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Astron. Astrophys. 331, 535--540 (1998) ASTRONOMY
AND
ASTROPHYSICS
Spatial distribution of the accretion luminosity
of isolated neutron stars and black holes in the Galaxy
S.B. Popov and M.E. Prokhorov
Sternberg Astronomical Institute, Universitetskii pr.13, 119899 Moscow, Russia
Received 6 June 1997 / Accepted 1 October 1997
Abstract. We present here a computer model of the spatial dis­
tribution 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 # 5kpc radius.
Key words: stars: neutron -- black hole physics -- Galaxy: stellar
content
1. 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 exam­
ple, in the ROSAT survey, but no one is identified). This article
will be concerned only about isolated compact objects, that will
simply designated as NS or BH.
During the last years, the spatial distribution and other prop­
erties of NS became of great interest, because NS can be ob­
served by the ROSAT satellite in soft X­rays 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 X­ray 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 ac­
cretion onto the surface of the NS. In this case the X­ray lumi­
nosity 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 be 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 dis­
tribution 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 luminos­
ity 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 gravi­
tational potential and the distribution of the ISM density. Pre­
liminary results of such computations for NS for #­ function
and maxwellian velocity distributions were presented in Popov
& Prokhorov (1998, paper I).
In Sect. 2 we briefly describe our model. In Sect. 3 the results
and a short discussion are presented. The last section contains
the conclusions.
2. The model
We solved numerically the system of differential equations of
motions in the Galactic potential, taken in the form (Paczynski
1990):
# i (R, Z) = GM i / # R 2 + [a i + (Z 2 + b 2
i ) 1/2 ] 2
# 1/2
with a quasi­spherical halo with a density distribution:
# = # 0
1 + (d/d 0 ) , d 2 = R 2 + Z 2 .
Here R and Z are the cylindrical coordinates, d the radius
in the quasi­spherical halo. The parameters of the potential are
given in the following table, # 0 being determined from the halo
mass, M 0 .

536 S.B. Popov & M.E. Prokhorov: Accretion luminosity of isolated neutron stars and black holes in the Galaxy
Fig. 1. The density distribution in particle per cubic centimeter in the R­Z plane.
Disk aD =0 b D =277 pc MD = 1.12 · 10 10 M#
Bulge aB =3.7 kpc b B =200 pc MB = 8.07 · 10 10 M#
Halo d 0 =277 pc M 0 = 5.0 · 10 10 M#
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, nHI and nH 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) = nHI + 2 · nH 2
nH 2
= n 2 (R) exp # -Z 2
2 · (70pc) 2
#
For 0 kpc # R # 3.4 kpc we assumed:
nHI = n 0 (R)exp # -Z 2
2 · (140 pc · R/2 kpc) 2 # ,
For 0 kpc # R # 2 kpc n 0 (R) was assumed to be uniform:
n 0 (R < 2kpc) = n(R = 2kpc)
Of course, this is not accurate for small R, so for the very cen­
tral 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 # R # 8.5 kpc
we assumed
nHI = 0.345 exp # -Z 2
2 · (212 pc) 2 # +
0.107 exp # -Z 2
2 · (530 pc) 2 # +
0.064 exp # -Z
403 pc
#
For 8.5 kpc # R # 16 kpc we assumed
nHI = n 3 (R) exp # -Z 2
2 · (530 pc · R/8.5 kpc) 2 #
n 0 (R), n 2 and n 3 (R) being taken from Bochkarev (1992).
The total gas density distribution in the R­Z plane used in
our computations is shown in Fig. 1.
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 an­
alytical approximation of the three­dimensional velocity distri­
bution of radio pulsars from Lyne & Lorimer (1994).
f LL (V ) #
x 0.19
(1 + x 6.72 ) 1/2 ,
V being the space velocity of the compact object, V char a char­
acteristic 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 distribu­
tion with a mean velocity of 350 km/s for V char =400 km/s.
This velocity distribution seems more likely than a #­ function
and a Maxwellian distribution, which we used in Paper I. Kick
velocities were taken equal for the NS and the BH. It is pos­
sible however that BH have lower kick velocities because of
their higher masses (see White and van Paradijs, 1996). One of

