5.2 Population synthesis results

A distinct approach to the analysis of binary star 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 [19719879] and generalized to binary systems of arbitrary mass in [226] (The Scenario Machine code). To achieve a sufficient statistical significance, such simulations usually involve a large number of binaries, typically of 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 reproduced6.

Presently, there are several population synthesis codes used for massive binary system studies, which take into account with different degree of completeness various aspects of binary stellar evolution (e.g., the codes by Portegies Zwart et al. [329Jump To The Next Citation Point469], Bethe and Brown [33], Hurley, Tout, and Pols [156], Belczynski et al. [24], Yungelson and Tutukov [423Jump To The Next Citation Point]). A review of applications of the population synthesis method to various types of astrophysical sources and further references can be found in [326467]. Some results of population synthesis calculations of compact binary mergers carried out by different groups are presented in Table 4.

Table 4: Examples of the estimates for Galactic merger rates of relativistic binaries calculated under different assumptions on the parameters entering population synthesis.

Authors Ref. NS + NS NS + BH BH + BH

Tutukov and Yungelson (1993) [420Jump To The Next Citation Point] × 10–4 × 10–5 × 10–6
Lipunov et al. (1997) [228Jump To The Next Citation Point] × 10–5 × 10–6 × 10–7
Portegies Zwart and Yungelson (1998) [329Jump To The Next Citation Point] × 10–5 10–6
Nelemans et al. (2001) [286Jump To The Next Citation Point] × 10–5 × 10–6
Voss and Tauris (2003) [437Jump To The Next Citation Point] × 10–6 × 10–7 10–5
O’Shaughnessy et al. (2005) [298Jump To The Next Citation Point] × 10–6 × 10–6 × 10–6
de Freitas Pacheco et al. (2006) [72] × 10–5

Actually, the authors of the studies mentioned in Table 4 make their simulations for a range of parameters. We list in the table the rates for the models which the authors themselves consider as “standard” or “preferred” or “most probable”. Generally, for the NS + NS merger rate Table 4 shows the scatter within a factor ∼ 4, which may be considered quite reasonable, having in mind the uncertainties in input parameters. There are two clear outliers, [420Jump To The Next Citation Point] and [437Jump To The Next Citation Point]. The high rate in [420Jump To The Next Citation Point] is due to the assumption that kicks to nascent neutron stars are absent. The low rate in [437Jump To The Next Citation Point] is due to the fact that these authors apply in the common envelope equation an evolutionary-stage-dependent structural constant λ. Their range for λ is 0.006 – 0.4, to be compared with the “standard” λ = 0.5 applied in most of the other studies. A low λ favours mergers in the first critical lobe overflow episode and later mergers of the first-born neutron stars with their non-relativistic companions7. 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 [329286Jump To The Next Citation Point] resulted in the absence of merging BH + BH systems, while a rather low M˙ assumed in [437] produced a high merger rate of BH + BH systems.

A word of caution should be said here. It is hardly possible to trace a detailed evolution of each binary, so one usually invokes the approximate approach to describe the change of evolutionary stages of the binary components (the so-called evolutionary track), their interaction, effects of supernovae, etc. Thus, fundamental uncertainties of stellar evolution mentioned above are complemented with (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 allstars, 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 of the order of factor two or three. This should always be born in mind when using the population synthesis calculations. However, we emphasize again the fact that the double NS merger rate, as inferred from binary pulsar statistics with account for the double pulsar observations [49182], is very close to the population syntheses estimates with a kick of about (250 – 300) kms.

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