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Gravitation & Cosmology, Vol. 5 (1999), Supplement, pp. 1--1
c
fl 1999 Russian Gravitational Society
NEUTRON STARS IN X­RAY PULSARS AND
THEIR MAGNETIC FIELDS
S.B. Popov 1 , V.M. Lipunov 1;2
1 Sternberg Astronomical Institute, Universitetskij pr. 13, 119899 Moscow
2 Physics Department, Moscow State University
Abstract
Estimates of the magnetic field of neutron stars in X­ray pulsars are obtained using
the hypothesis of the equilibrium period for disk and wind accretion and also from the
BATSE data on timing of X­ray pulsars using the observed maximum spin­down rate.
Cyclotron lines at energies – 100 keV in several Be­transient are predicted for future
observations.
We suggest a new method of estimating distances to X­ray pulsars and their magnetic
fields. Using observations of fluxes and period variations in the model of disk accretion
one can estimate the magnetic momentum of a neutron star and the distance to X­ray
pulsar. As an illustration the method is applied to the system GROJ1008­57.
1 Introduction
Among all astrophysical objects neutron stars (NSs) attract most attention of physicists. Now
we know more than 1000 NSs as radiopulsars and more than 100 NSs emiting X­rays, but the
Galactic population of these objects is about 10 8 -- 10 9 . So only a tiny fraction of one of the
most fascinating astrophysical objects is observed at present.
NSs can appear as sources of different nature: as isolated objects and as binary companions,
powered by wind or disk accretion from a secondary companion. X­ray pulsars are probably
one of the most prominent among binary sources, because there important parameters of NSs
can be determined.
Now we know more than 40 X­ray pulsars (see e.g. Bildsten et al. 1) , Borkus 2) ). Observations
of optical counterparts of X­ray sources give an opportunity to determine distances to these
objects and other parameters with relatively high precision, and with hyroline detections one
can obtain the value of magnetic field, B, of a NS. But lines are not detected in all sources of
that type and magnetic field can be estimated from period measurements (see e.g. Lipunov 4) ).
Precise distance measurements usually are not available immediately after X­ray discovery
(especially, if localization error boxes are large and X­ray sources have transient nature). In
that sense methods of simultaneous determination of field and distance basing only on X­ray
observations can be useful, and several of them were suggested by different authors previously.
Here we try to obtain estimates of the magnetic fields (and distances) of NSs in X­ray
pulsars from their period (and flux) variations.

2 High Energy Astrophysics: Accretion Theory
2 Estimates of the magnetic field
Magnetic fields of accreting NSs can be estimated using period variations or using the hypothesis
of the equilibrium period (see Lipunov 4) ). We use both of these methods.
For estimating of magnetic momentum of NSs using observed values of maximum spin­down
we use the following main equation:
dI!
dt
= \Gammak t
¯ 2
R 3
co
;
where I -- NS's momentum of inertia, ! = 2ú
p -- spin frequency, ¯ -- magnetic momentum,
R co =
i
GM
! 2
j 1=3
-- corotation radius. We used k t = 1=3, I = 10 45 g cm 2 , M = 1:4M fi .
We used graphs from (Bildsten et al. 1) ) to derive spin­up and spin­down rates and flux
changes measurements. Data on these graphs is shown with one day time resolution.
Equilibrium period can be written in different forms for disk and wind­fed systems. For the
first case we used the following equation:
p eq:disk = 2:7 ¯ 6=7
30
L \Gamma3=7
37
s: (1)
For wind­accreting systems we have:
p eq:wind = 10:4 L \Gamma1
37
T \Gamma1=6
10
¯ 30
s: (2)
Here L 37 -- luminosity in units 10 37 erg s \Gamma1 , T 10 -- orbital period in units 10 days, ¯ 30 -- magnetic
momentum in units 10 30 G cm 3 .
Estimates of the magnetic momentum, ¯, obtained with different assumptions are shown in
the table 1. Three values are shown: an estimate from spin­down obtained from the BATSE
data (Bildsten et al. 1) ); an estimate from the equilibrium period for wind­fed systems (eq. (2));
an estimate for disk­accreting systems (eq. (1)). Less probable values are marked with asterix.
