One of the earliest theories on the acceleration of cosmic rays was proposed by Enrico Fermi in 1949 . It became known as the "Second Order Fermi Mechanism". In this model, particles collide stochastically with magnetic clouds in the interstellar medium. Those particles involved in head-on collisions will gain energy (similar to a sling-shot process used to accelerate spacecrafts around planets), and those involved in tail-end collisions will lose energy. On average, however, head-on collisions are more probable. In this way, particles gain energy over many collisions.
This mechanism naturally predict a power law energy spectrum, but the power index depends on the local details of the model and would not give rise to a universal power law for cosmic rays arriving from all directions. This mechanism is also too slow and too inefficient to account for the observed UHE cosmic rays.
A more efficient version of Fermi Acceleration was proposed independently by a number of workers in the late 1970's [2-5]. In this model, particles are accelerated by a strong shock propagating through interstellar space. The following gives a schematic of the process as described in Prof. Longhair's book :
Consider the case of a strong shock propagating at a supersonic, but non-relativistic speed U through a stationary interstellar gas. Figure (a) at left shows the situation in the rest frame of the gas: the density, pressure, and temperature of the gas upstream and downstream of the shock front are r2, p2, T2 and r1, p1, T1, respectively.
When viewed in the rest frame of the shock front as in figure (b) below, particles are arriving from downstream with speed v1=U and exiting upstream at speed v2. Conservation of the number of particles implies the relation: r1v1=r2v2. In the case of strong shock we expect r2/r1=(g+1)/(g-1), where g is the usual ratio of heat capacities. For a fully ionized plasma, one expects g=5/3, leading to a velocity ratio of v1/v2=4.
First order Fermi acceleration naturally predicts a power law spectrum of DNA(E)/dE ~ E-2. While the power index of 2 does not agree with the measured index of ~3, this model predicts, for the first time, a power law spectrum with a unique spectral index that is independent of the details of the local environment. The mechanism requires only the presence of strong shocks, which are quite plausibly present in the suspected sources of cosmic rays.