Fermi acceleration

Fermi acceleration, sometimes referred to as diffusive shock acceleration (a subclass of Fermi acceleration ), is the acceleration that charged particles undergo when being repeatedly reflected, usually by a magnetic mirror (see also Centrifugal mechanism of acceleration). This is thought to be the primary mechanism by which particles gain non thermal energies in astrophysical shock waves. It plays a very important role in many astrophysical models, mainly of shocks including solar flares and supernova remnants.

There are two types of Fermi acceleration: first-order Fermi acceleration (in shocks) and second-order Fermi acceleration (in the environment of moving magnetized gas clouds). In both cases the environment has to be collisionless in order for the mechanism to be effective. This is because Fermi acceleration only applies to particles with energies exceeding the thermal energies, and frequent collisions with surrounding particles will cause severe energy loss and as a result no acceleration will occur.

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:
${\displaystyle {\frac {dN(\epsilon )}{d\epsilon }}\propto \epsilon ^{-p}}$
where the spectral index ${\displaystyle p\gtrsim 2}$ 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 ${\displaystyle \beta _{s}}$, the velocity of the shock divided by the speed of light.

...
Wikipedia