First Order Fermi Acceleration
Shock waves typically have moving magnetic inhomogeneities both preceding and following them. Consider the case of a charged particle traveling through the shock wave (from upstream to downstream). If it encounters a moving change in the magnetic field, this can reflect it back through the shock (downstream to upstream) at increased velocity. If a similar process occurs upstream, the particle will again gain energy. These multiple reflections greatly increase its energy. The resulting energy spectrum of many particles undergoing this process (assuming that they do not influence the structure of the shock) turns out to be a power law:
where the spectral index depends, for non-relativistic shocks, only on the compression ratio of the shock.
The term "First order" comes from the fact that the energy gain per shock crossing is proportional to, the velocity of the shock divided by the speed of light.
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