4.3 Stability

The stability of radial oscillations for non-rotating stars in general relativity is well understood. Especially, the stability of static spherically symmetric stars can be determined by examining the mass-radius relation for a sequence of equilibrium stellar models, see for example Chapter 24 in [150]. The radial perturbations are described by a Sturm–Liouville second order equation with the frequency of the mode being the eigenvalue 2 ω, then for real ω the modes will be stable while for imaginary ω they will be unstable [52], see also Chapter 17.2 in [188].

The stability of the non-radially pulsating stars (Newtonian or relativistic) is determined by the Schwarzschild discriminant

dp Γ 1p dρ S(r) = ---− --------, (60 ) dr ρ + p dr
where Γ 1 is the star’s adiabatic index. This can be understood if we define the local buoyancy force f per unit volume acting on a fluid element displaced a small radial distance δr to be
f ∼ − g (r )S (r)δr, (61 )
where g is the local acceleration of gravity. When S is negative in some region the buoyancy force is positive and the star is unstable against convection, while when S is positive the buoyancy force is restoring and the star is stable against convection. Another way of discussing the stability is through the so-called Brunt–Väisälä frequency N 2 = gS(r) which is the characteristic frequency of the local fluid oscillations. Following earlier discussions when 2 N is positive, the fluid element undergoes oscillations, while when N 2 is negative the fluid is locally unstable. In other words, in Newtonian theory stability to non-radial oscillations can be guaranteed only if S > 0 everywhere within the star [65]. In general relativity [78Jump To The Next Citation Point], this is a sufficient condition, and so if S > 0 the quasi-normal modes are stable. For an extensive discussion of stellar instabilities for both non-rotating and rotating stars (which are actually more interesting for the gravitational wave astronomy) refer to [177140192Jump To The Next Citation Point].

For completeness the same applies as outlined at the end of Section 3.3. A model calculation of Price and Husain [168], however indicated that the nearly Newtonian quasi-normal modes might be a basis for the fluid perturbations. Further mathematical investigation is needed to clarify this issue.

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