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The Neutron Star Census
S.B. Popov 1 , M. Colpi 2 , A. Treves 3 , R. Turolla 4 , V.M. Lipunov 1;5 and M.E. Prokhorov 1
ABSTRACT
The paucity of old isolated accreting neutron stars in ROSAT observations is used
to derive a lower limit on the mean velocity of neutron stars at birth. The secular
evolution of the population is simulated following the paths of a statistical sample of
stars for different values of the initial kick velocity, drawn from an isotropic Gaussian
distribution with mean velocity 0 Ÿ hV i Ÿ 550 km s \Gamma1 . The spin--down, induced by
dipole losses and the interaction with the ambient medium, is tracked together with
the dynamical evolution in the Galactic potential, allowing for the determination of
the fraction of stars which are, at present, in each of the four possible stages: Ejector,
Propeller, Accretor, and Georotator. Taking from the ROSAT All Sky Survey an upper
limit of ¸ 10 accreting neutron stars within ¸ 140 pc from the Sun, we infer a lower
bound for the mean kick velocity, hV i ¸ ? 200 \Gamma 300 km s \Gamma1 ; corresponding to a velocity
dispersion oe V ¸ ? 125 \Gamma 190 km s \Gamma1 . The same conclusion is reached for both a constant
(B ¸ 10 12 G) and a magnetic field decaying exponentially with a timescale ¸ 10 9 yr.
Such high velocities are consistent with those derived from radio pulsar observations.
Present results, moreover, constrain the fraction of low velocity stars, which could have
escaped pulsar statistics, to ¸ ! 1%.
Subject headings: Accretion, accretion disks --- stars: neutron --- X--rays: stars
1 Sternberg Astronomical Institute, Universiteskii Pr. 13, 119899, Moscow, Russia; e--mail: polar@xray.sai.msu.su
2 Dept. of Physics, University of Milan, Via Celoria 16, 20133 Milan, Italy; e--mail: colpi@uni.mi.astro.it
3 Dipartimento di Scienze, Universit`a dell'Insubria, Via Lucini 3, 22100, Como, Italy; e--mail: treves@uni.mi.astro.it
4 Dept. of Physics, University of Padova, Via Marzolo 8, 35131 Padova, Italy; e--mail: turolla@pd.infn.it
5 Dept. of Physics, Moscow State University; e--mail: lipunov@sai.msu.su
1

1. Introduction
Isolated neutron stars (NSs) are expected to be as
many as 10 8 --10 9 , a non--negligible fraction (¸ 1%) of
the total stellar content of the Galaxy. The number
of observed radio pulsars is now ¸ 1,000. Since the
pulsar lifetime is ú 10 7 yr, this implies that the bulk
of the NS population, mainly formed of old objects,
remains undetected as yet. Despite intensive searches
at all wavelengths, only a few (putative) isolated NSs
which are not radio pulsars (or soft fl repeaters) have
been recently discovered in the X--rays with ROSAT
(Walter et al. 1996; Haberl et al. 1996; Haberl et
al. 1998; Ne¨uhauser & Tr¨umper 1999; Schwope et
al. 1999). All these sources emit a thermal spec­
trum at ú 100 eV and the derived column densi­
ties place them at relatively close distances ( ¸ ! 140
pc) with luminosities L ¸ 10 30 --10 31 ergs \Gamma1 . For two
of them (RX J1856­3754, RX J0720­3125), extremely
faint optical counterparts (m V ¸ 26) have been firmly
established (Walter & Matthews 1997; Kulkarni &
van Kerkwijk 1998; Motch & Haberl 1998) while for
other sources plate searches placed limits down to
mV ¸ ? 26. Although the extreme X--ray to optical
flux ratio (? 10 3 ) makes the NS option rather ro­
bust, the exact nature of their emission is still con­
troversial. Up to now, two main possibilities have
been suggested, either relatively young NSs radiating
away their residual internal energy or much aged NSs
accreting the interstellar gas. Both options have ad­
vantages and drawbacks. Cooling atmosphere mod­
els fail to predict in a natural way the optical excess
observed in the best studied object, RX J1856­3754,
while this seems be a distinctive feature of accreting
atmospheres (Zane, Turolla & Treves 1999). On the
other hand, at least if the Bondi--Hoyle scenario ap­
plies, accretion is reduced for increasing star velocity
and for v ? 20 km s \Gamma1 it fails to produce the luminosi­
ties inferred from ROSAT data. When they become
available, proper motion measurements for some of
the five isolated NS candidates detected so far will
prove decisive in assessing the true nature of these
sources. Meanwhile, we feel that a more thorough
analysis of the statistical properties of NSs is of inter­
est and can be useful in providing indirect evidence
in favor or against the accretion scenario.
