Pulsar X-Ray and Gamma-Ray Pulse Profiles: Constraint on Obliquity and Observer Angles

We model the thermal X-ray profiles of Geminga, Vela, and PSR 0656+14, which have also been detected as γ-ray pulsars, to constrain the phase space of obliquity and observer angles required to reproduce the observed X-ray pulsed fractions and pulse widths. These geometrical constraints derived from the X-ray light curves are explored for various assumptions about surface temperature distribution and flux anisotropy caused by the magnetized atmosphere. We include curved spacetime effects on photon trajectories and the magnetic field. The observed γ-ray pulse profiles are double peaked with phase separations of 0.4-0.5 between the peaks. Assuming that the γ-ray profiles are due to emission in a hollow cone centered on the magnetic pole, we derive the constraints on the phase space of obliquity and observer angles, for different γ-ray beam sizes, required to produce the observed γ-ray peak phase separations. We compare the constraints from the X-ray emission to those derived from the observed γ-ray pulse profiles and find that the overlapping phase space requires both obliquity and observer angles to be smaller than 20°-30°, implying γ-ray beam opening angles of at most 30°-35°.