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 \θ_j \ 1/\Γ is considered, where \Γ is the Lorentz factor of the emitting shell. The shell, with comoving width \Δ r^\, is assumed to be illuminated for a comoving time \Δ t^\ and to radiate a broken power-law \ν L_\ν spectrum peaking at comoving photon energy pk,0\. Synthetic GRB pulses are calculated, and the relation between energy flux and internal comoving energy density is quantified. Curvature effects dictate that the measured \ν F_\ν flux at the measured peak photon energy ₚₖ is proportional to \^3ₚₖ in the declining phase of a GRB pulse. Possible reasons for discrepancy with observations are discussed, including adiabatic and radiative cooling processes that extend the decay timescale, a nonuniform jet, or 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.