Diffusive shock acceleration in $N$ dimensions
| Autor: | Ofir Arad, Assaf Lavi, Uri Keshet, Yotam Nagar |
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| Rok vydání: | 2020 |
| Předmět: |
Physics
High Energy Astrophysical Phenomena (astro-ph.HE) Spectral index 010504 meteorology & atmospheric sciences Astrophysics::High Energy Astrophysical Phenomena Isotropy FOS: Physical sciences Astronomy and Astrophysics Acceleration (differential geometry) Eigenfunction 01 natural sciences Shock (mechanics) Space and Planetary Science 0103 physical sciences Convection–diffusion equation Anisotropy Astrophysics - High Energy Astrophysical Phenomena Nuclear Experiment 010303 astronomy & astrophysics Energy (signal processing) 0105 earth and related environmental sciences Mathematical physics |
| DOI: | 10.48550/arxiv.2002.11123 |
| Popis: | 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. Comment: 14 pages, 7 figures, comments welcome |
| Databáze: | OpenAIRE |
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