Diffusive shock acceleration in $N$ dimensions. (arXiv:2002.11123v1 [astro-ph.HE])
<a href="http://arxiv.org/find/astro-ph/1/au:+Lavi_A/0/1/0/all/0/1">Assaf Lavi</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Arad_O/0/1/0/all/0/1">Ofir Arad</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Nagar_Y/0/1/0/all/0/1">Yotam Nagar</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Keshet_U/0/1/0/all/0/1">Uri Keshet</a>
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 $Ninmathbb{N}$ 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 $s_E(Ntoinfty)=2$ 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.
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 $Ninmathbb{N}$ 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 $s_E(Ntoinfty)=2$ 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.
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