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Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
Period distribution of old accreting isolated neutron stars
M.E. Prokhorov 1 , S.B. Popov 2 , and A.V. Khoperskov 3
1 Sternberg Astronomical Institute, Universitetski pr. 13, 119899 Moscow
e-mail: mystery@sai.msu.ru
2 Sternberg Astronomical Institute, Universitetski pr. 13, 119899 Moscow
e-mail: polar@sai.msu.ru
3 Volgograd State University, Department of Theoretical Physics, 40068 Volgograd;
Sternberg Astronomical Institute, Universitetski pr. 13, 119899 Moscow
e-mail: khoperskov@vlink.ru
Abstract. In this paper we present calculations of period distribution for old accreting isolated neutron stars
(INSs). After a few billion years of evolution low velocity INSs come to the stage of accretion. At this stage the
INS's period evolution is governed by magnetic braking and the angular momentum accreted. Since the interstellar
medium is turbulent the accreted momentum can either accelerate or decelerate the spin of an INS, therefore the
evolution of the period has a chaotic character. Our calculations show that in the case of constant magnetic eld
accreting INSs have relatively long spin periods (some hours and more, depending on the INS's spatial velocity,
its magnetic eld and the density of the surrounding medium). Such periods are much longer than the values
measured by ROSAT for 3 radio-silent isolated neutron stars. Due to their long periods INSs should have high
spin up/down rates, _
p, which should uctuate on a time scale of  1 yr.
Key words. neutron stars { magnetic elds { stars: magnetic eld { X-rays: stars { accretion
1. Introduction
The spin period is one of the most precisely determined
parameters of a neutron star (NS). Estimates of masses
(for isolated objects), radii, temperatures, magnetic elds
etc. are usually model dependent. That is why it is very
important to have a good picture of evolution of the only
model independent physical parameters of INSs | the
spin period and its derivative, as they are usually used
to determine other characteristics of NSs. Our aim is to
obtain the distribution of periods p for old accreting INSs
(AINSs), we also brie y discuss possible values of period
derivative _
p.
A lot has been done to understand the period evolution
of radio pulsars (see Beskin et al. 1993) and NSs in close
binaries (Ghosh & Lamb 1979, Lipunov 1992). AINS are
especially interesting from the point of view of period and
magnetic eld evolution since their history is not \pol-
luted" by huge accretion, as happens with their relatives
in close binary systems, where NSs can accrete up to 1 M
during extensive mass transfer.
In early 90 s there has been a great enthusiasm about
the possibility to observe a huge population of AINSs with
ROSAT (Treves & Colpi 1991, Blaes & Rajagopal 1991,
Blaes & Madau 1993, Madau & Blaes 1994). However,
Send oprint requests to: S. Popov
it has become clear that AINSs are very elusive
(Treves et al. 1998) due to their high spatial velocities
(Neuhauser & Trumper 1999, Popov et al. 2000a)
or/and magnetic eld properties (Colpi et al. 1998,
Livio et al. 1998).
Still, AINSs (and radio-silent INSs in gen-
eral) are a subject of interest in astrophysics (see
Caraveo et al. 1996, Treves et al. 2000 and references
therein). A few candidates seem to be observed by ROSAT
(Motch 2001), although it is also possible that these
sources (at least a part of them) can be better explained
by young cooling NSs (see Neuhauser & Trumper 1999,
Yakovlev et al. 1999, Walter 2001, Popov et al. 2000b).
Nevertheless, such sources should be very abundant at
the low uxes attainable by Chandra and XMM{Newton
observatories. In (Popov et al. 2000b) the authors obtain
that at  10 13 erg s 1 cm 2 AINSs become more
abundant than young cooling NSs, and the expected
number is about 1 source per square degree for uxes
> 10 16 erg s 1 cm 2 . So, the calculation of properties
of AINSs is of great importance now.
Previous estimates of the spin properties of AINSs
(Lipunov & Popov 1995a, Konenkov & Popov 1997) have
given only typical values of periods, but no realistic dis-
tributions of this parameter are calculated. \Spin equilib-
rium" of NSs with the interstellar medium (ISM) has been

