Initial binary distributions. From observations of spectroscopic binaries it is possible to derive the
formation rate of binary stars with initial masses ,
(with mass ratio
), orbital
semimajor axis
, and eccentricity
. According to [327
], the present birth rate of binaries in our
Galaxy can be written in factorized form as
One usually assumes a mass ratio distribution law in the form where
is a free
parameter; another often used form of the
-distribution was suggested by Kuiper [204]:
The range of is
. In deriving the above Equation (56
), the authors of [327] took
into account selection effects to convert the “observed” distribution of stars into the true one. An
almost flat logarithmic distribution of semimajor axes was also found in [4]. Integration of
Equation (56
) yields one binary system with
and
per year in the
Galaxy, which is in reasonable agreement with the Galactic star formation rate estimated by
various methods; the present-day star formation rate is about several
per year (see, for
example, [257, 401]).
Constraints from conservative evolution. To form a NS at the end of thermonuclear evolution, the
primary mass should be at least . Equation (56
) says that the formation rate of such binaries is
about 1 per 50 years. We shall restrict ourselves by considering only close binaries, in which mass transfer
onto the secondary is possible. This narrows the binary separation interval to
(see Figure 1
);
the birth rate of close massive (
) binaries is thus 1/50 × 2/5 yr–1 = 1/125 yr–1. The
mass ratio
should not be very small to make the formation of the second NS possible. The
lower limit for
is derived from the condition that after the first mass transfer stage, during
which the mass of the secondary increases,
. Here
and
the mass of the helium core left after the first mass transfer is
. This
yields
where we used the notation , or in terms of
:
An upper limit for the mass ratio is obtained from the requirement that the
binary system remains bound after the sudden mass loss in the second supernova
explosion5.
From Equation (45) we obtain
or in terms of :
Inserting in the above two equations yields the appropriate mass ratio range
,
i.e. 20% of the binaries for Kuiper’s mass ratio distribution. So we conclude that the birth rate of binaries
which potentially can produce double NS system is
.
Of course, this is a very crude upper limit – we have not taken into account the evolution of the binary
separation, ignored initial binary eccentricities, non-conservative mass loss, etc. However, it is not easy to
treat all these factors without additional knowledge of numerous details and parameters of binary evolution
(such as the physical state of the star at the moment of the Roche lobe overflow, the common
envelope efficiency, etc.). All these factors should decrease the formation rate of double NS. The
coalescence rate of compact binaries (which is ultimately of interest for us) will be even smaller – for
the compact binary to merge within the Hubble time, the binary separation after the second
supernova explosion should be less than (orbital periods shorter than
40 d) for
arbitrary high orbital eccentricity
(see Figure 3
). The model-dependent distribution of NS kick
velocities provides another strong complication. We also stress that this upper limit was obtained
assuming a constant Galactic star-formation rate and normalization of the binary formation by
Equation (56
).
Further (semi-)analytical investigations of the parameter space of binaries leading to the formation of
coalescing binary NSs are still possible but technically very difficult, and we shall not reproduce them here.
The detailed semi-analytical approach to the problem of formation of NSs in binaries and evolution of
compact binaries has been developed by Tutukov and Yungelson [419, 420].
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