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 solar minimum of 2009 has been identified as an exceptional event with regard to cosmic ray (CR)modulation, since conditions in the heliosphere have reached unprecedented quiet levels. This unique minimum has been observed by the Earth–orbiting satellite, PAMELA, launched in June, 2006, from which vast sets of accurate proton and electron preliminary observations have been made available. These simultaneous measurements from PAMELA provide the ideal opportunity to conduct an in–depth study of CR modulation, in particular charge–sign dependent modulation. In utilizing this opportunity, a three–dimensional, steady–state modulation model was used to reproduce a selection of consecutive PAMELA proton and electron spectra from 2006 to 2009. Thiswas done by assuming full drifts and simplified diffusion coefficients, where the rigidity dependence and absolute value of themean free paths for protons and electrons were sequentially adjusted below 3 GV and 300 MV, respectively. Care has been taken in calculating yearly–averaged current–sheet tilt angle and magnetic field values that correspond to the PAMELA spectra. Following this study where the numerical model was used to investigate the individual effects resulting from changes in the tilt angle, diffusion coefficients, and global drifts, it was found that all these modulation processes played significant roles in contributing to the total increase in CR intensities from 2006 to 2009, as was observed by PAMELA. Furthermore, the effect that drifts has on oppositely charged particles was also evident from the difference between the peak–shaped time profiles of protons and the flatter time profiles of electrons, as is expected for an A < 0 polarity cycle. Since protons, which drift into the heliosphere along the heliospheric current–sheet, haven’t yet reached maximum intensity levels by 2008, their intensities increased notably more than electrons toward the end of 2009. The time and energy dependence of the electron to proton ratios were also studied in order to further illustrate and quantify the effect of drifts during this remarkable solar minimum period. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2012.

The solar minimum of 2009 has been identified as an exceptional event with regard to cosmic ray (CR)modulation, since conditions in the heliosphere have reached unprecedented quiet levels. This unique minimum has been observed by the Earth–orbiting satellite, PAMELA, launched in June, 2006, from which vast sets of accurate proton and electron preliminary observations have been made available. These simultaneous measurements from PAMELA provide the ideal opportunity to conduct an in–depth study of CR modulation, in particular charge–sign dependent modulation. In utilizing this opportunity, a three–dimensional, steady–state modulation model was used to reproduce a selection of consecutive PAMELA proton and electron spectra from 2006 to 2009. Thiswas done by assuming full drifts and simplified diffusion coefficients, where the rigidity dependence and absolute value of themean free paths for protons and electrons were sequentially adjusted below 3 GV and 300 MV, respectively. Care has been taken in calculating yearly–averaged current–sheet tilt angle and magnetic field values that correspond to the PAMELA spectra. Following this study where the numerical model was used to investigate the individual effects resulting from changes in the tilt angle, diffusion coefficients, and global drifts, it was found that all these modulation processes played significant roles in contributing to the total increase in CR intensities from 2006 to 2009, as was observed by PAMELA. Furthermore, the effect that drifts has on oppositely charged particles was also evident from the difference between the peak–shaped time profiles of protons and the flatter time profiles of electrons, as is expected for an A < 0 polarity cycle. Since protons, which drift into the heliosphere along the heliospheric current–sheet, haven’t yet reached maximum intensity levels by 2008, their intensities increased notably more than electrons toward the end of 2009. The time and energy dependence of the electron to proton ratios were also studied in order to further illustrate and quantify the effect of drifts during this remarkable solar minimum period. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2012.

