1 Introduction

The monitoring of long-term and periodic variations both in pulse shape and slow-down rate of the isolated pulsar PSR B1828-11 shows strong Fourier power at periods of $\simeq$ 1000, 500, 250, and 167 d, with the strongest one at period $\simeq$500 d (Stairs et al. 2000). The close relationship between the periodic changes in the beam shape and the spin-down rate of the pulsar suggests the possibility of precession of the spin axis in a rotating body. The precession of the spin axis would provide cyclic changes in the inclination angle $\chi$ between the spin and magnetic symmetry axes. The result will be periodic variations both in the observed pulse-profile and spin-down rate of the pulsar.

Recently Jones & Andersson (2001) and Link & Epstein (2001) studied a freely precessing neutron star due to a small deformation of the star from spherical symmetry coupled to a torque such as magnetic dipole moment, gravitational radiation, etc., and explained some part of the observed data. Because of the strong periodicity at period $\simeq$504 d seen in the data, Jones & Andersson (2001) reasonably suggested that the actual free precession period is $P_{\rm pre}=1009$ d. A coupling between the magnetic dipole moment and star's spin axis can provide a strong modulation at period $P_{\rm pre}/2\simeq 504$ d, when the magnetic dipole is nearly orthogonal to the star's deformation axis. But their model could not explain the strong Fourier component corresponding to a period of $\simeq$250 d (see Stairs et al. 2000). The latter component has a significant contribution in the observed variations of period residual $\Delta p$, its derivative $\Delta \dot p$, and pulse shape. For this reason, Link & Epstein (2001) assumed that the strongest Fourier component ($\simeq$500 d) represent the actual free precession period. They found that for a small deformation parameter of $\epsilon=(I_3-I_1)/I_1\simeq 9\times 10^{-9}$, a free precession of the angular momentum axis around the symmetry axis of the crust could provide a period at $P_{\rm pre}\simeq 511$ d. Here I₁=I₂<I₃ are the principle moment inertia of the star. Further, they showed that a coupling of nearly orthogonal (fixed to the body of the star) magnetic dipole moment to the spin axis would provide the observed harmonic at period $\simeq$250 d. Their model has good agreement with observations in the pulse period, but as they mentioned, it failed to explain the Fourier component at period $\simeq$1000 d seen in the data (as well as 167 d).

The existence of precession in a neutron star is in strong conflict with the superfluid models for the neutron star interior structure. These models have successfully explained the glitch phenomena (with both pre- and post-glitch behavior) in most neutron stars in which the pinned vortices to the star crust become partially unpinned during a glitch (Alpar et al. 1984). As shown by Shaham (1977) and Sedrakian et al. (1999), the precession should be damped out by the pinned (even imperfect) vortices on a time scale of few precession periods. For example, PSR B1828-11 with typical degree of vortex pinning, $I_{\rm pinned}/I_{\rm star}\sim 1.4$% (indicated by pulsar glitches in stars that frequently glitch), would precess for $\ll$40 s, far shorter than the observed periods (Link & Epstein 2001). Here $I_{\rm star}$ is the total moment inertia of the star, while $I_{\rm pinned}$ is the portion of star's fluid moment inertia that is pinned to the crust.

The free precession description provides an effective decoupling between the internal superfluid and the crust. Recently, Link & Cutler (2002) studied the problem more carefully by considering dynamics of the pinned vortices in a free precessing star under both Magnus, $f_{\rm m}$, and hydrodynamics (due to the precession), $f_{\rm p}$, forces. They found that the precessional (free) motion itself prevents the vortex pinning process and keeps the vortices unpinned in the crust of PSR B1828-11 while precessing, for a force density (per unit length) $f_{\rm p}\sim 10^{16}$ dyn cm^-1. As a result, they found that partially pinned-vortex configuration cannot be static.

The effective core-crust decoupling causes the core and the crust to rotate at different rates (Sedrakian et al. 1999). The latter would increase core magnetic flux-tube displacement relative to the crust, and then sustain the magnetic stresses on the solid crust in forcing it to break (in platelets) and move as the star rotate (Ruderman 1991a,b). The stresses are strong enough to move the crustal lattice by continual cracking, buckling, or plastic flow to relative stresses beyond the lattice yield strength. As a result, because of the very high electrical conductivity of the crust ( $\sigma \sim 10^{26}$ s^-1), the foot points of external magnetic field lines move with the conducting plates in which the field is entangled. Furthermore, since the core magnetic flux tubes are frozen into the core's fluid, the precessing crust drags them and then increases core flux-tube displacements. Therefore, during precession of the crust such plate motion is unavoidable. In addition, as shown by Malkus (1963, 1968) the precessional motion of the star exerts torques to the core and/or crust resulting from shearing flow at the thin core-crust boundary region. These torques, so-called precessional torque, are able to sustain a turbulent hydromagnetic flow in the boundary region, and then increase local magnetic field strength. This would excite convective fluid motions in the core-crust boundary, increase magnetic stresses on the crust and cause it to break down in platelets.

In this paper, motivated by the above conjecture, we suggest that the magnetic field may vary somewhat with time, relative to the body axes of the star. The question that arises now is whether the whole observed Fourier spectrum of PSR B1828-11, can be consistently generated by a time-varying magnetic field during the course of free precession of the star. In other words, under what conditions will the observed cyclical changes in the timing data be produced by precession of the star's crust coupled to the magnetic dipole torque of a time-varying magnetic field. In Sect. 2 we address this question in detail. Following Link & Epstein (2001) we assume that the star precesses freely around the spin axis, but with period $P_{\rm pre}\simeq 1000$ d. Then we show that the magnetic torque exerted by a dipole moment may produce the other observed harmonics as seen in data, if the magnetic dipole vector rotates with a period close to $P_{\rm pre}$relative to the star's body axes. Section 3 is devoted to further discussion.