##
A Simple Model of Nonlinear Diffusive Shock Acceleration

Berezhko, E. G.; Ellison, D. C.
*Published in:*
ApJ, 526, 385

### Abstract

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 u_{0} with respect to a critical value
u^{*}_{0}~ηp^{1/4}_{max}, which is a function of the injection rate η and maximum
accelerated particle momentum p_{max}. For u_{0}__0,
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^{3/4}_{S0}
if M^{2}_{S0} >M_{A0}, or by
r_{tot}~1.5M^{3/8}_{A0}
in the opposite case (M_{S0} is the sonic Mach number and
M_{A0} is the Alfvén Mach number). If u_{0}>u*_{0}, the
shock, although still strong,
becomes almost unmodified and accelerated particle production
decreases inversely proportional to u_{0}.