Astron. Astrophys. 346, 91-100 (1999)

3. Results

The code described above allows to generate populations of compact object binaries and trace their statistical properties. In this paper we mainly concentrate on the dependence of these properties as a function of the kick velocity distribution. In Figs. 1 and 2 we present two example evolutionary paths leading to formation of compact object binaries: black hole neutron star binary and double neutron star systems.

Fig. 1a-g. An example evolutionary path leading to formation of a black hole neutron star binary. Units of mass are solar masses, and units of distance are astronomical units: a  initial binary; b  after tidal circularization when star 1 expanded to giant size; c  system after first mass transfer, star 1 explodes as supernova; d  system after first supernova; e  after tidal circularization when star 2 expanded to giant size; f  system after second mass transfer, star 2 explodes as supernova; g  system after second supernova: black hole neutron star binary.

Fig. 2a-e. An example evolutionary path leading to formation of a neutron star neutron star binary: a  initial binary; b  after tidal circularization when both stars expanded to giant sizes, system ejects common envelope; c  system after common envelope phase, star 1 explodes as supernova; d  system after first supernova, now star 2 explodes as supernova; e  system after second supernova: neutron star neutron star binary.

Let us first consider different evolutionary paths of the binaries depending on their initial parameters. We first consider a model with the Cordes & Chernoff (1997) kick velocity distribution. Tracing the evolution now depends on four parameters: M (primary initial mass), q (initial mass ratio), a (initial semi-major axis), and e (initial eccentricity). In order to visualize the evolutionary effects we fix two of them and present the types of systems obtained in the course of the binary evolution. In Fig. 3 we fix the initial orbital separation and eccentricity . The graphs are empty in the lower left part for which . This is the region for which the secondary is not massive enough to undergo a supernova explosion. However, in some systems for which the primary mass is , the mass of the secondary is increased by accretion when the primary goes through the giant phase. Most of the systems end up either by disruption in the first supernova explosion or by a spiral in of the neutron star to the secondary (top panels) The remaining systems may be disrupted in the second supernova explosion. The compact object binary population (bottom right panel) is bimodal. The neutron star neutron star binaries are formed from the systems with q very close to unity, while the black hole neutron star systems are primarily formed when q is intermediate. Thus the initial distribution of the mass ratio q has a strong influence on the production rates of these two types of compact object binaries. The number of systems shown in each panel does not correspond to the actual production rates. We present one thousand systems in each panel. The actual calculation produces 51% systems with the secondary not massive enough to undergo a supernova explosion, 40% systems torn after the first explosion, 6.2% of systems merged after the first explosion, 2.7% systems torn after the second explosion, and 0.35% of compact object binaries.

Fig. 3. Possible results of the evolution of a binary with the initial orbital separation and eccentricity for different values of the initial primary mass M and the initial mass ratio q. The top left panel shows the systems that were disrupted in the first supernova explosion. The solid line corresponds to the initial secondary mass . The top right panel shows the systems in which the neutron star, born in the first supernova explosion, merged with the companion. The bottom left panel shows the systems disrupted in the second supernova explosion. The crosses in the bottom right panel show the neutron star binaries and the filled circles are the black hole neutron star binaries. We present one thousand of each type of binaries. In this calculation we used the Cordes & Chernoff (1997) kick velocity distribution.

In Fig. 4 we present the results of the evolution of systems for which we fix the initial primary mass and the initial mass ratio, while varying the the initial orbital parameters a and e. The systems with small orbital separations and high eccentricity merge in the early phase of the evolution and are not considered in this paper. Disruption after the first supernova explosion may occur to all systems. The population of systems that end up merging after the first explosion, are disrupted in the second explosion, or form a compact object binary originate from the same region of the parameter space. One should note that because of the circularization of orbits already in the initial stages of the binary evolution the final population is not very sensitive to the initial eccentricity. On the other hand the distribution of the initial orbital separation is important, as the compact object binaries originate in systems with relatively small orbital separations (see bottom right panel in Fig. 4). One should note that all the compact object binaries shown in Fig. 4 are black hole neutron star binaries. Double neutron star systems are formed only when q is nearly unity in our simulations. As before the number of systems shown in each panel does not correspond to the actual production rates, and show one thousand systems in each panel. The actual calculation produces 12% systems born in contact, 43% systems with the secondary not massive enough to undergo a supernova explosion, 34% systems torn after the 1st explosion, 0.7% systems merged after the first explosion, 0.50% systems torn in the second explosion and only 0.15% mergers.

