Diffusive acceleration in relativistic shocks: particle feedback

The spectral index s of high-energy particles diffusively accelerated in a non-magnetized relativistic shock, such as in a gamma-ray burst afterglow, depends on the unknown angular diffusion function D, which itself depends on the particle distribution function f if acceleration is efficient. We develop a relaxation code to compute s and f for an arbitrary functional D that depends on f. A local D( f) dependence is motivated and shown, when rising (falling) upstream, to soften (harden) s with respect to the isotropic case, shift the angular distribution towards upstream (downstream) directions, and strengthen (weaken) the particle confinement to the shock; an opposite effect on s is found downstream. However, variations in s remain modest even when D is a strong function of f, so the standard, isotropic-diffusion results remain approximately applicable unless D is both highly anisotropic and not a local function of f. A mild, similar to 0.1 softening of s, in both 2D and 3D, when D( f) rises sufficiently fast, may be realized in ab initio simulations.

Automated summary

The spectral index s of high-energy particles diffusively accelerated in a non-magnetized relativistic shock depends on the unknown angular diffusion function D, which itself depends on the particle distribution function f. The local D( f) dependence is shown to have effects on s, the angular distribution, and the particle confinement at the shock, but variations in s remain modest even when D is a strong function of f.