Our fitting is performed based on the Markov Chain Monte Carlo (MCMC) method to produce posterior predictions for the model parameters. The MCMC method is widely used in finding the best set of parameters for a specified model, e.g., GRB 080413B (Geng et al. 2016); GRBs 100418A, 100901A, 120326A, and 120404A (Laskar et al. 2015). In our work, fitting the afterglows of the four bursts with the MCMC method is to test whether or not the late simultaneous bumps followed by a steep decay can be explained with an external-forward shock in a free-to-shocked wind environment. The posterior probability density functions for the physical parameters, i.e., Ek,iso,on, Γ0, θc, θjet/θc, θv/θc, p, ϵe, ϵB, A*, and Rtr, are presented in Figure 6, where only the fitting result of GRB 120326A is shown as an example. The optimal result from MCMC fitting is shown in Figure 5 with the blue line (XRT) and red line (optical), and the obtained parameters at the 1σ confidence level are reported in Table 1, where the values of the transition radius 






R


tr




 (i.e., 1.05 × 1017, 1.91 × 1017, 3.98 ×1017, and 6.31 × 1016 cm for GRBs 120326A, 100901A, 100814A, and 120404A, respectively) are consistent with those found in other bursts (e.g., Ramirez-Ruiz et al. 2001; Kong et al. 2010; Feng & Dai 2011; Li et al. 2020). It can be found that both the X-ray afterglow and the optical afterglow of these bumps can be well modeled with an external-forward shock in a free-to-shocked wind environment.

1.
The theoretical light curves do not well fit the transition behavior from the free-wind phase to the shocked-wind phase in the afterglows of GRB 120326A. This may imply that the density jump factor ξ from the free-wind medium to the shocked-wind medium may be less than 4, i.e., ξ  4. Figure 1 reveals that the external-forward shock propagating in a shocked-wind environment can yield a plateau before the late simultaneous bumps in the case of θv = 4θc. Then, we try to fit the afterglows of GRB 120326A with the emission of the external-forward shock propagating in a homogeneous medium, where the priors of 





log
10

(

E

k
,
iso
,
on




erg

-
1


)


, 






log


10




Γ


0




, θc, θjet/θc, θv/θc, p, 






log


10




ϵ


e




, 






log


10




ϵ


B




, and 






log


10




n


0




 are set as a uniform distribution in the range of (52, 55), (1.5, 3.0), (0.3, 8.0), (2.5, 4.5), (0.0, 8.0), (2.1, 2.9), (−3.2, −0.5), (−6.5, −2.0), and (0.0, 1.7) in our MCMC fittings, respectively. However, no well-fitting result is found. The fitting result with minimum reduced χ2 is shown with dashed lines in the top left panel of Figure 5. It can be found that the early plateau of GRB 120326A could not be well fitted based on the emission of the external-forward shock in a homogeneous medium.


2.
There seems to be a plateau at tobs ∼ 600 s in the optical afterglow of GRB 100814A, which may indicate an energy injection into the external shock. If so, our model could not well describe the afterglows of GRB 100814A in its early phase. This may be the reason for the deviation in our theoretical results relative to the observations in the early phase. In addition, the X-ray and optical afterglows of GRB 100814A show achromatic behavior, which could not be explained only by our scenario (i.e., an off-core observed structured jet propagating into a free-to-shocked wind environment) since the appearance of the late simultaneous bump/plateau in our scenario is achromatic. Other emission rather than that of the external-forward shock may contribute to the long X-ray shallow decay. In addition, our model could not well describe the X-ray emission at tobs ≳ 3 × 105 s in GRB 100901A, which seems to be another component in this phase.


3.
The fitting result of GRB 120404A shows that this burst is observed in the core. Based on the fitting result of this burst, the deceleration radius of the (0, 0) jet can be estimated and is 2.43 × 1016 cm, which is at around the transition radius 






R


tr


=
6.31
×


10


16




 cm of this burst.
Citation: Feng & Dai 2011
Citation function: Similarities