Relation between the Intrinsic and Observed Central Engine Activity Time: Implications for Ultra-long GRBs

The gamma-ray burst (GRB) central engine intrinsic activity time is usually described through either the ?-ray duration T90 or through a generalized burst duration that includes both the ?-ray emission and (when present) an extended flaring X-ray plateau. Here, we define a more specific operational description of , and within the framework of the internal?external shock model, we develop a numerical code to study the relationship between T90 and , as well as between and , for different initial conditions. We find that when s, late internal collisions or refreshed external collisions result in values of and larger than , usually by factors of 2?3. For s, the is always a good estimator for , while T90 can underpredict when the late central engine activity is moderate. We find a clear bimodal distribution for , based on our simulations as well as on the observational data for T90 and . We suggest that is a reliable measure for defining ?ultra-long? GRBs. Bursts with T90 of order 103 s need not belong to a special population, while bursts with s, where the late central engine activity is more moderate and shows up in X-rays, may represent a new population. These conclusions are insensitive to the initial conditions assumed in the models.