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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