An exact solution of cosmic ray modulation problem in a stationary composite heliosphere model

Abstract

A new theoretical approach to describe the physical processes of energy particle propagation is proposed. This approach is based on the analytically iterative method for solving closed cosmic ray (CR) modulation problems, which was proposed by Shakhov and Kolesnyk. First, we have applied the approach on a simple model of the heliosphere, wherein the diffusion coefficients κ for each region of CR modulation are constants. This approach produced a very good matching of the obtained solution and also provided a numerical solution and an analytical solution. Finally, a modern problem of CR modulation in a stationary composite model of the heliosphere was considered. This model includes an environment that contains adjacent spherically symmetric regions with different modes of propagation of the solar wind (SW) speed for each layer. The CR scattering is due to different factors for each layer of the environment, as characterized by relevant κ values that simultaneously have dependence on the momentum of the particle p and the particle speed $\upsilon$, i.e. $\kappa \propto p\upsilon$. The local interstellar spectra (LISs) are given by a power-law unmodulated spectrum with the slope of the initial spectrum α, i.e. LIS ∝ p−α. An exact solution of the problem of CR modulation for low-energy particles and high-energy particles was first derived and qualitatively compared against the Voyager 1 data.