Gm. Webb et al., MULTIDIMENSIONAL GREENS-FUNCTIONS AND THE STATISTICS OF DIFFUSIVE SHOCK ACCELERATION, The Astrophysical journal, 453(1), 1995, pp. 178-206
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 fu
nction as a probability distribution, various statistical means and va
riances describing the shock acceleration process are obtained. These
include the mean time [tau] for particles to be accelerated from momen
tum pi up to momentum p(i) the mean distance [zeta] traveled by partic
les parallel to the electric field at the shock, and the mean distance
[chi] traveled by particles along the shock surface in the (V, B) pla
ne (a new result). We also obtain new second-order moments of the shoc
k acceleration process, including the variances sigma(zeta zeta)(2),si
gma(chi chi)(2), sigma(zeta tau)(2),sigma(zeta chi)(2), and sigma(chi
tau)(2), as well as the only variance sigma(tau tau)(2) obtained in pr
evious analyses. The distance, [zeta], traveled by the particle parall
el to the electric field at the shock is related to the particle energ
y changes at the shock owing to curvature and grad B drift in the elec
tric held at the shock. The Green's function for the limiting case of
no cross-held diffusion (K-perpendicular to = 0) consists of the produ
ct of two Dirac delta distributions and the one-dimensional Green's fu
nction for diffusive shock acceleration obtained by previous authors.
One of the Dirac delta distributions expresses the fact that for K-per
pendicular to = 0 particles are trapped on the same field line project
ion on the plane spanned by the fluid velocity V and magnetic held B (
the x-y plane). The other Dirac delta distribution contains informatio
n on the energy changes of particles owing to drift at the shock. A st
udy of the means and variances of the probability distribution for a m
odel diffusion tenser is used to determine the dependence of the shock
acceleration process on the angle theta(1) between the upstream magne
tic field B-1 and shock normal n, as well as on other physical paramet
ers in the model. A discussion is given of the role of perpendicular d
iffusion (K(perpendicular to)not equal 0) at general oblique shocks. E
xamples of shock acceleration statistics are obtained for traveling in
terplanetary shocks. Other possible astrophysical applications are als
o discussed.