The maximum energy of accelerated particles in relativistic collisionless shocks

The afterglow emission from gamma-ray bursts (GRBs) is usually interpreted as synchrotron radiation from electrons accelerated at the GRB external shock that propagates with relativistic velocities into the magnetized interstellar medium. By means of multi-dimensional particle-in-cell simulations, we investigate the acceleration performance of weakly magnetized relativistic shocks, in the magnetization range 0 ≲ σ ≲ 10^-1. The pre-shock magnetic field is orthogonal to the flow, as generically expected for relativistic shocks. We find that relativistic perpendicular shocks propagating in electron-positron plasmas are efficient particle accelerators if the magnetization is σ ≲ 10^-3. For electron-ion plasmas, the transition to efficient acceleration occurs for σ ≲ 3 × 10 ^-5. Here, the acceleration process proceeds similarly for the two species, since the electrons enter the shock nearly in equipartition with the ions, as a result of strong pre-heating in the self-generated upstream turbulence. In both electron-positron and electron-ion shocks, we find that the maximum energy of the accelerated particles scales in time as ε_maxalpha; ^1/2. This scaling is shallower than the so-called (and commonly assumed) Bohm limit ε_maxα, and it naturally results from the small-scale nature of the Weibel turbulence generated in the shock layer. In magnetized plasmas, the energy of the accelerated particles increases until it reaches a saturation value ε_sat/ γ₀m_ic ² ∼ σ^-1/4, where γ₀ m_ic ² is the mean energy per particle in the upstream bulk flow. Further energization is prevented by the fact that the self-generated turbulence is confined within a finite region of thickness σ^-1/2 around the shock. Our results can provide physically grounded inputs for models of non-thermal emission from a variety of astrophysical sources, with particular relevance to GRB afterglows.