In table 1 we use the following notation: LMXRB­ Low Mass X­Ray Binary; HMSG ­ High
Mass SuperGiant; BeTR­ Be­transient source. Values, which were used for estimates with the
hypothesis of the equilibrium period: spin period, mean luminosity in units 10 37 erg s \Gamma1 , orbital
period in units 10 days can be found on the Web: http://xray.sai.msu.ru/~ polar.
More precise estimates can be made by fitting all observed values of spin­up and spin­down
rate together with flux measurements. When the distance to the source is know only the value
of the magnetic field should be fitted. And on the figure 1 we show such estimates for Her X­1.
We plot spin­up and spin­down rates as a function of the parameter, which is a combination
of the spin period and source's luminosity. Spin­up and spin­down values derived from the
BATSE data (Bildsten et al. 1) ) are plotted as black dots, and theoretical curves for different
values of the magnetic momentum are also shown. In ideal the best curve for the magnetic
momentum should exist, which fits all observational points. In reality points have some errors,
distance to the source in also know with some uncertainty, and simple model of spin­up and
spin­down can be only the first approximation.
3 Discussion and conclusions
We made estimates of the magnetic field of NSs in X­ray pulsars. Estimates which were
made with an assumption that p = p eq are rather rough. Obtained values depend (except

Neutron stars in X­ray pulsars and their magnetic fields S.B. Popov and V.M. Lipunov 3
Table 1: Spin­down and magnetic momentum estimates
X­RAY maximum Source Magnetic Magnetic Magnetic
PULSAR dp/dt Type momentum momentum momentum
observ. (spin­down), (wind), (disk),
(spin­down) 10 30 G cm 3 10 30 G cm 3 10 30 G cm 3
GRO 1744­28 BeTR 0.93 \Lambda 0.58
HER X­1 9:3 \Delta 10 \Gamma13 LMXRB 0.3 0.18
4U 0115+63 3:0 \Delta 10 \Gamma10 BeTR 5.17 0.32 \Lambda 1.26
CEN X­3 7:5 \Delta 10 \Gamma12 HMSG 0.82 1.8 4.42
4U 1627­67 4:1 \Delta 10 \Gamma11 LMXBR 1.9 2.82
2S 1417­624 BeTR 8.64 \Lambda 17.82
GRO 1948+32 5:4 \Delta 10 \Gamma9 BeTR 22.0
OAO 1657­415 1:5 \Delta 10 \Gamma7 HMSG 115.1 0.15 4.33
EXO 2030+375 BeTR 0.1 3.45
GRO 1008­57 3:2 \Delta 10 \Gamma8 BeTR 53.3
A 0535+26 BeTR 30.24 \Lambda 101.23
GX 1+4 6:4 \Delta 10 \Gamma8 LMXRB 75.5 167.3
VELA X­1 3:8 \Delta 10 \Gamma9 HMSG 18.5 4.03 88.15
4U 1145­61 3:3 \Delta 10 \Gamma7 BeTR 172.1 0.23 \Lambda 16.7
A 1118­616 5:1 \Delta 10 \Gamma7 BeTR 212.8 245.5
4U 1535­52 3:6 \Delta 10 \Gamma7 HMSG 56.4 17.37 299.3
GX 301­2 8:9 \Delta 10 \Gamma6 HMSG 281.9 8.34 200.5
uncertainties connected with the method itself) on unknown parameters of NSs, such as masses,
radii, moments of inertia. All of them were accepted to have ``standard'' values, and of course
it is only the first approximation. For example, our estimate for the source GRO 1744­28 is ¯ ¸
10 30 G cm 3 , and it is smaller than the estimate shown in (Borkus 2) ), which is B ¸ (2 \Gamma 5) \Delta 10 12
G (we mark, that the estimate obtained by Joss & Rappaport 3) is significantly lower than both:
Borkus and our estimates). But if one take ``non­standard'' value for R, these estimates of ¯
and B can be in good correspondence.
