Statistical analyses on the energies of X-ray plateaus and flares in gamma-ray bursts

Distinct X-ray plateau and flare phases have been observed in the afterglows of gamma-ray bursts (GRBs), and most of them should be related to central engine activities. In this paper, we collect 174 GRBs with X-ray plateau phases and 106 GRBs with X-ray flares. There are 51 GRBs that overlap in the two selected samples. We analyze the distributions of the proportions of the plateau energy $E_{\rm plateau}$ and the flare energy $E_{\rm flare}$ relative to the isotropic prompt emission energy $E_{\rm \gamma,iso}$. The results indicate that they well meet the Gaussian distributions and the medians of the logarithmic ratios are $\sim -0.96$ and $-1.39$ in the two cases. Moreover, strong positive correlations between $E_{\rm plateau}$ (or $E_{\rm flare}$) and $E_{\rm \gamma,iso}$ with slopes of $\sim 0.95$ (or $\sim0.80$) are presented. For the overlapping sample, the slope is $\sim 0.80$. We argue that most of X-ray plateaus and flares might have the same physical origin but appear with different features because of the different circumstances and radiation mechanisms. We also test the applicabilities of two models, i.e., black holes surrounded by fractured hyperaccretion disks and millisecond magnetars, on the origins of X-ray plateaus and flares.