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Gravitation & Cosmology, Vol. 7 (2001), Supplement, pp. 1{1
c
2001 Russian Gravitational Society
STOCHASTIC SPIN EVOLUTION OF NEUTRON
STARS
S.B. Popov 1 , M.E. Prokhorov 1 , A.V. Khoperskov 2 , V.M. Lipunov 1;3
1 Sternberg Astronomical Institute, Universitetski pr. 13, 119899 Moscow
2 Volgograd State University, Department of Theoretical Physics, 40068 Volgograd
3 Moscow State University, Department of Physics
Abstract
In this paper we present calculations of period distribution for old accreting isolated
neutron stars (INSs).
At the age about few billions years low velocity INSs come to the stage of accretion. At
that stage their period evolution is governed by magnetic braking and accreted angular
momentum. Due to turbulence of the interstellar medium (ISM) accreted momentum
can both accelerate and decelerate rotation of an INS and spin evolution has chaotic
character.
Calculations show that for constant magnetic eld INSs have relatively long spin periods,
 10 4 {10 5 s, depending on parameters of INSs and ISM density. Due to long periods
INSs have high spin up/spin down rates, which should uctuate on a time scale about
few years.
1 Introduction
Spin period is the most precisely determined parameter of a neutron star (NS). Estimates of
other parameters: masses (for isolated objects), radii, temperatures, magnetic elds etc. are
always model dependent. Because of that it is very important to have a clear picture of period
evolution as far as this parameter is usually used to determine other characteristics of NSs.
Here we try to obtain distribution of spin periods for old accreting INSs (AINSs).
AINSs are now a subject of interest in astrophysics (see Treves et al. 9) ). Probably few
candidates are observed by ROSAT (Motch 7) ).
In the next section we describe the model we use to obtain period distributions and show
results for the easiest case of \spin equilibrium". Then in the section 3 we present our main
results for the \non-equilibrium" case and brie y discuss them in the section 4. Details of
calculations can be found in Prokhorov et al. 8) .
2 \Spin equilibrium"
Previous attempts to calculate typical periods of AINSs were made by Lipunov & Popov 5)
and Konenkov & Popov 3) . In these papers the authors do not try to obtain distributions:
only characteristic periods of AINSs are derived. The authors assume that AINSs are in \spin

2 High Energy Astrophysics: Accretion Theory, Gamma Ray Bursts, Relativistic and Compact Objects
equilibrium", i.e. all AINSs in these estimates have enough time to reach the stage at which
magnetic braking is compensated by accretion of angular momentum. We use terms \spin equi-
librium" and \non-equilibrium" in quotation-marks as far as there is no real equilibrium: period
can signicantly uctuate. But the situation in general is similar to real period equilibrium in
close binaries (see Ghosh & Lamb 2) , Lipunov 4) ).
We start with the following equations:
d!
dt
= F + ; F = k t  2
IR 3
co
;  
_
MJ
I
: (1)
Here I { moment of inertia of a NS, ! = 2=p { spin frequency,  = BR 3
 { magnetic moment of
a NS, R co = (GM=! 2 ) 1=3
{ corotation radius, k t { constant of order of unity, and  { turbulent
torque, <  >= 0. J is determined as J =Min (v t RG ; v ARA ), where v t = 10 6 cm s 1 (RG=R t ) 1=3
{ turbulent velocity at R = RG , R t = 2
10 20 cm, RG = 2GM=v 2 , v 2 = v 2
s + v 2
sp , v s { sound
velocity, v sp { spatial velocity, RA =

 2 =2 _
M
p
GM
 2=7
{ Alvfen radius, and v A = (GM=RA ) 1=2
{ Keplerian velocity at the Alfven radius. For the most reasonable parameters J = v t RG .
Now we have to introduce a kind of \diusion coeфcient", D, because due to interstellar
medium (ISM) turbulence we have a kind of diusion in the space of frequencies. This coeфcient
can be approximatelly determined as D = (1=6)

_
MJ=I
 2
RG =v (Lipunov & Popov 5) ). Here
_
M =  R 2
G  v { is an accretion rate.
We can determine an average spin frequency in the following way:
! 2
turb =
1
Z
0
! 4 e V (j!j)=D d!=
1
Z
0
! 2 e V (j!j)=D d!; V (j ! j) =  2
3GMI j ! j 3 : (2)
Finally, p turb = 2=! turb . For J = v t
RG we can write it as:
p turb = 3:9
10 8  2=3
30
I 1=3
45
M 26=9
1:4
n 2=3 v 43=9
7
R 2=9
t 2;20
s: (3)
Here v 7
= v=(10 7 cm s 1 ), R t 2;20
= R t =(2
10 20 cm).
We note strong dependence of p turb and D on v. If information on p and _
p is available the
problem can be reversed, and one can obtain an estimate of the velocity of a AINS as it was
done for stellar wind accretion in X-ray pulsars by Lipunov & Popov 6) .
Probability plotted in Fig. 1 was calculated as:
f(!; v; ) / ( 2 =GMID)! 2 e V=D ; (4)
and then normalized. Here V and D are functions of v; ; n. Note, that peaks in Fig. 1 are
indeed sharp. It means, that if \spin equilibrium" can be reached, it is possible to use just
one typical value, p turb , for each set of parameters as it was done by Lipunov & Popov 5) and
Konenkov & Popov 3) .
3 \Non-equilibrium" calculations
In this section we calculate probability distribution for the \non-equilibrium" case. These
distributions are calculated for magnetic moments with Gaussian distribution in logarithmic
scale with central value lg( 0
) = 30:06 and  = 0:32 (see Colpi et al. 2001 1) for details on
magnetic evoluyion of NSs). Velocities of AINS are taken from the Maxwellian distribution
with a mean velocity 200 km s 1 .

