3 Fitting the afterglow of GRB 021211

Using the emission features of reverse shock and forward shock described above, we can fit the optical light curve of GRB 021211. Here we take the values z=0.8, $E=3.2\times 10^{51}$ ergs, and p=2.3.

For the forward shock emission, the observed optical flux is

$$\frac{F_{\rm\nu,f}(t)}{F_{\rm\nu,max,f}}=\left\{\begin{array}... ...} \;\;\;t_{\rm m,f}< t < t_{\rm c,f}. \end{array} \right .$$ (11)

Using Eqs. (3), (5) and (11) we can give the afterglow light curve from the forward shock, as shown in Fig. 1 by the dashed line. From fitting the observed data we can obtain the relation

$$\epsilon_{\rm B}\left(\frac{\epsilon_{\rm e}}{0.1}\right)^{8/5}n^{3/5} \sim 9.1\times 10^{-4} .$$ (12)

In addition, the observation implies that  $t_{\rm m, f}$ should be less than 100 s (if $t_{\rm m, f}>100$ s, there will be a bump in the afterglow light curve), and  $t_{\rm c, f}$ should be larger than 1 day (otherwise there will be a steepening of the light curve), so from Eqs. (3), (4) we have

$$\epsilon_{\rm B}^{1/4}\left(\frac{\epsilon_{\rm e}}{0.1}\right) \leq 0.14$$ (13)

$$\epsilon_{\rm B}n^{2/3} \leq 0.04.$$ (14)

For the reverse shock emission, for $t > t_{\rm A}$, we have the relations $\nu_{\rm m,r}(t)=\nu_{\rm m,r}(t_{\rm A})\left(\frac{t}{t_{\rm A}}\right)^{-73/48}$, $F_{\rm \nu,max,r}(t)=F_{\rm\nu,max,r}(t_{\rm A})\left(\frac{t}{t_{\rm A}}\right)^{-47/48}$, then the observed flux can be written as

$$F_{\rm\nu,r}(t) F_{\rm\nu,max,r}(t)\left [\frac{\nu}{\nu_{\rm m,r}(t)}\right ]^{-(p-1)/2}$$ = (15)

$$F_{\rm\nu,max,r}(t_{\rm A})\left [\frac{\nu}{\nu_{\rm m,r}(t_{\rm... ... ]^{-\frac{(p-1)}{2}}\left(\frac{t}{t_{\rm A}}\right)^{-\frac{73p+21}{96}}\cdot$$ = (16)

Using Eqs. (8), (10) and (16) we can give the afterglow light curve from the reverse shock, as shown in Fig. 1 by the dotted line. From fitting we can obtain the relation

$$\left(\frac{\gamma_0}{300}\right)^{23/10}\left(\frac{\gamma_{... ...{\rm e}}{0.1}\right)^{13/10}n^{33/40}t_{\rm A}^2 \sim 9.8.$$ (17)

Combining Eqs. (12), (14) and (17), we get

$$\left(\frac{\gamma_0}{300}\right)\left(\frac{\gamma_{\rm A}... ...sim 31\epsilon_{\rm B}^{-1/184}n^{-27/184}t_{\rm A}^{-20/23}$$ (18)

$$\epsilon_{\rm B}^{1/16}\left(\frac{\epsilon_{\rm e}}{0.1}\right) \geq 0.077.$$ (19)

Therefore Eqs. (12), (13) and (19) give the constraint on the parameters  $\epsilon _{\rm B}$, $\epsilon _{\rm e}$ and n. Figure 2 shows the relation between  $\epsilon _{\rm B}$, $\epsilon _{\rm e}$ and n. The dotted, dash-dotted, dashed and dot-dot-dashed lines represent n=0.1, 1, 10 and 0.0016 respectively. We find that the allowed values of  $\epsilon _{\rm B}$ and $\epsilon _{\rm e}$ lie in the region confined by two lines Lc1 (Eq. (19)) and Lc2 (Eq. (13)). It is obvious that n must be larger than 0.0016, and  $\epsilon _{\rm e}$ must be larger than 0.0077. If we take n=1, $\epsilon_{\rm e}=0.07$, then  $\epsilon_{\rm B}=1.6\times 10^{-3}$. We propose that more observations are needed in order to further estimate the values of  $\epsilon _{\rm B}$, $\epsilon _{\rm e}$ and n.

Figure 1: The optical light curve of GRB 021211. The dashed line is the emission of the forward shock, the dotted line represents the emission from reverse shock, and the solid line is the total flux. Data from: Price & Fox (2002a, 2002b), Park et al. (2002), Li et al. (2002), Kinugasa et al. (2002), McLeod et al. (2002), Wozniak et al. (2002), Levan et al. (2002).

Figure 2: The relation between $\epsilon _{\rm B}$, $\epsilon _{\rm e}$ and n given by Eqs. (12), (13) and (19). The dotted, dash-dotted, dashed and dot-dot-dashed lines represent n=0.1, 1, 10 and 0.0016 respectively. The allowed values of  $\epsilon _{\rm B}$ and  $\epsilon _{\rm e}$ lie in the region confined by two lines Lc1 (Eq. (19)) and Lc2 (Eq. (13)).

From Eq. (18) we see that the initial Lorentz factor $\gamma_0$ depends on  $\epsilon _{\rm B}$ and n very weakly, so as an approximation, and taking  $\gamma_{\rm A}\sim \gamma_0$, then we have  $\gamma_0\sim 9300t_{\rm A}^{-20/23}$. Since the duration is about 15 s and the first observation time is 65 s after the burst, so the value of $t_{\rm A}$ should lie between 15 s and 65 s, and therefore we can get the initial Lorentz factor $250<\gamma_0 <900$, which is consistent with the lower limit estimates base on the $\gamma$-$\gamma$ attenuation calculation (Fenimore et al. 1993).