External Shock in a Multi-bursting Gamma-Ray Burst: Energy Injection Phase Induced by the Later Launched Ejecta

Central engines of gamma-ray bursts (GRBs) may be intermittent and launch several episodes of ejecta separated by a long quiescent interval. In this scenario, an external shock is formed due to the propagation of the first launched ejecta into the circum-burst medium and the later launched ejecta may interact with the external shock at a later period. Owing to the internal dissipation, the later launched ejecta may be observed at a later time (tjet). In this paper, we study the relation of tb and tjet, where tb is the collision time of the later launched ejecta with the formed external shock. It is found that the relation of tb and tjet depends on the bulk Lorentz factor (Γjet) of the later launched ejecta and the density ( ) of the circum-burst medium. If the value of Γjet or is low, the tb would be significantly larger than tjet. However, the tb ∼ tjet can be found if the value of Γjet or is significantly large. Our results can explain the large lag of the optical emission relative to the γ-ray/X-ray emission in GRBs, e.g., GRB 111209A. For GRBs with a precursor, our results suggest that the energy injection into the external shock and thus more than one external-reverse shock may appear in the main prompt emission phase. According to our model, we estimate the Lorentz factor of the second launched ejecta in GRB 160625B.