As discussed above (Section 6),
galactic binary systems are assumed to be distributed logarithmically along
their semi-major axis A (Abt and Levy, 1976[2]),

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).