Curvature Effects in Gamma‐Ray Burst Colliding Shells

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Abstract
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 θj 1/Γ is considered, where Γ is the Lorentz factor of the emitting shell. The shell, with comoving width Δr', is assumed to be illuminated for a comoving time Δt' and to radiate a broken-power-law νLν spectrum peaking at comoving photon energy . 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 νFν flux at the measured peak photon energy pk be proportional to 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.