Editing Appendix/Ramblings/PatrickMotl (section)

==May 4 (from Joel)==

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Patrick,

My initial response to your 1 May email follows.  In an effort to make sure we are on the same page when referencing the structural properties and the stability properties of various configurations, my response includes links to various chapters/subsections of my online wiki-based book.

Thanks for pursuing this problem.  I am still very interested in Its solution.

All the best,
Joel
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===Chosen Initial Models===

I presume that when you constructed your pair of initial (spherical) models, you used the analytically prescribed properties found in my chapter titled, "[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|BiPolytrope with <math>n_c = 5</math> and <math>n_e=1</math>]]" and that when you reference the "stability boundary in <math>~\xi</math>" you are drawing from the summary (Table 3 and Figure 3) found near the end of this chapter in the subsection titled, "[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Stability Condition]]."  In particular, based on the information contained in this chapter, I presume that you are assuming that the marginally unstable model along the <math>~\mu_e/\mu_c = 1</math> sequence is the configuration having the properties &hellip;
<div align="center">
<math>~(\xi_i, q, \nu) = (2.416, 0.5952, 0.6830) \, ,</math>
</div>
and that your pair of initial models have values of <math>~\xi_i</math> that are somewhat greater than and somewhat smaller than 2.416.  If this does not properly describe your choice of initial models, please clarify.

===Instability Toward Convection===

In my chapter titled, "[[2DStructure/AxisymmetricInstabilities#Axisymmetric_Instabilities_to_Avoid|Axisymmetric Instabilities to Avoid]]," I discuss &#8212; among other things &#8212; the ''Schwarzschild Criterion.''  Toward the very end of that chapter, in a subsection titled, ''[[2DStructure/AxisymmetricInstabilities#Modeling_Implications_and_Advice|Modeling Implications and Advice]],'' I specifically describe how convection might arise when modeling spherical polytropic configurations.  Would you please read this short subsection and let me know whether you agree with my explanation or not?

In the context of this discussion of the Schwarzschild criterion (i.e., convective instabilities) in polytropes, my question to you is:  ''What value of the adiabatic index, <math>~\gamma</math>, did you use when you evolved the bipolytrope?''  Did you use the same index for the entire configuration, or did you use a value in the envelope that is different from the value used in the core?

If you used <math>~\gamma = 6/5</math> for the core, then the core should have uniform specific entropy and therefore would be (only) marginally stable against convection.  Likewise, if you used <math>~\gamma = 2</math> for the envelope, it should be (only) marginally stable against convection.  Incidentally, these <b>are</b> the two separate values of the adiabatic index that I used when I used a free-energy analysis to examine stability.

NOTE:  In yet [[SSC/Stability/MurphyFiedler85#Review_of_the_BiPolytrope_Stability_Analysis_by_Murphy_.26_Fiedler_.281985b.29|another chapter of my book]], I discuss the bipolytropic stability analysis that was performed by Murphy &amp; Fiedler (1985b).  They examined the stability of bipolytropes having the core/envelope polytropic indexes swapped, that is, their equilibrium models had, <math>~(n_c, n_e) = (1, 5)</math>.  Their stability analysis was performed while assuming that the adiabatic index was <math>~\gamma = 5/3</math> ''throughout the entire configuration.''  In a short subsection near the beginning of my chapter titled, "[[SSC/Stability/MurphyFiedler85#Aside_Regarding_Convectively_Unstable_Core|Aside Regarding Convectively Unstable Core]]," I have pointed out that the cores of these models should have all been convectively unstable, according to the Schwarzschild criterion.  Please read this short subsection and let me know if you agree with my stated expectation for those models.

===Other Approaches to Stability Analysis===
After you send me feedback on the comments &amp; questions I have presented, above, I have a good deal more to tell you about my analysis of the stability of our <math>~(n_c, n_e) = (5, 1)</math> bipolytropic configurations.  I am still a bit confused, but I'm pretty sure that the transition from stable to unstable configurations along the <math>~\mu_e/\mu_c = 1</math> occurs at a value of <math>~\xi_i \approx 1.67</math>, rather than the value of <math>~\xi_i = 2.416</math> that I obtained via the free-energy analysis.  But, as I've said, there is a lot to tell you about this.

===Other Suggestions===
You asked what alternative models might be examined.  It might be smarter to first evolve some simpler models, such as pressure-truncated polytropes &#8212; '''not''' bipolytropes.  But we should discuss this after I read your feedback on the above.