\documentstyle[12pt, aasms, epsf]{article}
\catcode`@=11
\def\eqalign#1{\null\,\vcenter{\openup\jot\m@th
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
\def\eqalignleft#1{\null\,\vcenter{\openup\jot\m@th
  \ialign{\strut$\displaystyle{##}$\hfil&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
\catcode`@=12

\def\FIG#1#2{%
\vbox to\vsize{%
\vss%
{\epsfxsize=\hsize
\centerline{\epsfbox{#1}}
\bigskip\bigskip
\centerline{\bf Figure {#2}}
}
\vss%
}}

%\received{ } %\accepted{ } %\journalid{}{} %\articleid{}{}

\begin{document}

%\thesaurus{1(1;2;3)}
\title {
Population synthesis of X-ray sources at the Galactic center
}


\author{V.M.Lipunov\altaffilmark{1}, L.M.Ozernoy\altaffilmark{2},
S.B.Popov\altaffilmark{1}, K.A.Postnov\altaffilmark{1},
and M.E.Prokhorov \altaffilmark{1}}


\affil{$^1$ Sternberg Astronomical Institute, Moscow University,
		119899 Moscow, Russia}
\altaffiltext{1}{e-mail: lipunov@sai.msu.su}
\altaffiltext{2}{e-mail: ozernoy@hubble.gmu.edu; ozernoy@stars.gsfc.nasa.gov}
%\and
\affil{$^2$
Dept. of Physics \& Astron. and Inst. for Comput. Sci. \&
Inform., George Mason Univ., \\ also  Laboratory for Astronomy
\& Solar Physics, NASA/GSFC}

\begin{abstract}
We are exploring the evolution of massive binary star
populations for a nuclear starburst  occuring in the conditions similar
to the Milky Way galaxy, in its central part, on a time scale of 10 Myr.
A  computer code  is applied  allowing for
computing, by using Monte Carlo simulations,  the evolution of a large
ensemble of binary systems, with proper accounting for spin evolution
of magnetized neutron stars (NSs). Our results include the number of
X-ray transients (NS + main sequence star), super-accreting black holes
(BHs), and binaries consisting of a BH + supergiant, all as functions
of time.

We find that by 7 Myr after such a starburst one expects $\approx$ 1 X-ray
source with a BH (Cyg X-1 type), $\approx$ 1 SS 433-like source
(BH in the regime  of superaccretion), and $\approx$ 37 transient sources
with a NS, all to be within the central 1 kpc or so. An interesting result that
can be considered as a specific starburst feature, is that the ratio of
the number of  systems like SS 433
to the  number of X-ray transients  is about 1:100,  compared
to 1:1000, characteristic  to the average ratio in the
galactic field. The ratio of the total number  of  X-ray  sources  containing a
BH (of Cyg X-1 $+$ SS 433 types) to  the number of X-ray transients with NSs
turns out to be a sensitive function of the  age
of  the starburst, and its computed value  $\approx$  0.04 is consistent
with observations.

\end{abstract}

{\it Subject headings}: Galaxy: center -- stars: neutron stars: evolution --
 stars: formation -- stars: statistics -- X-rays: stars

%
%____________________________________________________________________________
%

\section {Introduction}
Since IRS 7 at the Galactic center was identified as a luminous red
supergiant (Lebofsky et al. 1982), a lasting debate has been initiated
whether the nucleus of the Milky Way galaxy had experienced a recent
star formation episode. For more than a decade, many arguments both
{\it pro} and {\it contra} have been accumulated. One of the strongest
arguments for the presence of a young stellar population was the
finding of luminous HeI/HI emission line stars both in IRS 16 (Hall et al.
1982) and in the central stellar cluster (Krabbe et al. 1991). The
spectroscopic study of the AF star, one of the most remarkable
representatives of these stars, came up with the conclusion that they
are probably related to luminous, blue supergiants in a short-lived
phase of intense mass loss (Najarro et al. 1993). However Morris (1993)
proposed that the interpretation of the HeI/HI emission line stars in
terms of young, massive stars is not the only possibility. The finding of a
very high stellar density in the central star cluster (Eckart et al.
1993) is seemingly consistent with the other interpretations: the
HeI/HI stars could be either 10 ${\rm M_\odot}$ black holes (BHs) that have
collided with giants (Morris 1993) or could result from collisions and
mergers of less massive stars in the cluster (for discussion of these
possibilities, see Genzel et al. 1994).

Until recently, this conflicting arguments have remained insufficient
to resolve the issue. Ozernoy et al. (1993) suggested a new approach by
showing that the 10 KeV gas in the central 200 pc of the Galaxy (Koyama et
al. 1989) could be produced by multiple supernova explosions. Such supernovae,
if sequential, might be associated with a starburst whose parameters
are a scaled-down version of those of starburst galaxies (Ozernoy 1994).
Related evidence was suggested  by Sofue (1994) using the North Polar Spur
data; it remains to be seen, however, whether this spur is associated
with the Galactic center indeed rather than being a local feature. Tamblyn
\& Rieke (1993) and Schaerer (1994) proposed starburst models which would
be able to account for a young stellar population at the Galactic center,
although the origin of some peculiar hot stars still remains to
be explained (Tamblyn et al. 1996). A recent finding of a possible WR star at
a projected distance of 0.5 pc (Blum et al. 1995) seems to be one of yet
missing links to massive stars produced during the starburst.

