2008ApJ...683..900Q

We explore the influence of the full curvature effect on the flux of the early X-ray afterglows of gamma-ray bursts (GRBs) in cases where the spectrum of the intrinsic emission is a power law. We find that the well-known t^–(2+β) curve appears only when the intrinsic emission is extremely brief or the emission arises from exponential cooling. The timescale of this curve is independent of the Lorentz factor. The resulting light curve exhibits two phases if the intrinsic emission has a power-law spectrum and a temporal power-law profile of infinite duration. The first phase is a rapid decay in which the light curve is well described by the t^–(2+β) curve. The second phase is a shallow decay in which the power-law index of the light curve is obviously smaller than in the first phase. The start of the shallow phase is strictly constrained by, and can in turn set a lower limit on, the radius of the fireball. In the case of power-law emission that lasts for only a limited time, there will be a third phase after the t^–(2+β) curve and the shallow decay phase, which is much steeper than the shallow phase. As a sample application, we fit the Swift XRT data for GRB 050219A with our model and show that the curvature effect alone can roughly account for this burst. Although the fit parameters cannot be uniquely determined, because of the various choices in the fitting, a lower limit on the fireball radius of this burst can be estimated, which is ∼10¹⁴ cm.