Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://xray.sai.msu.ru/~polar/html/publications/snegiri/snegiri.ps Äàòà èçìåíåíèÿ: Mon Jul 19 19:38:32 1999 Äàòà èíäåêñèðîâàíèÿ: Sat Dec 22 05:05:09 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 5

Old isolated neutron stars in the Galaxy: evolution
and field decay
S. B. Popov 1 ) and D. Yu. Konenkov 2 )
1) Sternberg Astronomical Institute
119899, Moscow, Russia
e­mail: polar@xray.sai.msu.su
2) A.F.Ioffe Institute of Physics and Technology
194021 St.Petersburg, Russia,
e­mail: dyk@astro.ioffe.rssi.ru
July 19, 1999
Abstract
Old isolated neutron stars form a large population of the Galactic objects, and
at the present time most of them are unobserved. In this talk we briefly describe
their main properties and show some examples.
As an example we discuss possible evolution of the spin period and the magnetic
field of the X­ray source RX J0720.4­3125 assuming this source to be an isolated
neutron star accreting interstellar medium. Magnetic field of the source is estimated
to be 10 6 \Gamma 10 9 G, and it is difficult to explain observed spin period 8.38 s without
1

invoking hypothesis of the magnetic field decay. We used the model of ohmic decay
of the crustal magnetic field. The estimates of accretion rate (10 \Gamma14 \Gamma 10 \Gamma16 M fi =yr),
velocity of the source relative to interstellar medium (10 \Gamma 50 km/s), neutron star
age (2 \Delta 10 9 \Gamma 10 10 yrs) are obtained.
Key words: netron stars, evolution, magnetic field decay
1 Introduction
Among all astrophysical objects neutron stars (NS) (and black holes) attract most atten­
tion of physicists. NS were initially predict by Lev Landau immediatelly after neutron
was discovered in 1932. But as far as there were no ideas how to observe so small (about
10 km in diameter) astronomical objects they were not a subject of intense search, and
were found as radiopulsars occasionally in 1967 in England.
Now we know more than 1000 NS as radiopulsars and more than 100 NS emmiting X­
rays, but the Galactic population of these objects is about 10 8 -- 10 9 , so only a tiny fraction
of one of the most facinating astrophysical objects is observed. Studying of the rest of
population is crucially important to obtain initial parameters of NS, and to understand
their evolution (for example, there are some argument for the proposition, that not all
NS pass through the stage of radiopulsar in their young years; it can happen due to too
long initial periods, too high (magnetars) or too low magnetic fields).
Isolated neutron stars (INS) have received special attention in the last few years. The
idea of observing such objects in the X­ray range has emerged in soon after discovery of the
first X­ray sources. Later it was supposed that INSs accreting matter of the interstellar
medium (ISM) can be observed with the ROSAT satellite in the UV and X­ray ranges.
2

It was also proposed, that old INS can be observed in supernovae remnants, for example
in RCW103 (Popov, 1998), if they were components of binary systems.
The estimates of energy characteristics, spectra and possibilities of observing INSs
were made in several works (see, for example, B¨ohringer et al., 1987; Treves and Colpi,
1991; Blaes and Madau, 1993). A large number of works is devoted to the study of
the spatial distribution of INSs (see, for example, Blaes and Rajagopal, 1991). Spatial
distribution of accretion luminosity of INS (and black holes) was considered in (Popov,
Prokhorov, 1998).
ROSAT observations showed, that only few INS are bright enough to appear on the
X­ray sky. It can be a result of high initial velocities of INS, which they obtain during
supernova explosion, as it appear in population synthesis (Popov et al. 1999).
In 1996, Haberl et.al. reported the discovery of the pulsating source RX J0720.4--
3125 with the ROSAT satellite in the soft X­ray range. We shall use two observational
characteristics of this source (Haberl et.al., 1997): the period p = 8:38 s and the blackbody
temperature T = (79 \Sigma 4) eV.
Using the hypothesis that this source is an INS accreting matter from the ISM, we
estimate the accretion rate and the magnetic field strength of this source. We show that
the INS could not increase its period to the observed value over the Universe lifetime if we
assume that it was born with the present­day magnetic field strength. Finally we conclude
that the magnetic field of this NS had to decay and calculate the magnetorotational
evolution of the NS.
