Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

==Analysis by Yabushita (1975)==
It appears as though the first analysis that asked the same set of questions we have asked regarding the type of stability that is associated with turning points is the one published by [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y S. Yabushita (1975)] and titled, ''On the Structure and Stability of a Polytrope with an Isothermal Core''.

From the article's summary (abstract) alone &#8212; immediately following &#8212; we note the following key points;
* Yabushita (1975) builds an equilibrium sequence for <math>~(n_c, n_e) = (\infty, \tfrac{3}{2})</math> bipolytropes and identifies a maximum-mass turning point along the sequence.
* A solution of the relevant LAWE shows that the "dynamical instability sets in precisely at the mass peak," if the adopted adiabatic index is the same as the <math>~\gamma = d\ln\rho/d\ln r</math> calculated by the equation(s) of state governing the unperturbed state.

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[[File:Yabushita1975Summary.png|Summary (Abstract) from Yabushita (1975)]]
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Then we see, from the article's introductory paragraphs that Yabushita &hellip;
* Understands that it is useful to analyze stability from both a free-energy (virial theorem) standpoint and a linear stability (radial oscillations) analysis &#8212; see the following reproduction of the first introductory paragraph.
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[[File:Yabushita1975Intro1.png|700px|Introductory paragraph from Yabushita (1975)]]
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* Understands that a strong analogy can be drawn between the behavior of pressure-truncated and bipolytropic sequences &#8212;&nbsp; "<font color="darkgreen">This will enable one to see the effect of an envelope on the stability of isothermal gas spheres ([http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y Yabushita 1968]) which otherwise will extend to infinity.</font>"

It might be worth looking at the following publications:
* [http://adsabs.harvard.edu/abs/1967AnAp...30..975G M. Gabriel &amp P. Ledoux (1967)]:  Yabushita says, 
* [ S. Chandrasekhar (1972) ''General Relativity'']:  Yabushita says, <font color="darkgreen">It is Chandrasekhar (1972) who first showed the analogy between neutron star models and a gaseous sphere with an isothermal core and an envelope with constant density.</font>