Diffusive Shock Acceleration: Breakdown of Spatial Diffusion and Isotropy

We point out that particles accelerated in a nonrelativistic shock of compression ratio r do not necessarily attain the standard, p = (r + 2)/(r - 1) spectral index. Previous derivations of the spectrum, based on the approximations of spatial diffusion or negligible anisotropy, are shown to rely on unjustified implicit assumptions. We prove analytically that the standard result is nevertheless valid in the limit of an isotropic medium. For an anisotropic medium, the problem generally requires a numerical treatment; the standard result remains valid as long as the anisotropy is not too strong, but p can substantially deviate from the standard result for sufficiently anisotropic scattering, even in the small-angle scattering limit. Additional spectral modifications, for example, by motions of scattering modes at intermediate optical depths from the shock, are discussed.