Mapping the Surface of the Magnetar 1E 1048.1-5937 in Outburst and Quiescence Through Phase Resolved X-ray Spectroscopy

We model the pulse profiles and the phase resolved spectra of the anomalous X-ray pulsar 1E 1048.1-5937 obtained with XMM-Newton to map its surface temperature distribution during an active and a quiescent epoch. We develop and apply a model that takes into account the relevant physical and geometrical effects on the neutron star surface, magnetosphere, and spacetime. Using this model, we determine the observables at infinity as a function of pulse phase for different numbers and sizes of hot spots on the surface. We show that the pulse profiles extracted from both observations can be modeled with a single hot spot and an antipodal cool component. The size of the hot spot changes from $\approx 80^{\circ}$ in 2007, 3 months after the onset of a dramatic flux increase, to $\approx 30^{\circ}$ during the quiescent observation in 2011, when the pulsed fraction returned to the pre-outburst $\approx$ 65\% level. For the 2007 observation, we also find that a model consisting of a single 0.4 keV hot spot with a magnetic field strength of $1.8 \times 10^{14}$ G accounts for the spectra obtained at three different pulse phases but under predicts the flux at the pulse minimum, where the contribution to the emission from the cooler component is non-negligible. The inferred temperature of the spot stays approximately constant between different pulse phases, in agreement with a uniform temperature, single hot spot model. These results suggest that the emitting area grows significantly during outbursts but returns to its persistent and significantly smaller size within a few year timescale.