Cosmic Rays III . The cosmic ray spectrum between 1 GeV and 10 4 GeV and the radio emission from supernova remnants

Based on a conjecture about the diffusion tensor of relativistic particles perpendicular to the magnetic field at a shock, and considering particle drifts, we develop a theory to account for the cosmic ray spectrum between 1 GeV and 10 GeV. The essential assumption is that the mean free path perpendicular to the magnetic field is independent of energy and has the scale of the thickness of the shocked layer. We then use the basic concept that the cosmic ray particles are accelerated in a supernova shock that travels through the interstellar medium. Radio observations, with additional supporting evidence from optical and X-ray data, suggest that small additional and transient bowshocks in locally free expansion contribute to the injection, while it is the overall shock which accounts for injection as well as the main acceleration of energetic particles, and we obtain a reasonable picture of the physical process with such a concept. Physically important ingredients besides the presence of a strong shock are diffusion, drifts, convection, adiabatic cooling, the injection history, and the topology of the magnetic field, here assumed for simplicity to be homogeneous in the interstellar medium. The result is a spectrum, which for strong shocks in a gas with adiabatic index 5/3 yields a spectrum of E. Interstellar turbulence with a Kolmogorov spectrum then leads by leakage from the galactic disk to a spectrum which is E, as observed in cosmic rays, and as deduced from radio observations of the nonthermal emission from our Galaxy as well as that of all other well-observed galaxies. We argue that the ratio of cosmic ray electrons to protons is determined by the amount of expansion which takes place from the cessation of electron injection to the break-up of the shell by cooling instabilities. Since the highest particle energy reached derives from geometrical arguments, it depends on the charge of the nucleus and so higher Z elements are predicted to reach higher energies.