> at which a dynamical instability sets in.  The **precise** location can be identified by solving the relevant eigenvalue problem and searching for the model at which the fundamental-mode oscillation frequency goes to zero.
</li>
<li>
I have taught myself how to solve this type of eigenvalue problem; as [[#Good_Comparisons_With_Previously_Published_Studies|described briefly above]] &#8212; accompanied by a few animated-gifs &#8212; in other chapters of my wiki-book, I show that I can quantitatively match related, previously published analyses.
</li>
<li>
 I have tentative eigenvalue-analysis results for <math>~(n_c, n_e) = (5, 1)</math> bipolytropes; all marginally unstable models arise **sooner** along each sequence &#8212; that is at smaller values of <math>~\xi_i</math> (see Figure 7) &#8212; than was predicted by the free-energy analysis.
</li>
</ul>

Other things to discuss:
<ul>
<li>Instead of bipolytropic structures, it might make more sense to test (with 3D code) the stability of pressure-truncated, n = 5 polytropes. See Figure 6.</li>
<li>Analytic solutions of a limited &#8212; but physically quite interesting &#8212; sample of eigenvalue problems.</li>
</ul>

I am very interested in discussing all of this with you, if you are interested and when you have the time.  Just let me know.

Cheers,
Joel</td></tr>
</table>