A Comptonization Model for the Prompt Optical and Infrared Emission of GRB 041219A

Prompt optical emission from the $\gamma$-ray burst of GRB 041219A has been reported by Vestrand et al. There was a fast rise of optical emission simultaneous with the dominant $\gamma$-ray pulse, and a tight correlation with the prompt $\gamma$-ray emission has been displayed. These indicate that the prompt optical emission and $\gamma$-ray emission would naturally have a common origin. We propose that this optical component can be modeled by considering the Comptonization of $\gamma$-ray photons by an electron cloud. As a result of this mechanism, the arrival time of the optical photons is delayed compared with that of the $\gamma$-rays. We restrict that the lag time to be shorter than 10 s, within which the prompt optical emission is considered to vary simultaneously with the prompt $\gamma$-ray emission. Taking the observations of GRB 041219A into account, we derive the number density of the surrounding electron cloud required for Comptonization. The redshift of GRB 041219A is predicted to be $z\lesssim 0.073$ as well.