Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

Abstract

Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution functionFo(r,p), (r=heliocentric distance,p=momentum) can be determined in a regionra≤r≤rbwhen a source is specified throughout the region and the momentum spectrum is specified on the boundaries atraandrb. Evaluation requires a knowledge of the Green's function which corresponds to the solution for monoenergetic particles released at heliocentric radiusro, Examples of Green's functions are given for the casera=0,rb=∞ and derived for the cases of finiteraandrb. The diffusion coefficient κ is assumed of the form κ = κo(p)rb. The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions.