Characteristics of Profiles of Gamma-Ray Burst Pulses Associated with the Doppler Effect of Fireballs

In this paper we derive in great detail the formula for count rates of gamma-ray bursts (GRBs) in the framework of fireballs, in terms of the integral of time, where the Doppler effect of the expanding fireball surface is the key factor concerned. Effects arising from the limit on the time delay due to the limited emitting areas on the fireball surface and other factors are investigated. Our analysis shows that the formula for the count rate of fireballs can be expressed as a function of τ, which is the observation timescale relative to the dynamical timescale of the fireball defined by Rc/c, where Rc is the fireball radius measured at an associated local time. The profile of light curves of fireballs depends only on the relative timescale, entirely independent of the real timescale and the real size of the objects. It displays in detail how a cutoff tail or a turnover feature (called the cutoff tail problem) in the decay phase of a light curve can arise. This feature is a consequence of a hot spot on the fireball surface, moving toward the observer, and has been observed in a few cases previously. Local pulses suddenly dimming produce light curves bearing a certain decay form (called a standard decay form) and exhibiting a sharp feature at their peaks. Light curves arising from gradually dimming local pulses are smooth at their peaks, and their profiles in the decay phase will obviously deviate from the standard form when the width of the local pulse is large enough. It is observed that light curves arising from relatively short local pulses should be the same, entirely independent of the local pulse shape. The impact of the rest-frame radiation form and of the variance of the form on the profile of light curves is insignificant, while the impact on the magnitude of the light curves is obvious. By performing fits to the count-rate light curves of six sample sources, we show how to obtain some physical parameters from the observed profile of the count rate of GRBs and show that there do exist some GRBs for which the profiles of their count rate light curves can be described by the formula provided. In addition, the analysis reveals that the Doppler effect of fireballs could lead to a power-law relationship between the FWHM of pulses and energy, which has been observed previously by many authors.