The stability of the non-radially pulsating stars (Newtonian or relativistic) is determined by the Schwarzschild discriminant
where
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
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
which is the characteristic frequency of the local fluid
oscillations. Following earlier discussions when
is positive, the fluid element undergoes oscillations, while
when
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.
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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 Problems/Comments to livrev@aei-potsdam.mpg.de |