Fig. 4. Possible results of the evolution of a binary with the initial primary mass and the mass ratio . The top left panel shows the systems disrupted in the first supernova explosion, the top right panel shows the systems in which the neutron star, born in the first supernova explosion, merged with the companion. The bottom left panel shows the systems disrupted in the second supernova explosion. The bottom right panel shows the black hole neutron star binaries, and there are no neutron star binaries formed for this mass ratio (). We present one thousand of each type of binaries. In this calculation we used the Cordes & Chernoff (1997) kick velocity distribution.

We present the dependence of the production rates of different types of compact object binaries on the width of the kick velocity distribution in Fig. 5. The number of double neutron star systems that merge within the Hubble time increases with the kick velocity, while the production rate of black hole neutron star systems becomes smaller. These two rates are nearly equal when the kick velocity is roughly that given by Cordes & Chernoff (1997).

Fig. 5. Population of the compact object binaries in the four categories: double neutron star binaries that merge within the Hubble time (empty squares), and these that do not (empty circles), neutron star black hole binaries that merge within the Hubble time (filled squares), and these that do not (filled circles). The error bars represent the counting statistics of the simulation.

In Fig. 6 we present the distributions of the orbital parameters of compact object binaries for a few representative values of the width of the kick velocity distribution . Each panel contains one thousand compact object binaries. To understand these plots let us first consider the properties of the population of objects just before the second supernova explosion in the case km s^-1. Some of the systems are wide, with the eccentricity varying from zero to unity, however, most of the systems populate a region with eccentricities above . These systems originate in binaries with small initial mass ratios. A characteristic property of this group is a correlation between eccentricity and period. Objects in this group originate in systems with small initial value of q. The masses of the primary compact objects in these systems have a narrow distribution, the mass of the secondary after the accretion phase weakly depends on the initial mass before accretion - see Eq. 8, and is typically . The mass of the secondary just before the second supernova explosion, is the mass of the helium core of secondary star and is in the range . The orbital parameters of such system after an explosive mass loss (second supernova explosion) depend only on the total mass loss, see e.g. Pols & Marinus (1994). In our calculation the newly formed neutron star has a mass of , thus the relative mass loss is in the range . Circular systems that loose more than half of the mass () are unbound, while in eccentric systems more than 50% mass loss may be required to dissociate a binary. Other systems become eccentric and the eccentricity is given by . Hence the lowest eccentricity a system can have after the second supernova explosion in our simulations is . The relation between the new orbital period and the new eccentricity for different relative mass loss x is

where is the orbital period before the explosion. This relation defines the curved shape of the distribution in the diagram. The objects populating the right hand side of the plot, originate from systems with intermediate or large q. The intermediate q objects went through accretion onto the neutron star (regime II of a mass transfer described above) and therefore similar reasoning as above applies to them. However systems with high initial value of q results in wide binaries (periods larger than 100 days) with different eccentricities. The compact object binaries in Fig. 6 (top left panel) with eccentricities below 0.45 and orbital periods smaller then days originate in binaries of intermediate and large initial mass ratio.

Fig. 6. Distribution of the initial orbital compact binary systems parameters in the space spanned by the period P and eccentricity e, for four different Gaussian distributions of the kick velocity km s-1 - top left panel , 200 km s-1 - top right panel , 400 km s-1 - bottom left panel , and 800 km s-1 - bottom right panel . Each panel contains 1000 objects.

In the case of non zero kick velocity the systems with high q are very unlikely to survive. In fact there are only two such systems on the plot for km s^-1, and none for higher velocities. The escape velocity in long period systems is low, and if such systems existed after the first supernova event, they have are very likely disrupted in the second supernova explosion.

As the kick velocity is increased the shape of the distribution in the diagram also changes. In the case km s^-1 there is a small fraction of high eccentricity, low period systems. This region of the parameter space fills up as the kick velocity increases, see lower panel in Fig. 6 where the case km s^-1 and km s^-1 are shown. This is due to the fact that with the increasing kick velocity the long period systems are easier to disrupt and the fraction of surviving short period systems becomes larger. We note that the distributions shown in Fig. 6 are similar to those obtained previously (Portegies Zwart & Spreeuw, 1996; Tutukov & Yungelson, 1993).

