Morphology of Gamma-Ray Halos around Middle-Aged Pulsars: Influence of the Pulsar Proper Motion

Abstract: Recently, gamma-ray halos of a few degree extension have been detected around two middle-aged pulsars, namely, Geminga and PSR B0656+14, by the High Altitude Water Cherenkov observatory (HAWC). The gamma-ray radiation arise from relativistic electrons that escape the pulsar wind nebula and diffuse in the surrounding medium. The diffusion coefficient is found to be significantly lower than the average value in the Galactic disk. If so, given a typical transverse velocity of $300-500{\,\rm km /s}$ for a pulsar, its displacement could be important in shaping the morphology of its gamma-ray halos. Motivated by this, we study the morphology of pulsar halos considering the proper motion of pulsar. We define three evolutionary phases of pulsar halo to categorize its morphological features. The morphology of pulsar halos below 10$\,$TeV is double peaked or single peaked with an extended tail, which depends on the electron injection history. Above 10 TeV, the morphology of pulsar halos is nearly spherical, due to the short cooling timescale ($<50\,$kyr) for tens TeV electrons. We also quantitatively evaluate the separation between the pulsar and the center of the gamma-ray halo, as well as the influence of different assumptions for the pulsar characteristics and the injected electrons. Our results suggest that the separation between the center of the gamma-ray halo above 10$\,$TeV and the associated pulsar is usually too small to be observable by HAWC or LHAASO. Hence, our results provide a useful approach to constrain the origin of extended sources at very high energies.