A Simple Analytical Method Using Fokker-Planck Equation for Modeling Particle Acceleration At Astrophysical Shocks

Shocks are ubiquitous in astrophysical environments, and particle acceleration at such astrophysical shocks is related to high-energy phenomena. In particular, the acceleration mechanism and the time evolution of the particle distribution function have been extensively examined. This paper describes a simple analytic method using the one-dimensional Fokker-Planck equation in the test-particle regime. We aim to investigate the evolution of the particle distribution function in the shock upstream, which could be streaming toward Earth along the open magnetic field geometry. The behavior of the analytical solution is examined over a wide range of parameters representing shock structure, such as the shock Mach number, plasma beta, injection fraction into diffusive shock acceleration, and the scale of the upstream magnetic field. The behavior is associated with upstream turbulence for diffusive shock acceleration, as expected. Additionally, pre-accelerated particles could affect the time evolution of the particle distribution only when the radiative or advection losses are small enough for the pre-accelerated distribution to have a flatter power-law slope than the power-law slope based on shock acceleration theory. We also provide a formula for a spherically expanding shock and its relevant application to calculate high-energy emission due to hadronic interactions. We suggest that the simple analytic method could be applied to examine astrophysical shocks with a wide range of plasma parameters.