2 M.E. Prokhorov et al.: Periods of accreting isolated neutron stars
always assumed, i.e. the authors have considered the situ-
ation, when all INS have enough time for spin evolution,
and they have not considered NSs with relatively high spa-
tial velocities. In this paper we present full analysis of the
problem.
Period estimates are especially important as this pa-
rameter can be used to distinguish accreting INSs from
young cooling NSs and background objects.
We proceed as follows: In the next section we de-
scribe the model used to calculate the period distribu-
tion. In Section 3 we present our results. In this paper
we do not address the question of the total number of
AINS, for this data we refer to our previous calcula-
tions (Popov et al. 2000a, Popov et al. 2000b). Here we
only show period distributions for AINSs. In the last sec-
tion we give a brief discussion, derive typical parameters
of _
p for AINSs and summarize the paper.
2. Model
In this section we describe our model of spin evolution of
AINS in a turbulent ISM. We consider constant ISM den-
sity and isotropic Kolmogorov turbulence 1 (a Kolmogorov
spectrum is in good correspondence with most observa-
tions of interstellar, i.e. intercloud, turbulence, see for ex-
ample Falgarone & Philips 1990). The orientations of tur-
bulent cells at all scales are assumed to be independent.
Turbulent velocities at dierent scales relate to each other
according to the Kolmogorov law:
v t (r 1 )
v t (r 2 ) =

r 1
r 2
 1=3
:
Observations show (see Ruzmaikin et al. 1988 and ref-
erences therein) that the turbulent velocity at the scale of
R t ' 2
10 20 cm ' 70 pc is about v t ' 10 km s 1 . The
above scale is close to the thickness of the gas disk of the
Galaxy, and the above velocity | to the speed of sound
in the ISM. It corresponds to the largest cell size possi-
ble and the fastest movements (otherwise turbulence will
eфciently dissipate its energy in shocks).
As an AINS moves through the ISM it can capture
matter inside the so called Bondi (or accretion, or gravita-
tional capture) radius, RG = 2GM=v 2 . Here G is the grav-
itational constant, M is the mass of NS, v =
p
v 2
NS + v 2
s ,
vNS is the spatial velocity of a NS relative to ISM, v s is the
speed of sound (we assume v s = 10 km s 1 , M = 1:4 M
everywhere in the paper; v s can be dependent on the lu-
minosity of AINS (Blaes et al. 1995), but here we neglect
it, but plan to include this dependence in our future cal-
culations).
The accretion rate _
M at the conditions stated above is
equal to (Hoyle & Littleton 1939, Bondi & Hoyle 1944):
_
M = R 2
G nm p v ;
1 Any other more complicated model of turbulence requires
better knowledge of the ISM structure especially on small
scales. This information can be obtained from radio pulsar scin-
tillation observations (see for example Smirnova et al. 1998).
We plan to include this data in our future calculations.
here n is the number density of the ISM, m p is the mass of
proton. This accretion rate corresponds to the luminosity:
L = GM _
M=R  10 32 n (v=10 km s 1 ) 3 erg s 1
Based on population synthesis models
(Popov et al. 2000b) we can expect, that on average
AINS should have luminosities of about 10 29 erg s 1 .
Simple calculations show, that most of this energy will be
emitted in X-rays with a typical blackbody temperature
of about 0.1 keV (if due to signicant magnetic eld
accretion proceeds onto small polar caps then the temper-
ature would be higher up to 1 keV). Note, that the Bondi
rate is just the upper limit, in reality due to heating
(Shvartsman 1970a, Shvartsman 1971, Blaes et al. 1995)
and magnetospheric eects (Toropina et al. 2001) the
accretion rate can be lower.
Due to the turbulence the accreted matter carries non-
zero angular momentum:
j t = v t (RG )
RG = v t (R t )R 1=3
t R 4=3
G :
In this formula it is considered that cells of the size r = RG
are the most important, otherwise in the Eq.(1) below it
is necessary to introduce factor  6= 1.
If j t is larger than the Keplerian value at the mag-
netosphere boundary (i.e. at the Alfven radius RA =
( 2 =2 _
M
p
GM ) 2=7 , here  is the magnetic moment of a
NS) an accretion disk is formed around the AINS. In the
disk a part of the angular momentum is carried outwards,
and the NS accretes matter with Keplerian angular mo-
mentum j K = v K (RA )
RA , v K { Keplerian velocity. This
situation holds only for very low magnetic elds and low
spatial velocities of NSs, thus we do not take it into ac-
count in the present calculations.
We consider the lowest spatial velocity of NSs 2 to
be equal to 10 km s 1 . This value is of order of the
speed of sound. Therefore lower spatial velocities can
not change the accretion rate signicantly. Moreover such
low values are not very probable due to non-zero spa-
tial velocities of NSs progenitors (see Popov et al. 2000a,
Arzoumanian et al. 2001 for limits onto the fraction of low
velocity INSs derived by dierent methods). For several
cases below we consider v ' v NS .
During the time required to cross a turbulent cell of
the size RG ,
t = RG =vNS ' 11:8 (v=10 km s 1 ) 3 yrs,
the change in the angular momentum of a NS J is given
as:
j
J j = _
Mj t
t = nm p v t (R t )R 1=3
t R 13=3
G
' 1:23
10 39 g cm 2 s 1

