A stochastic approach to the modelling of cosmic rays in the heliosphere

Abstract

A spatially three-dimensional numerical modulation model based on solving a set of stochastic differential equations, corresponding to the Parker transport equation, is employed to study the galactic proton spectra measured by the PAMELA instrument during January 2010 (solar minimum) to January 2014 (solar maximum). The model is able to reproduce these spectra well and various sets of values are determined for the modulation parameters required to do so, for different assumptions of the solar magnetic polarity and drift coefficient. A consideration central to the theme of this thesis is the scaling down of drifts with turbulence, i.e. towards solar maximum conditions, and a simple scaling function is introduced to incorporate this solar activity dependent suppression of drift effects. It is indicated that the difference in modulation between successive six-monthly averaged spectra cannot be accounted for by merely changing the tilt angle and magnetic field magnitude as function of time, but that an additional change in the magnitude of the diffusion coefficient is necessary; an increase in the rigidity dependence of the diffusion coefficient below 4.2 GeV may also be required towards solar maximum conditions. The drift properties of this modulation model are investigated by making use of its illustrative capabilities so that contour plots are produced showing particle entry positions and energies. It is found that energy losses increase when drifts are turned on, a fact which could be explained by considering contour plots indicating the regions in modulation space where these particles spend most time during the modulation process. Finally, it is indicated that the model produces charge-sign dependent modulation that is consistent with those of other authors and that it can be used in future studies to reproduce data.