The stability of gaseous masses for radial and non-radial oscillations in the post-Newtonian approximation of general relativity

Abstract

The stability of gaseous masses with respect to radial as well as non-radial oscillations is considered in the framework of the post-Newtonian equations of hydrodynamics The onset of dynamical instability at a radius R determined by a formula of the type R=2GM/c² K/γ-4/3 (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 non-radial oscillations, it is shown that the classical criterion of Schwarzschild based on the discriminant S(r)=dp/dr-γp/ρdρ/dr is replaced by one based on the discriminant C(r) = S(r) + π/c² dp/dr (Γ-γ+1/Γ-1 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/c²Γ₃-Γ/Γ₃-1 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 post-Newtonian approximation.