As discussed by Lipunov (1992), isolated NSs can
be classified into four main types: Ejectors, Pro­
pellers, Accretors and Georotators. In Ejectors the
relativistic outflowing momentum flux is always larger
than the ram pressure of the surrounding material so
they never accrete and are either active radio pulsars
or dead pulsars, still spun down by dipole losses. In
Propellers the incoming matter can penetrate down to
the Alfven radius but no further because of the cen­
trifugal barrier, and, although stationary inflow can
not occur, the piling up of the material at the Alfven
radius may give rise to (supposedly short) episodes
of accretion (Treves, Colpi & Lipunov 1993 ; Popov
1994). Steady accretion is also impossible in Georo­
tators where (similarly to the Earth) the Alfven ra­
dius exceeds the accretion radius, so that magnetic
pressure dominates everywhere over the gravitational
pull. It is the combination of the star period, mag­
netic field and velocity that decides which type a given
isolated NS belongs to and, since both P , B and V
change during the star evolution, a NS can go through
different stages in its lifetime. This argument shows
that, at variance with what was assumed in earlier
investigations (Treves & Colpi 1991; Blaes & Madau
1993; Zane et al. 1995) stationary accretion onto an
isolated old NS (ONS hereafter) depends crucially on
its rotational period and magnetic field.
The dynamical evolution of NSs in the Galactic
potential was studied by several authors (Paczy'nski
1990; Blaes & Rajagopal 1991; Blaes & Madau 1993;
Madau & Blaes 1994; Zane et al. 1995) who computed
the present velocity distribution, one of the key pa­
rameters governing accretion. Until now, however, lit­
tle attention was paid to the NSs magneto--rotational
evolution. Only recently, this issue was discussed in
some detail by Livio Xu & Frank (1998) and Colpi et
al. (1998), who found that, for a given velocity dis­
tribution, the number of accreting sources depends
strongly on the extent of the magnetic field decay.
Goal of this investigation is to consider these two
issues simultaneously, coupling the dynamical and the
magneto--rotational evolution for the isolated NS pop­
ulation. In particular, we explore the effects induced
on the current census of NSs by varying the mean
value of the kick velocity. In earlier investigations
(Treves & Colpi 1991; Blaes & Madau 1993; Zane
et al. 1995), the velocity distribution at birth was
assumed either Gaussian with a mean velocity (in­
ferred on the basis of existent data, Narayan & Os­
triker 1990) of ¸ 70 km s \Gamma1 , or skewed towards higher
velocities (¸ 200 km s \Gamma1 ) but still rich in slow stars
(Paczy'nski 1990). In the last few years however,
proper motion studies and revised distance estimates
of pulsar subpopulations revealed that young neutron
2

stars have high mean velocities, ¸ 400 km s \Gamma1 ac­
cording to the original suggestion by Lyne & Lorimer
(1994) or ¸ 200 km s \Gamma1 on the basis of the more re­
cent analyses by Hansen & Phinney (1997), Cordes &
Chernoff (1997) and Cordes & Chernoff (1998). Al­
though little is known about the low--velocity tail of
the distribution and it is still unclear if pulsars are re­
ally representative of the entire NS population (Hart­
man 1997; Hartman et al. 1997), the possibility that
the low--velocity tail is underpopulated with respect
to what was previously assumed should be seriously
taken into account. It is our aim to revise the esti­
mates on the number of old accreting neutron stars
in the Galaxy, and in the solar vicinity in particu­
lar, in the light of these new data, in the attempt to
reconcile theoretical predictions with present ROSAT
limits (Ne¨uhauser & Tr¨umper 1999). This is of po­
tential importance as it may indirectly unveil how the
long term evolution of key parameters, such as the
magnetic field, the period, the velocity distribution
and the star formation rate relate to the properties
of neutron stars at birth and to their interaction with
the Galactic environment.
