Transport of relativistic electrons at shocks in shell-type supernova remnants: diffusive and superdiffusive regimes

Abstract

Context. Understanding the transport properties of energetic particles in the presence of magnetic turbulence is crucial for interpreting observations of supernova remnants (SNR) and for assessing the cosmic-ray acceleration mechanism. Aims. We aim at obtaining information on the transport regimes of energetic electrons upstream and downstream of SNR blast waves by studying the X-ray rims. Methods. We considered emission profiles expected when synchrotron energy losses dominate, normal diffusion (typically causing an exponential decay), and superdiffusion (causing a power-law decay) for a spherically symmetric model. Then we compared the model profiles, projected on the plane of the sky, with Chandra observations of supernova (SN)1006 and Tycho's SN. Results. Our study shows that downstream of the blast wave the observed profile is exponentially cutoff due to synchrotron energy losses in the amplified and shock compressed magnetic field present. Upstream of the blast wave and close to the shock, the observed profile of SN1006 is well reproduced by electron diffusion in the Bohm regime. However, the long X-ray tail far upstream of the shock is better explained by considering a change in the energetic electrons' transport regime, that appears to become superdiffusive and gives rise to a power law intensity profile on scales over which the magnetic field strength can be assumed to be constant. Similar results are obtained for Tycho, although in this case the nonlinear effects associated with particle acceleration appear to be stronger close to the shock. Conclusions. The analysis of Chandra observations of the X-ray thin rims of SN1006 and of Tycho's SN, SN 1572, can be well explained assuming normal diffusion in the vicinity of the blast wave and superdiffusion far upstream, similarly to what has been found for particles accelerated at interplanetary shocks. This suggests that anomalous transport is common in both interplanetary space and interstellar medium.

On the Fractional Diffusion-Advection Equation for Fluids and Plasmas

“…Compared to the case of normal diffusion, which would give an exponential profile, we can see that in the case of superdiffusion and advection the density profile obtained by considering only the left contribution to the flux is steeper close to the source of particles at x = 0, and more shallow than an exponential farther away. This is similar to the behavior found for energetic particles propagating upstream of interplanetary shock waves [12,13,45,51,55] and upstream of supernova remnant shocks [56,57].…”

Fractional Parker equation for the transport of cosmic rays: steady-state solutions

“…With these solutions, we have an analytical expression for the energetic particle density profile at every distance from the shock, which allows for a more detailed comparison with observations. This will be useful for the study of superdiffusive transport both at heliospheric shocks and at supernova remnant shocks (Perri et al 2016); in particular, the possibility to fit the entire upstream profile with the Mittag-Leffler functions will allow a more precise determination of the break in the power-law profile, see Eqs. (9) and (33), and of the corresponding acceleration time for energetic particles .…”

“…Several spacecraft observations have shown that the energetic particle profile upstream of interplanetary shocks is characterized by a power-law decay rather than an exponential decay, implying that transport can be superdiffusive (Perri & Zimbardo 2007, 2009aSugiyama & Shiota 2011). Recently, this power-law profile was also found for relativistic electrons upstream of supernova remnant shocks (Perri et al 2016). The finding of superdiffusion in space suggests that the Parker equation could be extended by introducing fractional derivatives for the first term on the right-hand side of Eq.…”

Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model

“…The key point is that the nonlinear diffusion model naturally yields a power-law dependence of the cosmic-ray density on the distance upstream of the shock. Such a dependence was argued to be the signature of a fractional advection-diffusion equation (e.g., Perri & Zimbardo 2007, 2009Perri et al 2016). Yet our results clearly demonstrate that, when it comes to explaining the anomalous diffusion scalings of cosmic-ray particles, a transport equation with fractional derivatives is not the only game in town.…”

“…Observations of energetic electrons and protons in interplanetary space (e.g., Zimbardo et al 2015) and relativistic electrons in supernova remnants (Perri et al 2016) imply that the transport of cosmic-ray particles in the presence of turbulent scattering can be superdiffusive. In the superdiffusive regime, the mean square displacement á ñ…”