First-order Fermi acceleration in the two-stream limit

A study of the first-order Fermi mechanism for accelerating cosmic-rays at relativistic and non-relativistic shocks is carried out by using the two-stream approximation. Exact steady-state analytic solutions illustrating the shock acceleration process in the test-particle limit in which monoenergetic (relativistic) seed particles enter the shock through an upstream free-escape boundary are obtained. The momentum spectrum of the shock accelerated particles consists of a series of Dirac delta distributions corresponding to particles that have undergone an integral number of acceleration cycles. Since particles in the model have a finite fixed escape probability from the shock and the particle momenta p are equally spaced in log p, the envelope of the delta functions series is a power law in momentum. The solutions are used to discuss time-dependent aspects of the shock acceleration process in terms of the finite cycle time, escape probability, and momentum change per cycle that can be deduced from the steady-state model. The length-scale over which the accelerated particles extend upstream of the shock is shown to depend upon the particle energy, with the higher energy particles extending further upstream. This effect is shown to be intimately related to the kinematic threshold requirement that the particle speed exceed the fluid speed in order for particles to swim upstream of the shock and participate in the shock acceleration process.