Stochastic spin evolution of neutron stars S.B. Popov et al. 3
10 3
10 4
10 5
10 6
Periods, s
0.0
0.2
0.4
0.6
0.8
1.0
log B=11
log B=12
log B=13
0.0
0.2
0.4
0.6
0.8
1.0
v=10
v=20
v=30
10 3
10 4
10 5
10 6
Periods, s
M=2.5
M=1.4
M=1
n=10
n=1
n=0.1
Figure 1: Period distributions in equilibrium for dierent parameters. In each of the four plots
we vary one parameter: n; v; M;B. Results are normalized to unity at the maximum.
We solved numerically the following dierential equation:
df=dt = A=! 2 @(! 4 f)=@! +D=! 2 @=@!

! 2 @f=@!

; A =  2 =GMI: (5)
After initial parameters of an INS are chosen from the distributions described above we
check if this NS can reach the accretion stage in 10 10 yrs. To do it we calculate time which
it spends as Ejector: t E  10 9 n 1=2 v 6  1
30
yrs; v 6 = v=10 6 cm s 1 . We neglect the stage of
Propeller, as far as for constant eld it is much shorter than the stage of Ejector (Lipunov &
Popov 5) ).
In Fig. 2 we present a curve for n = 1 cm 3 and selected INS parameters:  = 2
10 29 G
cm 3 , v sp = 10 km s 1 . In Fig. 3 we show our nal results for Maxwellian velocity distribution
and log-Gaussian magnetic eld distribution for two values of the ISM density.
4 Discussion
After an INS come to the stage of accretion it is controlled by two processes (see eq.1): magnetic
spin-down and turbulent spin-up/spin-down. Initially magnetic spin-down is more signicant,
but at some period, p cr , these two processes become comparable. For longer periods an INS
will be governed mainly by turbulent forces. One can obtain the following formula for p cr :
p 2
cr = (4 2  2 )=(GM _
MJ). An INS reach p cr in
t  10 5 {10 7 yrs after onset of accretion:

t = (I
p
GM)=(
p
_
MJ ).

4 High Energy Astrophysics: Accretion Theory, Gamma Ray Bursts, Relativistic and Compact Objects
10 2
10 3
10 10 5
Periods, s
0.1
1
10 2
10 3
10 10 5
10 6
10 7
10 8
Periods, s
0.0001
0.001
0.01
0.1
1
n=1
n=0.1
Figure 2: Period distribution for  = 2

10 29 G cm 3 , v sp = 10 km s 1 , n = 1 cm 3 .
Results are normalized to unity at the max-
imum.
Figure 3: Period distributions for n = 1
cm 3 and n = 0:1 cm 3 . Results are nor-
malized to unity at the maximum of the
highest curve.
For eld decay picture should be completely dierent (see for example Konenkov & Popov 3) ).
AINSs with decayed eld can appear as pulsating sources with periods about 10 s and _
p about
10 13 s/s. As an INS passes through turbulent cells a value and a sign of _
p will uctuate on a
time scale RG =v sp ' 11:8 v sp =(10 km s 1 ) yr.
Acknowledgments
This work was supported by RFBR (01-02-06265, 00-02-17164, 01-15(02)-99310).
SP and MP thank M. Colpi, A. Treves and R. Turolla for discussions.
References
1 Colpi M., Possenti A., Popov S.B., Pizzolato F.: 2001, in `Physics of Neutron Star In-
teriors", Eds. D. Blaschke, N.K. Glendenning, & A. Sedrakian (Springer{Verlag, Berlin),
(astro-ph/0012394)
2 Ghosh, P., & Lamb, F.K. 1979, ApJ 232, 256
3 Konenkov, D.Yu., & Popov, S.B. 1997, PAZh, 23, 569
4 Lipunov, V.M., 1992, \Astrophysics of Neutron Stars", Springer{Verlag (Berlin)
5 Lipunov, V.M., & Popov, S.B. 1995a, AZh, 72, 711
6 Lipunov, V.M., & Popov, S.B. 1995b, Astron. Astroph. Transactions, 8, 221
7 Motch, C. 2000, astro-ph/0008485
8 Prokhorov, M.E., Popov, S.B., & Khoperskov, A.V. 2001, astro-ph/0108503
9 Treves, A., Turolla, R., Zane, S., & Colpi, M. 2000, PASP 112, 297