3

2 Evolution of the spin period and the magnetic field
of the neutron star
2.1 Spin period
In the low­density plasma an INS may have four possible evolutionary states (Lipunov,
1987): ejector (young quickly spinning NS, spinning down due to losses of the kinetic
energy onto emission of the magnitodipolar waves), propeller (NS ejecting plasma from
alfven radius), accretor (NS spinning slow enough to allow plasma from ISM to accrete
onto its surface), and georotator (usially fastly moving NS, which is not important for our
purposes). Two critical periods PE and PA separate different stages of the NS evolution.
These periods as well as period derivatives on ejector and propeller stages can be estimated
using the formulae from (Lipunov, 1987). If p ! PE , then NS is at the ejector stage; if
PE ! p ! PA , we have the NS at the propeller stage; and if p ? PA and, then NS is an
accretor.
The change in the period of an accreting INS is due to its interaction with the turbu­
lized ISM. This introduces specific features into the problem of the period evolution. If
we adopt the hypothesis of spin acceleration of the NS in the turbulized ISM (Lipunov
and Popov, 1995), the new characteristic period emerges,
P eq = 3450k 1=3
t ¯ 2=3
30
I 1=3
45
—
M \Gamma2=3
\Gamma15 v 7=3
1 6
v \Gamma2=3
t 6
M \Gamma4=3
1:4
sec; (1)
where ¯ 30 is the magnetic dipole moment in units of 10 30 Gs cm 2 , I 45 is the moment of
inertia in units of 10 45 g cm 2 , ae \Gamma24 is the ISM density in units of 10 \Gamma24 g cm \Gamma3 , v1 6
is
the NS velocity relative to the ISM in units 10 6 cm/s, and v t 6
is the turbulent velocity in
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units of 10 6 cm/s. The period P eq corresponds to the NS rms rotation rate obtained from
the solution of the corresponding Fokker--Planck equation. Here take into account the
three­dimensional character of turbulence, i.e. the fact that the vortex can be oriented
not only in the equatorial plane but also at any angle to this plane.
2.2 Ohmic dissipation of the crustal magnetic field of the neu­
tron star
The magnetic field decay in the neutron star crust has been investigated in many papers
(see, Urpin, Muslimov, 1992; Urpin, Konenkov, 1997) at which we address the reader.
The model contains two parameters, which determine the field evolution: the impurity
parameter Q, and the density at the bottom of the layer initially occupied by the field,
ae 0 .
Figure 1 shows the evolution of the NS surface magnetic field for various parameters
ae 0 and Q. We perfomed our calculations for the model of the NS based on the Friedman--
Pandharipande (1981) equation of state in the star core, with the mass of the neutron
star M = 1:4M fi , the radius R = 10:6 km, and the crust thickness \DeltaR = 940 m (Van
Riper, 1988). It is apparent that the decay at the initial stage (t ! 10 6 yr) is determined
by the initial depth of the current layer and that the subsequent decay rate depends on
the impurity concentration, characterized by Q.
Accretion may affect the field evolution. First, it heats the NS crust and, therefore,
decreases the conductivity. Second, the flux of matter toward the star center arises; it
tends to bring the field to deeper layers. However, calculations show (Urpin et al., 1996)
that accretion with the rate —
M ! 10 \Gamma14 M fi yr \Gamma1 speeds up insignificantly the field decay.
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3 Estimates of parameters of RX J0720.4--3125
Using the hypothesis that we observe an accreting isolated NS and taking the observed
period p = 8:38 s and the temperature T = (79 \Sigma 4) eV (Haberl et. al., 1997), we can
obtain the constraints on the magnetic field, accretion rate, and luminosity.
1. If we really observe the accretor, then p ? PA (B; —
M ), where PA is defined by (2).