2. The Model
In this section we summarize the main hypothesis
introduced to track the evolution of single stars and
describe shortly the technique used to explore their
statistical properties, referring to Popov & Prokhorov
(1998), Popov & Prokhorov (1999) for details on spa­
tial evolution calculations.
2.1. Dynamical evolution
The dynamical evolution of each single star in the
Galactic potential is followed solving its equations of
motion. The potential, proposed by Miyamoto & Na­
gai (1975) and adopted later by Pac'zynski (1990), is
given by the superposition of a spherical halo and of
two flattened bulge/disk components. In Galactocen­
tric cylindrical coordinates R, Z, r =
p
R 2 + Z 2 , they
are expressed as
\Phi H (R; Z) = \Gamma GMC
r C
Ÿ
1
2 ln
`
1 + r 2
r 2
C
'
+
r C
r
tan \Gamma1
`
r
r C
'–
(1)
and
\Phi i (R; Z) = \Gamma GM i
p
R 2 + [a i + (Z 2 + b 2
i ) 1=2 ] 2
(2)
where the index i stands both for B (bulge) and D
(disk). The values of the parameters appearing in
equations (1) and (2) are summarized in table 1.
In their motion through the Galaxy, NSs interact
with the interstellar medium (ISM) when (and if)
they enter the Propeller or the Accretor phase. In
these two stages the accreting material affects signifi­
cantly the star spin because braking torques arise (see
x2.3). Since the period evolution depends on both the
star velocity and the local density of the interstel­
lar medium, any attempt to investigate the statistical
properties of the NS population should incorporate
a (detailed) model of the ISM geography. Unfortu­
nately the distribution of molecular and atomic hy­
drogen in the Galaxy is highly inhomogeneous and
only an averaged description is possible. Here we use
the analytical distributions from Bochkarev (1992)
and Zane et al. (1995) . Denoting with nHI and nH2
the neutral and molecular hydrogen number density,
the total proton density is given by
n(R; Z) = nHI + 2nH2 : (3)
The molecular hydrogen distribution is approximated
as
nH2 (R; Z) = n 2 (R) exp
Ÿ
\GammaZ 2
2 \Delta (70pc) 2
–
(4)
where n 2 (R) is tabulated in Bochkarev (1992). The
map of atomic hydrogen is more complex and for R Ÿ
3:4 kpc we assume
nHI = n 0 (R) exp
''
\GammaZ 2
2 \Delta (140 pc) 2
`
R
2 kpc
' 2
#
: (5)
The preceding expression becomes inaccurate below
¸ 1 kpc, where the gas distribution is dominated by a
rotating ring and dense clouds. However, since we do
not calculate the evolution of NSs born in the central
2 kpc, this is not going to be of any relevance (see
Zane et al. 1996a for a discussion of the diffuse X--ray
emission from accreting NSs in the Galactic Center).
The density of cold and warm HI at 3:4 kpc Ÿ R Ÿ
8:5 kpc can be fitted by
nHI = 0:345 exp
Ÿ
\GammaZ 2
2 \Delta (212 pc) 2
–
+
3

Table 1
Parameters for the various contributions to the Galactic potential
Halo Bulge Disk
r C = 6:0 kpc aB = 0:0 kpc aD = 3:7 kpc
-- b B = 277 pc b D = 200 pc
MC = 5:0 \Theta 10 10 M fi MB = 1:1 \Theta 10 10 M fi MD = 8:1 \Theta 10 10 M fi
0:107 exp
Ÿ
\GammaZ 2
2 \Delta (530 pc) 2
–
+ 0:064 exp
Ÿ
\GammaZ
403 pc
–
: (6)
while at radii larger than 8.5 kpc we use
nHI = n 3 (R) exp
''
\GammaZ 2
2 \Delta (530 pc) 2
`
R
8:5 kpc
' 2
#
(7)
where n 0 (R) and n 3 (R) are again taken from Bochkarev
(1992). The total hydrogen density distribution used
in our computations is shown in figure 1; the den­
sity in the galactic plane varies by more than one or­
der of magnitude, ranging from 0:1 cm \Gamma3 to 4:3 cm \Gamma3 :
Within a region of ¸ 140 pc around the Sun, the ISM
is underdense, and we take n = 0:07 cm \Gamma3 (see Zane
et al. 1996a).
