A Simple Model of Nonlinear Diffusive Shock Acceleration

We present a simple model of nonlinear diffusive shock acceleration (also called first-order Fermi shock acceleration) that determines the shock modification, spectrum, and efficiency of the process in the plane-wave, steady state approximation as a function of an arbitrary injection parameter, η. The model, which uses a three-power-law form for the accelerated particle spectrum and contains only simple algebraic equations, includes the essential elements of the full nonlinear model and has been tested against Monte Carlo and numerical kinetic shock models. We include both adiabatic and Alfvén wave heating of the upstream precursor. The simplicity and ease of calculation make this model useful for studying the basic properties of nonlinear shock acceleration, as well as providing results accurate enough for many astrophysical applications. It is shown that the shock properties depend upon the shock speed u₀ with respect to a critical value u ∝ ηp, which is a function of the injection rate η and maximum accelerated particle momentum p_max. For u₀ < u, acceleration is efficient and the shock is strongly modified by the back pressure of the energetic particles. In this case, the overall compression ratio is given by r_tot ≈ 1.3M if M > M_A0, or by r_tot ≈ 1.5M in the opposite case (M_S0 is the sonic Mach number and M_A0 is the Alfvén Mach number). If u₀ > u, the shock, although still strong, becomes almost unmodified and accelerated particle production decreases inversely proportional to u₀.