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

Abstract

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.

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“…This is understandable, since more turbulent of the interplanetary environment means a slower diffusion. The drift suppression factor K d is larger for A < 0, and smaller for A > 0, consistent with previous work [64,65]. Our fittings further show that probably no break of the diffusion coefficient is required when A < 0, given the very close values of α 1 and α 2 .…”

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“…Qualitatively, similar behaviour is shown as in Figure 2 but now it clearly illustrates how the time dependence differ below and above ~10 GV; below this value the MFPs decrease progressively and significantly from 2006-08 to 2013-15 but the opposite, and at a much smaller scale, occurs above this value. This seems counterintuitive, but similar behaviour (trends) using precise spectra from AMS-02, was reported also in [80,81,87] and implies that GCRs with R > 10 GV have somewhat smaller MFPs at solar minimum than at solar maximum; as such quite an interesting aspect of solar modulation at higher rigidity. In this context, it is well known that time trends in modulation is occurring differently at low and high rigidity during A > 0 and A < 0 cycles; see the discussion in [88,89], and references there-in, and recent modeling studies (e.g., [81]).…”

“…This seems counterintuitive, but similar behaviour (trends) using precise spectra from AMS-02, was reported also in [80,81,87] and implies that GCRs with R > 10 GV have somewhat smaller MFPs at solar minimum than at solar maximum; as such quite an interesting aspect of solar modulation at higher rigidity. In this context, it is well known that time trends in modulation is occurring differently at low and high rigidity during A > 0 and A < 0 cycles; see the discussion in [88,89], and references there-in, and recent modeling studies (e.g., [81]). An example of the time dependence of λ at the Earth is shown in Figure 4 for protons and anti-protons at 1 GV from July 2006 to May 2017.…”

“…Although observations (detection) of electrons and positrons at lower R in geospace are intricate [85,86], such data will be quite useful for future analysis. For λ D this time dependence is about the same at all rigidities, although somewhat less for R < ~1 GV than above; see also [81].…”

“…Mathematical expressions for the R-dependence of the three MFPs, and the drift scale, as shown in Figure 2, are not repeated here (they are quite lengthy and would take up a substantial amount of space). They were given extensively in [48,60,72], and many were also utilized in [47,[79][80][81].…”