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As is known, the observed distribution of massive stars in the direction of the Galactic center looks very peculiar: the vast majority of all massive stars are concentrated towards the central 1 pc or so (Genzel et al. 1994). The observed X-ray sources demonstrate concentration towards the center, although significantly less pronounced: more than half of them are occupying a region of 750 pc 750 pc in size (Fig. 2).
In order to compare the observed distribution of X-ray sources in the Galactic center with what would be expected from the population synthesis computations, we have considered two hypothetical scenarios for the location of the starburst. In Scenario I, the progenitors of X-ray binaries were formed in a region of 1 pc in size. The resulting X-ray systems were ejected into and scattered within the central 1 kpc or so due to the `kick' that accompanied the formation of those systems. In Scenario II, the starburst happened on a scale pc.
Scenario I: the starburst occured in a central region of 1 pc in size
The mass distribution at the Galactic center was taken in the form (Lacy et al. 1991):
It is straightforward to show by integrating the motion of a star in the potential well due to the above mass distribution that the distance reached by the star and its velocity are related by:
where (in 100 km/s) is the initial ejection velocity at a radius . This implies that, in order to reach r=1 kpc from pc even with a zero velocity, needs to be as high as 450 km/s. For pc, the required km/s is less but not by a substantial factor. Such high velocities cannot be reached by imposing a `kick' onto an initial velocity dispersion of the newly formed stars without destroying binaries. This is supported by the absence of fast moving massive X-ray binary systems in our Galaxy (say, with v>50 km/s). Therefore, explaining the observed wide distribution of X-ray binaries at the Galactic center within Scenario I looks unlikely.
Scenario II: the starburst occured in a central region on a scale pc
How could one explain the origin of a starburst well outside the central parsec, where no material appropriate for an extensive star formation is currently seen ?
A feasible mechanism that might trigger (recurrent) starbursts in the central region of the Galaxy - collisions between giant molecular clouds - has been proposed by Ozernoy (1995). Each collision between two GMCs occuring with an average time interval of gives rise to the dissipation of a substantial part of the angular momentum of each of the clouds; as a result, they end up on much lower orbits. Besides, after the collision and dissipation of internal turbulent motions the clouds become gravitationally unstable, they could fragment and experience star formation. Therefore, a `wave of star formation' could start at comparatively large distances from the Galactic center and gradually propagate towards the center, accompanied by the fall of the remnants of the clouds onto the center.
In order to quantify this scenario, let us assume that the collision between two molecular clouds occurs at a large distance (about 500-750 pc) from the center and produces a shock that initiates an instantaneous starburst. Suppose that the stars formed kept the initial internal velocity dispersion within the molecular clouds (say, 3 km/s). Since a substantial part of the transverse velocities of the clouds is lost in the collision, the remnant (stars + gas) will be falling towards the Galactic center. Due to the conservation of angular momentum, the velocity dispersion of the stars will be growing as and will reach its maximum when the cloud passes at its minimum approach from the Galactic center. If this distance is pc (i.e. comparable to the initial radii of the clouds), the velocity dispersion of stars reaches km/s, while the systematic velocity of the cloud acquired at the central potential well turns out to be km/s as calculated above. A combination of these large systematic and chaotic velocities is expected to be the major factor leading to the scattering of the formed stars in the area of about 750 pc around the center.
For binary stars able to produce X-ray sources, two more factors could contribute to this scattering: (i) ejection of mass during the supernova explosion even if the ejection was spherically-symmetric relative to the exploding star, and (ii) a `kick' that the binary acquires as a result of an asymmetry of the supernova explosion. As for (i), an estimation for the acquired velocity ranges between 20 and 100 km/s (e.g. Shore et al. 1994). As for (ii), even if we use a rather large estimate of 400 km/s for `kick' velocity, which is being currently under discussion in literature (Lyne & Lorimer 1994), then a massive binary acquires a recoil velocity by a factor of 10 less unless it is disrupted. Therefore both effects (i) and (ii), while occuring for binaries during their infall onto the Galactic center, could result even in a larger scattering than that for the single stars.
Figures 2a to 2c represent the results of some of our model simulations confronted with the distribution of the GRANAT sources and NS+Be systems (the latter are more numerous than BH-sources both in the GRANAT observations and in our calculations). Initialy all the binaries have small, stochastically oriented peculiar velocities with a Maxwellian distribution and dispersion of 3 km/s. At the moment of the SN explosion they acquire, due to a `kick' and mass ejection from the system, an additional velocity of about 75 km/s (see section 2.2), also stochastically oriented in space. We have explored several variants of the resulting (by T=6-8 Myr) spatial distribution of the binaries formed in an instantaneous starburst for different initial locations of the starburst and different rotational velocities about the Galactic center, , ranging from zero to the circular velocity, .
If the starburst's distance from the center is pc and is not too large, the resulting spatial distribution, as can be seen in Fig. 2a, turns out to be quite extended and more or less symmetric, consistent with the observed distribution. Fig. 2a shows the 2D-projection of a representative spatial distribution by T=7 Myr. The starburst is assumed to occur at the distance of pc from the center on the line of sight and its center's coordinates are projected to the point pc, pc. The initial, after cloud-cloud collision, rotational velocity of the cloud about the Galactic center is taken to be km/s. If the initial position of the starburst is not located on the line of sight, the results do not change appreciably unless the value of is large enough.
The larger , the larger is the asymmetry of the spatial distribution because the stars, under those circumstances, are not able to reach the center and thereby to increase enough their velocity dispersion. Hence those systems cannot move far enough from the rotation plane, and this results in a non-sphericity, representative examples of which are shown in Fig. 2b and Fig. 2c. A rapid rotation about the center shifts those binaries in the direction of rotation (to the right on the figures) so as to make the distribution asymmetric relative to the Y-axis (which is perpendicular to the rotation plane). The asymmetry illustrated by Fig. 2b is not well pronounced (and therefore might be still consistent with the observed distribution of X-ray sources). On the contrary, the asymmetry shown in Fig. 2c is evidently at odds with what is observed. The asymmetry would be much weaker if we choosed such initial conditions, that, after 7 Myr from the starburst's onset, the center of the binaries' distribution would be situated on the line of sight: beyond the Galactic center or (as in Fig. 2b) in front of it. A large, compared to what is observed, asymmetry left by Myr makes such cases, as in Fig. 2c, unlikely although available free parameters (age, could weaken the anisotropy relative to the Y-axis (but still leaving the non-sphericity). Another source of asymmetry are substantially larger (say, 700-900 pc), at which a value of T=6-8 Myr would not be enough for recently born stars to reach the central region.
In sum, Fig. 2a and 2b indicate that, by Myr, a starburst would produce a quasi-isotropic projected distribution of X-ray sources occupying a large region around the center with the size of several hundred parsecs, consistent with the data, for a rather wide range of initial conditions (initial distance from the Galactic center pc, not too large ).