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Draft version January 4, 2002
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LOG N { LOG S DISTRIBUTIONS OF ACCRETING AND COOLING ISOLATED
NEUTRON STARS
S.B. Popov 1 , M. Colpi 2 , M.E. Prokhorov 1 , A. Treves 3 and R. Turolla 4
Draft version January 4, 2002
ABSTRACT
We model populations of isolated neutron stars in the Galaxy following their orbital and
magneto-rotational evolution. It is shown that accretors become more abundant than coolers
at uxes below  10 13 erg cm 2 s 1 ; and one can predict that about one accreting neutron
star per square degree should be observed at the Chandra and Newton ux limits of  10 16
erg cm 2 s 1 : The soft ROSAT sources associated with isolated neutron stars can be relatively
young cooling objects only if the neutron star birth rate in the Solar vicinity during the last
 10 6 yr is higher than that inferred from radiopulsar observations.
Subject headings: accretion, accretion disks | stars: kinematics | stars: magnetic elds |
stars: neutron | stars: statistics | X{rays: stars
1. INTRODUCTION
Despite intensive observational campaigns, no ir-
refutable identication of an isolated accreting neu-
tron star (NS) has been presented so far. Six soft
sources have been found in ROSAT elds which are
most probably associated to isolated radioquiet NSs.
Present X-ray and optical data however do not allow
an unambiguous identication of the physical mech-
anism responsible for their emission. These sources
can be powered either by accretion of the interstellar
gas onto old ( 10 10 yr) NSs or by the release of inter-
nal energy in relatively young ( 10 6 yr) cooling NSs
(see Treves et al. 2000 and Motch 2000 for recent re-
views). The ROSAT candidates, although relatively
bright (up to  1 ct s 1 ), are intrinsically dim and
their inferred luminosity (L  10 31 erg s 1 ) is near
to that expected from either a close-by cooling NS
or from an accreting NS among the most luminous.
Their X-ray spectrum is soft and thermal, again as
predicted for both accretors and coolers (Zane, Tur-
olla & Treves 2000; Treves et al. 2000). Up to now
only two optical counterparts have been identied
(RXJ 1856, Walter & Matthews 1997, for which a
distance estimate of  60 pc has been very recently
obtained, Walter 2000, and RXJ 0720, Kulkarni &
Van Kerkwick 1998). In both cases an optical excess
over the low-frequency tail of the black body X-ray
spectrum has been reported. While detailed multi-
wavelength observations with next-generation instru-
ments may indeed be the key for assessing the true
nature of these sources, other, indirect, approaches
may be used to discriminate in favor of one of the
two scenarios proposed so far.
In this Letter we compute and compare the log N {
log S distribution of both accreting and cooling NSs,
to establish the relative contribution of the two pop-
ulations to the observed number counts. Previous
studies derived the log N { log S distribution of ac-
cretors (Treves & Colpi 1991; Madau and Blaes 1994;
Manning et al. 1996) assuming a NSs velocity distri-
bution rich in slow stars (v  < 100 km s 1 ). More
recent measurements of pulsar velocities (e.g. Lyne
& Lorimer 1994; Hansen & Phinney 1997) and upper
limits on the observed number of accretors in ROSAT
surveys (Danner 1998a, b) point, however, to a larger
NS mean velocity (see Treves et al. 2000 for a critical
discussion). Recently Neuhauser & Trumper (1999,
NT99 hereafter) compared the number count distri-
bution of the ROSAT isolated NS candidates with
those of accretors and coolers. Here we address these
issues in greater detail, also in the light of the lat-
est contributions to the modeling of the evolution of
Galactic NSs (Popov et al. 2000, P2000 hereafter).
In x2 we use a population synthesis model, devel-
oped in P2000, to trace the properties of the NS spin,
magnetic eld and accretion luminosity in the Galac-
tic potential using a detailed map of the interstellar
medium (ISM). Cooling NSs are explored separately
1 Sternberg Astronomical Institute, Universitetskii Pr. 13, 119899, Moscow, Russia; e{mail: polar@sai.msu.ru
2 Dipartimento di Fisica, Universita di Milano Bicocca, P.zza della Scienza 3, 20126 Milano, Italy; e{mail: colpi@uni.mi.astro.it
3 Dipartimento di Scienze, Universita dell'Insubria, Via Lucini 3, 22100, Como, Italy; e{mail: treves@mi.infn.it
4 Dipartimento di Fisica, Universita di Padova, Via Marzolo 8, 35131 Padova, Italy; e{mail: turolla@pd.infn.it
1

2 LOG N-LOG S DISTRIBUTIONS OF ISOLATED NEUTRON STARS
in x3 within a simpler model of local sources. Results
are discussed in x4.
