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

In this paper, we derive in a much detail the formula of count rates, interms of the integral of time, of gamma-ray bursts in the framework offireballs, where the Doppler effect of the expanding fireball surface is thekey factor to be concerned. Effects arising from the limit of the time delaydue to the limited regions of the emitting areas in the fireball surface andother factors are investigated. Our analysis shows that the formula of thecount rate of fireballs can be expressed as a function of $\tau $ which is theobservation 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 theobjects. It displays in detail how a cutoff tail, or a turn over, feature(called a cutoff tail problem) in the decay phase of a light curve can beformed. 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. Byperforming fits to the count rate light curves of six sample sources, we showhow to obtain some physical parameters from the observed profile of the countrate of GRBs. In addition, the analysis reveals that the Doppler effect offireballs could lead to a power law relationship between the $FWHM$ of pulsesand energy, which were observed previously by many authors.Comment: 38 pages, 10 figures; accepted for publication in ApJ (10 December 2004, v617