论文标题
$ n $尺寸的扩散冲击加速度
Diffusive shock acceleration in $N$ dimensions
论文作者
论文摘要
通常在两个空间维度（2D）中研究无碰撞冲击，以了解3D情况。我们分析了dimensions的任意数字$ n \ $ n \ $ n \的扩散冲击加速度。对于压缩比$ \ MATHCAL {r} $的非相关性冲击，加速粒子的频谱索引为$ s_e = 1+n/（\ Mathcal {r} -1）$;奇怪的是，对于任何$ n $，熟悉的$ s_e = 2 $（即每个对数粒子能量箱相等的能量）在单原子气体中产生强烈的冲击。 $ s_e $和沿任意相对论冲击的各向异性之间的精确关系被得出，并用于在各向同性角扩散的情况下获得$ s_e $的分析表达式，从而确认3D中的类似结果。特别是，此方法产生$ s_e =（1+ \ sqrt {13}）/2 \ simeq 2.30 $在$ n = 2 $的超相关电击限制中，以及$ s_e（n \ to \ infty）= 2 $，对于任何强烈的震动而言。各向同性扩散式传输方程的角本征函数将2D减少到椭圆余弦函数，从而为问题提供了严格的解决方案。上游的第一个函数已经产生非常准确的近似值。我们展示了如何使用这些结果和其他结果来促进3D中的冲击研究。
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.
