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 T_ce is usually described through either the γ-ray duration T₉₀ or through a generalized burst duration t_burst that includes both the γ-ray emission and (when present) an extended flaring X-ray plateau. Here, we define a more specific operational description of T_ce, and within the framework of the internal-external shock model, we develop a numerical code to study the relationship between T₉₀ and T_ce, as well as between t_burst and T_ce, for different initial conditions. We find that when T_ce ≲ 10⁴ s, late internal collisions or refreshed external collisions result in values of T₉₀ and t_burst larger than T_ce, usually by factors of 2-3. For T_ce ≳ 10⁴ s, the t_burst is always a good estimator for T_ce, while T₉₀ can underpredict T_ce when the late central engine activity is moderate. We find a clear bimodal distribution for T_ce, based on our simulations as well as on the observational data for T₉₀ and t_burst. We suggest that t_burst 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 t_burst > 10⁴ 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.