Curvature effects in gamma-ray burst colliding shells

An elementary kinematic model for emission produced by relativistic spherical colliding shells is studied. The case of a uniform blast-wave shell with jet opening angle theta(j) much greater than 1/Gamma is considered, where Gamma is the Lorentz factor of the emitting shell. The shell, with comoving width Deltar', is assumed to be illuminated for a comoving time Deltat' and to radiate a broken-power-law nuL(nu) spectrum peaking at comoving photon energy epsilon(pk,0)'. Synthetic gamma-ray burst (GRB) pulses are calculated, and the relation between energy flux and internal comoving energy density is quantified. Curvature effects dictate that the measured nuF(nu) flux at the measured peak photon energy epsilon(pk) be proportional to epsilon(pk)(3) in the declining phase of a GRB pulse. Possible reasons for discrepancies with observations are discussed, including adiabatic and radiative cooling processes that extend the decay timescale, a nonuniform jet, and the formation of pulses by external shock processes. A prediction of a correlation between prompt emission properties and times of the optical afterglow beaming breaks is made for a cooling model, which can be tested with Swift.