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wave background from Previous: Gravitational
waves from supernova
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[144]; Phinney, 1991[161]).
Theoretical estimates, however, give much higher values, of about 1 per
3000
- 10000 year (Lipunov et al., 1987a[120];
Tutukov and Yungelson, 1993a[196];
Lipunov et al., 1995e[129]).
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[128];
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)[116].
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)[182] (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
is 3
events per year (Lipunov et al., 1995a).
