In this third paper of a series, we discuss the physics of the population of accelerated particles in the precursor of an unmagnetized, relativistic collisionless pair shock. In particular, we provide a theoretical estimate of their scattering length $l_{ scatt}(p)$ in the self-generated electromagnetic turbulence, as well as an estimate of their distribution function. We obtain $l_{ scatt}(p) \simeq (\gamma_p /\epsilon_B)(p/\gamma_{ \infty} mc)^2 (c/\omega_p)$, with p the particle momentum in the rest frame of the shock front, $\epsilon_B$ the strength parameter of the microturbulence, $\gamma_p$ the Lorentz factor of the background plasma relative to the shock front and $\gamma_{ \infty}$ its asymptotic value outside the precursor. We compare this scattering length to large-scale PIC simulations and find good agreement for the various dependencies.
Categories: astro-ph.HE physics.plasm-ph