Sources of high-energy cosmic rays are presently unknown, but can be constrained in various ways. Some of these constraints can be graphically presented on the so-called Hillas diagram. Previous versions of this diagram determined the range of geometrical sizes and magnetic fields of potential astrophysical accelerators, taking into account geometrical criteria and radiation losses. In this work, we update the Hillas diagram for protons, taking into account the losses associated with the $p\gamma$ and photopair interactions, relating the allowed regions to the source electromagnetic luminosity. The strongest constraints are obtained for bright compact sources such as the central regions of active galactic nuclei.

• Received 8 December 2022
• Accepted 31 May 2023

DOI:https://doi.org/10.1103/PhysRevD.107.123018