On the surface brightness radial profile of the extended $\gamma$-ray sources

The morphology of the extended $\gamma$-ray source is governed by the propagation process of parent relativistic particles. In this paper, we investigate the surface brightness radial profile of extended $\gamma$-ray sources illuminated by cosmic ray protons and electrons, considering the radiation mechanisms, projection effects, and the response of instruments. We found that the parent particle species and the propagation process can cause considerable differences in the observed radial profiles. Thus, the surface brightness profile can be used as a unique tool to identify the radiation mechanism and the propagation process of the parent particles. In addition, We also discuss the possible implications regarding the latest discoveries from very/ultra-high energy $\gamma$-ray instruments like LHAASO and HAWC.