A diffusive description of the focused transport of solar energetic particles

Intensity- and anisotropy-time profiles as a powerful diagnostic tool for interplanetary particle transport conditions^⋆

¹ Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany
e-mail: sa@tp4.rub.de
² Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
³ Departament d’Astronomia i Meteorologia, Institut de Ciències del Cosmos (ICC), Universitat de Barcelona, Spain
⁴ Physics Department, University of California, Berkeley, CA 94720-7300, USA
⁵ School of Space Research, Kyung Hee University, Yongin, Gyeonggi, Korea

Received: 15 August 2011
Accepted: 9 September 2011

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  ∝ ${\mathit{E}}_{\mathrm{kin}}^{-3}$ in an energy range from  ~ 50   keV to  ~ 500   keV.