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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 $N\in\mathbb{N}$ of dimensions. For a non-relativistic shock of compression ratio $\mathcal{R}$, the spectral index of the accelerated particles is $s_E=1+N/(\mathcal{R}-1)$; this curiously yields, for any $N$, the familiar $s_E=2$ (i.e., equal energy per logarithmic particle energy bin) for a strong shock in a mono-atomic gas. A precise relation between $s_E$ and the anisotropy along an arbitrary relativistic shock is derived, and is used to obtain an analytic expression for $s_E$ in the case of isotropic angular diffusion, affirming an analogous result in 3D. In particular, this approach yields $s_E = (1+\sqrt{13})/2 \simeq 2.30$ in the ultra-relativistic shock limit for $N=2$, and $s_E(N\to\infty)=2$ 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.