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 that 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 edge effect is similar to 0.7 and is completed over one decade in observer time. For a preejected stellar wind (rho proportional to r(-2)), beta increases by -0.4 over two decades in time as a result of the edge effect, and the steepening of the light curve as a result of 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 a very narrow opening angle of a few degrees or less, a case where 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 timescale of 3 days, cannot be explained by only the sideways expansion and the edge effects in a jet in a uniform interstellar medium-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 similar to 5 degrees, and the energy in the explosion to be about 10(51) ergs.