Tail emission from a ring-like jet: its application to shallow decays of early afterglows and to GRB 050709

Similar to the pulsar, the magnetic axis and the spin axis of the gamma-ray burst source may not lie on the same line. This may cause a ring-like jet due to collimation of the precessing magnetic axis. We analyze the tail emission from such a jet, and find that it has a shallow decay phase with temporal index equal to -1/2 if the Lorentz factor of the ejecta is not very high. This phase is consistent with the shallow decay phase of some early X-ray afterglow detected by {\it{swift}}. The ring-like jet has a tail cusp with sharp rising and very sharp decay. This effect can provide an explanation for the re-brightening and sharp decay of the X-ray afterglow of GRB 050709.
6 pages, 2 figures. Accepted by ChJAA