When the axis of a gamma-ray burst （GRB） does not coincide with the spin axis of its source, there may result a ring-shaped jet. Using some refined jet dynamics, we calculate multi-wavelength afterglow light curves for such ring-shaped jets. In the R-band we find an obvious break in the afterglow light curve due to the beaming effect and the break is affected by many parameters, such as the electron energy fraction ζe, the magnetic energy fraction ζB^2, the width of ring △θ and the medium number density n. The overall light curve can be divided into three power-law stages, i.e., an ultra-relativistic stage, an after-break stage and a deep Newtonian stage. For each stage the power-law index is larger in the ring-shaped jet than in the corresponding conical jet.
Extensive multi-band afterglow data are available for GRB 980703. Especially,its radio afterglow was very bright and was monitored until more than 1000 days after the trigger time. Additionally,there is no obvious special feature,i.e.,no rebrightenings,no plateau,and no special steep decay or slow decay in the multi-band afterglow light curves. All these conditions make GRB 980703 a precious sample in gammaray burst research. Here we use the observational data of GRB 980703 to test the standard fireball model in depth. It is found that the model can give a satisfactory explanation to the multi-band and overall afterglow light curves. The beaming angle of GRB 980703 is derived as ～ 0.23 radian,and the circum-burst medium density is ～ 27 cm-3. The total isotropic equivalent kinetic energy of the ejecta is ～ 3.8 × 1052 ergs. A rest-frame extinction of AV ～ 2.5 mag in the host galaxy is also derived.
Multiple rebrightenings have been observed in the multiband afterglow of GRB 030329. In particular, a marked and quick rebrightening occurred at about t 1.2 × 10^5 s. Energy injection from late and slow shells seems to be the best interpretation for these rebrightenings. Usually it is assumed that the energy is injected into the whole external shock. However, in the case of GRB 030329, the rebrightenings are so quick that the usual consideration fails to give a satisfactory fit to the observed light curves. Actually, since these late/slow shells freely coast in the wake of the external shock, they should be cold and may not expand laterally. The energy injection then should only occur at the central region of the external shock. Considering this effect, we numerically re-fit the quick rebrightenings observed in GRB 030329. By doing this, we were able to derive the beaming angle of the energy injection process. Our result, with a relative residual of only 5% - 10% during the major rebrightening, is bet- ter than any previous modeling. The derived energy injection angle is about 0.035. We assume that these late shells are ejected by the central engine via the same mechanism as those early shells that produce the prompt gamma-ray burst. The main difference is that their velocities are much slower, so that they catch up with the external shock relatively late and are manifested as the observed quick rebrightenings. If this were true, then the derived energy injection angle can give a good measure of the beaming angle of the prompt γ-ray emission. Our study may hopefully provide a novel method to measure the beaming angle of gamma-ray bursts.
A long plateau phase and an amazing level of brightness have been observed in the X-ray afterglow of GRB 060729. This peculiar light curve is likely due to longterm energy injection in external shock. Here, we present a detailed numerical study of the energy injection process of magnetic dipole radiation from a strongly magnetized millisecond pulsar and model the multi-band afterglow observations. It is found that this model can successfully explain the long plateaus in the observed X-ray and optical afterglow light curves. The sharp break following the plateaus could be due to the rapid decline of the emission power of the central pulsar. At an even later time （～ 5×10^6 s）, an obvious jet break appears, which implies a relatively large half opening angle of θ ～ 0.3 for the GRB ejecta. Due to the energy injection, the Lorentz factor of the outflow is still larger than two even at 10^7 s after the GRB trigger, making the X-ray afterglow of this burst detectable by Chandra even 642 d after the burst.
