We analytically study diffusive particle acceleration in relativistic, collisionless shocks. We find a simple relation between the spectral index s and the anisotropy of the momentum distribution along the shock front. Based on this relation, we obtain s = (3βu - 2βuβ d2 + βd3)/(βu - βd) for isotropic diffusion, where βu (βd) is the upstream (downstream) fluid velocity normalized to the speed of light. This result is in agreement with previous numerical determinations of s for all (βu βd), and yields s = 38/9 in the ultrarelativistic limit. The spectrum-anisotropy connection is useful for testing numerical studies and constraining anisotropic diffusion results. It suggests that the spectrum is highly sensitive to the form of the diffusion function for particles traveling along the shock front.
ASJC Scopus subject areas
- General Physics and Astronomy