n

v t (R t )
10 km s 1

R t
2
10 20 cm
 1=3 
v
10 km s 1
 26=3
:
Accordingly, the change of the spin frequency reads:
j
!j = j
J j =I
2 The lowest measured velocity is >  50 km s 1
(Lyne & Lorimer 1994), please bear in mind that selection ef-
fects are very important here.

M.E. Prokhorov et al.: Periods of accreting isolated neutron stars 3
' 1:23
10 6 s 1 
n

v t (R t )
10 km s 1

R t
2
10 20 cm
 1=3 
v
10 km s 1
 26=3
I 1
45 ;
here I = I 45 10 45 g cm 2 | moment of inertia of a NS.
The orientation of
! is random, and it is isotropically
distributed on a sphere. The value of the frequency change
is strongly dependent on the spatial velocity of the NS:

!  v 26=3 , so the maximum value for v NS = 10 km s 1
is
!max ' 6
10 8 rad s 1 .
On this view, it follows that it is possible to describe
the spin evolution of an AINS as random movement in 3-D
space of angular velocities !. Since typical temporal and
\spatial" scales
t and
! are reasonably small (
t 
t gal ' 10 10 yrs,
!  !  10 1  10 7 rad s 1 ) we
consider the problem to be continuous. Therefore, we can
use dierential equations valid for continuous processes.
In this case the spin evolution of an AINS is described
by the diusion equation with the coeфcient D given by:
D = 
6

! 2

t ; (1)
here  is a coeфcient which takes into account cells with
sizes r 6= RG (everywhere in this paper we use  = 1).
Apart from the in uence of random turbulence, an
AINS is spinning down due to magnetic braking:
d!
dt
=  t
 2
IR 3
co
!
j!j + F t =  t
 2
IGM
! 2 !
j!j + F t : (2)
Here R co = (GM=! 2 ) 1=3 is the corotation radius,  t is
a constant of order unity, which takes into account de-
tails of the interaction between the magnetosphere and
the accreted matter (everywhere in this paper we assume
 t = 1), F t is the random (turbulent) force, which has a
zero average value: hF t i = 0.
On large time scales and for relatively short spin peri-
ods we can neglect the momentum of the accreted matter
and the initial period, p 0 , of the NS. In that case the so-
lution of Eq. (2) looks as:
! / t 1 ; p  2
! / t :
Magnetic braking produces a convective term in the
evolutionary equation for the spin frequency distribution,
f(!). As far as the initial distribution of spin vectors is
isotropic (magnetic braking does not change the orienta-
tion of !) and the turbulent diusion is isotropic too, we
obtain a spherically symmetric distribution in space of the
angular velocities, i.e.:
f(!) = f 3 (!)
(f 3 (!) describes the 3-D distribution of AINSs, its dimen-
sion is [s 3 ]).
From (1) and (2) we obtain the evolutionary equation:
@f 3
@t
= A
! 2
@
@!
! 4 f 3


+ D
! 2
@
@!

