Steepening of Afterglow Decay for Jets Interacting with Stratified Media

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Abstract
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 Γ2 when an observer sees the edge of the jet. In a uniform density medium, the increase in the power-law index (β) of the light curve as a result of this edge effect is ~0.7 and is completed over one decade in observer time. For a preejected stellar wind (ρ ∝ r-2), β 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 β 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 β 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 ~5°, and the energy in the explosion to be about 1051 ergs.
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