Editing 2DStructure/AxisymmetricInstabilities (section)

=Modeling Implications and Advice=
Here are some recommendations to keep in mind as you attempt to construct equilibrium models of self-gravitating astrophysical fluids.

<font color="purple">Rayleigh-Taylor instability:</font>&nbsp; In order to avoid constructing configurations that are subject to the Rayleigh-Taylor instability, be sure that lower density material is never placed "beneath" higher density material.  For example &hellip;
<ul><li>When building a bipolytropic configuration, a value for the (imposed) discontinuous jump in the mean-molecular weight will need to be specified at the interface between the envelope and the core.  If you choose a ratio, <math>~{\bar\mu}_e/{\bar\mu}_c</math>, that is greater than unity, the resulting equilibrium model will exhibit a discontinuous density jump that makes the density higher at the base of the envelope than it is at the surface of the core.  The core/envelope interface of this configuration will be unstable to the Rayleigh-Taylor instability, but you won't know that until and unless you examine the hydrodynamic stability of the configuration.</li></ul>

<font color="purple">Schwarzschild criterion:</font>&nbsp; In order to avoid constructing configurations that violate the Schwarzschild criterion, be sure that the specific entropy of the fluid is uniform (marginally stable) or increases outward throughout the equilibrium structure, where the word "outward" is only meaningful when referenced against the direction that the effective gravity points.  For example &hellip;
<ul><li>Suppose that you build a spherically symmetric polytropic configuration whose ''structural'' index is, <math>~n</math>, but then you want to test the stability of the configuration assuming that compressions/expansions of individual fluid elements occur along adiabats for which the adiabatic index is, <math>~\gamma \ne (n+1)/n</math>.  Although we generally think of polytropes as being homentropic configurations, if <math>~\gamma \ne (n+1)/n</math>, then different fluid elements will, in practice, evolve along adiabats that are characterized by ''different'' values of the specific entropy; that is, throughout the equilibrium model, <math>~\nabla s</math> will not be zero.  Whether the specific entropy increase (stable) or decreases (unstable) outward will depend on whether you select a value for the ''evolutionary'' <math>~\gamma</math> that is greater than (stable) or less than (unstable) <math>~(n+1)/n</math>.</li>

<li>According to [http://www.ucolick.org/~woosley/ay112-14/lectures/lecture8.4x.pdf Woosley's class lecture notes],
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<math>~\frac{d\ln \rho}{d\ln P}\biggr|_\mathrm{structure} = \frac{n}{n+1}</math>
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<math>~></math>
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<math>~\frac{1}{\gamma_g}</math>
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<math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; stable
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&nbsp;
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<math>~<</math>
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<math>~\frac{1}{\gamma_g}</math>
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<math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; unstable
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This is another way of expressing the same stability criterion for polytropes.
</li>
<li>
Examples:  &nbsp; An n = 1 polytope is ''unstable toward convection'' if expansions (or contractions) occur along an adiabat with  <math>~\gamma_g < 2</math>.  Alternatively, an n = 5 polytope is  ''unstable toward convection'' if expansions (or contractions) occur along an adiabat with <math>~\gamma_g < \tfrac{6}{5}</math>.
</li></ul>