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 ($\theta_{\rm j}$) 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 ($\theta_{\rm v}$) from the jet axis, and the jet is uniform or non-uniform (the energy per unit solid angle decreases smoothly away from the axis $\epsilon (\theta)\propto (\theta/\theta_{\rm c})^{-k}$). 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 $\gamma\sim (\theta_{\rm j}-\theta_{\rm v})^{-1}$, and the second one corresponds to the time when $\gamma\sim (\theta_{\rm j}+\theta_{\rm v})^{-1}$. However, for non-uniform jet, the things become more complicated. For the case $\theta_{\rm v}=0$, 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 $\gamma\sim\theta_{\rm c}^{-1}$ and $\gamma\sim\theta_{\rm j}^{-1}$ respectively, while for $k>8/(p+4)$ there should be only one break corresponds to $\gamma\sim\theta_{\rm c}^{-1}$, and this provides a possible explanation for some rapidly fading afterglows whose light curves have no breaks since the time at which $\gamma\sim\theta_{\rm c}^{-1}$ is much earlier than our first observation time. For the case $\theta_{\rm v}\neq 0$, our numerical results show that, the afterglow light curves are strongly affected by the values of $\theta_{\rm v}$, $\theta_{\rm c}$ and k. If $\theta_{\rm v}$ is close to $\theta_{\rm c}$ 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 $\theta_{\rm v}/\theta_{\rm c}$ 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 $\gamma\sim (\theta_{\rm v} + \theta_{\rm c})^{-1}$