Curvature Effects in Gamma‐Ray Burst Colliding Shells

An elementary kinematic model for emission produced by relativistic sphericalcolliding shells is studied. The case of a uniform blast-wave shell with jetopening angle $\theta_j \gg 1/\Gamma$ is considered, where $\Gamma$ is theLorentz factor of the emitting shell. The shell, with comoving width $\Deltar^\prime$, is assumed to be illuminated for a comoving time $\Delta t^\prime$and to radiate a broken power-law $\nu L_\nu$ spectrum peaking at comovingphoton energy $\e_{pk,0}^{\prime}$. Synthetic GRB pulses are calculated, andthe relation between energy flux and internal comoving energy density isquantified. Curvature effects dictate that the measured $\nu F_\nu$ flux at themeasured peak photon energy $\e_{pk}$ is proportional to $\e^3_{pk}$ in thedeclining phase of a GRB pulse. Possible reasons for discrepancy withobservations are discussed, including adiabatic and radiative cooling processesthat extend the decay timescale, a nonuniform jet, or the formation of pulsesby external shock processes. A prediction of a correlation between promptemission properties and times of the optical afterglow beaming breaks is madefor a cooling model, which can be tested with Swift.Comment: 13 pages, 5 figures, added back-of-envelope estimate of curvature relation, minor corrections, ApJ, in press, v. 614, 10 Oct 200