We use the outer gap model to explain the spectrum and the energy dependent light curves of the X-ray and soft γ-rayradiations of the spin-down powered pulsar PSR B1509-58. In the outer gap model, most pairs inside the gap are createdaround the null charge surface and the gap’s electric field separates the opposite charges to move in opposite directions.Consequently, the region from the null charge surface to the light cylinder is dominated by the outflow current and that fromthe null charge surface to the star is dominated by the inflow current. We suggest that the viewing angle of PSR B1509-58 onlyreceives the inflow radiation. The incoming curvature photons are converted to pairs by the strong magnetic field of the star.The X-rays and soft γ-rays of PSR B1509-58 result from the synchrotron radiation of these pairs. The magnetic pair creationrequires a large pitch angle, which makes the pulse profile of the synchrotron radiation distinct from that of the curvatureradiation. We carefully trace the pulse profiles of the synchrotron radiation with different pitch angles. We find that thedifferences between the light curves of different energy bands are due to the different pitch angles of the secondary pairs, andthe second peak appearing at E > 10 MeV comes from the region near the star, where the stronger magnetic field allows thepair creation to happen with a smaller pitch angle.

Anomalous X-ray pulsars (AXPs) are thought to be magnetars which are young isolated neutron stars with extremely strongmagnetic fields of >1014 Gauss. Their tremendous magnetic fields inferred from the spin parameters provide a huge energyreservoir to power the observed X-ray emission. High-energy emission above 0.3 MeV has never been detected despiteintensive search. Here, we present the possible Fermi Large Area Telescope (LAT) detection of γ-ray pulsations above 200MeV from the AXP, 1E 2259+586, which puts the current theoretical models of γ-ray emission mechanisms of magnetars intochallenge. We speculate that the high-energy γ-rays originate from the outer magnetosphere of the magnetar.