\documentstyle[epsf, rotate, referee]{l-aa}
\begin{document}
%\baselineskip= 2\baselineskip
\thesaurus{(02.01.2; 08.14.1; 09.07.1; 10.07.1; 13.25.3)}
\title {Spatial distribution of the accretion luminosity of
isolated neutron stars and black holes in the Galaxy}

\author{Sergei B.Popov  \and
  Mikhail E.Prokhorov
}

\date{}

\institute{
Sternberg Astronomical Institute,
 Universitetskii pr.13, 119899, Moscow, Russia
}

\maketitle

\markboth{S.B. Popov \& M.E. Prokhorov: Spatial distribution of the
accretion luminosity}{}


\begin{abstract}
    We present here a computer model of the
spatial distribution of the luminosity,
produced by old isolated neutron stars and black holes
accreting from the interstellar medium.
We show that the luminosity distributions in the Galaxy
have a ring structure, with a maximum at $\approx 5 kpc$ radius.
\end{abstract}

\keywords{neutron stars --
black holes --
Galaxy: stellar content}

\section{Introduction}

 Old isolated neutron stars (NS) and black holes (BH)
form a large populations of  galactic  objects (about $10^8$--$10^9$
objects in the Galaxy), but most of them are unobserved today.
Less than $10^3$  young NS appear as
radio pulsars, and no isolated BH has been observed 
(probably, some of them are detected, for example, in the
$ROSAT$ survey, but no one is identified).
This article  will be concerned  only about isolated compact objects,
than will simply designated as NS or BH.

During the last  years, the spatial distribution
and other properties of NS became of great interest, because
NS can be observed by the {\it ROSAT}
satellite in soft X-rays due to accretion from the interstellar medium (ISM)
(see, for example, Treves \& Colpi 1991).
Several sources of this type have been observed (Walter et al. 1996).
BH also can appear as similar X-ray sources
(Heckler \& Kolb 1996) with some differences in spectrum and
temporal behaviour (absence of pulsations, for example).
That is why  we try here to
obtain a picture of the distribution of the accretion luminosity of
these sources.

Fast rotation and/or a strong magnetic field can prevent accretion
onto the surface of the NS. In this case the X-ray luminosity will be
very low (except for transient sources  due to
the formation of an envelope around the NS:
see Popov 1994 and Lipunov \& Popov 1995).
Here we consider only accreting NS.
Most NS are in the stage of accretion, because their magneto-rotational
evolution usually finishes at this stage approximately
 $10^8$ years after  their birth.
The NS properties (periods etc.) in the stage of accretion  depend
upon the magnetic field decay (see Konenkov \& Popov 1997).
BH, of course, can  only been seen as accretors.

    In the articles of Gurevich et al. (1993)and of
Prokhorov \& Postnov  (1993, 1994) it was shown
that the population of NS  forms a ring (or toroidal) structure
in the Galaxy.
The distribution  of the ISM (see, for example,
Bochkarev 1992) also has a ring structure.
The maxima of both distributions roughly coincide.

Therefore, most of the NS (and probably
BH) are located in the dense regions of the ISM.
Thus the accretion luminosity in these regions should be high.
The results of  computer simulations of this situation are
presented in this paper.

The  trajectories of NS and BH were computed directly
for a specified initial velocity distribution,
the Galaxy gravitational potential and
the distribution of the ISM density.
Preliminary results of such computations for NS for
$\delta$-- function and maxwellian velocity distributions were
presented in Popov \& Prokhorov (1998, paper I).

In Section 2 we briefly describe our model. In Section 3
the results and a short discussion are presented. 
The last Section contains the conclusions.

\section{The Model}

We solved numerically the system of differential equations of motions
in the  Galactic potential, taken in the form (Paczynski 1990):

$$
    \Phi_i (R,Z)=GM_i/\left(R^2+[a_i+(Z^2+b_i^2)^{1/2}]^2\right)^{1/2}
$$
with a quasi-spherical halo with a density distribution:
$$
   \rho=\frac{\rho_0}{1+(d/d_0)}, \,\,\, d^2=R^2+Z^2.
$$

Here $R$ and $Z$ are the cylindrical coordinates, $d$ the radius in
the quasi-spherical halo.
The parameters of the potential are given in the following table,
$\rho_0$ being determined from the halo mass, $M_0$.