This yields the first constraint on B and —
M :
B ! 4:6 \Delta 10 9 M 5=6
1:4
R \Gamma3
6
—
M 1=2
\Gamma15 G; (2)
2. The pulsation of X­ray emission means that the accreting matter is funneled by
the magnetic field onto the polar caps. Using the known relation R cap =
q
(R=RA ) \Delta R;
connecting the polar cap radius R cap , the Alfven radius RA , and the NS radius, we can
estimate R cap and obtain another constraint from the condition R cap ! R:
B ? 3:4 \Delta 10 4 M 1=4
1:4
R \Gamma5=4
6
—
M 1=2
\Gamma15 G: (3)
3. And finally, we know the polar cap temperature. Assuming that all heat released
in the polar caps is radiated away we find the additional relation between B and —
M :
B = 2:5 \Delta 10 7 (T=79 eV) 7
R 4
6
M \Gamma3=2
1:4
—
M \Gamma5=4
\Gamma15 G: (4)
Combining inequalities (2) and (3) and condition (4) yields the allowed range of the B
and —
M values:
2 \Delta 10 5 ! B[G] ! 10 9 ;
6 \Delta 10 \Gamma17 ! —
M [M fi =yr] ! 4 \Delta 10 \Gamma14 : (5)
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Now it is easy to show that spin­down time to p = 8:38 s greatly exceeds the Hubble
time if one assumes that magnetic field of RXJ 0720.4­3125 has the same strength in the
past. This means that in the past the given INS had a larger magnetic field that has
decayed over the time of evolution.
Since the NS presently accretes the matter from the ISM, we can estimate its period
derivative:
—
p ú p=t su = p
—
Mv t RG
I!
! 10 \Gamma12 s=s:
4 The evolutionary tracks of neutron star on the B \Gamma
P diagram
The evolution of the spin period was calculated taking into account the decay of the mag­
netic field. Since the treatment of the spin­up rate in the turbulized ISM at the accretor
stage is not quite a simple matter, we applied to the description of the period evolution
the following simplifying model. When the NS gets to the accretor stage, the spin­down
momentum substantially exceeds the spin­up momentum because the acceleration occurs
in the turbulized ISM and there is no constant spin­up momentum, unlike in a binary
system. However, we can obtain an analog of the equilibrium period which corresponds
to the stationary solution of the Fokker--Planck equation (Lipunov, 1987). Therefore the
evolution of the spin period at this stage was treated as a stationary spindown from PA
to P eq after which the period was set to be P eq , which, in its turn, changed due to the
magnetic field decay.
We calculated the magnetic and spin evolution of the NS with mass M = 1:4M fi
for the accretion rates 10 \Gamma15 M fi yr \Gamma1 and 10 \Gamma16 M fi yr \Gamma1 , moving through the ISM with
7

density ae = 10 \Gamma24 g/cm 3 . Such accretion rates at the given ISM density correspond to
the velocities of motion 19 and 41 km/s. We assumed the NS to be born as an ordinary
pulsar with a short period (p 0 = 0:01 s) and a ''standard'' initial magnetic field B 0 = 10 13
G. The model of the ohmic decay of magnetic field allows one to obtain both the high and
low dissipation rate. However, in this case we should fit the parameters of the model to
obtain the accreting INS with a period of 8.38 s and a field of 2 \Delta 10 7 G for the accretion
rate —
M = 10 \Gamma15 M fi yr \Gamma1 or with a field of 4 \Delta 10 8 G for the accretion rate 10 \Gamma16 M fi yr \Gamma1
(4). Therefore, the field should decay with a moderate rate to give the NS enough time
to spin down to p = 8:38 s. Moreover, it is desirable at the initial stage of evolution
to provide the agreement with magnetic fields and periods of the observed radio pulsars.
Basing on these assumptions we can obtain the estimates of the parameters ae 0 and Q, as
well as the lower limit on the INS age.
The evolutionary tracks of the NS for the accretion rates —
M = 10 \Gamma15 M fi yr \Gamma1 and
—
M = 10 \Gamma16 M fi yr \Gamma1 are shown in panels a) and b) of the Fig.2.
