Superdiffusion of cosmic rays in the vicinity of their accelerators and the resulting $γ$-ray emission

Abstract

We study the distribution of cosmic rays (CRs) in the vicinity of their accelerators, assuming that the transport of CRs in the interstellar medium surrounding the accelerators is described by the superdiffusion, beyond the normal diffusion. We find that the superdiffusivity, which is characterized by the superdiffusion parameter $α$, impacts significantly the distribution of CRs. For impulsive injection, the CR distribution behaves a constant radial profile, except with a power-law tail for $α< 2$ or a Gaussian tail for $α=2$ at large distance, $r$, from the accelerators. For stationary injection, the radial profile of CR protons tends to being proportional to $r^{α- 3}$, while that of CR electrons can deviate from the $r^{α- 3}$ profile, due to their severe energy losses. We also compute the $γ$-ray emission, produced by the interactions of CRs with ambient gas and radiation fields, within 100 pc regions around the accelerators. We find that by investigating the $γ$-ray morphology, we can distinguish the superdiffusion from the normal diffusion with present and next-generation imaging air Cherenkov telescopes.