Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://xray.sai.msu.ru/~polar/html/publications/iau195/iau.ps Äàòà èçìåíåíèÿ: Wed Sep 1 19:55:42 1999 Äàòà èíäåêñèðîâàíèÿ: Sat Dec 22 04:42:52 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: http astrokuban.info astrokuban

Highly Energetic Physical Processes and Mechanisms for Emission
from Astrophysical Plasmas, IAU Symposium 195
ASP Conference Series, Vol. 3 \Theta 10 8 , 1999
P. C. H. Martens and S. Tsuruta, eds.
Population synthesis of old neutron stars in the Galaxy
S.B. Popov
Sternberg Astronomical Institute, Universiteskii Pr. 13, 119899,
Moscow, Russia; e--mail: polar@xray.sai.msu.su
M. Colpi
Dept. of Physics, Univ. of Milan, Via Celoria 16, 20133 Milan, Italy;
e--mail: monica@pccolpi.uni.mi.astro.it
A. Treves
Dipartimento di Scienze, Univ. dell'Insubria, Via Lucini 3, 22100,
Como, Italy; e--mail: treves@mi.infn.it
R. Turolla
Dept. of Physics, Univ. of Padova, Via Marzolo 8, 35131 Padova, Italy
V.M. Lipunov
Dept. of Physics, Moscow State Univ.; Sternberg Astronomical Institute,
Universiteskii Pr. 13, 119899, Moscow, Russia
M.E. Prokhorov
Sternberg Astronomical Institute, Universiteskii Pr. 13, 119899,
Moscow, Russia; e--mail: mystery@sai.msu.su
Abstract. The paucity of old isolated accreting neutron stars in ROSAT
observations is used to derive a lower limit on the mean velocity of neu­
tron 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 Georo­
tator. 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 . 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. Present results,
moreover, constrain the fraction of low velocity stars, which could have
escaped pulsar statistics, to ¸ ! 1%.
1

2 Popov et al.
1. Introduction
Isolated neutron stars (NSs) are expected to be as many as 10 8 --10 9 , a non--
negligible fraction of the total stellar content of the Galaxy. The number of
observed radio pulsars is now ¸ 1,000. Since the pulsar lifetime is ¸ 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, Wolk & Neuhauser
1996; Haberl et al. 1998; Neuhauser & Trumper 1999. The extreme X--ray to
optical flux ratio (? 10 3 ) makes the NS option rather robust, but the exact
nature of their emission is still controversial. 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 advantages and drawbacks. Standard cooling atmosphere models fail to
predict in a natural way the spectrum the best studied object, RX J1856­3754
(see Walter et al. this volume). If the Bondi--Hoyle scenario applies, accretion
is reduced for increasing star velocity and for v ? 20 km s \Gamma1 it fails to produce
the luminosities inferred from ROSAT data (see Walter et al. this volume).
Meanwhile, we feel that a more thorough analysis of the statistical properties
of NSs is of interest 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, Propellers, 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 or dead
pulsars, still spun down by dipole losses. In Propellers the incoming matter
can penetrate down to the Alfven radius, RA , but no further because of the
centrifugal barrier, and stationary inflow can not occur, but 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 Georotators where (similarly to the Earth) the Alfven radius
exceeds the accretion radius, so that magnetic pressure dominates everywhere
over the gravitational pull. It is the combination of the star period, magnetic
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.
While the dynamical evolution of NSs in the Galactic potential was studied
by several authors (sse e.g. Madau & Blaes 1994; Zane et al. 1995), little
attention was paid to the NSs magneto--rotational evolution. Recently, this
issue was discussed in some detail by Livio, Xu & Frank (1998) and Colpi et al.
(1998). Goal of this investigation is to consider these two issues simultaneously,
coupling the dynamical and the magneto--rotational evolution for the isolated
NS population.
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 estimates on the number of old accreting neutron stars in the
Galaxy in the light of these new data, in the attempt to reconcile theoretical
predictions with present ROSAT limits (Ne¨uhauser & Tr¨umper 1999).

