A three-dimensional (3D) steady-state electron modulation model based on Parker (1965) transport equation is applied to study the modelling of – 7 MeV galactic and Jovian electrons in the inner heliosphere. The latter is produced within Jupiter's magnetosphere which is situated at - 5 AU in the ecliptic plane. The heliospheric propagation of these particles is mainly described by the heliospheric diffusion tensor. Some elements of the tensor, such as the diffusion coefficient in the azimuthal direction, which were neglected in the previous two-dimensional modulation studies are investigated to account for the three-dimensional transport of Jovian electrons. Different anisotropic solar wind speed profiles that could represent solar minimum conditions were modelled and their effects were illustrated by computing the distribution of 7 MeV Jovian electrons in the equatorial regions. In particular, the electron intensity time-profile along the Ulysses spacecraft trajectory was calculated for these speed profiles and compared to the 3-10 MeV electron flux observed by the Kiel Electron Telescope (KET) on board the Ulysses spacecraft from launch (1990) up to end of its first out-of-ecliptic orbit (2000). It was found that the model solution computed with the solar wind profile previously assumed for typical solar minimum conditions produced good compatibility with observations up to 1998. After 1998 all model solutions deviated completely from the observations. In this study, as a further attempt to model KET observations more realistically, a new relation is established between the latitudinal dependence of the solar wind speed and the perpendicular polar diffusion. Based on this relation, a transition of an average solar wind speed from solar minimum conditions to intermediate solar activity and to solar maximum conditions was modelled based on the assumption of the time-evolution of large polar coronal holes and were correlated to different scenarios of the enhancement of perpendicular polar diffusion. Effects of these scenarios were illustrated, as a series of steady-state solutions, on the computed 7 MeV Jovian and galactic electrons in comparison with the 3-10 MeV electron observed by the KET instrument from the period 1998 up to the end of 2003. Subsequent effects of these scenarios were also shown on electron modulation in general. It was found that this approach improved modelling of the post-1998 discrepancy between the model and KET observations but it also suggested the need for a time-dependent 3D electron modulation model to describe modulation during moderate to extreme solar maximum conditions. / Thesis (M.Sc.)--North-West University, Potchefstroom Campus, 2004.

A three-dimensional (3D) steady-state electron modulation model based on Parker (1965) transport equation is applied to study the modelling of – 7 MeV galactic and Jovian electrons in the inner heliosphere. The latter is produced within Jupiter's magnetosphere which is situated at - 5 AU in the ecliptic plane. The heliospheric propagation of these particles is mainly described by the heliospheric diffusion tensor. Some elements of the tensor, such as the diffusion coefficient in the azimuthal direction, which were neglected in the previous two-dimensional modulation studies are investigated to account for the three-dimensional transport of Jovian electrons. Different anisotropic solar wind speed profiles that could represent solar minimum conditions were modelled and their effects were illustrated by computing the distribution of 7 MeV Jovian electrons in the equatorial regions. In particular, the electron intensity time-profile along the Ulysses spacecraft trajectory was calculated for these speed profiles and compared to the 3-10 MeV electron flux observed by the Kiel Electron Telescope (KET) on board the Ulysses spacecraft from launch (1990) up to end of its first out-of-ecliptic orbit (2000). It was found that the model solution computed with the solar wind profile previously assumed for typical solar minimum conditions produced good compatibility with observations up to 1998. After 1998 all model solutions deviated completely from the observations. In this study, as a further attempt to model KET observations more realistically, a new relation is established between the latitudinal dependence of the solar wind speed and the perpendicular polar diffusion. Based on this relation, a transition of an average solar wind speed from solar minimum conditions to intermediate solar activity and to solar maximum conditions was modelled based on the assumption of the time-evolution of large polar coronal holes and were correlated to different scenarios of the enhancement of perpendicular polar diffusion. Effects of these scenarios were illustrated, as a series of steady-state solutions, on the computed 7 MeV Jovian and galactic electrons in comparison with the 3-10 MeV electron observed by the KET instrument from the period 1998 up to the end of 2003. Subsequent effects of these scenarios were also shown on electron modulation in general. It was found that this approach improved modelling of the post-1998 discrepancy between the model and KET observations but it also suggested the need for a time-dependent 3D electron modulation model to describe modulation during moderate to extreme solar maximum conditions. / Thesis (M.Sc.)--North-West University, Potchefstroom Campus, 2004.

Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2007.

The three-dimensional (3D) steady-state electron modulation model of Ferreira (2002), based on Parker (1965) transport equation, is used to study the modulation of the 7 MeV galactic and Jovian electron anisotropies in the inner heliosphere. The Jovian electrons are produced in Jupiter's magnetosphere which is situated at ~ 5 AU in the ecliptic plane. The propagation of these particles is mainly described by the diffusion tensor applicable for the inner heliosphere. Some of the elements of the diffusion tensor are revisited in order to establish what contribution they make to the three-dimensional anisotropy vector and its components in the inner heliosphere. The 'drift' term is neglected since the focus of this study is on low-energy electrons. The effects on the electron anisotropy of different scenarios when changing the solar wind speed from minimum to maximum activity is illustrated. The effects on both the galactic and Jovian electron anisotropy of changing the polar perpendicular coefficient, in particular, are illustrated. It is shown that the computed Jovian electron anisotropy dominates the galactic anisotropy close to the Jovian electron source at ~5 AU, as expected, testifying to the validity of the3D-model. For the latitudinal anisotropy, the polar perpendicular diffusion plays a dominant role for Jovian electrons close to the source, with the polar gradient becoming the dominant factor away from the electron source. Of all three anisotropy components, the azimuthal anisotropy is dominant in the equatorial plane close to the source. It is found that there is a large azimuthal gradient close to the source because the low-energy electrons tend to follow the heliospheric magnetic field more closely than higher energy particles. The transition of the solar wind speed from minimum to intermediate to maximum solar activity condition was used to illustrate the modulation of the magnitude of the 7 MeV total anisotropy vector along the Ulysses trajectory. It was found that during the two encounters with the planet a maximum anisotropy of 38% was computed but with different anisotropy-timepeaks as the approach to Jupiter was different. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2007.