! 3 @f 3
@!

; (3)
here A =  2 =IGM


.
The boundary condition at ! = 0 can be derived from
the equality of the ow of the particles, which at this point
becomes zero:
@f 3
@!
    !=0
= 0 :
It is easy to nd a stationary solution of equation (3)
on the semi-innity axis 3 (!  0)
f st
3 = C st exp

 2
3 IGMD j!j 3

; (4)
here the normalization constant C st can be derived from
the condition 4
R
f st
3 (!)! 2 d! = N , N is the total num-
ber of AINSs in the distribution. The total number of NSs
in the Galaxy is uncertain, N  10 8 {10 9 . Population syn-
thesis calculations (Popov et al. 2000b) give arguments
for a higher total number about 10 9 . Here we do not ad-
dress this question. From the curves below, which repre-
sent relative period distribution of AINSs, one can deter-
mine absolute numbers of AINSs with each period value
if the total number of these objects is known.
The position of the maximum of f st
3 (p) depends on v,
, n and M . An increase of v and  shifts the maximum
to larger p, an increase of n and M shifts the maximum
to shorter p.
If we have to solve the problem for a constant starfor-
mation rate we can write the second boundary condition
as:
f 3 (!A ) = IGM
4 t  2

_
N
! 4
A
;
here _
N is the rate at which NSs come to the
stage of accretion, !A = 2=pA is the accretor
frequency, at that value for given v and n accre-
tion sets on, pA = 2 5=14 (GM) 5=7 ( 2 = _
M) 3=7 '
300  6=7
30 (v=10 km s 1 ) 9=7 n 3=7 s.
The function f 3 (!) is connected with distributions of
the absolute values of the angular velocity, f 1 (!), and of
the spin period, f(p), according to the formulae:
f 1 (!) = 4! 2 f 3 (!) ;
f(p) = 32
p 4 f 3

2
p

:
Figure 1 shows the evolution of f(p) as a function of
time for typical parameters of an AINS and the ISM.
For given v,  and n NSs enter the accretor stage when
they spin down to pA . Up to this value they evolve as
ejectors and propellers (Lipunov 1992, Colpi et al. 2001).
For pA < p < p cr the in uence of the turbulence is
small and AINS spin down according to p / t. Here p cr
denes the stage when spin changes due to magnetic brak-
ing and turbulent acceleration/deceleration become of the
same order of magnitude (see below the Discussion). One
3 As far as NS enter the stage of accretion with relatively
short periods !A  h!(f st
3 )i, the obtained distribution is dif-
fered from the real one only on high frequencies ! ' !A .

