A&A 399, 653-658 (2003)
Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195, Iran
Received 10 September 2002 / Accepted 27 November 2002
Analysis of both pulse timing and pulse shape variations of the isolated pulsar PSR B1828-11 shows highly correlated and strong Fourier power at periods 1000, 500, 250, and 167 d (Stairs et al. 2000). The only description based on a free precession of the star's rigid crust coupled to the magnetic dipole torque, explains the 500 d-component, as the fundamental Fourier frequency, with its harmonic 250 d-component (Link & Epstein 2001). In this paper, we study a time-varying magnetic field model and show that if the dipole moment vector rotates with a period nearly equal to the longest (assumed fundamental) observed period (1000 d) relative to the star's body axes, the resulting magnetic torque may produce the whole Fourier spectrum consistently. We also find the second and fourth harmonics at periods 500 and 250 d are dominant for small wobble angle and large field's inclination angle . We note that, although our model simply explains the observed pulse timing and pulse shape variations of PSR B1828-11, it also evokes some serious problems with our currently understanding from the physics of the neutron star structure.
Key words: pulsar: individual: PSR B1828-11 - stars: neutron - stars: magnetic fields
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 504 d seen in the data, Jones & Andersson (2001) reasonably suggested that the actual free precession period is d. A coupling between the magnetic dipole moment and star's spin axis can provide a strong modulation at period 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 250 d (see Stairs et al. 2000). The latter component has a significant contribution in the observed variations of period residual , its derivative , and pulse shape. For this reason, Link & Epstein (2001) assumed that the strongest Fourier component (500 d) represent the actual free precession period. They found that for a small deformation parameter of , a free precession of the angular momentum axis around the symmetry axis of the crust could provide a period at 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 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 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, % (indicated by pulsar glitches in stars that frequently glitch), would precess for 40 s, far shorter than the observed periods (Link & Epstein 2001). Here is the total moment inertia of the star, while 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, , and hydrodynamics (due to the precession), , 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) 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 ( 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 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 relative to the star's body axes. Section 3 is devoted to further discussion.
Consider a rigid, biaxial rotating star with
the principle axes
and corresponding principle
moment of inertia
The star's angular momentum
misaligned to the symmetry axis
by a wobble angle ,
In general, we assume that the stellar
magnetic field is
dipolar and changes with time as
To obtain the reported spectrum of PSR B1828-11, it is enough to consider
n=0 and 1 terms in Eq. (5) only.
terms will produce
the higher harmonics which will be discussed later. Simply a correct
behavior of the spin rate can be found by setting c1=0, c0=1, and
Therefore Eq. (5) reduces to (dropping constant terms)
In Fig. 1, by using Eq. (8) we fit the
data (Stairs et al. 2000)
with a precession period of
a wobble angle
and the inclination angle
between the magnetic dipole and star's symmetry axis. The
Fourier expansion coefficients are a0=.01 and a1=1. We
note that our
fit is indistinguishable from the one obtained by Link & Epstein
|Figure 1: Timing data for PSR B1828-11 (from Stairs et al. 2000). The top panel shows period residual (in ns) relative to the star's secular spin-down. The bottom panel gives the time derivative (in units of 10-15). The solid curves are the fit explained in the text.|
|Open with DEXTER|
It is interesting to note that Eq. (8) includes 1000, 500, 333, and 250 d Fourier components. Further, by considering terms in Eq. (5), one can find the higher harmonics in . These terms were missing in the Link & Epstein's model. In Table 1 we compare the time-varying magnetic field model with one suggested by Link & Epstein (2001), and the observations made by Stairs et al. (2000).
|Data||1000, 500, 250, 167||>|
|Time-varying mag. model||1000a, all harmonics||>|
|Link & Epstein's model||500a,||>|
The time-varying magnetic field model can also explain the observed timing data for PSR B1642-03. The analysis of timing data of PSR B1642-03, collected over a span of 30 years, exhibit strong Fourier power at periods 5000, 2500, and 1250 days (Shabanova et al. 2001). The suggested wobble and magnetic field inclination angles are and , respectively. Similar to PSR B1828-11, the spectra of PSR B1642-03 show wide spectral features at periods 2500 d and 1250 d. The pulse shape variations were not detected, probably, due to their small amplitudes. Furthermore, it is interesting to note that both PSR B1828-11 and PSR B1642-03 exhibit features around the sixth harmonic, , i.e. 167 d and 667 d, respectively. As seen from Eq. (5), by including the n= 2 term in the expansion, one can reasonably get these harmonics in . These terms were missing in previous studies.