S.B. Popov & M.E. Prokhorov: Accretion luminosity of isolated neutron stars and black holes in the Galaxy 537
Fig. 2. The accretion luminosity distribution in the R­Z plane for neutron stars for a characteristic kick velocity 200 km/s. The luminosity is in
ergs per second per cubic parsec. NNS = 10 9
Fig. 3. The accretion luminosity distribution in the R­Z plane for neutron stars for a characteristic kick velocity 400 km/s. The luminosity is in
ergs per second per cubic parsec. NNS = 10 9
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 accre­
tion luminosity. The accretion luminosity was calculated using
Bondi's formula:
L = # GM —
M
R lib
#
—
M = 2# # (GM ) 2 #(R, Z)
(V 2
s + V 2 ) 3/2 #
.
The sound velocity,V s , was taken to be 10 km/s everywhere.
We used a mass MNS = 1.4M# for NS and MBH = 10M#
for BH. # = nmH is the density, mH being the mass of the
hydrogen atom. The radii, R lib , where the energy is liberated,

538 S.B. Popov & M.E. Prokhorov: Accretion luminosity of isolated neutron stars and black holes in the Galaxy
Fig. 4. The accretion luminosity distribution in the R­Z plane for black holes for a characteristic kick velocity 200 km/s. The luminosity is in
ergs per second per cubic parsec. NBH = 10 8
Fig. 5. The accretion luminosity distribution in the R­Z plane for black holes for a characteristic kick velocity 400 km/s. The luminosity is in
ergs per second per cubic parsec. NBH = 10 8
were assumed to be equal to 10 km for NS and 90 km (i.e.
3 · R g , R g = 2GM/c 2 ) for BH. Calculations used a grid with a
cell size 100 pc in the R­direction and 10 pc in the Z­direction.
The luminosity is given on the figures in ergs per second per
cubic parsec.
For the normalization of our results we assumed that NNS =
10 9 and NBH = 10 8 in the considered volume of the Galaxy.
For a Salpeter mass function with #=2.35 the ratio of NS to BH
is about 10 if NS are formed from stars with masses between
10M# and # 45-50M# , and BH from stars with masses higher
than # 45-50M# . Motch et al. (1997) argued that NNS = 10 9
can be ruled out, NNS = 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,

S.B. Popov & M.E. Prokhorov: Accretion luminosity of isolated neutron stars and black holes in the Galaxy 539
23
25
27
29
­2x10 4 ­1x10 4 0 1x10 4 2x10 4
R, pc
Log
Luminosity
Fig. 6. Slice at Z=+5 pc for NS for a charac­
teristic kick velocity 200 km/s. NNS = 10 9 .
The accretion luminosity is in ergs per sec­
ond per cubic parsec. The solid line is a
smoothed curve.
that NNS = 10 9 is required to explain that the present heavy
element abundance in the Galaxy is about Z=0.02.
3. Results and discussion
In Figs. 2­5 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).
Fig. 6 shows the luminosity in the Galactic plane as a func­
tion 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 0­180 degrees, the left to
-- 180­360 degrees. The differences between the left and the
right parts of the curve give an indication of the accuracy of our
computations.
4. Concluding remarks
As can be seen from the figures, the distribution of the accretion
luminosity in R­Z plane forms a toroidal (ring) structure with
maximum at approximately 5 kpc.
As expected, BH give higher luminosity than NS, as they
have greater masses. But if the total number of BH is signifi­
cantly lower than the number of NS, their contribution to the
luminosity can be less than the contribution of NS. The to­
tal accretion lumiunosity of the Galaxy for NNS = 10 9 and
NBH = 10 8 is about 10 39
- 10 40 erg/s. For a characteristic ve­
locity 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 veloc­
ities 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, associ­
ated with accreting NS and BH.
Acknowledgements. The work was supported by the RFFI (95­02­
6053) and the INTAS (93­3364) 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. Pot­
tasch for his help.
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