The Maximum Energy of Shock-Accelerated Cosmic Rays

Identifying the accelerators of Galactic cosmic ray protons (CRs) with energies up to a few PeV ($10^{15}$ eV) remains a theoretical and observational challenge. Supernova remnants (SNRs) represent strong candidates, as they provide sufficient energetics to reproduce the CR flux observed at Earth. However, it remains unclear whether they can accelerate particles to PeV energies, particularly after the very early stages of their evolution. This uncertainty has prompted searches for other source classes and necessitates comprehensive theoretical modeling of the maximum proton energy, $E_{\rm max}$, accelerated by an arbitrary shock. While analytic estimates of $E_{\rm max}$ have been put forward in the literature, they do not fully account for the complex interplay between particle acceleration, magnetic field amplification, and shock evolution. This paper uses a multi-zone, semi-analytic model of particle acceleration based on kinetic simulations to place constraints on $E_{\rm max}$ for a wide range of astrophysical shocks. In particular, we develop relationships between $E_{\rm max}$, shock velocity, size, and ambient medium. We find that SNRs can only accelerate PeV particles under a select set of circumstances, namely, if the shock velocity exceeds $\sim 10^4$ km s$^{-1}$ and escaping particles drive magnetic field amplification. However, older, slower SNRs may still produce observational signatures of PeV particles due to populations accelerated when the shock was younger. Our results serve as a reference for modelers seeking to quickly produce a self-consistent estimate of the maximum energy accelerated by an arbitrary astrophysical shock.