Spatial distribution of secondary electrons' Synchrotron emission: property and implication

Galactic $\gamma$-ray sources can be produced by either high-energy protons via proton-proton collisions or electrons/positrons via inverse Compton scattering. Distinguishing between the hadronic and leptonic origin of $\gamma$-ray emission in Galactic sources remains challenging. Measurements of non-thermal X-ray spectra of these sources, which could originate from primary electrons in the leptonic scenario or secondary electrons/positrons in the hadronic scenario, have been suggested as an efficient way of discriminating between these scenarios. In this work, we investigate the morphology of the X-ray emission from secondary electrons/positrons. By calculating the surface brightness profile and the photon index profile of X-ray emission, we find that secondary electrons produce a distinctively flat X-ray surface brightness profile. Our results suggest that, in addition to the X-ray spectrum, the X-ray morphology is crucial to determine the radiation mechanism of ultrahigh-energy $\gamma$-ray sources and help to identify sources of PeV cosmic rays.