The propagation of solar cosmic-ray bursts

A model is presented in which we show analytically the three phases of anisotropy which occur during solar cosmic-ray events observed in the 7.5 MeV to 21 MeV kinetic-energy interval and reported by McCracken et al. (1971): (i) a highly anisotropic, near field-aligned, initial phase, (ii) a convective phase, and (iii) a late-time phase in which the anisotropy is approximately perpendicular to the mean interplanetary magnetic field. The model is based on the cosmic-ray particles being convectively transported out from the Sun, undergoing anisotropic diffusion along the interplanetary magnetic-field lines, and losing energy by adiabatic deceleration or by collision processes. The event is seen simply as a pulse moving outward from the Sun after a cosmic-ray burst with a negative density-gradient in front of it and a positive gradient behind. The convective phase (ii) occurs as the spatial peak moves past the observer and has a propagation speed V_dassociated with it; the anisotropy vector late in the decay phase (iii) is the result of a residual balance between the radial outward convection and the inward radial component of the anisotropic diffusion. The mathematical solutions are based upon a diffusion coefficient proportional to heliocentric radius and independent of energy and are thus rather special. However they yield formulae for the propagation speed of the convective phase and the direction in space of the long-time anisotropy which are useful as a guide to the dependence of these quantities on the solar wind speed V, the diffusion coefficient and the spectral index . In this model V_dincreases with V, , and ; and , the angle between the anisotropy vector at infinite time and the outward radial direction increases with /V and decreases as is increased. These predictions of the dependence of and V_dupon V, , and are open to observational verification.