A diffusive description of the focused transport of solar energetic particles

Abstract

The transport of solar energetic charged particles along the interplanetary magnetic field in the ecliptic plane of the sun can be described roughly by a one-dimensional diffusion equation. Large-scale spatial variations of the guide magnetic field can be taken into account by adding an additional term to the diffusion equation that includes the effect of adiabatic focusing. We solve this equation analytically by assuming a point-like particle injection in time and space and a spatial power-law dependence for the focusing length and the spatial diffusion coefficient. We infer the intensity-and anisotropy-time profiles of solar energetic particles from this solution. Through these the influence of different assumptions for the diffusion parameters can be seen in a mathematically closed form. The comparison of calculated and measured intensity-and anisotropy-time profiles, which are a powerful diagnostic tool for interplanetary particle transport, gives information about the large-scale spatial dependence of the focusing length and the diffusion coefficient. For an exceptionally large solar energetic particle event, which did occur on 2001 April 15, we fit the 27−512 keV electron intensities and anisotropies observed by the Wind spacecraft using the theoretically derived profiles. We find a linear spatial dependence of the mean free path along the guiding magnetic field. We also find the mean free path to be energy independent, which supports the theory of "velocity-dependent diffusion". This means that the intensity profiles for the discussed energies exhibit the same shape if they are plotted against the traveled distance and not against the time. In this case the profiles differ only in their maximum values and we can determine the energy spectra of the solar flare electrons out of the scaling factor we need to fit the data. The derived spectra exhibits a power-law dependence ∝E −3 kin in an energy range from ∼50 keV to ∼500 keV.

The telegraph equation for cosmic-ray transport with weak adiabatic focusing

“…Schlickeiser & Shalchi (2008) gave a systematic derivation of the diffusion approximation in a weak focusing limit, when the adiabatic focusing length is much greater than the turbulent scattering length of cosmic-ray particles. The diffusion approximation is the standard approximation for studying the cosmic-ray transport (e.g., Schlickeiser et al 2009;Artmann et al 2011;Tautz et al 2012, and references therein). Hence it is useful to know its limitations that are related to the fact that the signal propagation speed is infinite in the diffusion limit.…”

“…Momentum diffusion is neglected, which is typically justified for the transport of solar energetic particles (SEPs) in interplanetary space (e.g., Artmann et al 2011). A constant focusing length L is adopted, which is not a significant limitation as long as L does not change much within one scattering length λ.…”

“…Early observations suggested that the resulting particle transport is diffusive (Meyer et al 1956;Roelof 1969). Hence a simple description of the SEP transport can be provided by time-dependent solutions of a spatial diffusion equation with a diffusion coefficient determined by the spectrum of the turbulence (e.g., Kóta et al 1982;Artmann et al 2011, and references therein). Note for clarity that some SEP distributions are strongly anisotropic, in which case the particle transport is not diffusive but rather effectively scatter-free (e.g., Haggerty & Roelof 2002).…”

“…The formula gives a reasonably accurate fit to the observed SEP anisotropy profiles (Schlickeiser et al 2009;Artmann et al 2011). However, since |A| ≤ 3 by definition, an unsatisfactory feature, related to the infinite signal propagation speed of the diffusion equation, is that A → ∞ as t → 0.…”

“…Litvinenko & Noble (2013) used numerical simulations to emphasise the limitations of the diffusion approximation to focused particle transport, related to the fact that the propagation speed is infinite in the diffusion limit. Despite it limitations, the diffusion approximation is often used to model focused particle transport in astrophysics (e.g., Artmann et al 2011;Tautz et al 2012, and references therein).…”