\vskip 0.5cm

\noindent
%\begin{table*}
%\caption[]{}
\begin{tabular}{lccc}
Disk &$a_D$=0      &$b_D$=277 pc&$M_D=1.12\cdot 10^{10} M_{\odot}$\\
Bulge&$a_B$=3.7 kpc&$b_B$=200 pc&$M_B=8.07\cdot 10^{10} M_{\odot}$\\
Halo &             &$d_0$=277 pc&$M_0=5.0 \cdot 10^{10} M_{\odot}$\\
\end{tabular}
%\end{table*}

\vskip 0.5cm


 The density in our model is constant in time. 
The local density is calculated
using data and formulae from Bochkarev (1992) and Zane et al. (1995).
$n$ is total gas density,
$n_{HI}$ and $n_{H_2}$ are the densities of the neutral and molecular
hydrogen, $n_0(R)$, $n_2(R)$ and $n_3(R)$ are the values
of the densities for $Z=0$.

$$
   n(R, Z)= n_{HI}+2\cdot n_{H_2}
$$

$$
   n_{H_2}=n_2(R)\, exp \left[ \frac{ -Z^2}{2\cdot (70 pc)^2} \right]
$$

   For $0\, kpc \le R \le 3.4 \,  kpc $ we assumed: 

$$
   n_{HI}=n_0(R) exp \left[ \frac{-Z^2}{2\cdot (140 \, pc \cdot R/ 2
\, kpc)^2} \right], $$

    For $0 \, kpc \le R \le 2 \,  kpc \, \,  n_0(R) $ was  assumed to be
uniform:

$$
   n_0(R<2 kpc)=n(R=2 kpc)
$$

    Of course, this is not accurate for small R, so for the very central part
of the Galaxy our results are only a rough estimate (see Zane et al.
(1996) for detailed calculation of the NS emission from the Galactic
center region).   
For $3.4\,  kpc \le R \le 8.5 \, kpc$ we assumed 

$$
   n_{HI}=0.345\, exp \left[ \frac{-Z^2}{2\cdot (212 \, pc)^2} \right] +
$$
$$
          0.107\, exp \left[ \frac{-Z^2}{2\cdot (530 \, pc)^2} \right] +
$$
$$
          0.064\, exp \left[ \frac{-Z}{403 \, pc} \right]
$$

    For $ 8.5 \, kpc \le R \le 16 \, kpc$ we assumed

$$
   n_{HI}=n_3(R)\, exp \left[ \frac{-Z^2}{2\cdot (530 \, pc
\cdot R / 8.5 \, kpc)^2} \right]
$$
$n_0(R)$, $n_2$ and $n_3(R)$ being taken from Bochkarev (1992).

    The total gas density distribution in the $R$-$Z$ plane
used in our computations is shown on figure 1.

\begin{figure}
\epsfxsize=\hsize
\centerline{{\epsfbox{figure1.eps}}}
\caption{ The density distribution in  particle per
cubic centimeter in the $R$--$Z$ plane.}
\end{figure}

 In our model we assumed that the birthrate of NS and BH is
proportional to the square of the local density.
Stars were assumed to be 
born in the Galactic plane (Z=0) with circular velocities
plus additional isotropic kick velocities.

For the kick velocity distribution 
we used the formula from Lipunov et al. (1996).
This formula was constructed as an analytical approximation of the
three-dimensional velocity distribution of radio pulsars from
Lyne \& Lorimer (1994).

$$
  f_{LL}(V)\propto \frac{x^{0.19}}{(1+x^{6.72})^{1/2}},
$$
$V$ being the space velocity of the compact object,
$V_{char}$ a characteristic velocity, $x=V/V_{char}$ and
$f_{LL}$ the probability (see
the detailed description of the analytical approximation in  Lipunov et
al. (1996)). This formula reproduces the observed distribution
with a mean velocity of 350 km/s for $V_{char}$=400 km/s.
This velocity distribution seems more likely than a
$\delta$-- function and a Maxwellian distribution, which we
used in Paper I. Kick velocities were taken equal for the NS and the BH
It is possible however that BH have lower
kick velocities because of their higher masses (see White and
van Paradijs, 1996).
One of the reasons to make computations for $V_{char}$=200 km/s
was to explore this situation.

For each star we computed the exact trajectory and the
accretion luminosity.
The accretion luminosity was calculated using Bondi's formula:

$$
    L=\left(\frac{GM \dot M}{R_{lib}}\right)
$$

$$
   \dot M=2 \pi \left(
    \frac{(GM)^2 \rho(R,Z)}{(V_s^2+V^2)^{3/2}}\right)
$$.

    The sound velocity,$ V_s$, was taken to be 10 km/s everywhere.
We used a mass $M_{NS}=1.4 M_{\odot}$ for NS and
$M_{BH}=10 M_{\odot}$
for BH. 
$\rho =n  m_H$ is the density, 
$m_H$ being the mass of the hydrogen atom.  The radii,
$R_{lib}$, where the energy is liberated, were assumed to be equal to
10 km for NS and 90 km (i.e. $3\cdot R_g$, $R_g=2GM/c^2$)
for BH.  Calculations used a
grid with a cell size 100 pc in the R-direction and 10 pc in
the Z-direction.  The luminosity is
given on the figures in ergs per second per cubic parsec.