APS Conf. Ser. Style 3
2. The Model
In this section we summarize the main hypothesis introduced to track the evo­
lution of single stars and describe shortly the technique used to explore their
statistical properties, referring to Popov & Prokhorov (1998) for details on spa­
tial evolution calculations and to Konenkov & Popov (1997) and Lipunov &
Popov (1995) for details of magneto­rotational evolution.
2.1. Dynamical evolution
The dynamical evolution of each single star in the Galactic potential (taken in
the form proposed by Miyamoto & Nagai 1975) is followed solving its equations
of motion.
The period evolution depends on both the star velocity and the local density
of the interstellar medium, any attempt to investigate the statistical properties
of the NS population should incorporate a detailed model of the ISM geogra­
phy. Unfortunately the distribution of molecular and atomic hydrogen in the
Galaxy is highly inhomogeneous. Here we use the analytical distributions from
Bochkarev (1992) and Zane et al. (1995) for the hydrogen density n(R; Z).
Within a region of ¸ 140 pc around the Sun, the ISM is underdense, and we
take n = 0:07 cm \Gamma3 .
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.
Neutron stars at birth have a circular velocity determined by the Galactic
potential. Superposed to this ordered motion a kick velocity is imparted in a
random direction. We use here an isotropic Gaussian distribution with disper­
sion 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 .
2.2. Accretion physics and period evolution
The accretion rate was calculated according to the Bondi formula
—
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 (1)
where m p is proton mass, the sound speed V s is always 10 km s \Gamma1 and v 10
=
(V 2 + V 2
s ) 1=2 in units of 10 km s \Gamma1 . M and R denote the NS mass and radius,
which we take equal to 1:4 M fi and 10 km, respectively, for all stars.
All neutron stars are assumed to be born with a period P (0) = 0.02 s, and
a magnetic moment either ¯ 30 = 1 or ¯ 30 = 0:5, where ¯ 30 = ¯=10 30 Gcm 3 .
In the ejector phase the energy losses are due to magnetic dipole radiation.
When the gravitational energy density of the incoming interstellar gas exceeds
the outward momentum flux at the accretion radius, R ac ' 2GM=v 2 , matter
starts to fall in. This happens when the period reaches the critical value
PE (E ! P ) ' 10 ¯ 1=2
30
n \Gamma1=4 v 1=2
10
s: (2)
When P ? PE (E ! P ) the NS is in the propeller phase, rotational energy is
lost and the period keeps increasing at a rate taken from Shakura (1975).

4 Popov et al.
0.0 200.0 400.0 600.0
Velocity, km/s
1
10
100
Accretors
0.0 200.0 400.0 600.0
0.0
20.0
40.0
60.0
80.0
100.0
Ejectors
0.0 200.0 400.0 600.0
Velocity, km/s
0.00
0.20
0.40
0.60
Georotators
0.0 200.0 400.0 600.0
0.00
0.05
0.10
0.15
0.20
0.25
Propellers
Figure 1. Fractions of NSs in the different stages vs. the mean kick
velocity for ¯ 30 = 0:5 (open circles) and ¯ 30 = 1 (filled circles); typical
statistical uncertainty for ejectors and accretors is ¸ 1­2%.
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 : (3)
Note that the transitions P ! E and E ! P are not symmetric 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 increased the period up to
PA (P ! A) ' 420 ¯ 6=7
30
n \Gamma3=7 v 9=7
10
s : (4)
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
equilibrium period
P eq = 2:6 \Theta 10 3 v \Gamma2=3
(t)10
¯ 2=3
30
n \Gamma2=3 v 13=3
10
s (5)

APS Conf. Ser. Style 5
0.0 100.0 200.0 300.0 400.0 500.0
Velocity, km/s
10 0
10 1
10 2
10 3
10 4
10 5
10 6
Number
of
sources
Figure 2. Number of accreting NSs in the Solar vicinity for a constant
(¯ 30
= 1, filled circles; ¯ 30
= 0:5, opened circles) and decaying field
(t d = 2:2 \Theta 10 9 yr, triangles).
where v (t)
the turbulent velocity of the ISM (Lipunov & Popov 1995; Konenkov
& Popov 1997).
At the very low accretion rates expected 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 : (6)
3. Results and discussion
3.1. The NS census for a non--decaying field
We consider two representative values for the (costant) magnetic dipole moment,
¯ 30
= 0:5 and ¯ 30
= 1. The present fraction of NSs in the Ejector and Accretor
stages as a function of the mean kick velocity is shown in figure 1.