4 M.E. Prokhorov et al.: Periods of accreting isolated neutron stars
10 3
10 4
10 5
Periods, s
0.0001
0.001
0.01
0.1
1
Fig. 1. Evolution of the period distribution in time;  =
10 30 G cm 3 , n = 1 cm 3 , vNS = 10 km s 1 . Curves are plotted
for 4 dierent moments from 1:72
10 9 yrs to 9:8
10 9 yrs (tA
for the chosen parameters is equal to ' 1:7
10 9 yrs). An AINS
crosses the horizontal part from ' 10 2 s to 10 4 s in  6
10 7 yrs.
Curves were normalized to 1 in the maximum for the highest
curve.
can introduce another useful timescale,
t cr . This is the
time required to reach turbulent regime, p = p cr . These
considerations together with constant starformation rate
determine left part of the distribution where f(p) = const
(see Fig. 1).
As the period grows turbulence becomes more and
more important. Finally the rst AINSs reach the point
p = 1 (! = 0) and there the equilibrium component of
the distribution (described by Eq. (4) is formed quickly.
This component decreases in a power law fashion (f(p) /
p 4 ) at p  p turb (where p turb is determined by the width
of the distribution (4)), and decreases even faster than
exponentially at p  p turb . Please note that the number
of AINSs reaching equilibrium grows linearly with time.
As a result, the amplitude of the equilibrium component
of the distribution grows likewise. If the time required to
reach the accretor stage, t A , is large (t gal t A  t gal ) the
equilibrium component is not formed. For t A > t gal NS
never reach the stage of accretion.
Similar considerations can be applied also to binary X-
ray pulsars (see Lipunov 1992). In binaries NSs can reach
a real period equilibrium (Ghosh & Lamb 1979), and sta-
tionary solution should be used. In that case observations
of period uctuations can help to derive physical param-
eters of the systems, for example stellar wind velocity for
wind-fed pulsars (Lipunov & Popov 1995b).
3. Calculations and results
The main aim of this paper is to obtain period distribu-
tion for an AINS in a turbulent ISM. We assume that NS
are born with short ( 1 s) spin periods. The magnetic
moment distribution is taken to be in log-Gaussian form:
f() = 1
p
2m exp

(log  log  0 ) 2
2 2
m

; (5)
with log  0 = 30:06 and m = 0:32 (see Colpi et al. 2001
and references therein for a recent discussion and data on
magnetism in NSs). The magnetic eld is considered to be
constant (see, for example, Urpin et al. 1996 for calcula-
tions of magnetic eld decay and related discussion).
The velocity distribution (due to kick after supernova
explosion) is assumed to be Maxwellian:
f(vNS ) = 6
p

v 2
NS
v 2
m
exp

3
2
v 2
NS
v 2 m

(6)
with v m = 200 km s 1 which corresponds to
 v ' 140 km s 1 (see for example Lyne & Lorimer 1994,
Cordes & Cherno 1997, Hansen & Phinney 1997,
Lorimer et al. 1997, Cordes & Cherno 1998, and
Arzoumanian et al. 2001 for discussions on the kick
velocity of NSs).
We use two values of the ISM number density, n = 1
cm 3 and n = 0:1 cm 3 , as typical values for the Galactic
disk 4 .
All NS are divided into 30 groups in  interval from
10 28:6 to 10 31:6 G
cm 3 with the step 0.1 dec and 49 groups
in velocities ranging from 10 to 500 km s 1 with a step of
10 km s 1 .
For each group we calculate t A , the time necessary to
reach accretion:
t A = t E + t P ;
here t E and t P | durations of the ejector and propeller
stages correspondently.
In the stage of ejection a NS spins down due to
magneto-dipole losses:
t E ' 10 9 n 1=2
 v
10 km s 1
 
10 30 G
cm 3
 1
yrs.
The spin down in the propeller stage
(Shvartsman 1970b, Illarionov & Sunyaev 1975) is
not well known, but for a constant eld always t P < t E
(Lipunov & Popov 1995a) (see also results of numerical
calculations in Toropin et al. 1999), so we neglect t P in
the rest of the paper, i.e. we assume t A = t E .
For high spatial velocities after the ejector phase an
INS appears not as a propeller, but as a so-called georota-
tor (Lipunov 1992, see also Rutledge 2001). At this stage
RA > RG , so the surrounding matter does not feel the
gravitation of the NS, and its magnetosphere looks like
magnetospheres of the Earth, Jupiter etc. Partly because
of that eect we truncated our velocity distribution at
500 km s 1 .
For systems with t A < t gal = 10 10 yrs we solve Eq.
(3) on the time interval t gal t A . For angular velocities
4 This assumption is reasonable for relatively low velocity
INS, but only this fraction of the whole population is important
for us here.

