Morphology of Gamma-ray Halos around Middle-aged Pulsars: Influence of the Pulsar Proper Motion

Recently, \gray halos of a few degree extension have been detected around two middle-aged pulsars, Geminga and PSR B0656+14 by the High Altitude Water Cherenkov observatory (HAWC). The measurements of surface brightness profile of the pulsar halos suggest that the transport of particles therein is dominated by diffusion, and the diffusion coefficient is significantly lower than the average value in the Galactic disk. Since pulsars typically have a proper velocity of $400-500{\,\rm km s^{-1}}$, the displacement of these middle-aged pulsars due to the proper motion could be important in shaping the morphology of the pulsar halos. Motivated by this, we study the morphology of pulsar halos considering proper motion in the diffusion-dominated scenario. We find that the morphology of the pulsar halo can be basically classified into three evolutionary phases, depending on the velocity of the proper motion, cooling of the emitting electrons and the age of the pulsar. Generally, the morphology would appear highly asymmetric at $\lesssim 1\,$TeV while keeps more or less spherical at $\gtrsim 10\,$TeV for middle-aged pulsars. We also study the offset between the position of pulsars and the center of the intensity map of the corresponding halo. We find that proper motion can induce observable offsets seen by Fermi-LAT, HESS, HAWC and LHAASO from GeV up to a few TeV energies provided that the source is located within several kpc from Earth. It is more difficult to produce resolvable offset in the pulsar halo at higher energy due to more rapid cooling of emitting electrons. Our result can provide constraints on the origins of those extended sources at very high energies.