Constraints on thermal emission models of anomalous X-ray pulsars

Thermal emission from the surface of an ultramagnetic neutron star is believed to contribute significantly to the soft X-ray flux of the anomalous X-ray pulsars (AXPs). We compare the detailed predictions of models of the surface emission from a magnetar to the observed spectral and variability properties of AXPs. In particular, we focus on the combination of their luminosities and energy-dependent pulsed fractions. We use the results of recent calculations for strongly magnetized atmospheres in radiative equilibrium to obtain the angle-and energy-dependence of the surface emission. We also include in our calculations the significant effects of general relativistic photon transport to an observer at infinity as well as the effects of interstellar extinction. We find that the combination of the large pulsed fractions and the high inferred luminosities of AXPs cannot be accounted for by surface emission from a magnetar with two antipodal hot regions or a temperature distribution characteristic of a magnetic dipole. This result is robust for reasonable neutron star radii, for the range of magnetic field strengths inferred from the observed spin down rates, and for surface temperatures consistent with the spectral properties of AXPs. Models with a single hot emitting region can reproduce the observations, provided that the distance to one of the sources is ∼30% less than the current best estimate, and allowing for systematic uncertainties in the spectral fit of a second source. Finally, the thermal emission models with antipodal emission geometry predict a characteristic strong increase of the pulsed fraction with photon energy, which is apparently inconsistent with the current data. The energy-dependence of the pulsed fraction in the models with one hot region shows a wider range of behavior and can be consistent with the existing data. Upcoming high-resolution observations with Chandra and XMM-Newton will provide a conclusive test.