The purpose of this paper is to explore some consequences of the
putative starburst at the Galactic
center by elaborating the population synthesis model for the
late evolutionary stages of massive binary stars created in
the starburst, with emphasis on X-ray source statistics.  As is known,
X-ray observations of the Galactic center have
revealed a number of energetic X-ray sources located in the innermost
regions of the Galaxy (Pavlinsky et al. 1994, Churasov et al. 1994),
with a part of them being attributed, by their spectral
characteristics, to BH candidates.  These observations
demonstrate a substantially enhanced, compared to the average galactic value,
spatial density of X-ray binary systems in the central region of the
Galaxy. In this paper, we show that such a situation can be natural
consequence of binary stellar evolution if a starburst occurred a few
million years ago at the Galactic center.  We apply Monte Carlo
simulations to the evolution of a large ensemble of binary systems,
with proper accounting for the spin evolution of magnetized compact stars
(Lipunov 1992, Lipunov et al. 1994).
This method has been shown to be a powerful tool for studying
different products of stellar evolution both in spiral and elliptical
galaxies under a wide range of initial conditions and star formation
histories (see e.g.  Kornilov \& Lipunov 1984, Lipunov \& Postnov 1987,
Lipunov et al. 1994 and references therein).

In this paper, we will
focus on the most prominent, from the observational point of view,
representatives of the late stages of massive binary evolution, such as
X-ray transients (NS in a highly eccentric orbit around a main sequence
star, like A0535+26), super-accreting BHs (observationally seen as SS
433 if in pair with a Roche lobe filling secondary component), and BH +
supergiant binaries (like Cyg X-1, with an evolved supergiant
underfilling its Roche lobe). Different kinds of binary
pulsars which appear after a star formation burst
were considered previously (Lipunov et al.
1995a).  Here we calculate the numbers of the corresponding X-ray
sources and diccuss their spatial distribution.
We include in our study only massive
X-ray systems, because during the first 10 Myr only massive stars ($
M>15-20\, {\rm M}_\odot$) can leave the main sequence.
%
%____________________________________________________________________________
%
%\pagebreak
\section {The Model}

Monte-Carlo method for statistical simulation of binary evolution
was originally proposed by Kornilov \& Lipunov (1983a,b) for massive binaries
and developed later by Lipunov \& Postnov (1987) for low-massive binaries.
Dewey \& Cordes (1987) applied an analogous method
for analysis of radio pulsar statistics, and de Kool (1992)
investigated  by the Monte-Carlo method  the formation  of the galactic
cataclysmic variables. Recently Leitherer \& Heckman (1995) have modelled
star formation for elliptical and spiral galaxies, taking into
consideration only the single stars.

Monte-Carlo simulations of binary star evolution allows one to
investigate the evolution of a large ensemble of binaries  and to
estimate the number of binaries at different
evolutionary stages. Inevitable simplifications in the
analytical description of the binary evolution that we allow in our
extensive numerical calculations, make those numbers
approximate to a factor of 2-3.  However, the inaccuracy of direct
calculations  giving the numbers of different binary types
in the Galaxy (see e.g. Iben \& Tutukov 1984, Meurs \& van den Heuvel
1989, van den Heuvel 1994) seems to be comparable to what follows from the
simplifications in
the binary evolution treatment.  Moreover, no rotational evolution of
magnetized compact stars has been properly considered in those papers.

In our analysis of binary evolution, we use the `scenario machine', a
computer code that incorporates all current scenarios of binary
evolution (for a review, see van den Heuvel 1994)
and takes into account the influence of magnetic field of
compact objects on their observational appearance. A detailed description
of the computational techniques and input assumptions is summarized
elsewhere (Lipunov et al. 1995b), and here we list only principal
parameters and initial distributions.

We trace the evolution of binary systems during the first 10 Myr after their
formation in a star burst. Obviously, only
massive enough stars (with masses $\ge 8-10~ {\rm M}_\odot$) can evolve off
the main sequence during the time as short as this to yield compact
remnants (NSs and BHs).
Therefore we consider only massive binaries, i.e. those having the mass of
the primary (more massive) component in the range of $10-120$
${\rm M}_\odot$.

\subsection{Initial binary parameters}

To start the calculations, we choose  the distributions
of initial binary parameters: mass of the primary
ZAMS component, $M_1$; binary
mass ratio, $q=M_2/M_1<1$; and orbital separation, $a$. Initial
eccentricity is assumed to be zero.

