Diffusive Shock Acceleration in N Dimensions

Collisionless shocks are often studied in two spatial dimensions (2D) to gain insights into the 3D case. We analyze diffusive shock acceleration for an arbitrary number N I. of dimensions. For a nonrelativistic shock of compression ratio., the spectral index of the accelerated particles is sE = 1 + N (. - 1); this curiously yields, for any N, the familiar sE = 2 (i.e.,.equal energy per logarithmic particle energy bin) for a strong shock in a monatomic gas. A precise relation between sE and the anisotropy along an arbitrary relativistic shock is derived and is used to obtain an analytic expression for sE in the case of isotropic angular diffusion, affirming an analogous result in 3D. In particular, this approach yields sE = (1 + 13) 2. 2.30 in the ultrarelativistic shock limit for N.=.2, and sE (N. <yen>) = 2 for any strong shock. The angular eigenfunctions of the isotropic-diffusion transport equation reduce in 2D to elliptic cosine functions, providing a rigorous solution to the problem; the first function upstream already yields a remarkably accurate approximation. We show how these and additional results can be used to promote the study of shocks in 3D.