5 Excitation and Detection of 4 Quasi-Normal Modes of 4.2 Mode Analysis

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 tex2html_wrap_inline3050, then for real tex2html_wrap_inline2624 the modes will be stable while for imaginary tex2html_wrap_inline2624 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

equation700

where tex2html_wrap_inline3362 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 tex2html_wrap_inline3366 to be

equation708

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 tex2html_wrap_inline3374 which is the characteristic frequency of the local fluid oscillations. Following earlier discussions when tex2html_wrap_inline3376 is positive, the fluid element undergoes oscillations, while when tex2html_wrap_inline3376 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 In The Article], 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, 192Jump To The Next Citation Point In The Article].

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.



5 Excitation and Detection of 4 Quasi-Normal Modes of 4.2 Mode Analysis

image Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt
http://www.livingreviews.org/lrr-1999-2
© Max-Planck-Gesellschaft. ISSN 1433-8351
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