Dynamical evolution, light curves, and spectra of spherical and collimated gamma-ray burst remnants

We present an analytical approach to the dynamical evolution of fireballs or axisymmetric jets expanding into an external medium, with application to gamma-ray burst remnants. This method leads to numerical calculations of fireball dynamics that are computationally faster than hydrodynamic simulations. It is also a very flexible approach that can easily be extended to include more complex situations, such as a continuous injection of energy at the reverse shock and the sideways expansion in nonspherical ejecta. Some features of the numerical results for the remnant dynamical evolution are discussed and compared with the analytical results. We find that the ratio of the observer time at which the jet half-angle reaches twice its initial value to the time at which the light cone becomes wider than the jet is substantially smaller than predicted analytically. The effects that arise from the remnant's curvature and thickness further reduce this ratio, such that the afterglow light-curve break that is due to the remnant's finite angular extent overlaps the weaker break that is due to the jet's sideways expansion. An analysis of the effects of the curvature and thickness of the remnant on the afterglow light curves shows that these effects are important and should be taken into account for accurate calculations of the afterglow emission.