In our model we assume that the NS birthrate is
constant in time and proportional in magnitude to the
square of the local gas density (see for example Cox
1983; Firmani & Tutukov 1994). This implies that
NSs are preferentially produced in the molecular ring
region and migrate through the Galaxy during their
evolution, due to the large kick velocities acquired at
birth. Stars are assumed to be born in the Galactic
plane (Z = 0) and in the range 2 kpc ! R ! 16 kpc;
this limitation has no major influence on the results,
and on the number of sources found in the solar vicin­
ity in particular. In the present investigation all ef­
fects of NSs born in binary systems (see, e.g., Iben &
Tutukov 1998) has been neglected.
Neutron stars at birth have a circular velocity de­
termined by the Galactic potential. Superposed to
this ordered motion a kick velocity is imparted in a
random direction. The exact form of the kick dis­
tribution is still uncertain (see, e.g., Lipunov et al.
1996a; Iben & Tutukov 1998; Belczy'nski & Bulik
1999). Here we use an isotropic Gaussian distribu­
tion with dispersion oe V , simply as a mean to model
the true pulsar distribution at birth (see e.g. Cordes
1998). The mean velocity hV i = (8=ú) 1=2 oe V is varied
in the interval 0--550 km s \Gamma1 . The velocity dispersion
of the NS progenitors, ¸ 20 km s \Gamma1 , has been ne­
glected and runs with hV i of this order mimic a NSs
population with zero mean kick velocity at birth.
For each star the dynamical evolution has been fol­
lowed up to present time. The evolution of the spin
period P (as described in x2.3) is also computed, keep­
ing track of the different stages the neutron star ex­
periences (Ejector, Propeller, etc.). A number of rep­
resentative evolutionary paths have been calculated,
placing the stars on a uniform spatial grid with a ran­
domly orientated kick drawn from the specified Gaus­
sian distribution. Each evolutionary track is statisti­
cally independent and in order to mimic continuous
star formation in the galactic plane at various loca­
tions (the n 2 density law), a single track is sampled
in time and repeatedly used, shifting ahead the start­
ing time. Tracks beginning from a denser region have
accordingly a higher statistical weight. In the time
sampling procedure, a single track gives a contribu­
tion to the statistics equivalent to that of a set having
the same initial parameters and representing continu­
ous NS formation (we refer for details to the ``Scenario
Machine'' of Lipunov et al. 1996b).
A number of runs have been performed using dif­
ferent number of objects (up to a maximum of ¸ 10 3 )
and, in addition, tests have been carried out using
different seeds for the random orientation of kicks.
Statistical fluctuations on the number of stars in each
stage are typically of a few percent.
2.2. Accretion physics
The accretion rate was calculated according to the
Bondi formula
4

—
M = 2ú(GM ) 2 m p n(R; Z)
(V 2 + V 2
s ) 3=2
' 10 11 n v \Gamma3
10
g s \Gamma1 (8)
where m p is proton mass, the sound speed V s is al­
ways 10 km s \Gamma1 and v 10 = (V 2 + V 2
s ) 1=2 in units of
10 km s \Gamma1 . Here and in the following M and R de­
note the NS mass and radius, which we take equal to
1:4 M fi and 10 km, respectively, for all stars. Assum­
ing, for the sake of simplicity, a non--relativistic effi­
ciency ¸ GM=Rc 2 , the accretion luminosity is given
by
L = GM —
M
R : (9)
Although the spectrum emitted by accreting NSs may
differ rather substantially from a blackbody at the
star effective temperature (see Zampieri et al. 1995
for low--field spectra and Zane, Turolla & Treves 1999
for magnetized models) here we assume pure thermal
emission. Even in this simplified case, the effective
temperature depends on the star dipole field because
accretion occurs only at the two polar caps of radius
R cap = R
p
(R=RA ),
R cap = 9:5 \Theta 10 3 ¯ \Gamma2=7
30
n 1=7 v \Gamma3=7
10
cm ; (10)
here RA ' 1:1 \Theta 10 10 n \Gamma2=7 v 6=7
10
¯ 4=7
30
cm is the Alfven
radius and ¯ 30 is the star magnetic dipole moment in
units of 10 30 G cm 3 (see e.g. Zane et al. 1996a; Ko­
nenkov & Popov 1997). The reduced emitting area
produces harder spectra, with an effective tempera­
ture ¸ 3--4 times larger than in the unmagnetized
case
T eff ' 5 \Theta 10 8 ¯ 1=7
30
n 13=14 v \Gamma39=14
10
K ; (11)
Fig. 1.--- The hydrogen number density distribution
in the R--Z plane.
