Timing properties of magnetars

We study the pulse morphologies and pulse amplitudes of thermally emitting neutron stars with ultrastrong magnetic fields. The beaming of the radiation emerging from a magnetar was recently shown to be predominantly nonradial, with a small pencil and a broad fan component. We show that the combination of this radiation pattern with the effects of strong lensing in the gravitational field of the neutron star yields pulse profiles that show a qualitatively different behavior compared to that of the radially peaked beaming patterns explored previously. Specifically, we find that (i) the pulse profiles of magnetars with a single hot emission region on their surface exhibit 1-2 peaks, whereas those with an antipodal emission geometry have 1-4 peaks, depending on the neutron star compactness, the observer's viewing angle, and the size of the hot regions; (ii) the energy dependence of the beaming pattern may give rise to weakly or strongly energy-dependent pulse profiles and may introduce phase lags between different energy bands; (iii) the nonradial beaming pattern can give rise to high pulsed fractions even for very relativistic neutron stars; (iv) the pulsed fraction may not vary monotonically with neutron star compactness; (v) the pulsed fraction does not decrease monotonically with the size of the emitting region; (vi) the pulsed fraction from a neutron star with a single hot pole has, in general, a very weak energy dependence, in contrast to the case of an antipodal geometry. Comparison of these results to the observed properties of anomalous X-ray pulsars strongly suggests that they are neutron stars with a single hot region of ultrastrong magnetic field.