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Frequency distribution of galactic binaries

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 tex2html_wrap_inline12075 to tex2html_wrap_inline12077 . Accordingly, the initial frequency range covered by gravitational waves emitted by these binaries, tex2html_wrap_inline12079 , spans from tex2html_wrap_inline8945tex2html_wrap_inline12083 to tex2html_wrap_inline8945tex2html_wrap_inline12087  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 tex2html_wrap_inline9044 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 tex2html_wrap_inline8911 . 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 tex2html_wrap_inline12095 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 tex2html_wrap_inline89451 Hz) and NS (up to tex2html_wrap_inline89451 kHz) (LPP87). The rate of frequency change for a coalescing binary is


where tex2html_wrap_inline12103 , tex2html_wrap_inline12105 , tex2html_wrap_inline12107 are masses of the binary components, and tex2html_wrap_inline12109 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, tex2html_wrap_inline12115 yrtex2html_wrap_inline12117 . 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 tex2html_wrap_inline12119 defined by the matching condition


therefrom one finds


This distribution will break up at a frequency of about tex2html_wrap_inline12123  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 tex2html_wrap_inline89451 kHz. Thus the resulting form of the distribution is:


These simple qualitative considerations fully agree with numerical calculations (see Figure 40).

next up previous contents index
Next: Structure of the Galaxy Up: Model of the Galaxy Previous: Model of the Galaxy

Mike E. Prokhorov
Sat Feb 22 18:38:13 MSK 1997