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 underpredicts 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 .