The Very Early Gamma-ray Burst Afterglows Powered by Structured Jets
  If X-ray flashes (XRFs) and X-ray rich Gamma-ray Bursts(XRRGs) have the same origin with Gamma-ray Bursts (GRBs) but are viewed from larger angles of structured jets, their early afterglows may differ from those of GRBs. When the ultra-relativistic outflow interact with the surrounding medium, there are two shocks formed, one is a forward shock, the other is a reverse shock. In this paper we calculate numerically the early afterglow powered by uniform jet, Gaussian jet and power-law jet in the forward-reverse shock scenario. A set of differential equations are used to govern the dynamical evolution and synchrotron self-Compton effect has been taken into account to calculate the emission. In uniform jets, the very early afterglows of XRRGs and XRFs are significantly lower than GRBs and the observed peak times of RS emission are longer in interstellar medium environment. The RS components in XRRGs and XRFs are difficult to be detected. But in stellar wind, the reduce of very early flux and the delay of RS peak time are not so remarkable. In nonuniform jet(Gaussian jet and power-law jet), where there are emission materials on the line of sight, the very early light curve resembles isotropic-equivalent ejecta in general although the RS flux decay index shows notable deviation if the RS is relativistic(in stellar wind). 
T. Yan, D. M. Wei and Y. Z. Fan