GRB afterglow light curves from uniform and non-uniform jets

It is widely believed that gamma-ray bursts are produced by a jet-like outflows directed towards the observer, and the jet opening angle(θj) is often inferred from the time at which there is a break in the afterglow light curves. Here we calculate the GRB afterglow light curves from a relativistic jet as seen by observers at a wide range of viewing angles (θv) from the jet axis, and the jet is uniform or non-uniform(the energy per unit solid angle decreases smoothly away from the axis ǫ(θ) ∝ (θ/θc) ). We find that, for uniform jet(k = 0), the afterglow light curves for different viewing angles are somewhat different: in general, there are two breaks in the light curve, the first one corresponds to the time at which γ ∼ (θj − θv) , and the second one corresponds to the time when γ ∼ (θj + θv) . However, for non-uniform jet, the things become more complicated. For the case θv = 0, we can obtain the analytical results, for k < 8/(p + 4)(where p is the spectral index of electron energy distribution) there should be two breaks in the light curve correspond to γ ∼ θ c and γ ∼ θ −1 j respectively, while for k > 8/(p+4) there should be only one break corresponds to γ ∼ θ c , and this provides a possible explanation for some rapidly fading afterglows whose light curves have no breaks since the time at which γ ∼ θ c is much earlier than our first observation time. For the case θv 6= 0, our numerical results show that, the afterglow light curves are strongly affected by the values of θv, θc and k. If θv is close to θc and k is small, then the light curve is similar to the case of k = 0, except the flux is somewhat lower. However, if the values of θv/θc and k are larger, there will be a prominent flattening in the afterglow light curve, which is quite different from the uniform jet, and after the flattening a very sharp break will be occurred at the time γ ∼ (θv + θc) .