The 3D numerical modeling of the solar modulation of galactic protons and helium nuclei related to observations by PAMELA between 2006 and 2009

The global features of the modulation of galactic cosmic ray protons and helium nuclei in a very quiet heliosphere are studied with a comprehensive, three-dimensional, drift model and compared to proton and helium observations measured by PAMELA from 2006 to 2009. Combined with accurate very local interstellar spectra (VLIS) for protons and helium nuclei, this provides the opportunity to study in detail how differently cosmic ray species with dissimilar mass-to-charge ratio ( $A/Z$ ) are modulated down to a few GV. The effects at Earth of the difference in their VLIS’s and those caused by the main modulation mechanisms are illustrated. We find that both the PAMELA proton and helium spectra are reproduced well with the numerical model, assuming the same set of modulation parameters and diffusion coefficients. A comparative study of 3He2 (He-3) and 4He2 (He-4) modulated spectra reveals that they do not undergo identical spectral changes below 3 GV mainly due to differences in their VLIS’s. This result is important to uncover and investigate the effects on the proton to total helium ratio ( $p$ /He) caused by the difference in their VLIS’s and those by $A/Z$ . The computed $p$ /He displays three modulation regimes, reflecting the complex interplay of modulation processes in the heliosphere. At rigidities above ∼3 GV, the $p$ /He ratio at the Earth is found to deviate modestly from a value of ∼5.5, largely independent of the assumed modulation conditions. This result indicates that the PAMELA measurement of $p$ /He reveals at these rigidities the shapes of their VLIS’s. Below ∼0.6 GV, $p$ /He increases with decreasing rigidity from 2006 to 2009 and significant variations are predicted depending on the assumed solar modulation conditions. This result indicates that as modulation levels decreased from 2006 to 2009, the contribution of adiabatic energy changes dissipated faster for protons than for helium nuclei at the same rigidity mainly due to different slopes of their VLIS’s. The differences between modulation effects for protons and helium are found to be the consequence of how the combined interplaying modulation mechanisms in the heliosphere affect the modulated spectra based on their $A/Z$ and particularly on their VLIS’s.