Tracks 1 in both panels of Fig. 2 illustrate the evolution with the maximum possible
rate of magnetic field dissipation with time. The model parameters for the accretion rate
—
M = 10 \Gamma15 M fi yr \Gamma1 are: ae 0 = 3 \Delta 10 13 g/cm 3 , Q = 0:02, and v = 10 6 cm/s. In the first 10 6
yrs, the field decays by about 20 times, while the NS slows down to p ú 0:3 s. At that time
the NS represents a typical radio pulsar. Subsequently, due to the cooling, the field decay
slows down. In the next 8 \Delta 10 7 yrs the field decays to 4:5 \Delta 10 11 G, the period increases,
and the radiopulsar dies. However, the ejector stage continues for 1:2 \Delta 10 9 years. During
this stage, the field decays to 2:5 \Delta 10 10 G, the period increases to 2:1 s, and the NS goes
over to the propeller stage (1) which lasts ¸ 10 9 yrs. At this stage, further deceleration
of the NS occurs, according to (4), to p = 5:7 s, and the field decays to 3 \Delta 10 9 G. The
8

star goes over to the accretor stage and becomes the source of periodic X­ray radiation.
The core heating speeds up insignificantly the field decay (Urpin et.al., 1996). The field
decays to 2 \Delta 10 7 G over 4 \Delta 10 9 yrs. The period does not increase because the field is weak
enough. As soon as P eq becomes equal to the current period, the spin­up of the NS due
to the interaction with the ISM may occur. The period fluctuates around P eq , but we do
not estimate here the amplitude of these fluctuations.
Tracks 2 illustrate the evolution with slower field decay. Such a decay can be obtained,
for example, for the crust with lower impurity content. For —
M = 10 \Gamma15 M fi yr \Gamma1 , we set
Q = 0:01, while the other parameters remained the same as for track 1 in Fig. 2a. Tracks
1 and 2 coincide at the initial stage of evolution, when the dissipation rate does not depend
on the impurity concentration. However, in 10 7 yrs the tracks diverge. As a result, the
star resides at the ejector stage 2 \Delta 10 9 yrs going over to the propeller stage with p = 3
s and B = 4 \Delta 10 10 G. The spin­down rate at the propeller stage is much higher than in
the first case for three reasons: due to the longer spin period, the stronger magnetic field
during the NS transition from the ejector to the propeller stage, and to the lower rate of
the field decay caused by the lower value of the Q parameter. At the accretor stage, the
NS spin period decreases to 190 s over 2 \Delta 10 9 yrs, and when the field decays to 4 \Delta 10 8 G, the
effect of acceleration in the turbulized ISM may begin to act. The track in the absence of
such acceleration (v t = 0) is shown by the dashed line. In this case the final period turns
out to be about 200 s. The turbulent acceleration may decrease this period. However,
the self­consistent calculation of the evolution of the NS period at the accretor stage with
allowance for the magnetic field decay would require the solution of the Fokker--Planck
equation for the distribution function in the space of angular velocities, which is beyond
the scope of this work.
9

At the accretion rate —
M = 10 \Gamma16 M fi yr \Gamma1 we used for track 1 the following parameters:
ae 0 = 3 \Delta 10 13 g/cm 3 , Q = 0:01, and v t = 10 6 cm/s. The NS comes to the propeller stage
in 2:7 \Delta 10 9 yrs having a period of 3.1 s and a magnetic field of 2 \Delta 10 10 G. The transition
to the accretor stage occurs when the NS has a period of 4.9 s and a magnetic field of
7 \Delta 10 8 G. Deceleration at this stage does not occur because the magnetic field is low. The
full time taken for the field decay to 4 \Delta 10 8 G is 6 \Delta 10 9 yrs. In this case p = 5 s. The
second track in Fig. 2b was calculated for the greater initial depth of the current layer
corresponding to ae 0 = 6 \Delta 10 13 g/cm 3 . As a consequence, at the initial stage of evolution
the magnetic field decays with a slower rate, the ejector stage lasts 2:1 \Delta 10 9 yrs, and the
propeller stage lasts 1:9 \Delta 10 9 yrs. For the field 4 \Delta 10 8 G, the period is 63 s.
