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A long-duration gamma-ray burst (GRB) has been widely thought to arise fromthe collapse of a massive star, and it has been suggested that its ambientmedium is a homogenous interstellar medium (ISM) or a stellar wind. There aretwo shocks when an ultra-relativistic fireball that has been ejected during theprompt gamma-ray emission phase sweeps up the circumburst medium: a reverseshock that propagates into the fireball, and a forward shock that propagatesinto the ambient medium. In this paper, we investigate the temporal evolutionof the dynamics and emission of these two shocks in an environment with ageneral density distribution of $n\propto R^{-k}$ (where $R$ is the radius) byconsidering thick-shell and thin-shell cases. A GRB afterglow with one smoothonset peak at early times is understood to result from such external shocks.Thus, we can determine the medium density distribution by fitting the onsetpeak appearing in the light curve of an early optical afterglow. We apply ourmodel to 19 GRBs, and find that their $k$ values are in the range of 0.4 - 1.4,with a typical value of $k\sim1$, implying that this environment is neither ahomogenous interstellar medium with $k=0$ nor a typical stellar wind with$k=2$. This shows that the progenitors of these GRBs might have undergone a newmass-loss evolution.

Early Afterglows of Gamma-Ray Bursts in a Stratified Medium with a Power-Law Density Distribution