The distribution in orbital separations is taken as deduced from observations
(Abt 1983):
\begin{equation}
f(\log a) ={\rm const}\,,\qquad \max~\{10~ {\rm R}_\odot,~\hbox{Roche Lobe}~
(M_1)\} < \log a < 10^4~{\rm R}_\odot.
\end{equation}
As for the mass ratio distribution in binaries, the observational information
is poor. Meanwhile, from the evolutionary point of view, it is particularly
important how different are the initial masses of the components
(see, e.g., Trimble 1983). A customary `zero assumption'
is that the mass ratio distribution has a flat shape, i.e. the
binaries with a high mass ratio occur as frequently as those with equal
masses. As one cannot reliably establish such distribution directly from
observations (due to a number of selection effects), we have
parametrized it by a power law, assuming the primary mass distribution
to obey the Salpeter mass function:
%
\begin{equation}
\eqalignleft{
f(M_1) \propto  M_1^{-2.35}\,,&~10~{\rm M}_\odot < M_1< 120~{\rm M}_\odot
\,;       \cr
f(q)   \propto  q^{\alpha_q}\,,& q\equiv M_2/M_1<1 \,.   \cr
}
\end{equation}
A comparison of the observed X-ray source statistics with the predictions
of the current evolutionary scenarios shows (Lipunov et al. 1995b) that
the initial mass ratio should be strongly centered around unity,
so that we assume  $\alpha_q=2$ in the present calculations.

\subsection{Initial parameters of compact stars}

As we only deal with initially massive binary systems (their
primary's mass is higher than 10 ${\rm M}_\odot$), which are capable
of evolving off the main sequence during less than $10^7$ years,
the compact stars left behind stellar evolution are
(in the case of the most massive stars) NSs and BHs.

We assume that a NS with a mass of $1.4~ {\rm M}_{\odot}$ is formed
as result of the collapse of a star, whose core mass prior to collapse was
$M_*\sim (2.5-35)~{\rm M}_{\odot}$. This corresponds to an initial mass
range $\sim (10 - 60)~{\rm M}_{\odot}$, taking into account that a massive
star can lose more than $\sim (10-20)\%$ of its initial mass during the
evolution with a strong stellar wind (de Jager 1980).

We also take into account that the collapse of a massive star into a NS
can be asymmetrical, so that
an additional kick velocity, $v_{kick}$, presumably randomly oriented in
space, should be imparted to the newborn NS.
In the present calculations, the kick velocity was taken
to be 75~km/s
(for a more detailed study of kick velocity distributions,
see Lipunov et al. 1995c).

The magnetic field of a rotating NS
largely defines the evolutionary stage the star would have in
a binary system (Schwartzman 1970; Davidson \& Ostriker 1973;
Illarionov \& Sunyaev 1975). For this purpose,
in our calculations we use a general classification scheme for
magnetized objects elaborated by Lipunov (1992).

Briefly, the evolutionary stage that a rotating magnetized NS has in a binary
system depends on the star's spin period $P$ (or spin frequency
$\omega=2\pi/P$), its magnetic field strength $B$ (or, equivalently,
magnetic dipole moment $\mu=BR^3$, where $R$ is the NS
radius), and the physical parameters of the surrounding  plasma (such as
density $\rho$ and sound velocity $v_s$) supplied by the secondary star.
The latter, in turn, could be a normal optical main sequence (MS) star,
or red giant, or another compact star). In terms of Lipunov's formalism,
the NS evolutionary stage is determined by one or another inequality
between the following characteristic radii: the light cylinder
radius of the NS, $R_l=c/\omega$ ($c$ is the speed of light);
the corotation radius, $R_c=(GM/\omega^2)^{1/3}$; the gravitational capture
radius, $R_G=2GM/v^2$ (where $G$ is the Newtonian gravitational
constant and $v$ is the NS velocity relative to the surrounding
plasma); and the stopping radius $R_{stop}$. The latter is a
characteristic distance at which the ram pressure of the accreting matter
matches either the NS magnetosphere pressure
(this radius is called Alfven radius, $R_A$) or the pressure of relativistic
particles ejected by the rotating magnetized NS (this radius is called
Schwartzman radius, $R_{Sh}$).  For instance, if
$R_l>R_G$  then the NS is at the ejector stage (E-stage) and can
be observed as a radiopulsar; if $R_c<R_A<R_G$, then so-called propeller
regime is established and the matter is
expelled by the rotating magnetosphere; if $R_A<R_c<R_G$, we deal with
an accreting NS (A-stage), etc. These inequalities can easily be
translated into relationships between the spin period $P$ and
some critical period that depends on
$\mu$, the orbital parameters, and accretion rate $\dot M$ (the latter
relates $v$, $v_s$, $\rho$, and the binary's major semiaxis $a$ via the
continuity equation). Thus, the evolution of a NS in a binary
system is essentially reduced to the NS spin evolution
$\omega(t)$, which, in turn, is determined by the evolution of
the secondary component and orbital separation $a(t)$. Typically,
a single NS embedded into the interstellar medium evolves like $E\to P\to A$
(for details, see Lipunov \& Popov 1995). For a NS belonging to a
binary, the evolution complicates as the secondary
star evolves: for example, $E\to P\to A\to E$ (recycling), etc.

When the secondary component in a binary overfills its Roche lobe, the
rate of accretion onto the compact star can be high enough to reach the
Eddington luminosity $L_{Edd}\simeq 10^{38}~(M/{\rm M}_\odot)$ \hbox{erg/s}\,
at the $R_{stop}$; then a supercritical regime sets in (it is worth noting
that not only superaccretors but superpropellers and superejectors
can exist as well; see Lipunov 1992).

If a BH is formed in due course of the evolution, it can only appear
as an accreting or superaccreting X-ray source; other very interesting
stages such as BH + radiopulsar which may constitute a rather large
fraction of all binary pulsars after a starburst are
considered in Lipunov et al. (1995a).

