Solar Modulation of Galactic Cosmic Rays

Beginning with the equations of the convection-diffusion model including energy losses, it is shown that the streaming of galactic cosmic rays in the solar-wind cavity can be neglected above a few hundred MeV per nucleon. For the case in which the diffusion coefficient is separable into radially and rigidity dependent parts, an integral is derived which relates the intensity at points within the cavity to that at infinity through an energy-loss parameter. This parameter is a function of energy; it is defined in terms of observable quantities and is identified, tentatively, as the mean energy lost by the cosmic-ray particles in moving into the interplanetary region. The integral is formally equivalent to that obtained by using Liouvilie's theorem. Experimental data for the modulation of protons and helium ions in 1963-1965 and electrons and protons in 1965-1966 are shown to behave as predicted, down to kinetic energies of a few hundred MeV per nucleon. The diffusion coefficient was proportional to the magnitude of the charge of the cosmic-ray species during these periods, and the analytical results are formally equivalent to those obtained for a heliocentric force field proportional to the magnitude of the charge of the cosmic-ray species. Making use of the observed lower limits of validity, it is shown that the e-folding distance for the diffusion coefficient lies between 0.8 and 1.6 a.u., thus setting limits on the radial dependence of the scattering process. Estimates are given of the changes in energy loss in the periods 1963-1965 and 1965- 1966 and of the energy loss at solar minimum