X-Ray Afterglows from the Gamma-Ray Burst “Large-angle” Emission

Abstract

In this work, we derive basic analytical results for the timing and decay of the gamma-ray burst (GRB) counterpart and delayed afterglow light curves for a brief emission episode from a relativistic surface endowed with angular structure, consisting of a uniform core of size ${\theta }_{c}$ (Lorentz factor ${{\rm{\Gamma }}}_{c}$ and surface emissivity ${i}_{\nu ^{\prime} }^{{\prime} }$ are angle independent) and an axially symmetric power-law envelope (${\rm{\Gamma }}\sim {\theta }^{-g}$). In this "large-angle emission" model, radiation produced during the prompt emission phase (GRB) at angles $\theta \gt {\theta }_{c}$ arrives at the observer well after the burst (delayed emission). The dynamical time range of the very fast decaying GRB "tail" and of the flat afterglow "plateau" and the morphology of the GRB counterpart/afterglow are all determined by two parameters: the core's parameter ${{\rm{\Gamma }}}_{c}{\theta }_{c}$ and the envelope's Lorentz factor index g, leading to three types of light curves that display three post-GRB phases (type 1: tail, plateau/slow decay, post-plateau/normal decay), two post-GRB phases (type 2: tail and fast decay), or just one (type 3: normal decay). We show how X-ray light-curve features can be used to determine core and envelope dynamical and spectral parameters. Testing of the large-angle emission model is done using the Swift/XRT X-ray emission of two afterglows of type 1 (GRB 060607A, GRB 061121), one of type 2 (GRB 061110A), and one of type 3 (GRB 061007). We find that the X-ray afterglows with plateaus require an envelope Lorentz factor ${\rm{\Gamma }}\sim {\theta }^{-2}$ and a comoving-frame emissivity ${i}_{\nu ^{\prime} }^{{\prime} }\sim {\theta }^{2}$; thus, for a typical afterglow spectrum ${F}_{\nu }\sim {\nu }^{-1}$, the lab-frame energy release is uniform over the emitting surface.