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The GW from coalescing compact binaries composed of NS and BH are the best understood of all astrophysical GW sources. A conservative lower limit to the event rate of galactic binary neutron star coalescence of about 1 per 100000 year follows from double pulsar statistics studies (Narayan et al., 1991; Phinney, 1991). Theoretical estimates, however, give much higher values, of about 1 per 3000 - 10000 year (Lipunov et al., 1987a; Tutukov and Yungelson, 1993a; Lipunov et al., 1995e). Observational estimates are subject to (unknown) selection effects, whereas theoretical estimates based on evolutionary studies of binary stars use a number of poorly known parameters of binary evolution, such as the initial binary mass distribution exponent (see Section 7).
In Figure 47 we show the rates of coalescence of binary WD and NS in a model elliptical galaxy consisting of stars, depending on time elapsed since the (assumed instantaneous) star formation. Calculations were made with the adopted best values of key evolutionary parameters (Lipunov et al., 1995d; and Section 7). We note that the coalescence rate of double WD 1 per 100 year only slightly depends on the parameters, making these events an attractive mechanism for supernovae of type 1a observed in elliptical galaxies at nearly the same rates (Iben and Tutukov 1984a,b[72, 73]). Evolution of supernova rates in elliptical galaxies was first modeled by Lipunov and Postnov (1988). Unlike double WD, the coalescence rate of double NS evolves significantly with time.
Figure 47: Evolution of coalescence rates of WD+WD and NS+NS
binaries with time in a model elliptical galaxy with a mass of
(Lipunov et al., 1995a).
Having calculated the evolution of binaries in the model elliptical galaxy with a -function-like star formation, one can easily model evolution of a galaxy with an arbitrary time dependence of star formation rate. For example, a normal spiral galaxy can be approximated as having a constant star formation rate. In this case our calculations give a coalescence rate of double NS of 1 per 4000 year.
The next obvious step was to calculate the distribution of the binary NS coalescence event rates on the sky, using Tully's Nearby Galaxies Catalog. Since nothing is known about the initial binary distributions in other galaxies, we have assumed that the initial mass function of binary components and initial semi-major axes distribution f(a) of binaries in all galaxies are similar to those observed in our Galaxy:
The total number of binaries in a galaxy must be proportional to its total mass, which can be estimated by using an average mass-luminosity relation for galaxies of different morphological types. We adopted the M/L relation following Sil'chenko (1984) (see Table 14):
These values give estimations of the mass of the stellar component of galaxies (dark matter halo is not included) to an accuracy of factor 2. Star formation rates were taken to be constant for both spiral and irregular galaxies; for elliptical galaxies the star formation rate was assumed to be constant and nonzero only during the first billion years.
The resulting map of coalescence rate of binary NS is presented in Figure 48 in terms of events per one square degree per year. The event rate integrated over the whole sky is 3 per year. This is a very optimistic estimate for the LIGO experiments.
Figure 48: Coalescence rate of binary NS in terms of events per
one square degree per year in galactic coordinates. Integral event rate
events per year (Lipunov et al., 1995a).