R. D. Strauss

MODELING THE MODULATION OF GALACTIC AND JOVIAN ELECTRONS BY STOCHASTIC PROCESSES

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.

A stochastic differential equation code for multidimensional Fokker-Planck type problems

Abstract We present a newly developed numerical code that integrates Fokker–Planck type transport equations in four to six spatial dimensions (configuration plus momentum space) and time by means of stochastic differential equations. In contrast to other, similar approaches our code is not restricted to any special configuration or application, but is designed very generally with a modular structure and, moreover, allows for Cartesian, cylindrical or spherical coordinates. Depending on the physical application the code can integrate the equations forward or backward in time. We exemplify the mathematical ideas the method is based upon and describe the numerical realisation and implementation in detail. The code is validated for both cases against an established finite-differences explicit numerical code for a scenario that includes particle sources as well as a linear loss term. Finally we discuss the new possibilities opened up with respect to general applications and newly developed hardware.

Cosmic ray modulation beyond the heliopause: a hybrid modeling approach

Results from a newly developed hybrid cosmic ray (CR) modulation model are presented. In this approach, the transport of CRs is computed by incorporating the plasma flow from a magnetohydrodynamic model for the heliospheric environment, resulting in representative CR transport. The model is applied to the modulation of CRs beyond the heliopause (HP) and we show that (1) CR modulation persists beyond the HP, so it is unlikely that the Voyager spacecraft will measure the pristine local interstellar spectra of galactic CRs when crossing the HP. (2) CR modulation in the outer heliosheath could maintain solar-cycle-related changes. (3) The modulation of CRs in the outer heliosheath is primarily determined by the ratio of perpendicular to parallel diffusion, so that the value of the individual diffusion coefficients cannot be determined uniquely using this approach. (4) CRs can efficiently diffuse between the nose and tail regions of the heliosphere.

On the propagation times and energy losses of cosmic rays in the heliosphere

[1] We present calculations of the propagation times and energy losses of cosmic rays as they are transported through the heliosphere. By calculating these quantities for a spatially 1D scenario, we benchmark our numerical model, which uses stochastic differential equations to solve the relevant transport equation, with known analytical solutions. The comparison is successful and serves as a vindication of the modeling approach. A spatially 3D version of the modulation model is subsequently used to calculate the propagation times and energy losses of galactic electrons and protons in different drift cycles. We find that the propagation times of electrons are longer than those of the protons at the same energy. Furthermore, the propagation times are longer in the drift cycle when the particles reach the Earth by drifting inward along the heliospheric current sheet. The calculated energy losses follow this same general trend. The energy losses suffered by the electrons are comparable to those of the protons, which is in contrast to the generally held perception that electrons experience little energy losses during their propagation through the heliosphere.

ON ASPECTS PERTAINING TO THE PERPENDICULAR DIFFUSION OF SOLAR ENERGETIC PARTICLES

The multitude of recent multi-point spacecraft observations of solar energetic particle (SEP) events has made it possible to study the longitudinal distribution of SEPs in great detail. SEPs, even those accelerated during impulsive events, show a much wider than expected longitudinal extent, bringing into question the processes responsible for their transport perpendicular to the local magnetic field. In this paper, we examine some aspects of perpendicular transport by including perpendicular diffusion in a numerical SEP transport model that simulates the propagation of impulsively accelerated SEP electrons in the ecliptic plane. We find that (1) the pitch-angle dependence of the perpendicular diffusion coefficient is an important, and currently mainly overlooked, transport parameter. (2) SEP intensities are generally asymmetric in longitude, being enhanced toward the west of the optimal magnetic connection to the acceleration region. (3) The maximum SEP intensity may also be shifted (parameter dependently) away from the longitude of best magnetic connectivity at 1 AU. We also calculate the maximum intensity, the time of maximum intensity, the onset time, and the maximum anisotropy as a function of longitude at Earths orbit and compare the results, in a qualitative fashion, to recent spacecraft observations.

Modelling anomalous cosmic ray oxygen in the heliosheath

This paper discusses a numerical modulation model to describe anomalous cosmic ray acceleration and transport in the heliosheath, the portion of the heliosphere between the termination shock and the heliopause. The model is based on the well known Parker transport equation and includes, in addition to diffusive shock acceleration at the solar wind termination shock, momentum diffusion (Fermi II, stochastic acceleration) and adiabatic heating occurring in the heliosheath together with both a latitude dependent compression ratio and injection efficiency as inferred from hydrodynamic heliospheric models. The model is applied to the study of anomalous cosmic ray oxygen, with the resulting intensities compared to recent Voyager 1 spacecraft observations in the heliosheath. Comparison shows that the model is able to very satisfactorily reproduce these observations, which includes a modulated spectral form at the termination shock and subsequent unfolding into the heliosheath. It is concluded that a combination of momentum diffusion and adiabatic heating, under certain realistic assumption of the solar wind speed in the heliosheath, form a viable re-acceleration mechanism, or continuous acceleration process, to explain the very contentious anomalous cosmic ray observations in the heliosheath.

