Disk-fed magnetic neutron stars have been predicted to experience spin-up or spin-down episodes due to the torque exerted by the accretion disk. In the magnetically-threaded accretion disk model, first suggested by Ghosh & Lamb (1979a, 1979b), the spin-up rate is given by
where I is the moment of inertia of the neutron star, the accretion rate, G the gravity constant and the inner edge of the accretion disk. The dimensionless torque , which includes the torque contribution from both matter accretion and magnetic stress, is a function of the "fastness parameter" , where is the corotation radius.
With the mean spin-up rate and X-ray luminosity observed in SMC X-1, Eq. (1) has two sets of solutions: (1) the magnetic moment of the pulsar µ is less than a few with ; (2) µ is around with (Li & van den Heuvel 1997). If the X-ray intensity during the low state, which is lower than that during the high state by a factor of , is due to a reduction in the mass accretion rate, the condition implies . However, Gruber & Rothschild (1984; see also Levine et al. 1996; Wojdowski et al. 1998) have suggested the possibility of modulation of the observed X-ray intensity by a tilted, precessing, accretion disk like that in Her X-1 (Katz 1973), which would imply that the intrinsic X-ray luminosity or mass accretion rate of the pulsar could be quite steady. In this case the pulsar magnetic moment may lie between and , though a magnetic moment as high as seems less likely for SMC X-1, because of the following arguments. As seen in Fig. 2, the spin-up rate of the pulsar in 1980 s and 1990 s varied around its mean value by . The most straightforward explanation for this change is that the accretion rate has fluctuated by a similar (or a bit larger) factor 1. If , would be so close to that the pulsar would spin down when the accretion rate decreased by a small factor (less than 20%), contradicted with the steady spin-up observed.
The above arguments are based on the classical accretion torque models, which, however, has encountered difficulties in explaining the period evolution in the X-ray pulsar Cen X-3, which possesses many similarities with SMC X-1. Both pulsars are in a close binary system with a supergiant companion star overflowing its Roche-lobe, accreting from a disk (Tjemkes et al. 1986) at a high rate, with the X-ray luminosities close to or higher than the Eddington luminosity (Nagase 1989). Tidal torque between the distorted supergiant and the neutron star leads to an orbital decay at a similar rate in the two systems (cf. White et al. 1995). However, Cen X-3 exhibits a pulse period evolution quite different from SMC X-1. Prior to 1991, Cen X-3 had already been found to show a secular slow spin-up superposed with fluctuations and short episodes of spin-down. The more frequently sampled BATSE data show that Cen X-3 exhibits 10-100 d intervals of steady spin-up and spin-down at a much larger rate, and the long-term ( years) spin-up trend is actually the average consequence of the frequent transitions between spin-up and spin-down (Finger et al. 1994). Such spin behavior has been found in at least 4 out of 8 persistent X-ray pulsars observed with BATSE (Bildsten et al. 1997), and is difficult to explain in terms of classical accretion torque models, which would require finely tuned step-function-like changes in the mass accretion rate.
It is interesting to see whether the secular spin-up in SMC X-1 actually consists of transitions of short-term spin-up and spin-down, as in Cen X-3. If it is the case, this would indicate a larger instantaneous accretion torque, and hence a higher magnetic moment.
© European Southern Observatory (ESO) 1999
Online publication: April 12, 1999