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Method of estimating distances to X­ray pulsars and
their magnetic fields
S.B. Popov
Sternberg Astronomical Institute
119899, Russia, Moscow,
Universitetski pr. 13
e­mail: polar@xray.sai.msu.su
Abstract
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. Es­
timates of the distance: approximately 6 kpc, and the magnetic momentum:
approximately 4 \Delta 10 31 G \Delta cm 3 , are obtained.
1 Introduction
Neutron stars (NS) can appear as isolated objects (radiopulsars, old isolated
accreting NS, etc.) and as X­ray sources in close binary systems. The most
prominent of the last ones are X­ray pulsars, where important parameters of
NS can be determined.
Now more than 40 X­ray pulsars are known (see, for example, Bildsten et
al., 1997). Observations of optical counterparts give an opportunity to obtain
distances to these objects with high precision, and with hyroline detections
one can obtain the value of magnetic field of a NS. But lines are not detected
in all sources of that type (partly because they can lay out of the range of
necessary spectral sensitivity of devices, when field are high), and magnetic
field can be estimated from period measurements (see Lipunov, 1982, 1992).
Precise distance measurements usually are not available immediately after X­
ray discovery (especially, if error boxes, as for example in the BATSE case, are
large). So, methods of their determination basing only on X­ray observations
can be useful.
Here we propose a simple method to determine magnetic field and distance
to X­ray pulsar using only X­ray flux and period variations measurements.
1

2 Method
In Lipunov (1982) it was proposed to use maximum spin­up and spin­down
values to obtain limits on the magnetic momentum of X­ray pulsars in disk or
wind models, using known values of luminosity (method, based on maximum
spin­down, is very insensitive to uncertainties in luminosity and produces better
results).
In this short note we propose a rough simple method to determine magnetic
field without known distance and to determine distance itself. The method is
based on several measurements of period derivative, —
p, and X­ray pulsar's flux,
f . Fitting two parameters: distance, d, and magnetic momentum, ¯, one can
obtain good correspondence with the observed —
p and f , and that way produce
good estimates of distance and magnetic field.
Here we consider only disk accretion. In that case one can write (see Lipunov,
1982, 1992):
dI!
dt = —
M (GM fflR A ) 1=2
\Gamma k t
¯ 2
R 3
c
; (1)
where ! -- spin frequency of a NS, M -- its mass, I-- its moment of inertia, RA
­ Alfven radius, R c -- corotation radius. We use the following values: ffl = 0:45,
k t = 1=3 (see Lipunov, 1992). The first term on the right side represents
acceleration of a NS from an accretion disk, and the second term represents
deceleration. The form of the deceleration term is general, only typical radius
of interaction should be changed. It is equal to R c for accretors, R l ­ light
cylinder radius for ejectors, and RA for propellers (see the details in Lipunov
1992).
Lets rewrite eq. (1) in terms of period and its derivative:
—
p = 4ú 2 ¯ 2
3 G I M \Gamma (0:45) 1=2 2 \Gamma1=14 ¯ 2=7
I
(GM ) \Gamma3=7
h
p 7=3 L
i 6=7
R 6=7 ; (2)
where L = 4úd 2 \Delta f -- luminosity, f -- observed flux.
So, in eq. (2) we know all parameters (I, M , R etc.) except ¯ and d. Fitting
observed points with them we can obtain estimates of ¯ and d. If ¯ is known,
one can immediately obtain d from eq. (2) even from one determination of —
p
(in that case it is better to use spin­down value). Uncertainties mainly depend
on applicability of that simple model.
3 Illustration of the method
To illustrate the method, we apply it to the X­ray pulsar GRO J1008­57, dis­
covered by BATSE (Bildsten et al., 1997). It is a 93:5 s X­ray pulsar, with the
flux about 10 \Gamma9 erg cm \Gamma2 s \Gamma1 . A 33 day outburst was observed by BATSE in
August 1993. The source was also observed by EXOSAT (Macomb et al., 1994)
2

