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 M and angular momentum a). 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 [83Jump To The Next Citation Point89]:
[ ] 63-- 3∕10 M ω ≈ 1 − 100(1 − a) ≈ (0.37 + 0.19a), (67 )
4M [ 63 3∕10]−1 τ ≈ -------9∕10- 1 − ----(1 − a) ≈ M (1.48 + 2.09a). (68 ) (1 − a) 100
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 [6Jump To The Next Citation Point]. 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 [6Jump To The Next Citation Point]. Nevertheless, Andersson and Kokkotas [19Jump To The Next Citation Point], 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 7View Image 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:
⌊ ( ) ( ) ⌋1∕2 M 10 km 3 ωf(kHz ) ≈ 0.78 + 1.635⌈ 1.4M--- --R---- ⌉ . (69 ) ⊙
We can also find the following relation for the frequency of the first w-mode:
( ) [ ( ) ( )] ωw(kHz ) ≈ 10-km-- 20.92 − 9.14 --M---- 10-km-- . (70 ) R 1.4M ⊙ R
View Image

Figure 7: The left graph shows the numerically obtained f-mode frequencies plotted as functions of the mean stellar density. In the second graph the functional Rωw is plotted as a function of the compactness of the star (M and R are in km, ωf-mode and ωw -mode in kHz). The letters A, B, C, … correspond to different equations of state for which one can refer to [19Jump To The Next Citation Point].

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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