The full curvature effect expected in early X-ray afterglow emission of gamma-ray bursts

Abstract

We explore the influence of the full curvature effect on the flux of early X-ray afterglow of gamma-ray bursts (GRBs) in cases when the spectrum of the intrinsic emission is a power-law. We find that the well-known $t^{-(2+\beta)}$ curve is present only when the intrinsic emission is extremely short or the emission arises from an exponential cooling. The time scale of this curve is independent of the Lorentz factor. The resulting light curve would contain two phases when the intrinsic emission has a power-law spectrum and a temporal power-law profile with infinite duration. The first phase is a rapid decay one where the light curve well follows the $t^{-(2+\beta)}$ curve. The second is a shallow decay phase where the power-law index of the light curve is obviously smaller than that in the first phase. The start of the shallow phase is strictly constrained by the fireball radius, which in turn, can put a lower limit to the latter. In the case when the temporal power-law emission lasts a limited interval of time, there will be a third phase after the $t^{-(2+\beta)}$ curve and the shallow decay phase, which is much steeper than the shallow phase. As an example of application, we fit the XRT data of GRB 050219A with our model and show that the curvature effect alone can roughly account for this burst. Although fitting parameters can not be uniquely determined due to various choices of fitting, a lower limit of the fireball radius of this burst can be estimated, which is $\sim 10^{14}cm$.