The distribution of the newborn NSs in the initial magnetic dipole moment is
taken to be
%
\begin{eqnarray}
f(\log\mu)=~{\rm const}\,,~~
	  10^{28} \le \mu \le 10^{32}~ \hbox{G cm}^3\,,
\end{eqnarray}
%
and the initial rotational period of the NS
is assumed to be $1$~ms.

It is not clear as yet whether the NS magnetic field
decays or not (for a comprehensive review, see  Chanmugam
1992).  Below, we assume that the magnetic fields of
NSs decays exponentially on a timescale of $10^8$ years.  A
radiopulsar is assumed to be turned ``on'' until its period $P$
has reached a ``death-line'' value, $P_{death}$, defined from the relation
$\mu_{30}/P_{death}^2=0.4$, where $\mu_{30}$ is the dipole magnetic moment in
units of $10^{30}$ G~cm$^3$, and $P_{death}$ is taken in seconds.

The mass limit for NS (the Oppenheimer-Volkoff limit) is taken to be
$M_{OV}=2.5~ {\rm M}_\odot$, which corresponds to a hard equation of
state of the NS matter.

The most massive stars are assumed to collapse into a BH once
their mass before the collapse is $M>M_{cr}=35~ {\rm M}_\odot$ (which
would correspond to an initial mass of the ZAMS star as high as $\sim
60~ {\rm M}_\odot$ since a substantial mass loss due to a strong
stellar wind occurs for the most massive stars).  The BH mass is
calculated as $M_{bh}=k_{bh}M_{cr}$, where the parameter $k_{bh}$ is
taken to be 0.3, as follows from the studies of binaries NS + BH
(Lipunov et al.  1994).

\subsection{Other parameters of the evolutionary scenario}

We consider stars with a constant (solar) chemical composition.  The
process of mass transfer between the binary components is treated,
when appropriate, as a conservative one, i.e. the total angular momentum of
the binary system is considered to be constant. The non-conservativeness of
the mass transfer is treated via ``isotropic re-emission" mode
(Bhattacharya \& van den Heuvel 1991). If the rate of accretion from one star
to another is sufficiently high (e.g., the mass transfer occurs on a
timescale a few times shorter than the thermal Kelvin-Helmholz time for the
normal companion) or the compact object is engulfed by a giant companion,
the common envelope (CE) stage of the binary evolution  can set in (see
Paczy\'nski 1976; van den Heuvel 1983).

During the CE stage, an effective spiral-in of the binary
components occurs.  This complicated process is not fully understood
as yet, so we use the
conventional energy consideration to find the binary system characteristics
after the CE stage by introducing a parameter $\alpha_{CE}$ that
measures what fraction of the system's orbital energy goes, between the
beginning and the end of the spiralling-in process, into the
binding energy (gravitational minus thermal) of the ejected common
envelope. Thus,
%
\begin{equation}
\alpha_{CE}\left({GM_aM_c\over 2a_f}-{GM_aM_d\over 2a_i}\right) =
	{GM_d \left(M_d-M_c\right) \over R_d}\,,
\end{equation}
where $M_c$ is the mass of the core of the mass losing star of initial
mass $M_d$ and radius $R_d$ (which is simply a function of the initial
separation $a_i$ and the initial mass ratio $M_a/M_d$), and no
substantial mass growth for the accretor is assumed (see, however,
Chevalier 1993).  The less $\alpha_{CE}$, the closer
becomes binary after the CE stage. In the present calculations
we take $\alpha_{CE}=0.5$.

Other cases of non-conservative evolution (e.g., evolutionary stages with a
strong stellar wind or those where the loss of the binary angular momentum
occurs due to gravitational radiation or magnetic stellar wind) are
treated using the well known prescriptions (see e.g. Verbunt \& Zwaan
1981; Rappaport et al. 1982; Lipunov \& Postnov 1987).

\subsection{Parameters of the starburst}

We assume that an instantaneous starburst occured
in the region of the Galactic center, with half of the stars
(in number) being formed in binaries.
The total mass of the stars formed during the starburst
was taken to be $\sim 4\times 10^5~ {\rm M}_\odot$, as indicated by the
Tamblyn \& Rieke (1993) analysis.
As for the size of the region that experienced the starburst,
its particular value
is of no importance for evaluating the numbers of various systems, and we
address this issue in Sec. 3.2 while discussing the spatial
distribution of massive systems after the starburst.

In order to obtain statistically significant results, the evolution of
300,000 binary systems was computed. Then we normalized the figures so
as to be in agreement with the Tamblyn \& Rieke (1993) calculations of
the number of massive OB-stars (1,900 stars with $M>10~{\rm M}_\odot$)
that survived $\sim 6-8$ Myrs after the starburst
onset (this age is supported below by an additional argument).
In fact, the number of stars in IRS 16 estimated by Tamblyn \& Rieke (1993)
is uncertain within a factor of 2, but this is actually of no importance
as long as we only use relative numbers of different X-ray source species.
\newpage

\section {Results}

\subsection{Number of accreting X-ray binaries of selected types}

The numbers of three types of compact binaries containing accreting NSs and
BHs, which are most interesting from the point of view of their
observational appearances, have been calculated as a function of time:
{\it i)}
X-ray transient source containing a NS in an eccentric orbit around a main
sequence Be-star (the observed prototype: A0535+26),
{\it ii)} BH accreting at a highly supercritical rate from
the Roche lobe filling component (the assumed prototype: SS 433), and
{\it iii)} BH accreting from the stellar wind of an OB-supergiant
(the observed prototype: Cyg X-1). The evolution of the selected types of
X-ray binaries during the first 10 million years after the starburst onset is
presented in Fig. 1.

