Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt3 (section)

===Part III===
From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration.  Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem.  In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure <math>~(P_e)</math> constant, but also assuming that the configuration's structural form factors are invariable.  

This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state.  While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors.  In support of this argument, we point to the paper by {{ GW80full }} which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|chapter of this H_Book details the Goldreich &amp; Weber derivation]].  Goldreich &amp; Weber use [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations.  In &sect;IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis.  They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core."  Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable.  (Based on the Goldreich &amp; Weber discussion, we should also look at the published work of {{ Schwarzschild41full }}, who has evaluated radial modes, and of {{ Cowling41full }}, who has obtained eigenvalues of some low-order nonradial modes.)

In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors.  In order to test this underlying assumption, following [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)], it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation.  Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability.  Better yet, the ''sign'' of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.


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