Editing Appendix/Ramblings/PatrickMotl (section)

===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.