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With continuous measurements from space-borne cosmic-ray detectors such as AMS-02 and PAMELA, precise spectra of galactic cosmic rays over the 11 yr solar cycle have become available. For this study, we utilize proton and helium spectra below 10 GV from these missions from 2006 to 2017 to construct a cosmic-ray transport model for a quantitative study of the processes of solar modulation. This numerical model is based on Parker’s transport equation, which includes four major transport processes. The Markov Chain Monte Carlo method is utilized to search the relevant parameter space related to the drift and the diffusion coefficients by reproducing and fitting the mentioned observed spectra. The resulting best-fit normalized  χ 2  is mainly less than 1. It is found that (1) when reproducing these observations the parameters required for the drift and diffusion coefficients exhibit a clear time dependence, with the magnitude of the diffusion coefficients anticorrelated with solar activity; (2) the rigidity dependence of the resulting mean free paths varies with time, and their rigidity dependence at lower rigidity can even have a larger slope than at higher rigidity; (3) using a single set of modulation parameters for each pair of observed proton and helium spectra, most spectra are reproduced within observational uncertainty; and (4) the simulated proton-to-helium flux ratio agrees with the observed values in terms of its long-term time dependence, although some discrepancy exists, and the difference is mostly coming from the underestimation of proton flux.

A Numerical Study of the Solar Modulation of Galactic Protons and Helium from 2006 to 2017