Modulation of Galactic Cosmic Rays in the Inner Heliosphere, Comparing with PAMELA Measurements

We develop a numerical model to study the time-dependent modulation of galactic cosmic rays (GCRs) in the inner heliosphere. In the model a time-delayed modified Parker heliospheric magnetic field (HMF) and a new diffusion coefficient model, NLGCE-F, from Qin \& Zhang (2014), are adopted. In addition, the latitudinal dependence of magnetic turbulence magnitude is assumed as $\sim (1+\sin^2\theta)/2$ from the observations of Ulysses, and the radial dependence is assumed as $\sim r^S$, where we choose an expression of $S$ as a function of the heliospheric current sheet (HCS) tilt angle. We show that the analytical expression used to describe the spatial variation of HMF turbulence magnitude agrees well with the Ulysses, Voyager 1, and Voyager 2 observations. By numerically calculating the modulation code we get the proton energy spectra as a function of time during the recent solar minimum, it is shown that the modulation results are consistent with the PAMELA measurements.