Modeling the Modulation of Galactic and Jovian Electrons by Stochastic Processes

Abstract

We present a newly developed numerical modulation model to study the transport of galactic and Jovian electrons in the heliosphere. The model employs stochastic differential equations (SDEs) to solve the corresponding transport equation in five dimensions (time, energy, and three spatial dimensions) which is difficult to accomplish with the numerical schemes used in finite difference models. Modeled energy spectra for galactic electrons are compared for the two drift cycles to observations at Earth. Energy spectra and radial intensity profiles of galactic and Jovian electrons are compared successfully to results from previous studies. In line with general drift considerations, it is found that most 100 MeV electrons observed at Earth enter the heliosphere near the equatorial regions in the A > 0 cycle, while they enter mainly over the polar regions in the A < 0 cycle. Our results indicate that 100 MeV electrons observed at Earth originate at the heliopause with ∼600 MeV undergoing adiabatic cooling during their transport to Earth. The mean propagation time of these particles varies between ∼180 and 300 days, depending on the drift cycle. For 10 MeV Jovian electrons observed at Earth, a mean propagation time of ∼40 days is obtained. During this time, the azimuthal position of the Jovian magnetosphere varies by ∼1 •. At a 50 AU observational point, the mean propagation time of these electrons increases to ∼370 days with an azimuthal position change of Jupiter of ∼20 •. The SDE approach is very effective in calculating these propagation times.

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

“…For a pseudo-particle in position (r, θ, φ) and momentum p, the stochastic differential equations equivalent to Equation (1) have the form (Zhang 1999;Pei et al 2010b;Strauss et al 2011a;Kopp et al 2012)…”

Anisotropic diffusion of Galactic cosmic ray protons and their steady-state azimuthal distribution

“…Mentioning only a few examples, a starting point for heliospheric studies of this kind can be found in the paper by Zhang (1999) where he applied the method to CR modulation. More recently, Pei et al (2010) and Strauss et al (2011) have applied SDEs in a more comprehensive heliospheric model. Farahat et al (2008) employed SDEs for a CR propagation study in the Galaxy and, e.g., Marcowith & Kirk (1999), as well as Achterberg & Schure (2011) calculated the shock acceleration of energetic particles.…”

“…We only apply the backward method in this study, since it is well-suited to the given problem. For more details on the numerical scheme and especially on determining the root of the diffusion tensor, we refer the reader to Kopp et al (2012) and Strauss et al (2011) where the basis of the code used in this study is discussed in greater detail. Exemples of pseudo-particle trajectories are shown in Fig.…”

Solution of Heliospheric Propagation: Unveiling the Local Interstellar Spectra of Cosmic-ray Species

“…Since during the high activity period the heliospheric magnetic field is far from being considered regular, in this work we introduced a correction factor that suppresses any drift velocity at solar maximum. For the sake of completeness we should note that the presence of turbulence in the interplanetary medium should reduce the global effect of CR drift in the heliosphere (see, e.g., discussion in Minnie et al 2007) and this is usually incorporated introducing a drift suppression factor (see, e.g., Strauss et al 2011) that is effective at rigidity below 1 GV. We compute the CR propagation from the Termination Shock (TS) down to the Earth orbit using a Monte Carlo approach, i.e., the HelMod Monte Carlo code (Bobik et al 2012) that solves the two-dimensional Parker equation for CR transport through the heliosphere.…”

“…Jokipii et al 1977;Jokipii & Kopriva 1979;Potgieter & Moraal 1985;Boella et al 2001;Strauss et al 2011;Della Torre et al 2012;Bobik et al 2013a,b). A low solar activity period between cycles 23 and 24 was recently studied by the PAMELA instrument (see e.g.…”