The necessary equality relation between
requires the magnetic poles
at the surface of the star move with relative velocity
cm s-1 respect to the body axes.
On the other hand, the magnetic poles
d. Because of the very high electrical conductivity of the solid
s-1), this result would be hardly
Variation of the magnetic dipole vector with time in a neutron star (with
solid crust and no active convective zone)
may be understood through
the so-called neutron star crustal tectonics scenario which has been
proposed originally to explain magnetic dipole evolution and resulting
observable features in millisecond pulsars, low-mass X-ray binaries and
radio pulsars (Ruderman 1991a,b).
The solid crustal lattice of neutron star is subject to various strong
stresses. The pinned superfluid neutron star vortex lines exert a strong
force on the crustal lattice of nuclei which pin them
(Anderson & Itoh 1975; Ruderman 1976;
Alpar et al. 1984).
Further, the evolving core magnetic flux
tubes which pass through the crust, pull it strongly at the
base of the crust Srinivasan et al. (1990). In the
rapidly rotating weakly magnetized
neutron stars such as millisecond pulsars and low-mass X-ray binaries, the
lattice stresses from pinned vortices are dominant, while in the older
pulsars such as radio pulsars with
strong magnetic fields, the magnetic stresses from core flux-tube
displacement may become important. For PSR B1828-11 and PSR B1642-03 with
magnetic field strength
and effectively unpinned superfluid vortices (Link & Cutler 2002),
the latter case
is more appropriate. The quantized magnetic flux tubes in a core's type II
superconducting proton sea terminate at the base of the crust. These flux
tubes move in response to changes in the positions of neutron star's core
If the crust were to remain immobile the shear stress, S(B), on the base
highly conducting crust (from core magnetic flux tube motion) could
grow to reach
It is worth noting that the required torque variation may as well be that due to the internal torque, by the partially pinned vortices during the precessional motion of the crust. The internal torques would arise from different coupled components of the star which move with different velocities, e.g. the mutual friction torque which arise from different velocities of vortex lines and superfluid. These torques are able to sustain hydromagnetic shear flows and turbulences in the core-crust boundary, excite the fluid convection motions, and cause magnetic field variations (Malkus 1963, 1968). Further, they affect the motion of the neutron star crust, for example, by tilting away its angular velocity vector from alignment with star principle axis (Sedrakian et al. 1999). To find a clear picture of dynamics of magnetic field in a precessing neutron star, one has to consider the effect of the internal torques. This is currently under investigation (Rezania 2003). Finally, in this paper we showed that a time-varying magnetic field model is able to explain consistently the timing analysis of both PSR B1828-11 and PSR B1642-03, if the field's symmetry axis rotates with a rate nearly equal to their precession rates, relative to the star's body axes. Unfortunately, at this stage, the large speed of the magnetic poles at the surface of the star required by this model is difficult to accept. Further studies on the evolution and dynamics of magnetic fields in precessing stars (especially the plate tectonics model) seem necessary. These will be left for future investigations.
I would like to thank S. M. Morsink , B. Link, and S. Sengupta for their careful reading the manuscript and useful discussions. It is a pleasure to thank M. Jahan-Miri for stimulating discussions during the course of this work and draw my attention to the plane tectonics model. The author is also grateful to I. Stairs for useful discussions and providing the timing data for PSR B1828-11. I would like to thank Yousef Sobouti and Roy Maartens for continuing encouragement. This research was supported by the Natural Sciences and Engineering Research Council of Canada.