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.

The interest in the role of the solar wind termination shock (TS) and heliosheath in cosmic ray (CR) modulation studies has increased sigm6cantly as the Voyager 1 and 2 spacecraft approach the estimated position of the TS. For this work the modulation of galactic CR protons, anti-protons, electrons with a Jovian source, positrons, Helium, and anomalous protons and Helium, and the consequent charge-sign dependence, are studied with an improved and extended two-dimensional numerical CR modulation model including a TS with diffusive shock acceleration, a heliosheath and drifts. The modulation is computed using improved local interstellar spectra (LIS) for almost all the species of interest to this study and new fundamentally derived diffusion coefficients, applicable to a number of CR species during both magnetic polarity cycles of the Sun. The model also allows comparisons of modulation with and without a TS and between solar minimum and moderate maximum conditions. The modulation of protons and Helium with their respective anomalous components are also studied to establish the consequent charge-sign dependence at low energies and the influence on the computed p/p, e-/p, and e-/He. The level of modulation in the simulated heliosheath, and the importance of this modulation 'barrier' and the TS for the different species are illustrated. From the computations it is possible to estimate the ratio of modulation occurring in the heliosheath to the total modulation between the heliopause and Earth for the mentioned species. It has been found that the modulation in the heliosheath depends on the particle species, is strongly dependent on the energy of the CRs, on the polarity cycle and is enhanced by the inclusion of the TS. The computed modulation for the considered species is surprisingly different and the heliosheath is important for CR modulation, although 'barrier' modulation is more prominent for protons, anti-protons and Helium, while the heliosheath cannot really be considered a modulation 'barrier' for electrons and positrons above energies of ~150 MeV. The effects of the TS on modulation are more pronounced for polarity cycles when particles are drifting primarily in the equatorial regions of the heliosphere along the heliospheric current sheet to the Sun, e.g. the A < 0 polarity cycle for protons, positrons, and Helium, and the A > 0 polarity cycle for electrons and anti-protons. This study also shows that the proton and Helium LIS may not be known at energies <~ 200 MeV until a spacecraft actually approaches the heliopause because of the strong modulation that occurs in the heliosheath, the effect of the TS, and the presence of anomalous protons and Helium. For anti-protons, in contrast, these effects are less pronounced. For positrons, with a completely different shape LIS, the modulated spectra have very mild energy dependencies <~ 300 MeV, even at Earth, in contrast to the other species. These characteristic spectral features may be helpful to distinguish between electron and positron spectra when they are measured near and at Earth. These simulations can be of use for future missions to the outer heliosphere and beyond. / Thesis (Ph.D. (Physics))--North-West University, Potchefstroom Campus, 2004.

A two dimensional hydrodynamic model describing the solar wind interaction with the local interstellar medium, which surrounds the solar system, is used to study the heliosphere both as a steady-state- and dynamic structure. The finite volume method used to solve the associated system of hydrodynamic equations numerically is discussed in detail. Subsequently the steady state heliosphere is studied for both the case where the solar wind and the interstellar medium are assumed to consist of protons only, as well as the case where the neutral hydrogen population in the interstellar medium is taken into account. It is shown that the heliosphere forms as three waves, propagating away from the initial point of contact between the solar wind and interstellar matter, become stationary. Two of these waves become stationary at sonic points, forming the termination shock and bow shock respectively. The third wave becomes stationary as a contact discontinuity, called the heliopause. It is shown that the position and geometry of the termination shock, heliopause and bow shock as well as the plasma flow characteristics of the heliosphere largely depend on the dynamic pressure of either the solar wind or interstellar matter. The heliosphere is modelled as a dynamic structure, including both the effects of the solar cycle and short term variations in the solar wind observed by a range of spacecraft over the past ~ 30 years. The dynamic model allows the calculation of an accurate record of the heliosphere state over the past ~ 30 years. This record is used to predict the time at which the Voyager 2 spacecraft will cross the termination shock. Voyager 1 observations of 10 MeV cosmic ray electrons are then used in conjunction with a cosmic ray modulation model to constrain the record of the heliosphere further. It is shown that the dynamic hydrodynamic model describes the heliosphere accurately within a margin of error of ±0.7 years and ±3 AU. The model predicts that Voyager 2 crossed the termination shock in 2007, corresponding to preliminary results from observations indicating that the crossing occurred in August 2007. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2008.

We investigate the physics of charged-particle acceleration at spherical shocks moving into a uniform plasma containing a turbulent magnetic field with a uniform mean. This has applications to particle acceleration at astrophysical shocks, most notably, to supernovae blast waves. We numerically integrate the equations of motion of a large number of test protons moving under the influence of electric and magnetic fields determined from a kinematically defined plasma flow associated with a radially propagating blast wave. Distribution functions are determined from the positions and velocities of the protons. The unshocked plasma contains a magnetic field with a uniform mean and an irregular component having a Kolmogorov-like power spectrum. The field inside the blast wave is determined from Maxwell's equations. The angle between the average magnetic field and unit normal to the shock varies with position along its surface. It is quasi-perpendicular to the unit normal near the sphere's equator, and quasi-parallel to it near the poles. We find that the highest intensities of particles, accelerated by the shock, are at the poles of the blast wave. The particles "collect" at the poles as they approximately adhere to magnetic field lines that move poleward from their initial encounter with the shock at the equator, as the shock expands. The field lines at the poles have been connected to the shock the longest. We also find that the highest-energy protons are initially accelerated near the equator or near the quasi-perpendicular portion of the shock, where the acceleration is more rapid.

The heliosphere and solar magnetic field play an important role in protecting the solar system from harmful, high-energy Galactic radiation. Until recently, the magnetic field had been assumed to be passive, carried outwards by the solar wind. The influence of the solar magnetic field on the plasma has just begun to be understood. Among the consequences, the magnetic field could cause the heliotail to be short, collimating the flow into two lobes instead of the classical long, comet-like tail. In this dissertation, I investigate the role certain aspects of the magnetic field have on the heliosphere and detail how interstellar neutral particles alter its effect on the environment. From the observation by Voyager 1 (V1) and Voyager 2 (V2), it is clear that the plasma environment in the outer heliosphere is not fully understood. I present the first time-dependent model of the outer heliosphere that includes solar-cycle variations of the magnetic field strength. I find that the model can accurately predict the plasma environment at V2 but cannot describe all features observed at V1, suggesting additional processes are present. The effect of including the heliospheric current sheet (HCS) on large-scale modeling of the heliosphere is also studied. The inherent numerical dissipation in the HCS reduces the magnetic field strength in the heliosheath; however, the two-lobe structure of the heliotail remains. Neutral hydrogen has also been shown to greatly affect the location of the heliospheric boundaries. The large mean free path of these neutrals requires them to be described kinetically. To understand how the neutrals affect the influence of the solar magnetic field, I developed the Solar-wind with Hydrogen Ion Exchange and Large-scale Dynamics (SHIELD) model, a kinetic-magnetohydrodynamic model of the outer heliosphere. The model couples a 3D Monte-Carlo model to the magnetohydrodynamic solver. SHIELD reproduces the results of similar models, namely a higher filtration of neutrals into the heliosphere when compared to a fluid description of the atoms. When SHIELD is applied to the heliotail, the two-lobe structure persists even with kinetic neutrals. These results show that the solar magnetic field plays a crucial role in determining the heliospheric structure.

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.

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