M.E. Prokhorov et al.: Periods of accreting isolated neutron stars 5
10 2
10 3
10 4
10 10 6
10 7
10 8
Periods, s
0.0001
0.001
0.01
0.1
1
Fig. 2. The period distribution for populations of AINSs evolv-
ing in an ISM with number density n = 1 cm 3 (upper solid
curve) and n = 0:1 cm 3 (the second dotted curve). Lower
curves show results for the low velocity part of the AINS pop-
ulation for n = 0:1 cm 3 (v < 60; 30; 15 km s 1 correspon-
dently, see Discussion below). Curves are normalized to 1 at
the maximum of the highest curve.
we use grid with 200 cells from ! = 0 to !A . A conser-
vative implicit scheme is used. The derived distributions
f(p; ; v) (for dierent groups) are summed together with
weights from Eqs. (5) and (6).
The nal distribution for n = 1 cm 3 and n = 0:1
cm 3 are shown on Fig. 2.
We note, that the nal distribution signicantly diers
from the one for a single set of parameters (n; ; v) as
can be seen from Fig. 1 and Fig. 2. In the nal one we
a see power-law cuto (f / p 4 ) at long periods (p >
10 7 s) similar to the cuto in the Fig. 1. Then at the
intermediate periods (10 4 < p < 10 7 s) the curve has
a power-law ascent due to summation of individual curve
maxima. Finally a sharp cuto at short periods (p < 10 4 s)
is present because dierent INSs enter the accretor stage
with dierent periods: pA = pA (n; ; v).
4. Discussion and conclusions
After an INS comes to the stage of accretion (p > pA ) its
spin is controlled by two processes (see Eq.2): magnetic
spin-down and turbulent spin-up/spin-down. A schematic
view of such evolution process is shown in Fig. 3.
We can describe them with characteristic timescales:
t mag and t turb . As one can calculate, t mag  p and
t turb  p 1 . Here we evaluate t as p= _
p, and _
p turb =
p 2 _
Mj=(2I). Here we again note, that j t can be larger that
the Keplerian value of angular momentum at the magne-
tosphere, j K = v K (RA ); RA . So we write j instead of j t
as we did before, now j = min(jK ; j t ).
_
pmag = 2 2 =(IGM) (7)
t
lg p
t t
p
p
p
p
p
Propeller
e cr
0
e
A
cr
turb
dt= R G /v
lg
Fig. 3. Schematic view of the period evolution of an INS. A
NS starts with p = p0 , then spins down according to magneto-
dipole formula up to pE , then a short propeller stage (marked
by a circle) appears and lasts down to p = pA . At the situation
illustrated in the gure tE  tA .
On the stage of accretion at rst magnetic braking is more
important, after at t = tcr turbulence starts to in uence sig-
nicantly spin evolution of a NS period stars to uctuate with
a typical time scale RG=vNS close to mean value noted p turb
on the vertical axis.
Initially (immediately after p = pA ) magnetic spin-
down is more signicant:
t mag = IGM
2 2 p; (8)
but at some period, p cr , these two timescales should be-
come of the same order, and for longer periods an INS will
be governed mainly by turbulent forces.
One can obtain the following formula:
p 2
cr = 4 2  2
GM _
Mj
: (9)
This period lies between pA and p turb . An INS reaches
p cr in
t cr  10 5 {10 7 yrs after onset of accretion:

t cr = I
p
GM

q
_
Mj
: (10)
Here
t cr is calculated approximately as p cr = _
pmag .
The period evolution of an AINS can be described in
the following way (Fig. 3): after the INS comes to the stage
of accretion it spins down for  10 5 10 7 yrs up to p cr ,
then the evolution is mainly controlled by the turbulence,
and period uctuates with typical value p turb , which is

