2016ApJ...831..111M

We present time-resolved spectral analysis of the steep decay segments of 29 bright X-ray flares of gamma-ray bursts (GRBs) observed with the Swift/X-ray telescope, and model their light curves and spectral index evolution behaviors with the curvature effect model. Our results show that the observed rapid flux decay and strong spectral index evolution with time can be well fitted with this model, and the derived characteristic timescales (t_c) are in the range of 23 ∼ 264 s. Using an empirical relation between the peak luminosity and the Lorentz factor derived from the prompt gamma-rays, we estimate the Lorentz factors of the flares (Γ_X). We obtain Γ_X = 17 ∼ 87 with a median value of 52, which is smaller than the initial Lorentz factors of prompt gamma-ray fireballs. With the derived t_c and Γ_X, we constrain the radiating regions of 13 X-ray flares, yielding R_X = (0.2 ∼ 1.1) x 10¹⁶ cm, which are smaller than the radii of the afterglow fireballs at the peak times of the flares. A long evolution feature from prompt gamma-ray phase to the X-ray epoch is found by incorporating our results with a sample of GRBs whose initial Lorentz factors are available in the literature, i.e., Γ∝[t_p/(1+z)]^{–0.69±0.06}. These results may shed light on the long-term evolution of GRB central engines.