Modeling Gamma-Ray Burst X-Ray Flares within the Internal Shock Model

X-ray afterglow light curves have been collected for over 400 Swift gamma-ray bursts with nearly half of them having X-ray flares superimposed on the regular afterglow decay. Evidence suggests that gamma-ray prompt emission and X-ray flares share a common origin and that at least some flares can only be explained by long-lasting central engine activity. We have developed a shell model code to address the question of how X-ray flares are produced within the framework of the internal shock model. The shell model creates randomized GRB explosions from a central engine with multiple shells and follows those shells as they collide, merge and spread, producing prompt emission and X-ray flares. We pay special attention to the time history of central engine activity, internal shocks, and observed flares, but do not calculate the shock dynamics and radiation processes in detail. Using the empirical E_p - E_iso (Amati) relation with an assumed Band function spectrum for each collision and an empirical flare temporal profile, we calculate the gamma-ray (Swift/BAT band) and X-ray (Swift/XRT band) lightcurves for arbitrary central engine activity and compare the model results with the observational data. We show that the observed X-ray flare phenomenology can be explained within the internal shock model. The number, width and occurring time of flares are then used to diagnose the central engine activity, putting constraints on the energy, ejection time, width and number of ejected shells. We find that the observed X-ray flare time history generally reflects the time history of the central engine, which reactivates multiple times after the prompt emission phase with progressively reduced energy...