Hyperaccreting Black Hole as Gamma-Ray Burst Central Engine. II. Temporal evolution of central engine parameters during Prompt and Afterglow Phases

A hyperaccreting stellar-mass black hole has been proposed as the candidate central engine of gamma-ray bursts (GRBs). The rich observations of GRBs by \textit{Fermi} and \textit{Swift} make it possible to constrain the central engine model by comparing the model predications against data. This paper is dedicated to studying the temporal evolution of central engine parameters for both prompt emission and afterglow phases. We consider two jet launching mechanisms, i.e., $\nu\bar{\nu}$ annihilations and the Blandford-Znajek (BZ) processe, and obtain analytical solutions to these two models. We then investigate the black hole central engine parameters, such as the jet power, the dimensionless entropy $\eta$, and the central engine parameter $\mu_0=\eta (1+\sigma_0)$ (where $\sigma_0$ is the initial magnetization of the engine) at the base of the jet. The black hole may be spun up by accretion, or spun down by the BZ process, leaving imprints in GRB lightcurves. Usually, a BZ jet is more powerful and is likely responsible for the late time central engine activities. However, an initially non-spinning black hole central engine may first launch a thermal "fireball" via neutrino annihilations, and then launch a Poynting-flux-dominated jet via the BZ process. Multiple flares, giant bumps, and plateaus in GRB afterglows can be well produced as the result of late time accretion onto the black hole.