Steepening of Afterglow Decay for Jets Interacting with Stratified Media

We calculate light-curves for Gamma-Ray Burst afterglows when material ejected in the explosion is confined to a jet which propagates in a medium with a power-law density profile. The observed light-curve decay steepens by a factor of $\Gamma^2$ when an observer sees the edge of the jet. In a uniform density medium the increase in the power-law index ($\beta$) of the light-curve as a result of this {\it edge effect} is $\sim0.7$ and is completed over one decade in observer time. For a pre-ejected stellar wind ($\rho \propto r^{-2}$) $\beta$ increases by $\sim0.4$ over two decades in time due to the edge effect and the steepening of the light-curve due to the jet sideways expansion takes about four decades in time. Therefore, a break in the light-curve for a jet in a wind model is unlikely to be detected even for very narrow jets of opening angle of a few degrees or less, in which case the lateral expansion occurs at early times when the afterglow is bright. The light-curve for the afterglow of GRB 990510, for which an increase in $\beta$ of approximately 1.35 was observed on a time scale of 3 days, cannot be explained only by the sideways expansion and the edge effects in a jet in a uniform ISM -- the increase in $\beta$ is too large and too rapid. However, the passage of the cooling or synchrotron peak frequencies through the observing band at about 0.1 -- 1 day together with jet edge effect explains the observed data. The jet opening angle is found to be $\sim 5^o$ and the energy in the explosion to be less than about $10^{50}$ erg.