For the normalization of our results we assumed that 
$N_{NS}=10^9$  and $N_{BH}=10^8$  in the considered volume
of the Galaxy. For a Salpeter mass function with $\alpha$=2.35
the ratio of NS to BH is about 10 if NS are formed from 
stars with masses between $10M_{\odot}$ and $\approx 45-50 M_{\odot}$,
and BH from stars with masses higher than $\approx 45-50
M_{\odot}$.  Motch et al. (1997) argued that $N_{NS}=10^9$ can be
ruled out,  $N_{NS}=10^8$ being a more probable value, but for the
calculations of the distribution the total  number is not so
important, and for other numbers of compact objects
the results (i.e. the value of the luminosity) can be easily scaled.
It should be mentioned, as suggested by the unknown 
referee,
that $N_{NS}=10^9$ is required to explain that the present heavy
element abundance in the Galaxy is about Z=0.02.


\begin{figure}
\epsfxsize=\hsize
\centerline{{\epsfbox{figure2.eps}}}
\caption{ The accretion luminosity distribution in the $R$--$Z$ plane for
neutron stars for a characteristic kick velocity 200 km/s.
The luminosity is  in ergs per second per cubic parsec.
$N_{NS}=10^9$}
\end{figure}

\begin{figure}
\epsfxsize=\hsize
\centerline{{\epsfbox{figure3.eps}}}
\caption{ The accretion luminosity distribution in the $R$--$Z$ plane for
neutron stars for a characteristic  kick velocity 400 km/s.
The luminosity is in ergs per second per cubic parsec.
$N_{NS}=10^9$}
\end{figure}

\begin{figure}
\epsfxsize=\hsize
\centerline{{\epsfbox{figure4.eps}}}
\caption{The  accretion luminosity distribution in the $R$--$Z$ plane for
black holes for a characteristic  kick velocity 200 km/s.
The luminosity is in ergs per second per cubic parsec.
$N_{BH}=10^8$}
\end{figure}

\begin{figure}
\epsfxsize=\hsize
\centerline{{\epsfbox{figure5.eps}}}
\caption{The  accretion luminosity distribution in the $R$--$Z$ plane for
black holes for a characteristic  kick velocity 400 km/s.
The luminosity is  in ergs per second per cubic parsec.
$N_{BH}=10^8$}
\end{figure}



\section{Results and discussion}


    On figures 2--5 we show as a radial cut through the Galactic disk
the results for two characteristic
values of the velocity distribution for NS and BH.
The scales for $R$ and $Z$ axes 
are different in order to show clearly the
structure in $Z$ direction. Differences between the luminosity
distribution for $Z>0$ and $Z<0$ demonstrate the accuracy of the
statistical computations (curves were not smoothed).

Figure 6 shows the luminosity in the Galactic plane as a function of radius
for a characteristic  kick  velocity $V_{char}$= 200 km/s.
The figure is not completely symmetric. 
The right hand part corresponds to the
azimutal angles 0--180 degrees, the left to --- 180--360 degrees.
The differences between the left and the right parts of the curve
give an indication of the accuracy of our computations.



\begin{figure}
\epsfxsize=14cm
\centerline{\rotate[r]{\epsfbox{figure6.eps}}}
\caption{Slice at Z=+5 pc for NS for a characteristic  kick velocity
200 km/s. $N_{NS}=10^9$. The accretion  luminosity is in ergs per second
per cubic parsec.  The solid line is a smoothed curve.  }
\end{figure}



\section{Concluding remarks}

    As  can be seen from the figures, the distribution of
the accretion luminosity  in R-Z plane
forms a toroidal (ring) structure with  maximum
at approximatelly 5 kpc.

 As expected, BH give  higher luminosity than  NS,
as they have greater masses.
But if the total number of BH is significantly lower than the number
of NS, their contribution
to the luminosity can be less than the contribution of NS.
The total accretion lumiunosity of the Galaxy for $N_{NS}=10^9$
and $N_{BH}=10^8$ is about $10^{39}-10^{40}${\it erg/s}.
For a characteristic velocity of 200 km/s  the maximum of the
distribution is situated approximately at  5.0 kpc for NS  and  at  4.8
kpc for BH.  For NS with a characteristic velocity of 400 km/s maximum is
located at 5.5 kpc, and for BH at 5.0 kpc.  This result is also
expected because of the higher masses of the BH.

    The toroidal structure of the luminosity distribution
of NS and BH is an interesting and important feature
of the Galactic potential.
As one can expect, for low characteristic kick velocities and for BH
we have a higher luminosity.

    As we made very general assumptions, we argue, that such a distribution
is not unique  for our Galaxy, and all spiral galaxies can have such
a distribution of the accretion luminosity, associated with accreting NS
and BH.

\begin{acknowledgements}
    The work was supported by the RFFI (95-02-6053) and
the INTAS (93-3364) grants.
The work of S.P. was also supported by the ISSEP.
We thank Dr. I.E. Panchenko,
the unknown referee 
and Dr. J. Lequeux, who made a lot
of suggestions to improve the article (especially the quality of the
language) and Prof. S.R. Pottasch for his help.
\end{acknowledgements}


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\end{document}