We show several examples in table 2. NSs radii are calculated from the following simple
formula:
R = (2¯=B) 1=3
:
Here ¯ are taken from table 1, and values of B are taken from Nagase 5) , Borkus 2) and Wang 8) .
As one can see from the table for several sources measured B are not in correspondence with
our calculated ¯, and radii of NSs are too big. Mostly these cases are long period wind­fed
pulsars like GX 301­2, where formation of temporal reverse disk is possible for the cases of fast
spin­down, so there maximum spin­down can be not the best field estimate, and estimates from
the equilibrium period for wind­accretion case are in better correspondence with observations.
In more clear cases (Her X­1, GRO 1744­28), where we are sure, that accretion is of the disk
type, our estimates from maximum spin­down are in good correspondence with observations.
And we predict for the cases of Be­transients, where disk accretion is working for sure, that
in 2S 1417­624, GRO 1948+32, GRO 1008­57, A 1118­616 and 4U 1145­61 observations of
cyclotron lines at energies – 100 keV are possible in future.
Observations of period and flux variations can be used also for simultaneous determination
of magnetic field of a NS and distance to the X­ray source (Popov 6) ).

4 High Energy Astrophysics: Accretion Theory
-10.0 -9.0 -8.0 -7.0
-10.0
-11.0
-12.0
-13.0
-14.0
spindown
-10.0 -9.0 -8.0 -7.0
-14.0
-13.0
-12.0
-11.0
-10.0
-9.0
spinup
Figure 1: Dependence of period derivative, —
p, on the parameter p 7=3 f , f -- observed flux, for
Her X­1. Both axis are in logarithmic scale. Observations (Bildsten et al., 1997) are shown
with black dots. Five curves are plotted for different values of the magnetic field. Solid curve:
¯ = 0:1 \Delta 10 30 G \Delta cm 3 . Dot­dashed curve: ¯ = 0:15 \Delta 10 30 G \Delta cm 3 . Long dashed curve: ¯ =
0:2 \Delta 10 30 G \Delta cm 3 . Dotted curve: ¯ = 0:3 \Delta 10 30 G \Delta cm 3 . Dashed curve: ¯ = 1 \Delta 10 30 G \Delta cm 3 . All
curves are plotted for the distance d = 4 kpc.
The method is based on several measurements of period derivative, —
p, and X­ray pulsar's
flux, f . Fitting distance, d, and magnetic momentum, ¯, one can obtain good correspondence
with the observed p; —
p and f , and that way produce good estimates of distance and magnetic
field (see also another way of estimating of these parameters based on the equilibrium period
and spin­up measurements applied to GRO1744­28 in (Joss & Rappaport 3) ).
Lets consider only disk accretion due to application of our method to the system, in which
most probably accretion is of the disk type. In that case one can write (see Lipunov 4) ):
—
p =
4ú 2 ¯ 2
3 G I M \Gamma
p
0:45 2 \Gamma1=14 ¯ 2=7
I (GM) \Gamma3=7
h
p 7=3 L
i 6=7
R 6=7 ; (3)

Neutron stars in X­ray pulsars and their magnetic fields S.B. Popov and V.M. Lipunov 5
Table 2: Magnetic fields, magnetic momentum and radii
X­RAY Magnetic Magnetic Neutron
PULSAR momentum field star
(calcul.), (observ.), radius,
10 30 G cm 3 10 12 G km
GRO 1744­28 0.58 ¸ (2 \Gamma 5) ¸ (8:3 \Gamma 6:1)
HER X­1 0.3 3 5.8
4U 0115+63 5.17 1.1 21.1
A 0535+26 101.23 11 26.4
VELA X­1 18.5 2.3 25.2
4U 1535­52 56.4 1.9 39
GX 301­2 281.9 3.5 54.4
where L = 4úd 2 \Delta f -- luminosity, f -- the observed flux.
