

Although the perturbation equations in the exterior of a star are
similar to those of the black-hole and the techniques described
earlier can be applied here as well, special attention must be
given to the interior of the star where the perturbation
equations are more complicated.
For the time independent case the system of equations (51
,
52
,
53
) inside the star reduces to a 4th order system of ODEs [79]. One can then even treat it as two coupled time independent
wave equations
. The first equation will correspond to the fluid and the second
equation will correspond to spacetime perturbations. In this way
one can easily work in the Cowling approximation (ignore the
spacetime perturbations) if the aim is the calculation of the QNM
frequencies of the fluid modes (f,
p,
g, ...) or the Inverse Cowling Approximation [22] (ignore the fluid perturbations) if the interest is in
w
-modes. The integration procedure inside the star is similar to
those used for Newtonian stars and involves numerical integration
of the equations from the center towards the surface in such a
way that the perturbation functions are regular at the center of
the star and the Lagrangian variation of the pressure is zero on
the surface (for more details refer to [141
,
130
]). The integrations inside the star should provide the values of
the perturbation functions on the surface of the star where one
has to match them with the perturbations of the spacetime
described by Zerilli's equation (21
,
23
,
24
).
In principle the integrations of the wave equation outside the
star can be treated as in the case of the black holes. Leaver's
method of continued fraction has been used in [119,
138], Andersson's technique of integration on the complex
r
plane was used in [21] while a simple but effective WKB approach was used by Kokkotas
and Schutz [130
,
211
].
Finally, there are a number of additional approaches used in
the past which improved our understanding of stellar oscillations
in GR. In the following paragraphs they will be discussed
briefly.
-
Resonance Approach.
This method was developed by Thorne [199], the basic assumption being that there are no incoming or
outgoing waves at infinity, but instead standing waves. Then by
searching for resonances one can identify the QNM frequencies.
The damping times can be estimated from the half-width of each
resonance. This is a simple method and can be used for the
calculations of the fluid QNMs. In a similar fashion
Chandrasekhar and Ferrari [59] have calculated the QNM frequencies from the poles of the
ratio of the amplitudes of the ingoing and outgoing waves.
-
Direct Numerical Integration.
This method was used by Lindblom and Detweiler [141] for the calculation of the frequencies and damping times of
the
f
-modes for various stellar models for thirteen different
equations of state. In this case, after integration of the
perturbation equations inside the star, one gets initial data
for the integration of the Zerilli equation outside. The
numerical integration is extended up to ``infinity'' (i.e. at a
distance where the solutions of Zerilli equations become
approximately simple sinusoidal ingoing and outgoing waves),
where this solution is matched with the asymptotic solutions of
the Zerilli equations which describe ingoing and outgoing
waves. The QNM frequencies are the ones for which the amplitude
of the incoming waves is zero. This method is more accurate
than the previous one at least in the calculation of the
damping times as has been verified in [19
], but still is not appropriate for the calculation of the
w
-modes.
-
Variational Principle Approach.
Detweiler and Ipser [78
] derived a variational principle for non-radial pulsational
modes. Associated with that variational principle is a
conservation law for the pulsational energy in the star. The
time rate of change of that pulsational energy, as given by the
variational principle, is equal to minus the power carried off
by gravitational waves. This method was used widely for
calculating the
f
[74] and
g
-modes [87] and in studies of stability [78,
75].
-
WKB.
This is a very simple method but quite accurate, and contrary
to the previous three methods it can be used for the
calculation of the QNM frequencies of the
w
-modes (this was the first method used for the derivation of
these modes). In practice one substitutes the numerical
solutions of the Zerilli equation with their approximate WKB
solutions and identifies the QNM frequencies as the values of
the frequency for which the amplitude of the incoming waves is
zero [130
,
211].
-
WKB-Numerical.
This is a combination of direct integration of the Zerilli
equation and the WKB method. The trick is that instead of
integrating outwards, one integrates inwards (using initial
data at infinity for the incoming wave solution); this
procedure is numerically more stable than the outward
integration. On the stellar surface one needs of course not the
solution for incoming waves but the one for outgoing ones. But
through WKB one can derive approximately the value of the
outgoing wave solution from the value of the incoming wave
solution. In this way errors are introduced, nevertheless the
results are quite accurate [130].


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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
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