4 Discussion and conclusion

The current afterglow observations usually detect radiation several hours after the burst, at this stage the Lorentz factor is independent of the initial Lorentz factor, thus these observations do not provide useful information on the initial extreme relativistic motion. The initial Lorentz factor is a very important quantity for constraining the GRB models since it specifies how "clean'' the fireball is. Therefore to detect the early afterglow of GRBs is very important, since it can provide the information on the initial Lorentz factor. It is fortunately that the early afterglow of GRB 021211 have been observed, by fitting its optical light curve we obtain its initial Lorentz factor $250<\gamma_0 <900$, this value seems reasonable since it is widely believed that the initial fireball Lorentz factor should be larger than 100 in order to avoid photon-photon attenuation. Of course, further constraint on the value of $\gamma_0$ needs more early afterglow observations.

GRB 990123 is the first burst for which its optical flash was observed, the peak flux was about 1 Jy in R-band. After that many efforts have been made to try to find the optical flash from other GRBs, but only upper limits are given (Akerlof et al. 2000). Here we also note that the optical flux of GRB 021211 is substantially fainter than 990123 at similar epochs. Why GRB 990123 is so bright? One reason may be that GRB 990123 is a very bright burst, so its reverse shock emission is also very strong. On the other hand, from fitting we note that the values of  $\epsilon _{\rm B}$ and  $\epsilon _{\rm e}$ of GRB 021211 are somewhat smaller, which leads to the fact that the typical synchrotron frequency of reverse shock is well below the optical band, so the early afterglow (or optical flash) is weak. While for GRB 990123 the typical synchrotron frequency of reverse shock is close to the optical band (Sari & Piran 1999a; Kobayashi & Zhang 2003).

For smaller values of $\epsilon _{\rm B}$ and  $\epsilon _{\rm e}$, not only the typical synchrotron frequency of reverse shock is small, but also the typical synchrotron frequency of forward shock is small, so the time  $t_{\rm m, f}$ when the typical frequency of forward shock crosses the optical band is also small, for GRB 021211, the observations required  $t_{\rm m,f}\leq 100$ s. The late time afterglow for $t > t_{\rm m, f}$ is $F_{\nu}=F_{\rm\nu, max, f}\left(t/t_{\rm m,f}\right)^{-3(p-1)/4}$, so for smaller value of  $t_{\rm m, f}$, the observed optical flux should be much fainter than those with larger values of  $t_{\rm m, f}$, so we suggest that the so-called dark bursts whose afterglow have not been observed might be due to their very small values of  $t_{\rm m, f}$.

The early afterglow of GRB 021211 shows that there is a early break in its optical light curve, before the break time the flux declined with a power-law index of about -1.6, while at later time the flux decayed with a slope of about -1. Although the reverse shock model predicts that the optical flux should decay with a power-law index of about -2, here we show that the superposition of both the forward shock and the reverse shock emission can well account for the observed light curve. Therefore we suggest that this early break might be a common feature in early optical afterglow, and before the break time the slope of flux decline may be flatter than -2 since it contains the contribution from both the reverse shock and the forward shock emission.

Acknowledgements

This work is supported by the National Natural Science Foundation (grants 10073022 and 10225314) and the National 973 Project on Fundamental Researches of China (NKBRSF G19990754).