A rough estimate of the formation rate of compact binaries can be obtained ignoring many details of binary evolution. To do this, we shall use the observed initial distribution of binary orbital parameters and assume the simplest conservative mass transfer () without kick velocity imparted to the nascent compact stellar remnants during SN explosions.
[604*, 811], the present birth rate of binaries in our Galaxy can be written in factorized form as 1. Another often used form of the -distribution was suggested by Kuiper :
The range of is . In deriving the above Eq. (62*), Popova et al.  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 .
Taking Eq. (62*) at face value, assuming 100% binarity, the mass range of the primary components to , a flat distribution over semimajor axes (contact at ZAMS) (border between close and wide binaries), for close binaries, and for and for for wide binaries with , as accepted in Tutukov and Yungelson’s BPS code IBiS [791*], we get per yr, which is several times higher than modern estimates of the current Galactic SFR. However, if per year is used as a constant average SFR for 13.5 Gyr, we get the right mass of the Galactic disc. Clearly, Eq. (62*) is rather approximate, since most of the stellar mass resides in low-mass stars for which IMF, , , binary fraction, etc., are poorly known. However, if we consider only solar chemical composition stars with (which can evolve off the main-sequence in the Hubble time), we get, under the “standard” assumptions in the IBiS-code, e.g., a WD formation rate of 0.65 per yr, which is reasonably consistent with observational estimates (see Liebert et al. ), the SN II + SN Ib/c rate about 1.5/100 per yr, which is consistent with the inferred Galactic rate ) or with the pulsar formation rate . We also get the “proper” rate of WD + WD mergers with Superchandrasekhar total mass for SN Ia (a few per thousand years).
62*) tells us 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 5*); the birth rate of close massive () binaries is thus . 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 :21 From Eq. (51*) 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 that can potentially produce a NS binary 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 NS binaries. The coalescence rate of compact binaries (which is ultimately of interest to 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 constant Galactic star-formation rate and a normalization of the binary formation by Eq. (62*).
Further (semi-)analytical investigations of the parameter space of binaries leading to the formation of coalescing NS binaries are still possible but technically very difficult, and we shall not reproduce them here. The detailed semi-analytical approach to the problem of the formation of NSs in binaries and the evolution of compact binaries has been developed by Tutukov and Yungelson [787, 788*].
A distinct approach to the analysis of stellar binary evolution is based on the population synthesis method – a Monte Carlo simulation of the evolution of a sample of binaries with different initial parameters. This approach was first applied to model various observational manifestations of magnetized NSs in massive binary systems [377, 378, 147] and generalized to binary systems of arbitrary mass in  (The Scenario Machine code). To achieve a sufficient statistical significance, such simulations usually involve a large number of binaries, typically on the order of a million. The total number of stars in the Galaxy is still four orders of magnitude larger, so this approach cannot guarantee that rare stages of the binary evolution will be adequately reproduced.22
Presently, there are several population synthesis codes used for massive binary system studies, which take into account with different degrees of completeness various aspects of stellar binary evolution (e.g., the codes by Portegies Zwart, Nelemans et al. [607*, 872], Bethe and Brown , Hurley, Tout, and Pols [294*], Belczynski et al. , Yungelson and Tutukov [791*], De Donder and Vanbeveren ). A review of applications of the population synthesis method to various types of astrophysical sources and further references can be found in [602, 869*]. Some results of population-synthesis calculations of compact-binary mergers carried out by different groups are presented in Table 6.
|Authors||Ref.||NS + NS||NS + BH||BH + BH|
|Tutukov and Yungelson (1993)||[788*]||3 × 10–4||2 × 10–5||1 × 10–6|
|Lipunov et al. (1997)||[429*]||3 × 10–5||2 × 10–6||3 × 10–7|
|Portegies Zwart and Yungelson (1998)||[607*]||2 × 10–5||10–6|
|Nelemans et al. (2001)||[518*]||2 × 10–5||4 × 10–6|
|Voss and Tauris (2003)||[815*]||2 × 10–6||6 × 10–7||10–5|
|De Donder and Vanbeveren (2004)||[134*]||3 × 10–3 – 10–5||3 × 10–5||0|
|O’Shaughnessy et al. (2005)||||7 × 10–6||1 × 10–6||1 × 10–6|
|de Freitas Pacheco et al. (2006)||||2 × 10–5|
|Dominik et al. (2012)||[157*]||(0.4 – 77.4)||(0.002 – 10.6)||(0.05 – 29.7)|
|× 10–6||× 10–6||× 10–6|
|Mennekens and Vanbeveren (2013)||[479*]||10–7 – 10–5||10–6 – 10–5||0|
Actually, the authors of the studies mentioned in Table 6 make their simulations for a range of parameters. We list in the table the rates for the models the authors themselves consider as “standard” or “preferred” or “most probable”, calculated for solar metallicity (or give the ranges). Generally, for the NS + NS merger rate Table 6 shows the scatter within a factor 4, which may be considered quite reasonable, having in mind the uncertainties in input parameters. There are several outliers, [788*], [815*], and [479*]. The high rate in [788*] is due to the assumption that kicks to nascent neutron stars are absent. The low rate in [815*] is due to the fact that these authors apply in the common envelope equation an evolutionary-stage-dependent structural constant . Their range for is 0.006 – 0.4, to be compared with the “standard” applied in most of the other studies. Low favors mergers in the first critical lobe overflow episode and later mergers of the first-born neutron stars with their non-relativistic companions23. A considerable scatter in the rates of mergers of systems with BH companions is due, mainly, to uncertainties in stellar wind mass loss for the most massive stars. For instance, the implementation of winds in the code used in [607, 518*] resulted in the absence of merging BH + BH systems, while a rather low assumed in [815*] produced a high merger rate of BH + BH systems. We note an extreme scatter of the estimates of the merger rate of NS + NS binaries in [157*]: the lowest estimate is obtained assuming very tightly bound envelopes of stars (with parameter ), while the upper estimate – assuming completely mass-conservative evolution. The results of the Brussels group [134, 479*] differ from StarTrack-code results [157*] and other codes in predicting an insignificant BH-BH merging rate. This is basically due to assumed enhanced mass loss in the red supergiant stage (RSG) of massive star evolution. In this scenario, unlike, e.g., Voss and Tauris’ assumptions , the allowance for the enhanced mass loss at the Luminous Blue Variable (LBV) phase of evolution for stars with an initial mass leads to a significant orbital increase and hence the avoidance of the second Roche-lobe overflow and spiral-in process in the common envelope, which completely precludes the formation of close BH binary systems merging within the Hubble time. The Brussels code also takes into account the time evolution of Galactic metallicity enrichment by massive single and binary stars. A more detailed comparison of different population synthesis results of NS + NS, NS + BH and BH + BH formation and merging rates can be found in [2*].
A word of caution. It is hardly possible to trace the detailed evolution of each binary, so approximate descriptions of evolutionary tracks of stars, their interaction, effects of supernovae, etc. are invoked. Thus, fundamental uncertainties of stellar evolution mentioned above are complemented by (i) uncertainties of the scenario and (ii) uncertainties in the normalization of the calculations to the real Galaxy (such as the fraction of binaries among all stars, the star formation history, etc.). The intrinsic uncertainties in the population synthesis results (for example, in the computed event rates of binary mergers etc.) are in the best case not less than (2 – 3 ). This should always be borne in mind when using population synthesis calculations. However, we emphasize again the fact that the NS binary merger rate, as inferred from pulsar binary statistics with account for pulsar binary observations [77, 336, 363], is very close to the population syntheses estimates assuming NS kicks of about (250 – 300) km s–1.