typical temperatures are around 100 eV for V ¸ 100
km s \Gamma1 for non­evolving magnetic field, and a little
bit lower for evolving.
2.3. Period evolution
All neutron stars are assumed to be born with a
period P (0) = 0.02 s, and a magnetic moment ei­
ther ¯ 30 = 1 or ¯ 30 = 0:5. Different distributions of
the initial periods, like the one recently proposed by
Spruit & Phinney (1998), were also tested and pro­
duced very similar results.
The ejector regime begins with the pulsar phase
and proceeds also after the breakdown of the coher­
ence condition when the star becomes a dead, or
silent, pulsar. In this phase the energy losses are due
to magnetic dipole radiation and the period increases
in time according to
P = P (0) + 3 \Theta 10 \Gamma4 ¯ 30 t 1=2 s (12)
where t is in yr. When the gravitational energy den­
sity of the incoming interstellar gas exceeds the out­
ward momentum flux at the accretion radius, R ac '
2GM=v 2 , matter starts to fall in. For this condition
to be met, the period must have reached a critical
value
PE (E ! P ) ' 10 ¯ 1=2
30
n \Gamma1=4 v 1=2
10
s (13)
which is attained by dipole braking (for a constant
field) in a time
t E ' 10 9 ¯ \Gamma1
30
n \Gamma1=2 v 10 yr : (14)
When P ? PE (E ! P ) matter can penetrate down
to the Alfven radius, but the interaction with the ro­
tating magnetosphere prevents accretion to go any
further because of the centrifugal barrier. The NS is
now in the propeller phase, rotational energy is lost
to the ISM and the period keeps increasing at a rate
dP
dt '
—
M R 2
A P
I ' K P ff s s \Gamma1 : (15)
Here we take K = 2:4 \Theta 10 \Gamma14 ¯ 8=7
30
n 3=7 v \Gamma9=7
10
, ff = 1
(Shakura 1975) and I = 10 45 g cm 2 is the star moment
of inertia.
It should be stressed, however, that our expres­
sion for the propeller spin--down is just an approx­
imation. The propeller physics is very complicated
and its through understanding requires a full 2--D or
5

even 3--D MHD investigation of accretion onto a ro­
tating dipole (see e.g. Toropin et al. 1999; see also
other spin--down formulae for that stage in Lipunov
& Popov 1995). The propeller spin--down, as mod­
eled by equation (15), is very efficient and acts on a
typical timescale
t P ' 1:3 \Theta 10 6 ¯ \Gamma8=7
30
n \Gamma3=7 v 9=7
10
yr : (16)
Numerical simulations (Toropin et al. 1999; Toropin,
private communication) seem indeed to confirm that
spin--down in the propeller phase is very fast due to
the large mass expulsion rate. Note that the present
expression for the propeller torque is somewhat differ­
ent from that adopted by Treves et al. (1993; 1998).
As the star moves through the inhomogeneous ISM a
transition from the propeller back to the ejector phase
may occur if the period attains the critical value
PE (P ! E) ' 3 ¯ 4=5
30
v 6=7
10
n \Gamma2=7 s : (17)
Note that the transitions P ! E and E ! P are
not symmetric because the two periods (13) and (17)
derive from the same physical condition but evaluated
at different radii, as first discussed by Shvartsman in
the early '70s.
