User:Tohline/SSC/FreeEnergy/Equilibrium Sequence Instabilities
Supporting Derivations for Free-Energy PowerPoint Presentation
The derivations presented here are an extension of our accompanying free-energy synopsis. These additional details proved to be helpful while developing an overarching PowerPoint presentation.
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General Free-Energy Expression
Coincidence Between Points of Secular and Dynamical Instability
From the accompanying graphical display of equilibrium sequences, it seems that a <math>~\nu_\mathrm{max}</math> turning point will only exist in five-one bipolytropes for <math>~\mu_e/\mu_c</math> less than some value — call it, <math>~(\mu_e/\mu_c)_\mathrm{begin}</math> — which is less than but approximately equal to <math>~\tfrac{1}{3}</math>. As we move along any sequence for which <math>~\mu_e/\mu_c < (\mu_e/\mu_c)_\mathrm{begin}</math>, in the direction of increasing <math>~\ell_i</math>, it is fair to ask whether the system becomes dynamically unstable (at <math>~[x_\mathrm{eq}]_\mathrm{crit}</math>) before or after it encounters the point of secular instability marked by <math>~\nu_\mathrm{max}</math>.
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