Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

==Statement of the Problem==
We can construct detailed force-balance (DFB) models of spherically symmetric, ''isolated'' polytropes over the entire range of polytropic indexes:  <math>~\infty \ge n \ge 0</math>.  Given that, <math>~\gamma = (n+1)/n</math>, this corresponds to the range of adiabatic exponents, <math>~1 \le \gamma \le \infty</math>.  These configurations are ''known'' to be dynamically stable if <math>~n < 3</math>; that is, if <math>~\gamma > 4/3</math>.  How do we know the answer to this stability question?
*  An ''analytic'' free-energy analysis tells us that <math>~\partial^2 \mathfrak{G}/\partial R^2 > 0</math> (i.e., the system is dynamically unstable) for models having polytropic indexes, <math>~n > 3</math>;
*  Linear stability analyses, performed numerically, tell us that the square of the system's fundamental-mode eigenfrequency is negative for models having <math>~\gamma < 4/3</math>.


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