Accretion onto the star surface occurs when the
corotation radius R co = (GM P 2 =4ú 2 ) 1=3 becomes
larger than the Alfven radius (and RA ! R ac , see
below). This implies that braking torques have in­
creased the period up to
PA (P ! A) ' 420 ¯ 6=7
30
n \Gamma3=7 v 9=7
10
s : (18)
As soon as the NS enters the accretor phase, torques
produced by stochastic angular momentum exchanges
in the ISM slow down the star rotation at the equi­
librium period
P eq = 2:6 \Theta 10 3 v \Gamma2=3
(t)10
¯ 2=3
30
n \Gamma2=3 v 13=3
10
s (19)
where v (t) the turbulent velocity of the ISM (Lipunov
& Popov 1995; Konenkov & Popov 1997).
Actually the condition that P – PA is not suf­
ficient to guarantee that matter is captured at the
accretion radius. At the very low accretion rates ex­
pected for fast, isolated NSs, it could be that the
Alfven radius is larger than the accretion radius. The
condition RA ! R ac translates into a limit for the
star velocity
v ! 410 n 1=10 ¯ \Gamma1=5
30
km s \Gamma1 : (20)
Above this velocity the NS behaves as a georotator
and it experiences a torque which is computed in the
same way as in the propeller stage. The star can
enter the georotator phase only from the propeller or
accretor stage.
Figure 2 illustrates the various possible stages for
a NS 10 10 yr old as a function of the star velocity
and magnetic field for a constant ambient density of
1 cm \Gamma3 . The propeller region in figure 2 is very small,
as a consequence of the extremely efficient spin--down
implied by equation (15) with our choice of the pa­
rameters ff and K.
3. Results and discussion
In this section, we present the results of our nu­
merical simulations both in the cases of a constant
and of a decaying magnetic field.
3.1. The NS census for a non--decaying field
Here we consider two values for the magnetic dipole
moment (¯ 30 = 0:5 and ¯ 30 = 1) as representative of
the constant NS magnetic field. The present frac­
tion of NSs in the Ejector and Accretor stages as a
function of the mean kick velocity is shown in figure
3. Statistical errors are typically ¸ 2%. Propellers
and Georotators are much less abundant and never
exceeds ¸ 1% of the total number. In addition, their
fractions oscillate intrinsically because of the changes
in the velocity and in the ISM density along the star
trajectory. These effects add further noise to the sta­
tistical fluctuations and, given the small number of
sources in these two states, prevent any definite con­
clusion about the dependence of the fractions from the
mean kick velocity. We note that the fraction of Ac­
cretors increases with the field strength, in agreement
with the findings of Livio Xu & Frank (1998) and
Colpi et al. (1998). Highly magnetized NSs suffer, in
fact, a more severe spin--down during the Ejector and
Propeller phases and reach the accretion phase earlier
in their history. For large enough mean kick veloci­
ties (? 400 km s \Gamma1 for ¯ 30
= 1 and ? 300 km s \Gamma1
for ¯ 30 = 0:5) the fraction of accretors becomes very
small and again results become statistically uncertain.
We note also that the numerical simulation produces
a picture of the present status of the Galactic NS pop­
ulation which is consistent with that emerging from
figure 2. In fact, if we assume that the velocity gives
a measure of the average kick, we see, with reference
to ¯ 30 = 1, that Accretors are more abundant than
6

Ejectors up to ¸ 100 km s \Gamma1 , Propellers are extremely
rare and Georotators nearly absent. This is precisely
what figure 3 shows.
To obtain an estimate of the number of accreting
sources in the Solar vicinity (taken to be a sphere of
radius 140 pc centered on the Sun), we used all the
stars contained in a torus of the same radius located
at 8 kpc from the center of the Galaxy. The result was
then rescaled to the volume of interest and is shown in
figure 4. Here, and in the following the total number
of Galactic NSs was assumed to be 10 9 . As expected,
the local density is a sensitive function of the kick
velocity and varies between n ¸ 8:5 \Theta 10 \Gamma3 pc \Gamma3 and
n ¸ 6 \Theta 10 \Gamma4 pc \Gamma3 for hV i = 0 and 190 km s \Gamma1
respectively. These figures should be compared with
n ¸ 1:4 \Theta 10 \Gamma3 (N=10 9 ) pc \Gamma3 as derived by Paczy'nski
(1990) for a zero mean velocity and with n ¸ 3 \Gamma 7 \Theta
10 \Gamma4 (N=10 9 ) found by Blaes & Madau (1993) and
Zane et al. (1995) for an initial mean velocity of ¸ 60
km s \Gamma1 . The values of the local density of isolated NSs
are thus consistent with these previous estimates.
