Abstract

We present analytic calculations of synchrotron radiation from the forward and the reverse shock of gamma-ray burst blast waves, in the thick-shell scenario (i.e. when the reverse shock is relativistic). We show that this scenario can naturally account for the plateau phase, observed early in the afterglows of about half the bursts detected by Swift. We generalize our approach to include power-law luminosity of the central engine and show that when radiation from both regions (forward and reverse shock) is taken into account, a wide range of possibilities emerge, including chromatic and achromatic breaks, frequency-dependent spectral evolution during the injection break and widely varying decay indices in different bands. For both the forward and the reverse shock, we derive formulas for the spectral parameters and the observed flux in different power-law segments of the spectrum, as a function of observer time. We explore the F_b-t_b relation (between the observed time of the end of the plateau phase and the flux at that point) in the framework of the presented model and show that model predictions favour the reverse shock as the dominant source of emission in both optical and X-rays. As case studies, we present simultaneous fits to X-ray and optical/IR afterglow data of GRB 080928 and GRB 090423. We identify the end of the plateau phase with the cessation of energy injection and infer the corresponding upper limits to central-engine activity, which are about 1 h for the former and 1.5 h for the latter. We conclude that smooth energy injection through the reverse shock is a plausible explanation for the plateau phase of gamma-ray burst afterglows. During that phase, radiation from the reverse shock is likely to be important, or even dominant, and should be taken into account when fitting model parameters to observations.