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A long-duration gamma-ray burst (GRB) has been widely thought to arise from the collapse of a massive star, and it has been suggested that its ambient medium is a homogenous interstellar medium (ISM) or a stellar wind. There are two shocks when an ultra-relativistic fireball that has been ejected during the prompt gamma-ray emission phase sweeps up the circumburst medium: a reverse shock that propagates into the fireball, and a forward shock that propagates into the ambient medium. In this paper, we investigate the temporal evolution of the dynamics and emission of these two shocks in an environment with a general density distribution of n∝R–k (where R is the radius) by considering thick-shell and thin-shell cases. A GRB afterglow with one smooth onset peak at early times is understood to result from such external shocks. Thus, we can determine the medium density distribution by fitting the onset peak appearing in the light curve of an early optical afterglow. We apply our model to 19 GRBs and find that their k values are in the range of 0.4-1.4, with a typical value of k ∼ 1, implying that this environment is neither a homogenous ISM with k = 0 nor a typical stellar wind with k = 2. This shows that the progenitors of these GRBs might have undergone a new mass-loss evolution. [ABSTRACT FROM AUTHOR]

Abstract: Gamma-ray bursts are most luminous explosions in the universe. Their ejecta are believed to move towards Earth with a relativistic speed. The interaction between this “relativistic jet” and a circumburst medium drives a pair of (forward and reverse) shocks. The electrons accelerated in these shocks radiate synchrotron emission to power the broad-band afterglow of GRBs. The external shock theory is an elegant theory, since it invokes a limit number of model parameters, and has well predicted spectral and temporal properties. On the other hand, depending on many factors (e.g. the energy content, ambient density profile, collimation of the ejecta, forward vs. reverse shock dynamics, and synchrotron spectral regimes), there is a wide variety of the models. These models have distinct predictions on the afterglow decaying indices, the spectral indices, and the relations between them (the so-called “closure relations”), which have been widely used to interpret the rich multi-wavelength afterglow observations. This review article provides a complete reference of all the analytical synchrotron external shock afterglow models by deriving the temporal and spectral indices of all the models in all spectral regimes, including some regimes that have not been published before. The review article is designated to serve as a useful tool for afterglow observers to quickly identify relevant models to interpret their data. The limitations of the analytical models are reviewed, with a list of situations summarized when numerical treatments are needed. [Copyright &y& Elsevier]