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

Abstract

Context. 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. Aims. 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. Methods. 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. Results. 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 investigation of the particle energy spectra indicates that a shock acceleration theory based on superdiffusive transport better explains observed spectral index values. Conclusions. This study, together with the analysis of energetic particles upstream of shock waves, allows us to fully determine the transport properties of accelerated particles, even in the case of superdiffusion. This represents a new powerful tool to understand the transport and acceleration processes at astrophysical shocks.

Superdiffusive transport in laboratory and astrophysical plasmas

“…In the above equation, n is the particle number density, C is a constant, 0 < < 1 and µ = 2 implies subdiffusion, while = 1 and 1 < µ < 2 implies superdiffusion; when fractional derivatives on both time and space are used, the anomalous regimes are characterized by ↵ = 2 /µ (see Zaslavsky 2002;Perrone et al 2013;Bovet et al 2014a;Stern et al 2014, for more details). This relation is different from the one given above for Lévy walks, even for = 1, but it can be shown that the same relation is obtained when the finite extent of the integration domain over x is taken into account (Zumofen & Klafter 1993;Perri et al 2015).…”

“…Litvinenko & Effenberger (2014) verified their results with complementary solution methods using a weak diffusion approximation and a Fourier-series solution on a finite domain (Stern et al 2014). Furthermore, the relation ↵ = 3 µ is recovered for the time dependence of the variance of particle displacement when a finite particle speed is considered (see also the discussion in Perri et al 2015). An additional, complementary solution method for fractional diffusion equations is based on a generalization of the equivalence between stochastic differential equations (SDEs) and the Fokker-Planck equation (Gardiner 2009).…”

“…Space plasmas are in an intermediate situation, since a number of plasma properties can be measured in situ by spacecraft. In this context, Perri & Zimbardo (2007 and Perri et al (2015) have developed some diagnostic tools to extract the transport properties of energetic particles accelerated at heliospheric shocks by the measured intensity profiles of the energetic particles.…”

“…In particular, for ↵ > 1, harder spectral indices than those predicted by DSA are obtained. The above expression of has been applied to the interpretation of energetic ion spectra at the termination shock by Perri & Zimbardo (2012a) and to electron spectra at interplanetary shocks by Perri et al (2015).…”

Acceleration of Solar Energetic Particles at a Fast Traveling Shock in Non-uniform Coronal Conditions

“…Thus, it appears that a power-law intensity decrease upstream is a direct consequence of Alfvén wave excitation by SEPs undergoing DSA at quasi-parallel shocks (see also Afanasiev [19] for a similar discussion). Therefore, at quasiparallel shocks upstream wave excitation provides an alternative explanation for such power laws thought by some to be produced by anomalous diffusion of energetic particles across traveling shocks [24].…”