2. ACCRETING ISOLATED NEUTRON STARS
The census model adopted here follows closely that
developed in P2000. All NSs are born in the Galac-
tic plane, their birth rate is taken to be proportional
to the square of the ISM density and is constant in
time. Initially a NS has a circular velocity xed by
its position in the Galactic plane and determined by
the Galactic potential. To this, an additional kick ve-
locity with random orientation and modulus selected
from a Maxwellian distribution is imparted.
Time resolution is adapted to follow in detail the
dierent evolutionary phases the star experiences.
NSs are born in the ejector (E) phase and then, as
braking due to electromagnetic torques increases the
period, come to the propeller (P) phase. Further
spin-down, produced by accretion torques, may drive
the star in the accretion (A) phase when matter can
penetrate the magnetosphere reaching the stellar sur-
face (see e.g. Lipunov 1992). Accretion is then as-
sumed to proceed at the local Bondi rate and the
large scale ISM distribution is mapped as in P2000.
The small scale structure of the ISM ( 100AU-1 pc,
see e.g. Meyer & Lauroesh 1999; Treves et al. 2000),
has not been taken into account as it would produce
an unacceptable increase of the computing time, lead-
ing to a worse statistics. A detailed mapping of the
ISM on all scales is fundamental in assessing the ob-
servability of single, luminous sources (see Zane et
al. 1996), but the eects produced by its multiphase
structure on the evolution of the entire NS population
are likely to average out.
Here we consider only the simplest picture, where
the star magnetic eld is constant and the dipole mo-
ment  has a log-gaussian distribution, similar to that
observed in radiopusars (hlog i = 30:06,  = 0:32 in
CGS units).
The magneto-rotational evolution is followed using
essentially the same approach as in P2000. In partic-
ular, the typical values of the transition periods are
 25 s and  1500 s for E! P and P! A transitions,
respectively. As soon as the star enters the accretor
stage, the period is set to the value typical of an ac-
cretor embedded in a turbulent medium (Lipunov &
Popov 1995; Konenkov & Popov 1997).
The statistics of accretors was improved by a factor
> 10 relative to P2000 by evolving only stars from the
low-velocity tail (v < 100 km s 1 ) of a Maxwellian
with hvi = 300 km s 1 . This assumption appears
justied since the time spent in the ejector phase in-
creases with velocity, t E  10 10  1
30 n 1=2 v 100 yr : The
present choice for hvi follows from radiopulsar obser-
vations and current limits on the detection of accret-
ing NSs (P2000). Results can be easily scaled to any
hvi > 200 km s 1 .
Nearly all highly magnetized isolated NSs in the
velocity range considered here become accretors, and
in calculating the log N { log S distribution we ob-
tain good statistics for uxes < 10 11 erg cm 2 s 1 .
At larger uxes the number of sources is too small
to give reliable counts, but the extrapolation log N 
(3=2) log S can be safely used for log S  > 11.
Our main results are presented in Figures 1 and 3,
and refer to a total of 10 9 NSs in the Galaxy. The
log N { log S distribution of accretors is computed
for sources within a distance of 5 kpc, together with
the distributions of velocity, accretion rate and ef-
fective temperature (calculated adopting as eective
surface the polar cap area). Isolated NSs with elds
< 0:5  10 11 G never appear as accretors in any run.
The brightest accretors have luminosities  10 32 erg
s 1 , but for the majority of them L  10 29 10 30 erg
s 1 . Our sources are relatively hot, with a mean ef-
fective temperature 300-400 eV and for them absorp-
tion can be neglected. We predict, on average, about
1 source per square degree, in the energy range 0.5-2
keV, for limiting uxes about 10 16 {10 15 erg cm 2
s 1 . This implies  3  10 4 sources in the whole sky.
Note that they are signicantly concentrated toward
the Galactic plane with a center-anticenter asymme-
try.