The density of isolated NSs projected onto the
Galactic plane is ¸ 2:4 \Theta 10 5 (N=10 9 ) kpc \Gamma2 for aver­
age initial velocities ¸ 200 km s \Gamma1 and a scaleheight
of 200 pc. This figure is close to the one deduced
by Ne¨uhauser & Tr¨umper (1999) from radio pulsars
observations (Lyne et al. 1998) with the assumption
of a pulsar lifetime ¸ 10 7 yr. We would like to note,
however, that a projected density ¸ 10 5 kpc \Gamma2 is now
obtained for N = 10 9 instead of 10 8 (as in Madau
& Blaes 1994; see also Ne¨uhauser & Tr¨umper 1999)
because of the larger mean velocity at birth. A to­
tal number ¸ 10 9 appears to be consistent with the
nucleosynthesis and chemical evolution of the Galaxy,
while 10 8 is derived from radio pulsars observations.
For the time being, it is still uncertain if all NSs expe­
rience an active radio pulsar phase, due to low initial
magnetic fields or long periods, or to the fall--back in
the aftermath of the supernova explosion (see Colpi
et al. 1996). Radio pulsars are observed only in a
fraction of SNRs (see e.g. Kaspi 1996; Frail 1998),
and even some of these coincidences are doubtful, so
there is a serious possibility that the total number
of NSs derived from radio pulsar statistics is only a
lower limit.
In order to compare the expected number of accret­
ing ONSs with the ROSAT All Sky Survey (RASS) re­
sults, we evaluated the number of those ONSs, within
140 pc from the Sun, producing an unabsorbed flux
of 10 \Gamma13 erg cm \Gamma2 s \Gamma2 at energies ¸ 100 eV (the con­
Fig. 2.--- The different stages of old NSs at present
as a function of the star velocity and magnetic dipole
moment.
Fig. 3.--- Fractions of NSs in the different stages vs.
the mean kick velocity for ¯ 30 = 0:5 (triangles) and
¯ 30 = 1 (diamonds); typical statistical uncertainty is
¸ 2%.
Fig. 4.--- Total number of NSs in the Solar vicinity
(R ! 140 pc) for ¯ 30 = 1.
7

tribution from sources at larger distances is found to
be negligible). This flux limit should correspond to
a RASS count rate of 0.01 cts s \Gamma1 for a blackbody
spectrum, once the interstellar absorption and the
ROSAT response function are accounted for. The re­
sults are illustrated in figure 5 (at 300 km s \Gamma1 and
above statistical errors are dominant). As expected,
the dependence of the number of visible sources on
the kick velocity is rather strong. The main result is
that for mean velocities below 200 km s \Gamma1 the num­
ber of ONSs with a flux above the RASS detection
limit would exceed 10. Most recent analyses on the
number of isolated NSs in the RASS (Ne¨uhauser &
Tr¨umper 1999) indicate that the upper limit is below
10. This implies a lower limit for the mean kick ve­
locity at birth of ¸ 200 km s \Gamma1 for a total number of
stars ¸ 10 9 .
This result is in agreement with the estimates de­
rived from pulsar statistics. An important aspect is
that our results exclude the possible presence of a
consistent low--velocity population at birth, which ex­
ceeds that contained in the gaussian with hV i ? 200
km s \Gamma1 (¸ 1% for hV i ! 70 km s \Gamma1 ).
3.2. The NS census for a decaying field
The time evolution of the magnetic field in isolated
NSs is still a very controversial issue and no firm con­
clusion has been established as yet. A strong point
is that radio pulsar observations (see e.g. Lyne et al.