So, with some small uncertainty in the equation above we know all parameters (I, M , R
etc.) except ¯ and d. Fitting observed points with them we can obtain estimates of ¯ and d.
Uncertainties mainly depend on applicability of that simple model.
To illustrate the method, we apply it to the X­ray pulsar GRO J1008­57, discovered by
BATSE (Bildsten et al. 1) ). It is a 93:5 s X­ray pulsar, with the BATSE flux about 10 \Gamma9 erg
cm \Gamma2 s \Gamma1 . The source was identified with a Be­system with ¸ 135 d orbital period.
On figure 2 we show observations (as black dots) and calculated curves for the disk model
on the plane —
p -- p 7=3 f , where f -- observed flux (logarithms of these quantities are shown).
Curves were plotted for different values of the source distance, d, and NS magnetic momentum,
¯. Spin­up and spin­down rates were obtained from graphs in Bildsten et al. 1) .
If one uses maximum spin­up, or maximum spin­down values to evaluate parameters of
the pulsar, then one can obtain values different from the best fit (they are also shown on the
figure): d ú 8 kpc, ¯ ú 37:6 \Delta 10 30 G\Delta cm 3 for maximum spin­up, and two values for maximum
spin­down: d ú 4 kpc, ¯ ú 37:6 \Delta 10 30 G\Delta cm 3 and the one close to our best fit (two similar
values of maximum spin­down were observed for different fluxes, but we mark, that formally
maximum spin­down corresponds to the values, which are close to our best fit). It can be used
as an estimate of the errors of our method: accuracy is about the factor of 2 in distance, and
about the same value in magnetic field, as can be seen from the figure.
Determination of magnetic field (and, probably, distance) only from X­ray observations
can be very useful in uncertain situations, for example, when only X­ray observations without
precise localizations are available.
Acknowledgments
PSB thanks prof. Joss for discussions. The work was supported by the RFBR (98­02­16801)
and the INTAS (96­0315) grants.
References
1 Bildsten, L. et al., 1997, ApJ Suppl. 113, 367
2 Borkus, V.V., 1998, PhD dissertation, Space Research Institute, Moscow
3 Joss, P.C., & Rappaport, S., 1997, in IAU Coll. 163 proc., Eds. D.T. Wickramasinghe et al.,
ASP Conference series, 121, 289

6 High Energy Astrophysics: Accretion Theory
-6.0 -5.0 -4.0 -3.0
-9.0
-8.0
-7.0
spindown
4_37.6
8_37.6
5.8_10
5.8_45
5.8_37.6
-6.0 -5.0 -4.0 -3.0
-9.2
-8.2
-7.2
-6.2
spinup
Figure 2: Dependence of period derivative, —
p, on the parameter p 7=3 f , f -- observed flux, for
GRO 1008­57. Both are axis in logarithmic scale. Observations (Bildsten et al., 1997) are
shown with black dots. Five curves are plotted for disk accretion for different values of distance
to the pulsar and NS magnetic momentum. Solid curve: d = 4 kpc, ¯ = 10 \Delta 10 30 G \Delta cm 3 . Dashed
curve: d = 8 kpc, ¯ = 10 \Delta 10 30 G \Delta cm 3 . Long dashed curve: d = 8 kpc, ¯ = 45 \Delta 10 30 G \Delta cm 3 .
Dot­dashed curve (the best fit): d = 5:8 kpc, ¯ = 37:6 \Delta 10 30 G \Delta cm 3 . Dotted curve: d = 4 kpc,
¯ = 45 \Delta 10 30 G \Delta cm 3 .
4 Lipunov, V.M., 1992, ``Astrophysics of Neutron Stars'', Springer­Verlag
5 Nagase, F., 1992, in Ginga Memorial Symposium (ISAS Symp. on Astroph.) eds. F.Makino
& F.Nagase, 1
6 Popov S.B., 1999, Astr. Astroph. Trans. (in press), astro­ph/9906012
7 Rappaport, S, & Joss, P.C., 1997, ApJ 486, 435
8 Wang, Y.­M., 1996, ApJ 465, L111