Acceleration of ultrahigh-energy cosmic rays in the early afterglows of gamma-ray bursts: Concurrence of jet dynamics and wave-particle interactions
The origin of ultrahigh-energy cosmic rays (UHECRs) remains a mystery. It has been suggested that UHECRs can be produced by the stochastic acceleration in relativistic jets of gamma-ray bursts (GRBs) at the early afterglow phase. We develop a time-dependent model for proton energization by cascading compressible waves in GRB jets while considering the concurrent effect of the jet's dynamics and the mutual interactions between turbulent waves and particles. Considering the fast mode of a magnetosonic wave as the dominant particle scatterer and assuming the interstellar medium for the circumburst environment, our numerical results suggest that protons can be accelerated up to 1019 eV during the early afterglow. An estimation shows that ultrahigh-energy nuclei can easily survive photodisintegration in the external shocks in most cases, thus allowing the acceleration of 1020 eV cosmic-ray nuclei in the proposed frame. The spectral slope can be as hard as d N /d E ∝E0, which is consistent with the requirement for the interpretation of the intermediate-mass composition of the UHECRs as measured by the Pierre Auger Observatory.