1998) seem to rule out fast decay with typical times
less than ú 10 Myr, but this does not exclude the
possibility that B decays over much longer timescales
(t d ¸ 10 9 \Gamma 10 10 yr). For this reason we have in­
vestigated to what extent the decay of the B--field
influences the results presented in the previous sec­
tion. We refer here only to a very simplified picture
in which B(t) = B(0) exp (\Gammat=t d ) and no attempt is
made to justify this law on a physical ground. In this
respect we just mention that detailed models predict
both exponential (quite similar to the one assumed
here) and non--exponential decay (e.g. a power--law,
Urpin & Konenkov 1997). Calculations have been
performed for t d = 1:1 \Theta 10 9 yr, t d = 2:2 \Theta 10 9 yr
and ¯ 30 (0) = 1. Since no bottom field was specified,
the magnetic moment becomes very low for stars born
¸ 10 10 yr ago (¸ 10 26 G cm 3 and ¸ 10 28 Gcm 3 for
the two values of t d respectively).
Results are summarized in figure 6. As it is ex­
pected (see Colpi et al. 1998), the number of Pro­
pellers is significantly increased with respect to the
non--decaying case, while Ejectors are now less abun­
dant. Georotators are still very rare, ¸ ! 1%, and are
not shown in figure 6. The fraction of Accretors is
approximately the same for the two values of t d , and,
at least for low mean velocities, is comparable to that
of the non--decaying field while, at larger speeds, it
seems to be somehow higher. This shows that the
fraction of Accretors depends to some extent on how
the magnetic field decays. We know that a core field
decaying initially but then freezing produces an un­
derabundance of Accretors relative to a constant field
(Livio Xu & Frank (1998) and Colpi et al. (1998))
because NSs persist in the Ejector phase never ap­
proaching the Propeller or the Accretor phase. By
contrast, a fast and progressive decay of B (t d ! 10 9
yr) would lead to an overabundance of Accretors be­
cause this situation is similar to ``turning off'' the
magnetic field, i.e., quencing any magnetospheric ef­
fect on the accreting matter (see Popov & Prokhorov
1999 (in press)).
Summarizing, we can conclude that, although both
the initial distribution and the subsequent evolution
of the magnetic field strongly influences the NS census
and should be accounted for, the lower bound on the
average kick derived from ROSAT surveys is not very
sensitive to B, at least for not too extreme values of
t d and ¯(0), within this model.
4. Conclusions
In this paper we have investigated how the present
distribution of neutron stars in the different stages
(Ejector, Propeller, Accretor and Georotator) de­
pends on the star mean velocity at birth. On the
basis of a total of ¸ 10 9 NSs, the fraction of Ac­
cretors was used to estimate the number of sources
within 140 pc from the Sun which should have been
detected by ROSAT. Most recent analysis of ROSAT
data indicate that no more than ¸ 10 non--optically
identified sources can be accreting ONSs. This im­
plies that the average velocity of the NS population
at birth has to exceed ¸ 200 km s \Gamma1 , a figure which
is consistent with those derived from radio pulsars
statistics. We have found that this lower limit on the
mean kick velocity is substantially the same either
for a constant or a decaying B--field, unless the decay
timescale is shorter than ¸ 10 9 yr. Since observable
accretion--powered ONSs are slow objects, our results
exclude also the possibility that the present velocity
distribution of NSs is richer in low--velocity objects
8

Fig. 5.--- Number of accreting NSs in the Solar vicin­
ity above the ROSAT All Sky Survey detection limit
for a constant (¯ 30 = 1, diamonds) and decaying field
(t d = 2:2 \Delta 10 9 yrs, triangles).
Fig. 6.--- Fractions of NSs in the different stages vs.
the average kick velocity for a decaying field with an
e--folding time t d = 2:2 \Theta 10 9 yrs (triangles) and t d =
1:1 \Theta 10 9 yrs (diamonds).
with respect to a Maxwellian. The paucity of accret­
ing ONSs seem therefore to lend further support in
favor of neutron stars as very fast objects.
Acknowledgments
Work partially supported by the European Com­
mission under contract ERBFMRX­CT98­0195. The
work of S.P., V.L and M.P. was supported by grants
RFBR 98­02­16801 and INTAS 96­0315. S.P. and
V.L. gratefully acknowledge the University of Milan
and of Insubria (Como) for support during their vis­
its.
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