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Recently, gamma-ray halos of a few degree extension have been detected aroundtwo middle-aged pulsars, namely, Geminga and PSR B0656+14, by the High AltitudeWater Cherenkov observatory (HAWC). The gamma-ray radiation arise fromrelativistic electrons that escape the pulsar wind nebula and diffuse in thesurrounding medium. The diffusion coefficient is found to be significantlylower than the average value in the Galactic disk. If so, given a typicaltransverse velocity of $300-500{\,\rm km /s}$ for a pulsar, its displacementcould be important in shaping the morphology of its gamma-ray halos. Motivatedby this, we study the morphology of pulsar halos considering the proper motionof pulsar. We define three evolutionary phases of pulsar halo to categorize itsmorphological features. The morphology of pulsar halos below 10$\,$TeV isdouble peaked or single peaked with an extended tail, which depends on theelectron injection history. Above 10 TeV, the morphology of pulsar halos isnearly spherical, due to the short cooling timescale ($<50\,$kyr) for tens TeVelectrons. We also quantitatively evaluate the separation between the pulsarand the center of the gamma-ray halo, as well as the influence of differentassumptions for the pulsar characteristics and the injected electrons. Ourresults suggest that the separation between the center of the gamma-ray haloabove 10$\,$TeV and the associated pulsar is usually too small to be observableby HAWC or LHAASO. Hence, our results provide a useful approach to constrainthe origin of extended sources at very high energies.

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