Chandrasekhar, S. (1965) The stability of gaseous masses for radial and nonradial oscillations in the postNewtonian approximation of general relativity Astrophysical Journal, 142 . pp. 15191540. ISSN 0004637X

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Official URL: http://adsabs.harvard.edu/abs/1965ApJ...142.1519C
Related URL: http://dx.doi.org/10.1086/148434
Abstract
The stability of gaseous masses with respect to radial as well as nonradial oscillations is considered in the framework of the postNewtonian equations of hydrodynamics The onset of dynamical instability at a radius R determined by a formula of the type R=2GM/c^{2} K/γ4/3 (where K is a constant) is confirmed in case the "ratio of the specific heats" γ = (a log P/ia log p). (where the subscript s denotes that the derivative is with respect to constant entropy) is a constant. An expression for K is derived which does not involve any knowledge of the equilibrium configuration beyond the Newtonian framework; and the values of K appropriate to the poly tropes are also listed. With respect to the onset of instability for nonradial oscillations, it is shown that the classical criterion of Schwarzschild based on the discriminant S(r)=dp/drγp/ρdρ/dr is replaced by one based on the discriminant C(r) = S(r) + π/c^{2} dp/dr (Γγ+1/Γ1 dΓ/dr/dρ/ρdr), where π is the internal energy (per unit volume) and Γ is a ratio defined by the relation ρπ = p/(Γ  1). An alternative form for C(r), namely, C(r) = S(r) [1+π/c^{2}Γ_{3}Γ/Γ_{3}1 d(log p)/dr/d(log ρ)/dr], where r3 = 1 + (a log Tla log p)s, shows that the condition for the occurrence of convective instability is unaltered in the postNewtonian approximation.
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