Diffusive acceleration in relativistic shocks: particle feedback

ABSTRACT The spectral index s of high-energy particles diffusively accelerated in a non-magnetized relativistic shock, such as in a γ-ray burst afterglow, depends on the unknown angular diffusion function $\mathcal {D}$, 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 $\mathcal {D}$ that depends on f. A local $\mathcal {D}(f)$ 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 $\mathcal {D}$ is a strong function of f, so the standard, isotropic-diffusion results remain approximately applicable unless $\mathcal {D}$ is both highly anisotropic and not a local function of f. A mild, ∼0.1 softening of s, in both 2D and 3D, when $\mathcal {D}(f)$ rises sufficiently fast, may be realized in ab initio simulations.