The acceleration of thermal protons at parallel collisionless shocks: Three-dimensional hybrid simulations

We present three-dimensional hybrid simulations of collisionless shocks that propagate parallel to the background magnetic field to study the acceleration of protons that forms a high-energy tail on the distribution. We focus on the initial acceleration of thermal protons and compare it with results from one-dimensional simulations. We find that for both one- and three-dimensional simulations, particles that end up in the high-energy tail of the distribution later in the simulation gained their initial energy right at the shock. This confirms previous results but is the first to demonstrate this using fully three-dimensional fields. The result is not consistent with the "thermal leakage" model. We also show that the gyrocenters of protons in the three-dimensional simulation can drift away from the magnetic field lines on which they started due to the removal of ignorable coordinates that exist in one- and two-dimensional simulations. Our study clarifies the injection problem for diffusive shock acceleration.