6 M.E. Prokhorov et al.: Periods of accreting isolated neutron stars
determined by the properties of the surrounding ISM and
spatial velocity of an INS.
We note, that we do not take into account any selec-
tion eects. For example, as period and luminosity both
depend on the velocity of an INS, they are correlated.
Taking into account the lower ux limits attainable by the
present day satellites it is possible to calculate the proba-
bility to observe an accreting INS with some period. Also
periods (and luminosities) are correlated with the position
of an INS in the Galaxy.
So, it is reasonable to make calculations which include
all these eects in order to make better predictions for
observations. We plan to unite our population synthesis
calculations with detailed calculations of spin evolution
later. In this paper we present distributions as if all ac-
creting INSs can be observed.
For the main part of its life the spin period of a low-
velocity AINS is governed by turbulent forces. The char-
acteristic timescale for them can be written as: t turb =
I!=( _
Mj), and it is equal to 10 4 -10 5 yrs for typical param-
eters.
If the eld decays the picture should be-
come completely dierent (see for example
Konenkov & Popov 1997, Wang 1997), and observa-
tions of AINSs can put important limits onto models of
magnetic eld decay in NS (Popov & Prokhorov 2000).
For decaying elds AINSs can appear as pulsating
sources with periods about 10 s and _
p about 10 13 s/s
(Popov & Konenkov 1998). The value and sign of _
p will
uctuate as an INS passes through the turbulent cell on
a time scale RG =vNS , which is about a year for typical
parameters. Irregular uctuations of _
p on that time
scale can be signicant indications for the accretion (vs.
cooling) nature of observed luminosity.
Roughly _
p can be estimated from the expression:
j _
pj  p 2
_
Mj
2I  v 17=3 : (11)
Here we neglect magnetic braking in Eq. (2). This equa-
tion for _
p is valid for the turbulent regime of spin evolu-
tion, i.e. for p  p turb . For a given value of the spin period
_
p uctuates between +p 2 _
Mj
2I and p 2 _
Mj
2I , and depends on
n, v, but not on  (if j = j t , not j = j A ). In this picture
the _
p distribution in the interval specied above (Eq.11)
is at.
The behavior of p and _
p of AINSs in molecular clouds
can be dierent (Colpi et al. 1993), especially for low spa-
tial velocities of NSs. Some of our assumptions in that case
are not valid, and the results cannot be applied directly.
But we note, that passages through molecular clouds are
relatively rare and short, they cannot signicantly in u-
ence the general picture of AINSs spin evolution.
The period distribution which can be obtained from
observation (for example from ROSAT data) can be dif-
ferent from the two upper curves in Fig. 2. Such surveys
are ux-limited, so they include (in the case of AINS) the
most luminous objects. But they form only a small frac-
tion of the whole population.
To illustrate this in Fig. 2 we also plot distributions
for low velocity objects (v < 60; 30; 15 km s 1 ). In the
case of xed a ISM density an upper limit to the value
of the space velocity corresponds to a lower limit to the
accretion luminosity of an AINS. Clearly, the brighter the
source, the shorter (on average) its spin period. Even in
such groups with relatively short periods their values are
far from typical periods of ROSAT INSs,  5 20 s. So,
these objects cannot be explained by accretors with con-
stant eld B  10 12 G.
Calculations of period distributions for decaying mag-
netic eld, for populations of INSs with a signicant frac-
tion of magnetars and for an accretion rate dierent from
the standard Bondi-Hoyle-Littleton value (due to heating
and in uence of a magnetosphere) will be done in a sepa-
rate paper.
In conclusion we stress readers attention on the main
results of the paper:
| we obtained spin period distributions for AINSs with
constant magnetic eld and \pulsar" properties (mag-
netic elds, initial periods and velocity distributions).
| these distributions are shown in Fig. 2. They have a
broad maximum at very long periods,  10 5 | 10 6 s.
In that case the observed objects should not show any
periodicity.
| the periods of these objects should uctuate on a time
scale RG =vNS  1 yr.
Acknowledgements. This work was supported by grants of the
RFBR 01-02-06265, 00-02-17164, 01-15(02)-99310.
AK thanks Sternberg Astronomical Institute for hospital-
ity. SP and MP thank Monica Colpi, Roberto Turolla and Aldo
Treves for discussions and Universities of Como, Milano and
Padova for hospitality.
We thank Vasily Belokurov and the referee of the paper for
their comments on the text and useful suggestions.
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