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

Abstract

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 , are adopted. In addition, the latitudinal dependence of magnetic turbulence magnitude is assumed as ∼ (1 + sin 2 θ)/2 from the observations of Ulysses, and the radial dependence is assumed as ∼ 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.

Modulation of Galactic Cosmic Rays in the Inner Heliosphere over Solar Cycles

“…However, the dissipation rate of turbulence energy given by the TRST model is also slower than that predicted by WKB theory. It has been shown by Qin & Shen (2017) that the TRST model agrees well with the magnetic turbulence magnitude observed by Ulysses, Voyager 1 and Voyager 2 measurements.…”

“…Figure 6 shows an example of the modulated proton energy spectra near solar maximum in 1990 (red line) and near solar minimum in 1997 (blue line). Black line represents the GCR source used in this work (Qin & Shen 2017). Blue and sky blue solid circles mean IMP 8 observations in 1997 from Webber & Higbie (2003) and BESS magnetic spectrometer observations in 1997 from Shikaze et al (2007), respectively.…”

“…However, the Fisk type HMF is difficult to handle to some extent, and its existence is also debated (e.g., Zurbuchen et al 1997;Roberts et al 2007;Sternal et al 2011), see the review in Potgieter (2017). Following Qin & Shen (2017), we use the modification proposed by Jokipii & Kota (1989) in this work, and the modified expression can be written as (Langner 2004)…”

“…It is shown that this modification can significantly enhance the magnetic field intensity and reduce the drift velocity at large radial distance in polar regions of the heliosphere without influence the equatorial regions obviously (Bobik et al 2012(Bobik et al , 2013Qin & Shen 2017).…”

“…where j 0 = 1.17 × 10 4 m −2 s −1 sr −1 (GeV/nuc) −1 to fit Voyager 2 observations (Qin & Shen 2017), p 0 = In this work, we use the time-backward Markov stochastic process method which is proposed by Zhang (1999) to solve the well known Parker transport equation. The Parker transport equation can be written in terms of a set of equivalent stochastic differential equations (SDEs) given by (Zhang 1999;Pei et al 2010b;Strauss et al 2011;Kopp et al 2012)…”