Relation between the intrinsic and observed central engine activity time: implications for ultra-long GRBs

The GRB central engine intrinsic activity time $T_{\rm ce}$ is usually described through either the $\gamma$-ray duration $T_{90}$ or through a generalized burst duration $t_{\rm burst}$ which includes both the $\gamma$-ray emission and (when present) an extended flaring X-ray plateau. Here, we define a more specific operational description of $T_{\rm ce}$, and within the framework of the internal-external shock model, we develop a numerical code to study the relationship between $T_{90}$ and $T_{\rm ce}$, as well as between $t_{\rm burst}$ and $T_{\rm ce}$, for different initial conditions. We find that when $T_{\rm ce}\lesssim 10^4$ s, late internal collisions or refreshed external collisions result in values of $T_{\rm 90}$ and $t_{\rm burst}$ larger than $T_{\rm ce}$, usually by factors of $2-3$. For $T_{\rm ce}\gtrsim 10^4$ s, the $t_{\rm burst}$ is always a good estimator for $T_{\rm ce}$, while $T_{90}$ can underpredict $T_{\rm ce}$ when the late central engine activity is moderate. We find a clear bimodal distribution for $T_{\rm ce}$, based on our simulations as well as on the observational data for $T_{90}$ and $t_{\rm burst}$. We suggest that $t_{\rm burst}$ is a reliable measure for defining "ultra-long" GRBs. Bursts with $T_{90}$ of order $10^3$ s need not belong to a special population, while bursts with $t_{\rm burst} > 10^4$ 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.