Fig. 1a shows the number of X-ray transient sources consisting of
a NS in an eccentric orbit around a massive secondary that
acquired enough angular momentum during the first mass exchange
to become a rapidly rotating Be-star. This occurs for
the binaries whose components have comparable initial masses and are not too
distantly separated to avoid the common envelope stage
formation during the first mass exchange (the last condition is always
satisfied for the massive binaries at the Galactic center).
To become an X-ray transient, the NS must accrete matter from the
secondary Be-star, at least during the periastron passages.
Since the duration of the accretion depends upon the orbital eccentricity,
rotational period, and magnetic field of the NS, not all the transients are
in the accretion stage at the same time, therefore the observed number of
these sources can be a few times less compared to Fig. 1a.
Fig. 1b shows the evolution of X-ray binaries with a BH, like Cyg X-1.
The number of such sources depends strongly on time,
giving $\approx 1$ source by $t=7$ Myr.
The evolution of a superaccreting BH of SS 433 type is presented in
Fig. 1c.

It is tempting to speculate that the
presence of a radio jet in the well known X-ray source 1E 1740-2942 (Churasov
et al. 1994) may indicate that it belongs to superaccreting BH binaries.
It was shown that this source
can hardly be a system like Cyg X-1 (Heindl et al. 1993), and there are
arguments (Mirabel 1991) that it can be an isolated BH,
accreting matter from the surrounding molecular cloud.
At the moment, however, there are no direct observations which could
prove that this object is an isolated BH.

The ratio of numbers of BH containing binaries (of both SS 433 and
Cyg X-1 type) to X-ray transients with Be-stars is plotted in Fig. 1d.
This ratio is remarkably sensitive to the time elapsed after
the starburst and therefore it can be used for an independent estimation of
the age of the X-ray binaries at the Galactic center. Absolute numbers
of different systems seem to be worse age-indicators as they are subjected
to different poorly known selection effects.

\subsection{Spatial distribution of X-ray binaries at
the Galactic center after the starburst}

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
$\approx$ 750 pc $\times$ 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 {\it Scenario I}, the progenitors of X-ray binaries were formed in a
region of $\sim$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 {\it Scenario II},
the starburst happened on a scale $\gg 1$ pc.

\begin{center}
{\it Scenario I: the starburst occured in a central region of $\sim$1 pc
in size}
\end{center}

 The mass
distribution at the Galactic center was taken in the form (Lacy et al. 1991):
\begin{equation}
M(r)=3.4\times 10^6~ {\rm M}_\odot\,
\left(1+\frac{R}{1\, {\rm pc}}\right)\, .
\end{equation}
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:
\begin{equation}
\ln {r\over  r_0}={1\over 2.94}\left(v_0^2-v^2\right)~,
\end{equation}
where $v_0$ (in 100 km/s) is the initial ejection velocity at a radius $r_0$.
This implies that, in order to reach $r=1$ kpc from $r_0=1$ pc even with
a zero velocity, $v_0$ needs to be as high as 450 km/s. For $r_0=10$ pc, the
required $v_0=368$ 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.

\begin{center}
{\it Scenario II: the starburst occured in a central region on a scale
$\gg 1$ pc }
\end{center}

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 $\sim 2\cdot 10^8~{\rm yr}$ 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 $r^{-1}$ and will reach its maximum
when the cloud  passes at its minimum approach from the Galactic center.
If this distance is $\sim 10$ pc (i.e. comparable to the initial radii of the
clouds), the velocity dispersion of stars reaches $\sim 150-225$
km/s, while the systematic velocity of the cloud acquired at the central
potential well turns out to be $v_0\sim 300-400$ 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 {\it GRANAT} sources and NS+Be systems (the latter are more numerous than
BH-sources both in  the {\it 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, $v_{\phi}$, ranging
from zero to the circular velocity, $v_{circ}$.

If the starburst's distance from the center is $r_0\sim 500$ pc and 
$v_{\phi_0}$
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
$r_0=500$ pc from the center on the line of sight and its center's coordinates
are projected to the point $X_0=0$ pc, $Y_0=0$ pc.
The initial, after cloud--cloud collision, rotational velocity of the cloud
about the Galactic center is taken to be $v_{\phi_0}= 50$ 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 $v_{\phi_0}$
is large enough.

The larger $v_{\phi_0}$, 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 $T\sim 7$ Myr makes such cases, as in Fig. 2c,
unlikely although  available free parameters
(age, $v_{\phi_0}, r_0, X_0,~ {\rm and}~ Y_0$ could weaken the anisotropy
relative to the $Y$-axis (but still leaving the non-sphericity).
Another source of asymmetry are substantially larger $r_0$ (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  $T\simeq 7$ 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 $r_0\sim 500$ pc,
not too large $v_{\phi_0}$).

