Simple analytic solutions appropriate for galactic cosmic-ray modulation

Abstract

Exact analytic solutions, for the galactic cosmic-ray density, to the standard time-independent spherically symmetric convection-diffusion-adiabatic deceleration equation governing the transport of cosmic rays in the interplanetary medium are presented. It is assumed that the solar-wind speed is constant and that the spatial diffusion coefficient is given by a power law, independent of distance from the sun, in accordance with Pioneer 10 and 11 measurements and recent calculations. One solution depicts with decreasing distance from the sun the energy loss and intensity decrease of an interstellar spectrum which is peaked in momentum. From this solution others are constructed which (1) are extremely simple analytically; (2) are sufficiently general both to yield excellent agreement with observations near earth and to demonstrate quantitatively the sensitivity of derived interstellar spectra on the forms assumed in the inner solar system, particularly at low energies; and (3) promise to replace the standard convection-diffusion solution often utilized in data analysis and modulation estimates, while providing a useful tool for future research involving cosmic-ray modulation.