The Voyager 1 spacecraft is now about 25 AU beyond the heliospheric termination shock and soon it should encounter the outer boundary of the heliosphere, the heliopause. This is set to be at 120 AU in the modulation model used for this study. This implies that Voyager 1, and soon afterwards also Voyager 2, should be able to measure the heliopause spectrum, to be interpreted as the lowest possible local interstellar spectrum, for low energy galactic electrons (1 MeV to 120 MeV). This could give an answer to a long outstanding question about the spectral shape (energy dependence) of the galactic electron spectrum at these low energies. These in situ electron observations from Voyager 1, until the year 2010 when it was already beyond 112 AU, are used for a comparative study with a comprehensive three dimensional numerical model for the solar modulation of galactic electrons from the inner to the outer heliosphere. A locally developed steady state modulation model which numerically solves the relevant heliospheric transport equation is used to compute and study modulated electron spectra from Earth up to the heliopause. The issue of the spectral shape of the local interstellar spectrum at these low energies is specifically addressed, taking into account modulation in the inner heliosheath, up to the heliopause, including the effects of the transition of the solar wind speed from supersonic to subsonic in the heliosheath. Modulated electron spectra from the inner to the outer heliosphere are computed, together with radial and latitudinal profiles, focusing on 12 MeV electrons. This is compared to Voyager 1 observations for the energy range 6–14 MeV. A heliopause electron spectrum is computed and presented as a new plausible local interstellar spectrum from 30 GeV down to 10 MeV. The comparisons between model predictions and observations from Voyager 1 and at Earth (e.g. from the PAMELA mission and from balloon flights) and in the inner heliosphere (e.g. from the Ulysses mission) are made. This enables one to make conclusions about diffusion theory applicable to electrons in the heliosphere, in particular the rigidity dependence of diffusion perpendicular and parallel to the local background solar magnetic field. A general result is that the rigidity dependence of both parallel and perpendicular diffusion coefficients needs to be constant below P < 0.4 GV and only be allowed to increase above this rigidity to assure compatibility between the modeling and observations at Earth and especially in the outer heliosphere. A modification in the radial dependence of the diffusion coefficients in the inner heliosheath is required to compute realistic modulation in this region. With this study, estimates of the intensity of low energy galactic electrons at Earth can be made. A new local interstellar spectrum is computed for these low energies to improve understanding of the modulation galactic electrons as compared to previous results described in the literature. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2012.

The Voyager 1 spacecraft is now about 25 AU beyond the heliospheric termination shock and soon it should encounter the outer boundary of the heliosphere, the heliopause. This is set to be at 120 AU in the modulation model used for this study. This implies that Voyager 1, and soon afterwards also Voyager 2, should be able to measure the heliopause spectrum, to be interpreted as the lowest possible local interstellar spectrum, for low energy galactic electrons (1 MeV to 120 MeV). This could give an answer to a long outstanding question about the spectral shape (energy dependence) of the galactic electron spectrum at these low energies. These in situ electron observations from Voyager 1, until the year 2010 when it was already beyond 112 AU, are used for a comparative study with a comprehensive three dimensional numerical model for the solar modulation of galactic electrons from the inner to the outer heliosphere. A locally developed steady state modulation model which numerically solves the relevant heliospheric transport equation is used to compute and study modulated electron spectra from Earth up to the heliopause. The issue of the spectral shape of the local interstellar spectrum at these low energies is specifically addressed, taking into account modulation in the inner heliosheath, up to the heliopause, including the effects of the transition of the solar wind speed from supersonic to subsonic in the heliosheath. Modulated electron spectra from the inner to the outer heliosphere are computed, together with radial and latitudinal profiles, focusing on 12 MeV electrons. This is compared to Voyager 1 observations for the energy range 6–14 MeV. A heliopause electron spectrum is computed and presented as a new plausible local interstellar spectrum from 30 GeV down to 10 MeV. The comparisons between model predictions and observations from Voyager 1 and at Earth (e.g. from the PAMELA mission and from balloon flights) and in the inner heliosphere (e.g. from the Ulysses mission) are made. This enables one to make conclusions about diffusion theory applicable to electrons in the heliosphere, in particular the rigidity dependence of diffusion perpendicular and parallel to the local background solar magnetic field. A general result is that the rigidity dependence of both parallel and perpendicular diffusion coefficients needs to be constant below P < 0.4 GV and only be allowed to increase above this rigidity to assure compatibility between the modeling and observations at Earth and especially in the outer heliosphere. A modification in the radial dependence of the diffusion coefficients in the inner heliosheath is required to compute realistic modulation in this region. With this study, estimates of the intensity of low energy galactic electrons at Earth can be made. A new local interstellar spectrum is computed for these low energies to improve understanding of the modulation galactic electrons as compared to previous results described in the literature. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2012.