Momentum Spectrum of Cosmic Radiation

If in a variable magnetic field charged high energy particles are injected at a certain point, they are accelerated but at the same time they diffuse outwards from the point of injection. Near the point of injection the density in a volume element of six-dimensional phase-space is increased by the acceleration process but decreased by the diffusion, and the momentum spectrum is obtained from the stationary state when these two effects are equal. The theory gives the power spectrum of (51) or (54), in which the value of the exponent depends in a very insensitive way on the rate of acceleration and the rate of diffusion.

The spectrum depends on the distance from the source of injection and the average energy density is orders of magnitude smaller than the density near the source of injection.

The theory is in agreement with the “local” theory of Cosmic Radiation, but it is difficult to reconcile it with galactic or extragalactic theories.