These two relations can be inverted and thus from the
``observed'' frequency and the damping time we can derive the
parameters of the oscillating black hole. In practice, the noise
of the detector will contaminate the signal but still (depending
on the signal to noise ratio) we will get a very accurate
estimate of the black hole parameters. A similar set of empirical
relations cannot be derived in the case of neutron star
oscillations since the stars are not as ``clean'' as black holes,
since more than one frequency contributes. Although we expect
that most of the dynamical energy stored in the fluid
oscillations will be radiated away in the
f
-mode, some of the
p
-modes may be excited as well and a significant amount of energy
could be radiated away through these modes [5]. As far as the spacetime modes are concerned we expect that
only the curvature modes (the standard
w
-modes) will be excited, but it is possible that the radiated
energy can be shared between the first two
w
-modes [5
]. Nevertheless, Andersson and Kokkotas [19
], using the properties of the various families of modes (f,
p, and
w), managed to create a series of empirical relations which can
provide quite accurate estimates of the mass, radius and equation
of state of the oscillating star, if the
f
and the first
w
-mode can be observed. In figure
7
one can see an example of the relation between the stellar
parameters and the frequencies of the
f
and the first
w
-mode for various equations of state and various stellar models.
There it is apparent that the relation between the
f
-mode frequencies and the mean density is almost linear, and a
linear fitting leads to the following simple relation:
We can also find the following relation for the frequency of the first w -mode:
From tests performed using polytropic stars to provide data for the above relations it was seen that these equations predict the masses and the radii of the polytropes usually with an error less than 10%. There is work underway towards extracting the parameters of the star from a noisy signal [126].
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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 |