Multidimensional Green's functions and the statistics of diffusive shock acceleration

G. M. Webb, G. P. Zank, C. M. Ko, D. J. Donohue

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A three-dimensional, time-dependent Green's function for the diffusive shock acceleration of energetic charged particles at general oblique MHD shocks, without losses, is derived. By interpreting the Green's function as a probability distribution, various statistical means and variances describing the shock acceleration process are obtained. These include the mean time 〈τ〉 for particles to be accelerated from momentum pi up to momentum p, the mean distance 〈ζ〉 traveled by particles parallel to the electric field at the shock, and the mean distance 〈χ〉 traveled by particles along the shock surface in the (V, B) plane (a new result). We also obtain new second-order moments of the shock acceleration process, including the variances σ2ζζ, σ2χχ, σ2ζτ, σ2ζχ, and σ2χτ, as well as the only variance σ2ττ obtained in previous analyses. The distance, 〈ζ〈, traveled by the particle parallel to the electric field at the shock is related to the particle energy changes at the shock owing to curvature and grad B drift in the electric field at the shock. The Green's function for the limiting case of no cross-field diffusion (K = 0) consists of the product of two Dirac delta distributions and the one-dimensional Green's function for diffusive shock acceleration obtained by previous authors. One of the Dirac delta distributions expresses the fact that for K = 0 particles are trapped on the same field line projection on the plane spanned by the fluid velocity V and magnetic field B (the x-y plane). The other Dirac delta distribution contains information on the energy changes of particles owing to drift at the shock. A study of the means and variances of the probability distribution for a model diffusion tensor is used to determine the dependence of the shock acceleration process on the angle θ1 between the upstream magnetic field B1 and shock normal n, as well as on other physical parameters in the model. A discussion is given of the role of perpendicular diffusion (K ≠ 0) at general oblique shocks. Examples of shock acceleration statistics are obtained for traveling interplanetary shocks. Other possible astrophysical applications are also discussed.

Original languageEnglish
Pages (from-to)178-206
Number of pages29
JournalAstrophysical Journal
Issue number1
StatePublished - 1 Nov 1995


  • Acceleration of particles
  • Methods: analytical
  • Shock waves


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