6 Popov et al.
0.0 200.0 400.0 600.0
Velocity, km/s
1
10
100
Accretors
0.0 200.0 400.0 600.0
0.0
20.0
40.0
60.0
80.0
100.0
Ejectors
0.0 200.0 400.0 600.0
Velocity, km/s
10 -6
10 -4
10 -2
10 0
Georotators
0.0 200.0 400.0 600.0
0.00
10.00
20.00
30.00
40.00
Propellers
Figure 3. 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 (open circles) and t d = 1:1 \Theta 10 9 yrs (filled circles).
Here, and in the following the total number of Galactic NSs was assumed to
be 10 9 . A total 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. It is uncertain if all NSs experience 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. 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 accreting ONSs with the
ROSAT All Sky Survey (RASS) results, we evaluated the number of those ONSs,
within 140 pc from the Sun, producing an unabsorbed flux of 10 \Gamma13 erg cm \Gamma2 s \Gamma1
or higher at energies ¸ 100 eV. The results are illustrated in figure 2. The main
point is that for mean velocities below 200 km s \Gamma1 the number of ONSs with a
flux above the RASS detection limit would exceed 10. Most recent analysis on
the number of isolated NSs in the RASS (Ne¨uhauser & Tr¨umper 1999) indicate
that the upper limit is below 10.
An important aspect is that our results exclude the possible presence of a
consistent low--velocity population at birth, which exceeds that contained in the
gaussian with hV i ? 200 km s \Gamma1 (¸ 1% for hV i ! 70 km s \Gamma1 ).

APS Conf. Ser. Style 7
10 26 10 27 10 28 10 29 10 30 10 31
Bottom magnetic momentum
10 -2
10 -1
10 0
10 1
10 2
Ejector
time,
Gyrs
td=1.d8 yrs
td=1.d9 yrs
td=1.d10 yrs
Figure 4. Ejector time vs. bottom magnetic momentum for two
values of initial magnetic momentum and different decay timescales,
t d = 10 8 , 10 9 , and 10 10 yr.
3.2. The NS census for a decaying field
We refer here only to a very simplified picture of the field decay in which B(t) =
B(0) exp (\Gammat=t d ). 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. Results are shown in figure 3.
For some values of t d and bottom field most of NS can stay at the Ejector
stage, and the number of accretors and Propellers would not be increased. We
show this analytical estimates graphically in figure 4, where the Ejector time,
TE , is plotted vs. bottom magnetic momentum for constant velocity and ISM
density (n = 1 cm \Gamma3 , v = 10 km/s), different t d and two values of the initial
magnetic momentum, 10 30 and 10 31 G cm 3 (see Popov & Prokhorov 1999).
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.

8 Popov et al.
4. Conclusions
In this paper we have investigated how the present distribution of neutron stars
in the different stages (Ejector, Propeller, Accretor and Georotator) depends on
the star mean velocity at birth. On the basis of a total of ¸ 10 9 NSs, the fraction
of Accretors 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 implies 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 with respect
to a Maxwellian. The paucity of accreting ONSs seem therefore to lend further
support in favor of neutron stars as very fast objects.
Acknowledgments. Work partially supported by the European Commis­
sion 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 sup­
port during their visits. S.P. also acknowledge organizers of the IAU 195.
References
Bochkarev, N.G. 1992, Basics of the ISM Physics (Moscow University Press)
Colpi, M., Turolla, R., Zane, S., & Treves, A. 1998, ApJ 501, 252
Cordes, J.M. 1998, in Neutron Stars and Pulsars, eds. Shibazaki, N. et al.
(Tokyo: Universal Academy Press)
Haberl, F., Motch, C., & Pietsch, W. 1998, Astron. Nachr. 319, 97
Konenkov, D.Yu., & Popov, S.B. 1997, PAZh, 23, 569
Lipunov, V.M. 1992, Astrophysics of Neutron Stars (Springer & Verlag)
Lipunov, V.M., & Popov, S.B. 1995, AZh, 71, 711
Livio, M., Xu, C., & Frank, J. 1998 ApJ, 492, 298
Madau, P., & Blaes, O. 1994, ApJ, 423, 748
Miyamoto, M., & Nagai, R. 1975, Pub. Astr. Soc. Japan, 27, 533
Ne¨uhauser, R., & Tr¨umper, J.E. 1999, A&A, 343, 151
Popov, S.B. 1994, Astron. Circ., N1556, 1
Popov, S.B., & Prokhorov, M.E. 1998, A&A, 331, 535
Popov, S.B., & Prokhorov, M.E. 1999, (in press)
Shakura, N.I. 1975, PAZh, 1, 23
Treves, A., Colpi, M., & Lipunov, V.M. 1993, A&A, 269, 319
Walter, F.M., Wolk, S.J., & Ne¨uhauser, R. 1996, Nature, 379, 233
Zane, S., Turolla, R., Zampieri, L., Colpi, M., & Treves, A. 1995, ApJ, 451, 739