100 1000
Polar cap temperature eV
0
0.2
0.4
0.6
0.8
1
fraction
-16 -14 -12 -10
log S
10 -3
10 -1
10 1
10 3
10 5
N
(>S)
Polar caps
Total flux
10 7 10 8 10 9 10 10 10 11 10 12
Accretion rate g/s
0
0.2
0.4
0.6
0.8
1
fraction
0 50 100 150
Velocity km/s
0
0.2
0.4
0.6
0.8
1
fraction
-3/2
-1
Fig. 1.| Upper left panel: the log N { log S distribu-
tion for accretors within 5 kpc from the Sun. The two
curves refer to emission from the entire star surface and
to polar cap emission in the range 0.5-2 keV; two straight
lines with slopes -1 and -3/2 are also shown for compari-
son. From top right to bottom right: the velocity, eective
temperature and accretion rate distributions of accretors;
all distributions are normalized to their maximum value.

Popov et al. 3
3. COOLING NEUTRON STARS
Soon after their birth in a type II supernova event,
cooling NSs emit X-ray luminosities as high as  10 34
erg s 1 for a few thousands years. Before disappear-
ing from the X-ray sky, they shine as faint soft sources
(L  10 32 erg s 1 ) for about a million year (e.g.
Yakovlev et al. 1999).
In the determination of the logN-logS, the cooler
sample should in principle be drawn from the same
parent population as accretors. However, as their to-
tal number is quite low ( 10 4 of the total number
of NS in the Galaxy) owing to the short duration of
the cooling phase, the results would be aected by
the very poor statistics. Notice however that coolers
can travel only a distance  400 pc during their life-
time and have softer spectra than accretors, so they
appear mostly as a local population of sources (due
to interstellar absorption).
For this reason, we prefer to approximate their spa-
tial distribution around the sun as homogeneous with
a scale height of  450 pc. We take cooling NSs as
\standard candles" with L = 10 32 erg s 1 , giving
o a black-body spectrum with eective temperature
 50 eV. The duration of the cooling phase is taken
to be 10 6 yr, according to the \slow cooling" sce-
nario (see NT99). We used two typical values for
the total NS spatial density: 0:33  10 3 pc 3 and
3:310 3 pc 3 . The former estimate follows from ra-
diopulsar statistics and was used by NT99, while the
latter corresponds to a total population of 10 9 NSs, as
implied by constraints on nucleosynthetic yields and
historic supernova rates (Arnett et al. 1989; van den
Bergh & Tammann 1991). Accordingly, coolers have
a mean density of 0:33 3:3  10 7 pc 3 :
The ISM structure is treated in a very simple way.
The ISM density inside the Local Bubble of radius
r l = 140 pc (Sfeir et al. 1999) is n = 0:07 cm 3 . At
larger distances the interstellar gas is assumed to be
uniformly distributed in a disk of half-thickness 450
pc with n = 1 cm 3 .
Results are shown in Figure 2, where we compare
the observed distribution of ROSAT candidates with
that of coolers, as predicted by the present model (in-
cluding absorption) and by NT99, the latter obtained
for a total NS number N tot = 2  10 8 . As it is appar-
ent, coolers dominate over accretors at large uxes
(for the same N tot ), essentially because of their much
higher luminosity (10 32 vs. 10 29 10 30 erg s 1 ).
This simple model reproduces the most important
feature, the \ attening" of the log N { log S distribu-
tion outside the Local Bubble. Such strong attening
can help to explain the observed data, if one assumes
that most of the brightest isolated NSs are identied
(Schwope et al. 1999).
The value of the ux at which the distribution at-
tens out (the \knee") depends on the size of the Lo-
cal Bubble r l : The ISM densities and the duration of
the cooling time aect instead the number of sources.
Since the cooling time can be an important parame-
ter, the comparison of theoretical and observed dis-
tributions could, in principle, help in discriminating
between \fast" and \slow" cooling models.
10 -2
10 -1
10 0
10 1
counts/s
10 -2
10 -1
10 0
10 1
10 2
10 3
N
(>S)
per
steradian
ROSAT points
absorbed coolers; lg N=9
absorbed coolers; lg N=8
line from NT99
Fig. 2.| The log N{logS for coolers. The upper curve
(lled circles) correspond to a NS density of 3:3  10 3
pc 3 and the lower one (open circles) to 3:3  10 4 pc 3 ;
here r l = 140 pc.
