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We present an analytical approach to the dynamical evolution of fireballs or axially symmetric 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 simpler than hydrodynamic simulations. It is also a very flexible approach, that can be easily extended to include more complex situations, such as a continuous injection of energy at the reverse shock, and the sideways expansion in non-spherical ejecta. Numerical results for the dynamic evolution are discussed and compared to the analytical solutions. The light curves computed from beamed ejecta show that, due to the increased swept up matter and the time delay of the large angle emission, the sideways expansion of the remnant gas does not lead to a dimming of the afterglow, and mitigates the subsequent downturn from the transition into the non-relativistic regime

Dynamical Evolution, Light Curves, and Spectra of Spherical and Collimated Gamma‐Ray Burst Remnants