In Fig. 7 we present the cumulative distributions of the lifetimes of compact object binaries for the same set of kick velocities as in Fig. 6. In the case of no kick velocity km s^-1 the distribution is bimodal: the systems with small q merge typically within the Hubble time (which we take to be 15 Myr). However the systems with higher q remain in wide orbits and their merger times exceed the Hubble time. With the increasing kick velocity only the tightly bound systems survive. The lifetime of a system scales with the fourth power of the semi-major axis, and therefore the median lifetime decreases with increasing kick velocity. In the case of km s^-1 the median lifetime is  years, and it decreases by a factor of four when the kick velocity is doubled. For the highest kick velocity velocity km s^-1 the median lifetime is only  years. One should note however, that the distribution of is very skewed, and we present it on a logarithmic axis. Thus there is a long tail extending to about the Hubble time. Most of the mergers take place in the Hubble time and only a small fraction of the total population lasts longer.

Fig. 7. Cumulative distributions of the - the lifetimes of compact object binaries for four different kick velocities: the solid line for km s-1, the short dashed line for km s-1, the long dashed line for km s-1, and the dash dotted line for km s-1. Each line was plotted from a distrbution of 1000 objects.

Finally in Fig. 8 we present - the fraction of all binaries with the primary star more massive than that produce a pair of compact objects in a binary system. We calculate for a large range of the kick velocity distribution widths. To calculate each point in Fig. 8 we calculated one thousand of compact binaries. This fraction falls down very approximately exponentially with the increasing kick velocity. We have fitted a modified exponential to this dependence

The fit is shown by the dashed line in Fig. 8 and is accurate to about 6%. Note that Eq. (13) can be used to determine the merger fraction for any kick velocity distribution that can be expressed as a linear combination of three dimensional Gaussians. Eq. (13) can be used together with Fig. 5 to obtain the production rates of any type of compact object binaries as a function of the width of the kick velocity distribution.

Fig. 8. Fraction of binary compact objects that merge within the Hubble time (15 Gyrs), as a function of the width of the kick velocity distribution. We also show the fit discussed in the text. Each calculation produced 1000 merging binaries.

The actual compact object merger rate in the Galaxy can be calculated given the observed supernova rate, the fraction of stars that exist in binaries, and assuming some form of the star formation history. Assuming that there is one supernova explosion every fifty years in the Galaxy and that binary fraction is 50%, we denote the supernova rate as  year^-1, and the binary fraction as , where and are factors of the order of unity.

In the simplest case when we assume that the star forming process has been going at the same rate throughout the history of the Galaxy we obtain the compact object merger rate

where is expressed in percents. Taking values of from our Fig. 8, we may calculate number of expected compact objects gravitational merging events, which for, let say, km s^-1 will be 0.0002 per year which yields approximately 1 event per 5000 years per Galaxy.

Since the compact object merger rate is directly proportional to it also depends exponentially on the kick velocity distribution width! The assumption about the constant star formation throughout the history of the Galaxy is in fact not really crucial. As we have seen most of the mergers take place within a few times years even for rather small velocity kicks, therefore in reality we are only concerned about the star forming history in the last years. The star forming rate in the Galaxy has most probably been constant over the last  years (Miller & Scalo, 1979).

The calculation of the detection rate in gravitational wave detectors (Abramovici et al., 1992) requires a number of additional assumptions, like for example the galaxy density in our local and far neighborhood etc., for a discussion see e.g. Phinney (1991); Chernoff & Finn (1993). Regardless of the assumptions the detection rate is proportional to the compact object merger rate in the Galaxy, provided that stellar populations are similar in other galaxies. When calculating the expected rates one has to take into account the masses of the systems that merge. The volume of the space in a flux limited sample of events scales as (Tutukov & Yungelson, 1993). Thus mergers of heavy objects like a neutron star and a black hole, or a pair of black holes are visible in a larger volume and may yield a similar observational rate despite the fact that they are not as frequent as the double neutron star mergers.

© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999
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