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 ${\theta }_j\gg 1\mathrm{/}\mathrm{\Gamma }$ is considered, where $\mathrm{\Gamma }$ is the Lorentz factor of the emitting shell. The shell, with comoving width $\Delta r^\prime$,isassumedtobeilluminatedforacomovingtime$\Delta t^\prime$andtoradiateabrokenpower-law$\nu L_\nu$ spectrum peaking at comoving photon energy ${\text{e}}_{pk,0}^{\mathrm{\prime }}$. Synthetic GRB pulses are calculated, and the relation between energy flux and internal comoving energy density is quantified. Curvature effects dictate that the measured $\nu F_{\nu }$ flux at the measured peak photon energy ${\text{e}}_{pk}$ is proportional to ${\text{e}}_{pk}^3$ 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.