\newpage
\section {Discussion and conclusions}

We compare our results with the {\it GRANAT} X-ray observations of the
Galactic center (Churazov et al. 1994, Pavlinsky et al. 1994). Besides Sgr
A$^\star$, 11 more
X-ray sources have been reported to be observed in the central region
of the Galaxy ($5^{\circ}\times 5^{\circ}$ across, which corresponds to
the linear size of 750 pc $\times$ 750 pc, assuming the 8.5 kpc
distance to the Galactic center). Two of these sources were
classified as BH-candidates by their hard power-law tails in the X-ray
spectrum (similar to Cyg X-1), and the other 9 sources are X-ray
transients probably containing NS.

As we noted above, the BH-candidates/X-ray transients ratio is a good
indicator of the time passed after the beginning of the starburst.  The
number of X-ray sources containing NS increases with time and becomes
approximately constant after 6 Myr.  The number of binary systems with
accreting BHs decreases approximately exponentially
with a characteristic timescale of $\approx 1.5$ Myr.
This is due to the fact that massive stars with
$M > 35~ {\rm M}_{\odot}$ as BH progenitors rapidly evolve and their
number strongly decreases after several Myr. The computed BH/NS ratio
is $\approx 0.04$ at the age of 7 Myr, which should be considered as a
lower limit to the true ratio because we are not able to observe all
the X-ray transients simultaneously.

As only a few BH-candidates have been found in the region so far
(Churazov et al. 1994, Pavlinsky et al. 1994),
some uncertainties in the estimated age of the starburst result from a
poor statistics. Yet, they are partially
reduced as we use relative numbers of the systems. In our calculations
employing the metallicity, we used the  solar abundance. As the
metallicity at the Galactic center is, by a factor of two, above the solar
one, it also can slightly change our results. Still, given the adopted
assumptions, our calculations of the absolute numbers of the systems of
different types have an accuracy of
 $\approx$20\%;
relative numbers have been calculated even with a better accuracy.
Bearing in mind that not all of our assumptions are realistic,
we feel that our absolute numbers are uncertain within a factor of 2-3 or so.

It is instructive to compare our results for the Galactic center
starburst with what might be expected for a {\it continuous} star formation,
with the average rate characteristic for the whole Galaxy. For this
continuous star formation model, we would expect to observe in the
region of $ 5^\circ\times 5^\circ $ around the Galactic center about 10\%
of the total number of X-ray sources in the Galaxy.  Accounting for
the projection effect, the fraction of the X-ray sources in the central
375 pc would be even smaller, $\sim$4 \%.  Meanwhile in the entire
Galaxy we currently observe only one SS 433-type source (SS 433 itself), one
Cyg X-1-type source (Cyg X-1 itself), and about 10 X-ray transients are being
discovered every year. Hence for the Galactic center one would expect,
after several years of observations with X-ray satellites, to reveal
about 5 X-ray transients and, most likely,  no SS 433-type or Cyg
X-1-type sources at all (the probability of their appearance is very
low). Therefore, the continuous star formation model is able to
explain neither the absolute nor relative numbers of the X-ray sources
actually observed in the Galactic center. In contrast, the
starburst model presented above seems to be quite successful in this respect.

As for the the size of the starburst region, the situation seems to be less
certain. Still, as we have shown in Sec. 3.2, the deepness of the central
potential well does rule out a scenario in which the starburst occurs in the
central 1 pc or so and then a `kick' during the stage of neutron star
formation ejects the binary system till the distances $\sim$1 kpc
within which the observed X-ray sources are concentrated.
We have discussed another scenario
according to which the starburst occurs on a scale $\gg 1$ pc as a result
of collision of two molecular clouds (Ozernoy 1995). In this approach,
the high velocities acquired by the infalling gas in the galactic potential
well are kept by the forming massive stars, which enables them to be
scattered up to 1 kpc or so. A potential problem with this scenario is that,
in a contrast to the observed distribution of X-ray sources on a scale like
this, the observed distribution of their progenitors, the massive stars, is
apparently seen much more concentrated towards the central 1 pc. Yet it is not
excluded that accounting for the unknown selection effects might
weaken/remove this problem. However, even in this case, the very
presence of several dozens of hot, massive stars in the central 1 pc
needs to be explained. In the frame of Scenario II
envisioned in Sec. 3.2, which involves the scale $\gg 1$ pc for the
starburst, a possible explanation would be as follows:
For the newborn stars forming as a result of collisions of the two GMCs,
 dispersion of their velocities increases,
owing to the conservation of angular momentum, up to the free fall  velocity in
due course of the  infall onto the center. A fraction of the material kept
 in the gaseous form after the dissipation might fall into the center
and produce massive stars whose velocity dispersion would not exceed 100
km/s. Further numerical modelling to test this would be highly desirable.
If successful, this scenario would combine formation of massive stars close to
the center of the Galaxy with an opportunity to observe the massive binary
successors, the X-ray systems, at very large distances from the center.

