### 5.4 Parameter estimation

For astronomy it is important not only to observe various astronomical phenomena but also to
try to mine information from these observations. From the observations of solar and stellar
oscillations (of normal stars) astronomers have managed to get details of the internal structure of
stars. In our days the GONG program [110] for detailed observation of the solar seismology
is well underway. This has suggested that, in a similar way, information about neutron star
parameters (mass, radius), and internal structure or the mass and the rotation rate of black holes
can be found, using the theory of QNMs. It will be instructive to briefly examine the case of
oscillating black holes since they are much “cleaner” objects than stars. From the normal mode
analysis of black hole oscillations we can get a spectrum which is related to the parameters of the
black hole (mass and angular momentum ). In particular, for the frequency of the
first quasi-normal mode (which as we have stated previously is the most important one for the
gravitational wave detection) the following approximate relations have been suggested [83, 89]:
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 -mode, some of
the -modes may be excited as well and a significant amount of energy could be radiated
away through these modes [6]. As far as the spacetime modes are concerned we expect that
only the curvature modes (the standard -modes) will be excited, but it is possible that the
radiated energy can be shared between the first two -modes [6]. Nevertheless, Andersson and
Kokkotas [19], using the properties of the various families of modes (, , and ), 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 and the first -mode can be
observed. In Figure 7 one can see an example of the relation between the stellar parameters and
the frequencies of the and the first -mode for various equations of state and various
stellar models. There it is apparent that the relation between the -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 -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].