Critical self-organization of astrophysical shocks

There are two distinct regimes of the first-order Fermi acceleration of shocks. The first is a linear (test-particle) regime in which most of the shock energy goes into thermal and bulk motions of the plasma. The second is an efficient regime in which the shock energy goes into accelerated particles. Although the transition region between them is narrow, we identify the factors that drive the system toward a self-organized critical state between those two regimes. Using an analytic solution, we determine this critical state and calculate the spectra and maximum energy of accelerated particles.