2024MNRAS.535.2482Y

Long-duration gamma-ray bursts (GRBs) are thought to be from core collapse of massive stars, and a rapidly spinning magnetar or black hole may be formed as the central engine. The extended emission in the prompt emission, flares, and plateaus in X-ray afterglow, are proposed to be as the signature of central engine re-activity. However, the direct evidence from observations of identifying the central engines remains an open question. In this paper, we systemically search for long-duration GRBs that consist of bumps in X-ray afterglow detected by Swift/XRT and find that the peak time of the X-ray bumps exhibit bimodal distribution (defined as 'early' and 'late' bumps) with division line at $t=7190$ s. Although we cannot rule out that such a bimodality arises from selection effects. We proposed that the long-duration GRBs with an early (or late) bumps may be originated from the fall-back accretion onto a new-born magnetar (or black hole). By adopting Monte Carlo Markov Chain (MCMC) method to fit the early (or late) bumps of X-ray afterglow with the fall-back accretion of magnetar (or black hole), it is found that the initial surface magnetic field and period of magnetars for most early bumps are clustered around $5.88\times 10^{13}$ G and 1.04 ms, respectively. Meanwhile, the derived accretion mass of black hole for late bumps is in the range of $[4\times 10^{-4}, 1.8\times 10^{-2}]\,{\rm M}_{\odot }$, and the typical fall-back radius is distributed range of $[1.04, 4.23]\times 10^{11}$ cm, which is consistent with the typical radius of a Wolf-Rayet star. However, we also find that the fall-back accretion magnetar model is disfavoured by the late bumps, but the fall-back accretion of black hole model cannot be ruled out to interpret the early bumps of X-ray afterglow.