and ASCA (Day et al., 1995). ROSAT made possible to localize the source with
high precision (Petre & Gehrels, 1994), and it was identified with a Be­system
(Coe et al., 1994) with ¸ 135 d orbital period (Shrader et al. 1999). We use here
only 1993 outburst, described in Bildsten et al. (1997).
The authors in Bildsten et al. (1997) show flux and frequency history of the
source with 1 day integration. In the maximum of the burst errors are rather
small, and we neglect them. Points with large errors were not used.
We used standard values of NS parameters: I = 10 45 g cm 2 , moment of
inertia; R = 10 km, NS radius; M = 1:4M fi , NS mass.
On figures 1­2 we show observations (as black dots) and theoretical curves (in
disk model, see Shrader et al. 1999, who proposed a disk formation during the
outbursts, in contrast with Macomb et al. (1994), who proposed wind accretion)
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, ¯.
The best fit (both for spin­up and spin­down) gives d ú 5:8 kpc and ¯ ú
37:6 \Delta 10 30 G \Delta cm 3 . It is shown on both figures. The distance is in correspondence
with the value in Shrader et al. (1999), and such field value is not unusual for
NS in general and for X­ray pulsars in particular (see, for example, Lipunov,
1992 and Bildsten et al., 1997). Tests on some other X­ray pulsars with know
distances and magnetic fields also showed good results.
4 Discussion and conclusions
The method is only approximate and depends on several assumptions (disk
accretion, specified values of M; I; R, etc.). Estimates of ¯, for example, can be
only in rough correspondence with observations of magnetic field B, if standard
value of the NS radius, R = 10 km is used (see, for example, the case of Her X­1
in Lipunov 1992). Non­standard values of I and M can also make the picture
more complicated.
The method can be, in principal, generalized for applications to wind­accreting
systems, and to disk­accreting systems with complicated time behavior (when,
for example, —
p changes appear with nearly constant flux, or even when —
p changes
are uncorrelated with flux variations).
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 shown on the figures): 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 spin­
down 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
3

-7.0 -6.0 -5.0 -4.0 -3.0 -2.0
-10.0
-9.0
-8.0
-7.0
-6.0
spindown
4_10
8_10
4_45
8_45
5.8_37.6
-7.0 -6.0 -5.0 -4.0 -3.0 -2.0
-10.0
-9.0
-8.0
-7.0
-6.0
spinup
Figure 1: Dependence of period derivative, —
p, on the parameter p 7=3 f , f -- ob­
served flux. Both axis are 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, ¯ = 37:6 \Delta 10 30 G \Delta cm 3 . Dashed curve: d = 8 kpc,
¯ = 37:6 \Delta 10 30 G \Delta cm 3 . Long dashed curve: d = 5:8 kpc, ¯ = 10 \Delta 10 30 G \Delta cm 3 .
Dot­dashed curve: d = 5:8 kpc, ¯ = 45 \Delta 10 30 G \Delta cm 3 . Dotted curve (the best
fit): d = 5:8 kpc, ¯ = 37:6 \Delta 10 30 G \Delta cm 3 .
4

-7.0 -6.0 -5.0 -4.0 -3.0 -2.0
-10.0
-9.0
-8.0
-7.0
-6.0
spindown
4_37.6
8_37.6
5.8_10
5.8_45
5.8_37.6
-7.0 -6.0 -5.0 -4.0 -3.0 -2.0
-10.0
-9.0
-8.0
-7.0
-6.0
spinup
Figure 2: Dependence of period derivative, —
p, on the parameter p 7=3 f , f -- ob­
served flux. 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 .
5

figures.
In some very uncertain situations, for example, when only X­ray observations
without precision localization are available, our method can give (basing on
several observational points, not one!, as, for example, in the case of maximum
spin­down determination of magnetic momentum), rough, but useful estimates
of important parameters: distance and magnetic momentum.
Acknowledgments
It is a pleasure to thank prof. V.M. Lipunov for numerous discussions and
suggestions and drs. I.E. Panchenko, M.E. Prokhorov and K.A. Postnov for
useful comments. The work was supported by the RFBR (98­02­16801) and the
INTAS (96­0315) grants.
References
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[2] Coe, M.J., et al., 1994, MNRAS, 270, L57
[3] Day, C.S.R., Zylstra, G.J., White, N.E., Nagase, F., Corbet, R.H.D., &
Petre, R., 1995, BAAS 186, 4802
[4] Lipunov, V.M., ``Astrophysics of Neutron Stars'', 1992, Springer­Verlag
[5] Lipunov, V.M., 1982, AZh 59, p. 888
[6] Macomb, D.J., Shrader, C.L., & Schultz, A.B., 1994, ApJ 437, p. 845
[7] Petre, R., & Gehrels, N., 1994, A&A 282, L33
[8] Shrader, C.L., Sutaria, F.K., Singh, K.P., & Macomb, D.J., 1999, ApJ 512,
p. 920
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