Doppler Effect Of Fireballs On The Light Curve Of Gamma-Ray Bursts

The history and progress of the observational and theoretical studies on gamma-ray bursts (GRBs) and their afterglows are reviewed, various theoretical models are introduced, and some problems of these models are discussed. Doppler effect of fireballs on the light curve of gamma-ray bursts is investigated.We derive in a much detail the formula of count rates, in terms of the integral of time, of gamma-ray bursts in the framework of fireballs, where the Doppler effect of the expanding fireball surface is the key factor to be concerned. Effects arising from the limit of the time delay due to the limited regions of the emitting areas in the fireball surface and other factors are investigated. Our analysis shows that the formula of the count rate of fireballs can be expressed as a function of r which is the observation time scale relative to the dynamical time scale of the fireball. The profile of light curves of fireballs depends only on the relative timescale, entirely independent of the real time scale and the real size of the objects. It displays in detail how a cutoff tail or a turn over feature in the decay phase of a light curve can be formed. This feature is a consequence of a hot spot in the fireball surface, moving towards the observer, and was observed in a few cases previously. Local pulses suddenly dimming would produce light curves exhibiting a sharp feature at their peaks. Light curves arising from gradually dimming local pulses would be smooth at their peaks. It is observed that light curves arising from relatively short local pulses would be the same, entirely independent of the shape of local pulses. Impacts of the rest frame radiation form and the variance of the form on the profile of light curves are insignificant, while that on the magnitude of the light curves would be obvious. By performing fits to the count rate light curves of six sample sources, we show that there do exist some GRBs that 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 were observed previously by many authors.Time profiles of many gamma-ray bursts consist of distinct pulses, which provides a possibility of characterizing the temporal structure of these bursts. We employ a simple model of highly symmetric fireballs to analyze the effect of the expansion speed on the light curve arising fromdifferent forms of local pulses. The relationship between the ratio, r, of the FWHM width of the rising phase of the light curve to that of the decaying phase and the Lorentz factor r is investigated. The analysis shows that, when the rest frame radiation form is ignored, temporal profiles of the light curve arising from pulses of fireballs will not be affected by the expansion speed (that is, r is almost a constant) as long as the fireball expands relativistically. When the rest frame radiation form (the Band function) is taken into account, there will be a break in the curves of r-logT. The location of the break depends mainly on the adopted value of the rest frame peak frequency vOp. One would reach almost the same result when a uniform jet is considered. In addition, we utilize a sample of 48 individual GRB pulses to check the relationship between the ratio r and the expansion speed r. We find no significant correlation between them, which is consistent with the theoretical analysis.