Frederic Effenberger

Solar energetic particle access to distant longitudes through turbulent field-line meandering

Context. Current solar energetic particle (SEP) propagation models describe the effects of interplanetary plasma turbulence on SEPs as diffusion, using a Fokker-Planck (FP) equation. However, FP models cannot explain the observed fast access of SEPs across the average magnetic field to regions that are widely separated in longitude within the heliosphere without using unrealistically strong cross-field diffusion. Aims. We study whether the recently suggested early non-diffusive phase of SEP propagation can explain the wide SEP events with realistic particle transport parameters. Methods. We used a novel model that accounts for the SEP propagation along field lines that meander as a result of plasma turbulence. Such a non-diffusive propagation mode has been shown to dominate the SEP cross-field propagation early in the SEP event history. We compare the new model to the traditional approach, and to SEP observations. Results. Using the new model, we reproduce the observed longitudinal extent of SEP peak fluxes that are characterised by a Gaussian profile with

The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

\sigma=30-50^\circ

Parameter estimation of superdiffusive motion of energetic particles upstream of heliospheric shocks

, while current diffusion theory can only explain extents of 11

The Diffusion Approximation versus the Telegraph Equation for Modeling Solar Energetic Particle Transport with Adiabatic Focusing. I. Isotropic Pitch-angle Scattering

^\circ

Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport

with realistic diffusion coefficients. Our model also reproduces the timing of SEP arrival at distant longitudes, which cannot be explained using the diffusion model. Conclusions. The early onset of SEPs over a wide range of longitudes can be understood as a result of the effects of magnetic field-line random walk in the interplanetary medium and requires an SEP transport model that properly describes the non-diffusive early phase of SEP cross-field propagation.

The IBEX ribbon as a signature of the inhomogeneity of the local interstellar medium

The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.

Pitch-angle scattering in magnetostatic turbulence. I. Test-particle simulations and the validity of analytical results

In-situ spacecraft observations recently suggested that the transport of energetic particles accelerated at heliospheric shocks can be anomalous, i.e. the mean square displacement can grow non-linearly in time. In particular, a new analysis technique has permitted the study of particle transport properties from energetic particle time profiles upstream of interplanetary shocks. Indeed, the time/spatial power laws of the differential intensity upstream of several shocks are indicative of superdiffusion. A complete determination of the key parameters of superdiffusive transport comprises the power-law index, the superdiffusion coefficient, the related transition scale at which the energetic particle profiles turn to decay as power laws, and the energy spectral index of the shock accelerated particles. Assuming large-scale spatial homogeneity of the background plasma, the power-law behaviour can been derived from both a (microscopic) propagator formalism and a (macroscopic) fractional transport equation. We compare the two approaches and find a relation between the diffusion coefficients used in the two formalisms. Based on the assumption of superdiffusive transport, we quantitatively derive these parameters by studying energetic particle profiles observed by the Ulysses and Voyager 2 spacecraft upstream of shocks in the heliosphere, for which a superdiffusive particle transport has previously been observed. Further, we have jointly studied the electron energy spectra, comparing the values of the spectral indices observed with those predicted by the standard diffusive shock acceleration theory and by a model based on superdiffusive transport. For a number of interplanetary shocks and for the solar wind termination shock, for the first time we obtain the anomalous diffusion constants and the scale at which the probability of particle free paths changes to a power-law...

The Telegraph Approximation for Focused Cosmic-Ray Transport in the Presence of Boundaries

The diffusion approximation to the Fokker-Planck equation is commonly used to model the transport of solar energetic particles in interplanetary space. In this study, we present exact analytical predictions of a higher order telegraph approximation for particle transport and compare them with the corresponding predictions of the diffusion approximation and numerical solutions of the full Fokker-Planck equation. We specifically investigate the role of the adiabatic focusing effect of a spatially varying magnetic field on an evolving particle distribution. Comparison of the analytical and numerical results shows that the telegraph approximation reproduces the particle intensity profiles much more accurately than does the diffusion approximation, especially when the focusing is strong. However, the telegraph approximation appears to offer no significant advantage over the diffusion approximation for calculating the particle anisotropy. The telegraph approximation can be a useful tool for describing both diffusive and wave-like aspects of the cosmic-ray transport.

A Hitch-hiker’s Guide to Stochastic Differential Equations

Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we solve analytically a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.

Simulations of 3D Magnetic Merging: Resistive Scalings for Null Point and QSL Reconnection

Context. An ew hypothesis is offered to explain the so-called ribbon feature appearing in the all-sky flux maps of energetic neutral atoms presently observed withthe IBEXspacecraft, namely that theribbon isa consequenc eo finhomogeneities inthelocal interstellar medium. Aims. The study aims at a detailed presentation of this hypothesis and its implications for the interpretation of ENA measurements with IBEX. Methods. Theoretical considerations regarding three different topics, namely IBEX measurements related to high-energy neutral atoms, Lyman-α observations (made with the Voyager 1 spacecraft) related to low-energy neutral atoms, and astronomical data related to the structure of the interstellar medium, are critically discussed in order to corroborate the hypothesis. Results. It is found that inhomogeneities in the local interstellar medium can explain not onl yt he IBEX ribbon andouter heliospheric Lyman-α observations, but can also account for the interstellar Lyman-α absorption that could only with difficulty be fully attributed to the hydrogen wall in the outer heliosheath if the heliospheric bow shock would indeed be absent. Conclusions. The IBEX observations of the ribbon provide a unique opportunity to learn more about the nature of the interstellar medium surrounding the heliosphere.