One may argue that the region of $750\times 750$ pc in size around the
Galactic center is broad enough so as to be contaminated by X-ray binaries
originating in the adjacent regions. However, the fraction of X-ray binaries
of such type among the `field' stars (i.e. not associated with the starburst
of interest) is much lower.  Yet, one could imagine in the Galactic
center another starburst of a similar age but on a much larger scale,
compared to what is considered above, which would of course somewhat change
our results; however,
no evidence for such a burst is known so far (for a comprehensive review
of available evidence for, and constraints to, possible recurrent starbursts
in the cental regions of the Galaxy, see Hartmann 1995).  We notice that
the proposed scenario of the origin of X-ray sources in a comparatively
compact starburst has a clear signature: the velocity of an X-ray source is
(statistically) expected to be the larger the closer the source is located
to the Galactic center.

The results of our modelling seem to be also relevant for studying the star
formation regions in other galaxies, including starburst galaxies. In
the latter, short episodes of violent star formation with a time scale
of $\sim$10 Myr have been suggested to occur recurrently some billion years
(Coziol \& Demers 1994).  As follows from our modelling of the population
synthesis of X-ray sources, production of a few 10$^5$ stars in a starburst
has to be accompanied by the formation of about 10 hard X-ray sources
at the starburst age of 6-8 Myr (and a larger number of X-ray sources at
earlier times).

To summarize, {\it the statistics of X-ray binaries, especially the ratio
of the number of systems containing a BH to the number of X-ray transients
with a NS, is a sensitive function of the starburst age on a time scale of
2 to 10 Myrs.} As an application, a relatively large fraction of accreting
BH candidates among the observed X-ray sources at the Galactic center
could be naturally explained if a starburst indeed occurred
$\sim$ 6-8 Myr ago (Tamblyn \& Rieke 1993).

%_________________________________________________________________________

{\it Acknowledgements}.
This work was partly supported by a COSMION grant
and by Grant No. JAP100 from the International Science Foundation and the
Russian Government.
V.L., S.P., K.P. and M.P. acknowledge useful discussions with I.E. Panchenko
and S.N. Nazin. We are thankful to the anonymous referee whose comments
helped us to improve the paper.
%
%_________________________________________________________________________
%

