The problem of the electromagnetic acceleration of cosmic rays to high energies by turbulent magnetic fields within the confines of our galaxy is considered. The model of the magnetic field used is essentially that proposed by Fermi, in which the field is assumed to be fairly regular and to run along the spiral arm of the galaxy. The magnetic field plays a dual role, storing or trapping the cosmic rays, and accelerating them when the field is not static. The fluctuating part of the magnetic field is described statistically in terms of a spectral decomposition of the field into hydromagnetic waves of different wave lengths moving in the direction of the primary field.
Two main problems are of concern: (a) the energy distribution of the high energy particles, and (b) the angular distribution of these particles.A partial differential equation of the diffusion type is derived which describes statistically the behavior of an ensemble of particles undergoing accelerating and decelerating interactions (betatron interactions) with the varying magnetic field. In addition to accelerating the particles, the betatron interactions change the component of momentum parallel to the field in a way which depends on the energy change. In addition to these processes, the differential equation accounts for interactions with inhomogeneities in the field whose scale is small compared to the helix radius, as well as removal of particles by nuclear collisions and by diffusion of particles out of the region of the magnetic field.Solutions to the steady-state diffusion equation show that a power-law energy spectrum results for the high energy particles. The exponent in the power-law spectrum is related to the parameters describing the magnetic field, the mean-square velocity of the turbulent medium, and the mean time for loss of particles by nuclear collisions and diffusion out of the spiral arm. An approximate form of the space-dependent steady-state diffusion equation is solved to estimate the mean time for escape of the particles by diffusion, and to relate this parameter to the length of the spiral arm and to the parameters describing the magnetic field. The results also show that the angular distribution of the particles is inextricably tied up with the energy spectrum, with the degree of anisotropy being determined by the relative effectiveness of the scattering by small scale inhomogeneities which tend to make the distribution isotropic, and the betatron processes which tend to make the distribution highly anisotropic with most of the particles lying in very steep spirals.
It appears from the results that a set of parameters describing the magnetic field can be found which are astronomically plausible, and which give results for the power-law exponent and the anisotropy within the range of values observed experimentally.
