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As discussed above (Section 6), galactic binary systems are assumed to be distributed logarithmically along their semi-major axis A (Abt and Levy, 1976),
with minimal and maximal separations ranging from to . Accordingly, the initial frequency range covered by gravitational waves emitted by these binaries, , spans from to Hz.
Emission of gravitational waves makes the system shrink with time (on average), so that the ``blue'' end of the initial frequency distribution expands to higher frequencies. Theoretically, it can reach as high frequency as 1 kHz, corresponding to binary NS coalescence (see Figure 40).
Figure 40: Number of binary stars per frequency decade. The total
number of stars in the Galaxy is
. Numbers <1 are mathematical expectations. Analytical estimation (equation
15.3.5) is shown by the solid line (Lipunov
et al., 1995a).
As the observed distribution of binary periods is flat (equation (15.3.1)), the normalization constant is the total number of galactic binaries per frequency decade:
It is clear that under a stationary star formation rate (which is a good approximation of the situation in our Galaxy) after some time the blue end of the GWB will be fully determined by coalescing binary white dwarfs (up to 1 Hz) and NS (up to 1 kHz) (LPP87). The rate of frequency change for a coalescing binary is
where , , are masses of the binary components, and expressed in solar masses. The stationary continuity equation provides us with the spectral density of the number of stars
where f is a coalescence rate of the binaries, yr . Then the number of stars per decade is
The red end of the GWB will not change and a characteristic ``break'' will appear in the spectrum. This break occurs at the frequency defined by the matching condition
therefrom one finds
This distribution will break up at a frequency of about Hz corresponding to the limiting frequency of binary WD. Only the binary NS or BH will form the blue end of the distribution up to 1 kHz. Thus the resulting form of the distribution is:
These simple qualitative considerations fully agree with numerical calculations (see Figure 40).