\begin{thebibliography}{}
\bibitem{}Abt, H., 1983, ARAA 21, 343
\bibitem{}Bhattacharia, D. \& van den Heuvel, E.P.J. 1991, Phys. Rep. 203, 1
\bibitem{}Blum, R.D., Sellgren, K., \& DePoy, D.L. 1995, ApJ 440, L17
\bibitem{}Chanmugam, G. 1992, ARA\&A 30, 143
\bibitem{}Chevalier, R.A.     1993, ApJ 411, L33
\bibitem{}Churazov, E. et al. 1994, in ``Frontiers of Space and Ground Based
Astronomy", eds. W.Wamsteker et al., Kluwer Acad. Press (in press)
\bibitem{}Coziol, R. \& Demeres, S. 1994, Observatoir du Mont Magantic,
preprint No. 080
\bibitem{}Davidson, K. \& Ostriker, J.P. 1973,  ApJ 179, 585
\bibitem{}Dewey, R.J. \& Cordes,  J.M. 1987, ApJ. 321, 780
\bibitem{}Eckart, A. et al. 1993, ApJ 407, L 77
\bibitem{}Genzel, R., Hollenbach, D. \& Townes, C.H. 1994, Rep. on Progr. in
Phys. 57, 417
\bibitem{}Hall, D.N.B. et al. 1982, ApJ 262, L 53
\bibitem{}Hartmann, D.H. 1995, ApJ 447, 646
\bibitem{}Heindl, W.A. et al. 1993, ApJ 408, 507
\bibitem{}Iben, I., Jr. \& Tutukov, A.V. 1984, ApJ 284, 719
\bibitem{}Illarionov, A.F. \& Sunyaev, R.A. 1975,  A\&A   39, 185 %
\bibitem{}de Jager, C. 1980,  ``The Brightest Stars'',
D.Reidel, Dordrecht %
\bibitem{}de Kool, M. 1992,  A\&A, 261, 188
\bibitem{}Kornilov, V.G. \& Lipunov, V.M. 1983a,  AZh, 60, 284
\bibitem{}Kornilov, V.G. \& Lipunov, V.M. 1983b,  AZh, 60, 574
\bibitem{}Kornilov, V.G. \& Lipunov, V.M. 1984,  AZh, 61, 686
\bibitem{}Koyama, K. et al. 1989, Nat 339, 603
\bibitem{}Krabbe, A. et al. 1991, ApJ 382, L 19
\bibitem{}Lacy, J.H., Achterman, J.M., \& Serabyn, E. 1991 ApJ 380, L71
\bibitem{}Lebofsky, M.J. et al., 1982, ApJ 263, 736
\bibitem{}Leitherer, C. \& Heckman, T.M. 1995, ApJS 96, 9
\bibitem{}Lipunov, V.M. 1992, ``Astrophysics of Neutron Stars", Springer
Verlag, Heidelberg
\bibitem{}Lipunov, V.M. 1994, Mem. Soc. Astr. It. 65, 21
\bibitem{}Lipunov, V.M. \& Popov, S.B. 1995, AZh 72, 711
\bibitem{}Lipunov, V.M. \& Postnov, K.A. 1987, Ap.\& Sp.Sci. 145, 1
\bibitem{}Lipunov, V.M., Postnov, K.A., Prokhorov, M.E.
\& Osminkin, E.Yu.  1994,  ApJ  423, L 121
\bibitem{}Lipunov, V.M., Postnov, K.A. \& Prokhorov, M.E. 1995a,
ApJ 441, 776
\bibitem{}Lipunov, V.M., Postnov, K.A. \& Prokhorov, M.E. 1995b,
A\&A (in press)
\bibitem{}Lipunov, V.M., Postnov, K.A. \& Prokhorov, M.E. 1995c,
ApJ (submitted)
\bibitem{}Lyne, A.G. \& Lorimer, D.R. 1994, Nature 369, 127
\bibitem{}Minos, J.C. \& Hernquist, L. 1994, ApJ 431, L9
\bibitem{}Mirabel, I.F. et al. 1991, A\&A  251, L43
\bibitem{}Morris, M. 1993, ApJ 408, 496
\bibitem{}Najarro, F. et al. 1994, A\&A 285, 573
\bibitem{}Ozernoy, L.M. 1994, in ``Multi-Wavelength Continuum Emission of
AGN", eds. T.J.-L. Courvoisier \& A. Blecha, Kluwer, p. 351
\bibitem{}Ozernoy, L.M. 1995, in ``The Unsolved Problems of the Milky Way
Galaxy" (IAU Symp. No. 169), ed. L. Blitz, Kluwer, p. 000
\bibitem{}Ozernoy, L.M., Titarchuk, L. \& Ramaty, R. 1993, in ``Back to the
Galaxy", Eds. S. Holt et al., AIP Conf. Proc. 278, 73
\bibitem{}Paczy\'nski, B. 1976, in ``Structure and Evolution of Close
Binary Systems", IAU Symp. No. 73, Eds. Egglton, P.P., Mitton, S.A., \&
Whealan, J.A.J. Reidel, Dordrecht, p. 75                             %
\bibitem{} Pavlinsky, M.N., Grebenev, S.A., \& Sunyaev, R.A. 1994,
ApJ  425, 110
\bibitem{}Rappaport, S.A., Joss P.C., \& Webbink, R.F. 1982, ApJ  254, 616 %
\bibitem{}Schaerer 1994, in ``The Unsolved Problems of the Milky Way 
Galaxy", (IAU Symp. No. 169), ed. L. Blitz, Kluwer, p. 000
\bibitem{}Shore, S.N., Livio, M., \& van den Heuvel, E.P.J. 1994,
``Interacting Binaries". Springer Verlag, Berlin, Ch. 3
\bibitem{}Sofue, Y. 1994, ApJ 431, L 91
\bibitem{}Schwartzman, V.F. 1970, Radiofizika 13, 1850         %
\bibitem{}Tamblyn, P. \& Rieke, G.H. 1993, ApJ 414, 573
\bibitem{}Tamblyn, P., Rieke, G.H., Hanson, M.M., Close, L.M., McCarthy,
D.W., \& Rieke, M.J. 1996, ApJ 456, 206
\bibitem{}Trimble, V. 1983,  Nature  303, 137                  %
\bibitem{}van den Heuvel, E.P.J. 1983, in  ``Accretion Driven Stellar
    X-ray Sources", Eds. Lewin, W.H.G. \& van den Heuvel, E.P.J., Cambridge
    Univ. Press, p. 303                                            %
\bibitem{}van den Heuvel, E.P.J. 1994, in ``Interacting Binaries",
    Eds. Shore, S.N., Livio, M., \& van den Heuvel, E.P.J., Berlin, Springer,
    p. 442                                                          %
\bibitem{}Verbunt, F. \& Zwaan, C. 1981,  A\&A  100, L7          %
\end{thebibliography}

\clearpage
\section{fugures}
\begin{figure}
%\caption
{Figure 1: The evolution of X-ray binaries of selected types
after the starburst at the Galactic center.}
\end{figure}

\begin{figure}
%\caption
{Figure 2a: 2-D projection of the spatial distribution.
Circles:   X-ray transients, squares: GRANAT sources.
 The starburst occured
at the point with coordinates $X=0$ pc, $Y=0$ pc, i.e.
on the line of sight (see the text). $v_{{\phi}_{0}}=50$ km/s.}
\end{figure}

\begin{figure}
%\caption
{Figure 2b: 2-D projection of the spatial distribution.
Circles: X-ray transients, squares: GRANAT sources.
 The starburst occured
at the point with coordinates $X=500$ pc, $Y=0$ pc. $v_{{\phi}_{0}}=v_{circ}$.}
\end{figure}

\begin{figure}
%\caption
{Figure 2c: 2-D projection of the spatial distribution.
Circles: X-ray transients, squares: GRANAT sources. The starburst occured
at the point with coordinates $X=0$ pc, $Y=0$ pc, i.e.
on the line of sight. $v_{{\phi}_{0}}=v_{circ}$.}
\end{figure}

\clearpage

\FIG{figure1.eps}{1}

\FIG{figure2a.eps}{2a}

\FIG{figure2b.eps}{2b}

\FIG{figure2c.eps}{2c}

\end{document}