4. DISCUSSION
The statistical analysis presented in the previous
sections shows that both scenarios, accretors and
coolers, have diфculties in explaining, under usual
assumptions, the observed properties of isolated NS
ROSAT candidates. The main problem is to pro-
duce enough sources that are, at the same time: i)
relatively bright (high count rates,  > 0:1 cts s 1 );
ii) close (low absorption NH  < 10 20 cm 2 ); iii) soft
(T eff  50 100 eV); iv) slowly rotating (for two
candidates, RX J0720 and RX J0420, the detected
periods are 8.4 and 22.7 s).
Polar cap accretion for a non-decaying magnetic
eld, B  10 12 G, cannot produce enough bright
sources and the emitted spectrum is likely to be
harder, see Figure 1. Moreover, accretors are ex-
pected to have much larger periods, P  > 10 3 s.
On the other hand, relatively young cooling NSs
can be spun down to periods  20 s in < 10 6 yrs
(when they are still hot) only for B  > 10 14 G, since
P  15 B 14 (t=10 6 yr) 1=2 s for dipole losses. If all NSs
experience an initial radiopulsar phase, their total
number in the Galaxy would be about (1 3)  10 8
(Lyne et al. 1998). In this case there will be not

4 LOG N-LOG S DISTRIBUTIONS OF ISOLATED NEUTRON STARS
bright enough isolated NSs in the Solar vicinity in
both scenarios.
If N tot  10 9 , coolers can explain the large number
of bright close-by sources and their typical temper-
atures (see Figure 2). However, the problem of re-
producing the observed periods still remains, unless
the progenitors of the ROSAT pulsating candidates
are NSs born with long periods, which never passed
through the pulsar phase; alternatively they could
be magnetars with ultra-high elds, as suggested by
Heyl & Kulkarni (1998). If proved correct, the \fast"
cooling scenario, in which the duration of the active
X-ray phase is shorter, would make again problematic
to account for the observed bright sources.
In the accretion scenario, increasing the total num-
ber of NSs is also a need. As already shown by Livio
et al. (1998) and Colpi et al. (1998), magnetic eld
decay can explain the low temperatures and periods.
Decay can both increase and decrease number of ac-
creting isolated NSs (Popov, & Prokhorov 2000). But
a complete statistical analysis is still to come.
10 -3
10 -2
10 -1
10 0
10 1
counts/s
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
N
(>S)
per
steradian
ROSAT points
absorbed coolers
RBS limits
Polar caps accretion
P2000 point (r<140 pc)
-14 -13 -12 -11
-1
-3/2
Fig. 3.| Comparison of the log N { log S distri-
butions for accretors and coolers together with observa-
tional points, the naive logN { logS from P2000 and the
ROSAT Bright Survey (RBS) limit (Schwope et al. 1999).
The scale on the top horizontal axes gives the ux in erg
cm 2 s 1 .
In any case, using \standard" assumptions on the
velocity, spin period and magnetic eld parameters,
the accretion scenario can not explain the observed
properties of the six ROSAT candidates.
Note that the number of accretors (of all luminosi-
ties) in the Galaxy is found to be about two orders
of magnitude larger than the number of young cool-
ers: few percents versus 0.01% of the total number
of isolated NSs. But on average accretors are 3 or-
ders of magnitude fainter than a typical cooler. A
key result of our statistical analysis is that accretors
should eventually become more abundant than cool-
ers at uxes below 10 13 erg cm 2 s 1 . This is illus-
trated in Figure 3 in which we summarize the main
outcome of our simulations.
ACKNOWLEDGMENTS
We wish to thank J. Lattimer, E. van den Heuvel,
V.M. Lipunov, and S. Campana for useful discus-
sions. SBP and MEP also thank the University of
Insubria for nancial support and the Universities
of Milano-Bicocca, Padova and the Brera Observa-
tory (Merate) for their kind hospitality. The work of
SBP and MEP was supported through grant RFBR
00-02-17164 and NTP Astronomy grants 1.4.4.1.
and 1.4.2.3. Support from the European Commis-
sion under contract ERBFMRXCT98-0195 and from
MURST under contract COFIN98021541 is acknowl-
edged.
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