Stability

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

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

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 [78

], 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 [177 , 140 , 192

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.