The effects of magnetic field modifications on the solar modulation of cosmic rays with a SDE-based model

Abstract A numerical model for the solar modulation of cosmic rays, based on the solution of a set of stochastic differential equations (SDEs), is used to illustrate the effects of modifying the heliospheric magnetic field, particularly in the polar regions of the heliosphere. SDE-based models are well suited for such studies so that new insights are gained. To this end, the differences in the modulation brought about by each of three choices for the heliospheric magnetic field, i.e. the unmodified Parker field, the Smith–Bieber modified field, and the Jokipii–Kota modified field, are studied as typical well-known cases. It is illustrated that although both these modifications change the Parker field satisfactorily in the polar regions of the heliosphere, the Smith–Bieber modification is more effective in reducing cosmic ray drift effects in these regions. The features of these two modifications, as well as the effects on the solar modulation of cosmic rays, are illustrated qualitatively and quantitatively. In particular, it is shown how the Smith–Bieber modified field is applied in a cosmic ray modulation model to reproduce observational proton spectra from the PAMELA mission during the solar minimum of 2006–2009. These SDE-based results are compared with those obtained in previous studies of this unusual solar minimum activity period and found to be in good qualitative agreement.

The heliospheric transport and modulation of multiple charged anomalous oxygen revisited

Context. Since the crossings of the solar wind termination shock by the Voyager 1 and 2 spacecraft, much speculation has surrounded the acceleration mechanism and region where the anomalous cosmic ray component is accelerated. A peculiar, and mostly overlooked feature of the observed anomalous oxygen spectrum near the termination shock, is the power law form of the roll-over (cut-off )a t the high energy range of this spectrum. Aims. We investigate, using a numerical model, why this deviation from the expected exponential form of the cut-off part of the anomalous oxygen spectrum occurs, and if the observed power law form can be explained in terms of the acceleration of multiple charged anomalous oxygen. Methods. Multiple charged anomalous cosmic rays are incorporated in a numerical model, based on the standard Parker transport equation, including acceleration at the solar wind termination shock. This is done by specifying an energy dependent charge state, constrained by observations. Results. Comparing computational results with spacecraft observations, it is found that the inclusion of multiply charged anomalous cosmic rays in the modulation model can explain the observed spectrum of anomalous oxygen in the energy range from 10−70 MeV per nucleon. The more effective acceleration of these multiple charge anomalous particles at the solar wind termination shock causes a significant deviation from the usual exponential cut-off spectrum to display instead a power law decrease up to 70 MeV per nucleon where galactic oxygen starts to dominate. In addition, the model reproduces the features of multiple charged oxygen at Earth so that a good comparison is obtained between computations and observations.

Solving Parker’s transport equation with stochastic differential equations on GPUs

Abstract The numerical solution of transport equations for energetic charged particles in space is generally very costly in terms of time. Besides the use of multi-core CPUs and computer clusters in order to decrease the computation times, high performance calculations on graphics processing units (GPUs) have become available during the last years. In this work we introduce and describe a GPU-accelerated implementation of Parker’s equation using Stochastic Differential Equations (SDEs) for the simulation of the transport of energetic charged particles with the CUDA toolkit, which is the focus of this work. We briefly discuss the set of SDEs arising from Parker’s transport equation and their application to boundary value problems such as that of the Jovian magnetosphere. We compare the runtimes of the GPU code with a CPU version of the same algorithm. Compared to the CPU implementation (using OpenMP and eight threads) we find a performance increase of about a factor of 10–60, depending on the assumed set of parameters. Furthermore, we benchmark our simulation using the results of an existing SDE implementation of Parker’s transport equation.

Cosmic ray anisotropies near the heliopause

Context. The Voyager 1 spacecraft became the first man-made probe to cross the heliopause into the local interstellar medium and measure the galactic environment, including charged particle intensities, in situ. Aims. We qualitatively explain the observed anisotropies of galactic and anomalous cosmic rays in the interstellar medium. Methods. A pitch-angle-dependent numerical model was constructed and applied to the study of both heliospheric (anomalous cosmic rays and termination shock particles) and galactic cosmic rays near the heliopause region. Results. In accordance with the observations, the model is able to reproduce the observed anisotropic nature of both particle populations. In the interstellar medium, the heliospheric particle distribution shows a peak at pitch angles near 90 ◦ , while for galactic particles, their distribution shows a deficiency at these pitch-angle values. Conclusions. The observed anisotropies are related to the pitch-angle dependence of the perpendicular diffusion coefficient, and if this dependence is chosen appropriately, the anisotropies observed by Voyager 1 can be explained naturally.