5 Conclusion
The observed period and the temperature of the X­ray source RX J0720.4-- 3125 can be
accounted for by using the hypothesis for the ISM accretion onto an old INS. We showed
that the NS magnetic field is low in this case (B ! 10 9 G). The time taken for deceleration
to p = 8:38 s with such a magnetic field exceeds the age of the Universe. We supposed
that the NS was born with higher strenght of magnetic field and that the field strength has
substantially decreased during the evolution. Using the model of ohmic dissipation of the
magnetic field in the NS crust, we calculated the possible evolution of the NS on the B \Gamma P
diagram. The observed rotational period can be obtained at Q = 0:01 \Gamma 0:05. However,
the evolution of the period depends on the field dissipation rate and, therefore, on the
parameters of the decay model. Thus, the twofold change in the impurity parameter Q
caused the spin period to change more than by the order of magnitude at the accretor
10

stage (Fig. 2a, tracks 1 and 2). For this reason, observations of periods of old INSs may
become an important test for the validity of the model of evolution of neutron stars.
Decay of the magnetic field may affect the estimate of the total number of the observed
accreting INSs. In particular, because the formation of a periodic X­ray source at the
propeller stage calls for rather strong magnetic field, its decay may reduce the number of
sources of this kind. At the same time, the absence of pulsations for other INS candidates
may suggest that their field has already decayed to such extent that it cannot funnel the
plasma toward the INS polar caps.
Due to decay, depending upon its timescale and minimum (bottom) magnetic field,
relative numbers of different stages of INS (Ejectors, Propeller or Accretors) can be in­
creased or decreased (Popov et al., 1999). Here we didn't addressed the problem of the
whole population's properties, but constructed the evolutionary track of the single object.
Acknowledgements
The authors express their gratitude to M.E. Prokhorov, V.A.Urpin, V.M. Lipunov, M.
Colpi, R. Turolla, A. Treves and D.G. Yakovlev. The work of D. Konenkov was supported
by the Russian Foundation for Basic Research (grant 97­02­18096a), and by INTAS (grant
96­0154), the work of S. Popov -- by the Russian Foundation for Basic Research (grant
98­02­16801), and by INTAS (grant 96­0315).
6 References
Blaes, O., Madau, P., ``Can we observe accreting, isolated neutron stars?'' ApJ 403, 690
(1993).
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Blaes, O., Rajagopal, M., ``The statistics of slow interstellar accretion onto neutron stars''
ApJ 381, 210 (1991).
B¨ohringer, H., Morfill, G.E. and Zimmermann, H.U., ``Characteristics of old neutron stars
in dense interstellar clouds'', ApJ, 313, 218 (1987).
Friedman, B., Pandharipande, V.R., Nucl. Phys., A361, 502 (1981).
Geppert, U., Urpin. V., Konenkov, D., ``Wind accretion and magnetorotational evolution
of neutron stars in close binaries'' A&A, 307, 807 (1996).
Haberl F., Motch, C., Buckley, D.A.H., Zickgraf, F.­J., Pietsch, W., , ``RXJ0720.4­3125:
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Konenkov D., Popov S., ``RX J0720.4­3125 as a possible example of the magnetic field
decay in neutron stars'' Astronomy Letters 23, 200 (1997).
Lipunov, V, ``Astrophysics of Neutron Stars'', Moscow: Nauka, (1987).
Lipunov, V., Popov, S.B., ``Period evolution of isolated neutron stars'' AZh, 72, 711
(1995).
Popov, S., ''On the nature of the compact X­ray source inside RCW 103''. Astron.
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Urpin, V., Geppert, U., Konenkov, D., ``Magnetic and spin evolution of neutron stars in
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Figure captions.
Figure 1.
The evolution of the surface magnetic field of the isolated neutron star for the standard
cooling scenario. Curves 1, 2, and 3 correspond to the initial depths of the current layer,
10 11 , 10 12 , and 10 13 g cm \Gamma3 , respectively. The solid curves correspond to Q = 0.001; the
dashed curves, to Q = 0.01; the dot­dashed curves, to Q = 0.1.
Figure 2.
The evolutionary tracks of the neutron star for the accretion rates —
M = 10 \Gamma15 M fi yr \Gamma1
(a) and —
M = 10 \Gamma16 M fi yr \Gamma1 (b). The model parameters are described in the text. The
dashed lines correspond to p = PE ; the dot­dashed lines, to p = PA . The dashed line
in Fig. 2a shows for the second track the neutron star evolution with no acceleration
in the turbulized intestellar medium. The numbers near the marks in tracks denote the
logarithm of the neutron star age in years. The observed radio pulsars are indicated by
dots (Taylor et al., 1993).
13