The three-dimensional (3D) steady-state electron modulation model of Ferreira (2002), based on Parker (1965) transport equation, is used to study the modulation of the 7 MeV galactic and Jovian electron anisotropies in the inner heliosphere. The Jovian electrons are produced in Jupiter's magnetosphere which is situated at ~ 5 AU in the ecliptic plane. The propagation of these particles is mainly described by the diffusion tensor applicable for the inner heliosphere. Some of the elements of the diffusion tensor are revisited in order to establish what contribution they make to the three-dimensional anisotropy vector and its components in the inner heliosphere. The 'drift' term is neglected since the focus of this study is on low-energy electrons. The effects on the electron anisotropy of different scenarios when changing the solar wind speed from minimum to maximum activity is illustrated. The effects on both the galactic and Jovian electron anisotropy of changing the polar perpendicular coefficient, in particular, are illustrated. It is shown that the computed Jovian electron anisotropy dominates the galactic anisotropy close to the Jovian electron source at ~5 AU, as expected, testifying to the validity of the3D-model. For the latitudinal anisotropy, the polar perpendicular diffusion plays a dominant role for Jovian electrons close to the source, with the polar gradient becoming the dominant factor away from the electron source. Of all three anisotropy components, the azimuthal anisotropy is dominant in the equatorial plane close to the source. It is found that there is a large azimuthal gradient close to the source because the low-energy electrons tend to follow the heliospheric magnetic field more closely than higher energy particles. The transition of the solar wind speed from minimum to intermediate to maximum solar activity condition was used to illustrate the modulation of the magnitude of the 7 MeV total anisotropy vector along the Ulysses trajectory. It was found that during the two encounters with the planet a maximum anisotropy of 38% was computed but with different anisotropy-timepeaks as the approach to Jupiter was different. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2007.

The transport of cosmic rays in the heliosphere is studied by making use of a newly developed modulation model. This model employes stochastic differential equations to numerically solve the relevant transport equation, making use of this approach’s numerical advantages as well as the opportunity to extract additional information regarding cosmic ray transport and the processes responsible for it. The propagation times and energy losses of galactic electrons and protons are calculated for different drift cycles. It is confirmed that protons and electrons lose the same amount of rigidity when they experience the same transport processes. These particles spend more time in the heliosphere, and also lose more energy, in the drift cycle where they drift towards Earth mainly along the heliospheric current sheet. The propagation times of galactic protons from the heliopause to Earth are calculated for increasing heliospheric tilt angles and it is found that current sheet drift becomes less effective with increasing solar activity. Comparing calculated propagation times of Jovian electrons with observations, the transport parameters are constrained to find that 50% of 6 MeV electrons measured at Earth are of Jovian origin. Charge-sign dependent modulation is modelled by simulating the proton to anti-proton ratio at Earth and comparing the results to recent PAMELA observations. A hybrid cosmic ray modulation model is constructed by coupling the numerical modulation model to the heliospheric environment as simulated by a magneto-hydrodynamic model. Using this model, it is shown that cosmic ray modulation persists beyond the heliopause. The level of modulation in this region is found to exhibit solar cycle related changes and, more importantly, is independent of the magnitude of the individual diffusion coefficients, but is rather determined by the ratio of parallel to perpendicular diffusion. / PhD (Space Physics), North-West University, Potchefstroom Campus, 2013

The transport of cosmic rays in the heliosphere is studied by making use of a newly developed modulation model. This model employes stochastic differential equations to numerically solve the relevant transport equation, making use of this approach’s numerical advantages as well as the opportunity to extract additional information regarding cosmic ray transport and the processes responsible for it. The propagation times and energy losses of galactic electrons and protons are calculated for different drift cycles. It is confirmed that protons and electrons lose the same amount of rigidity when they experience the same transport processes. These particles spend more time in the heliosphere, and also lose more energy, in the drift cycle where they drift towards Earth mainly along the heliospheric current sheet. The propagation times of galactic protons from the heliopause to Earth are calculated for increasing heliospheric tilt angles and it is found that current sheet drift becomes less effective with increasing solar activity. Comparing calculated propagation times of Jovian electrons with observations, the transport parameters are constrained to find that 50% of 6 MeV electrons measured at Earth are of Jovian origin. Charge-sign dependent modulation is modelled by simulating the proton to anti-proton ratio at Earth and comparing the results to recent PAMELA observations. A hybrid cosmic ray modulation model is constructed by coupling the numerical modulation model to the heliospheric environment as simulated by a magneto-hydrodynamic model. Using this model, it is shown that cosmic ray modulation persists beyond the heliopause. The level of modulation in this region is found to exhibit solar cycle related changes and, more importantly, is independent of the magnitude of the individual diffusion coefficients, but is rather determined by the ratio of parallel to perpendicular diffusion. / PhD (Space Physics), North-West University, Potchefstroom Campus, 2013