Study on the X-ray Flare of gamma-ray burst based on the fallback accretion model

ABSTRACT Approximately half of Swift gamma-ray bursts (GRBs) consists the X-ray flare in their X-ray afterglows. Up to now, the physical origin of X-ray flares remains a mystery. In this paper, we present a theoretical model to interpret the X-ray flares observed in long GRBs. In this model, the shockwave from a supernova explosion may break up outer materials of the progenitor stars into many clumps. The clumpy material with insufficient kinetic energy would fall back on to the black hole accretion disc at later times and releases energy through the Blandford–Znajek mechanism. This energy contributes to the X-ray flare emission of GRBs through internal dissipation processes. Based on Swift/XRT observational data up to 2024 December, we selected 66 long GRBs with X-ray flares. By applying our model to this sample, we find that the model can effectively interpret the X-ray flares within the framework of the progenitor star’s density profile. According to the fitting results, it can be inferred that the X-ray flares may originate from the clumpy material with peak densities of $10^{-4}{\!-\!}10^{4}\ \mathrm{g\ cm^{-3}}$ and radii spanning $10^{9}{\!-\!}10^{11}\ \mathrm{cm}$ . This radial range aligns with the typical radii of Wolf–Rayet stars. The mass of the clumpy material that produces the X-ray flare segment is ${\sim} 10^{-4} {\rm M}_{\odot }- {\rm M}_{\odot }$, consistent with the observations ejecta mass of GRB-SNe. Furthermore, our analysis reveals that the ratio of the angular velocity to the local Keplerian angular velocity, $f_{\Omega }$, is as low as ${\sim} 0.2$, which indicates that the progenitor star of long GRB was rotating slowly in the final moment before collapse. In summary, our results provide additional support for the origin of long GRBs from the collapse of massive stars.