Steepening of Afterglow Decay for Jets Interacting with Stratified Media

We calculate light-curves for Gamma-Ray Burst afterglows when materialejected in the explosion is confined to a jet which propagates in a medium witha power-law density profile. The observed light-curve decay steepens by afactor of $\Gamma^2$ when an observer sees the edge of the jet. In a uniformdensity medium the increase in the power-law index ($\beta$) of the light-curveas a result of this {\it edge effect} is $\sim0.7$ and is completed over onedecade 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 effectand the steepening of the light-curve due to the jet sideways expansion takesabout four decades in time. Therefore, a break in the light-curve for a jet ina wind model is unlikely to be detected even for very narrow jets of openingangle of a few degrees or less, in which case the lateral expansion occurs atearly 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 beexplained only by the sideways expansion and the edge effects in a jet in auniform ISM -- the increase in $\beta$ is too large and too rapid. However, thepassage of the cooling or synchrotron peak frequencies through the observingband at about 0.1 -- 1 day together with jet edge effect explains the observeddata. The jet opening angle is found to be $\sim 5^o$ and the energy in theexplosion to be less than about $10^{50}$ erg.Comment: 6 pages including 3 figures, submitted to ApJ lette