2 Pulse width evolution obtained from time histories

A visual inspection of the BATSE catalog of multiple-peaked time histories reveals that peaks usually have about the same duration at the beginning of the burst as near the end of the burst. Our aim is to characterize and measure the pulse shape as a function of arrival time. The aligned peak method measures the average pulse temporal structure, each burst contributes to the average by aligning the largest peak ([Mitrofanov 1997]). We used all 53 bursts from the BATSE 4B Catalog that were longer than 20 s and brighter than 5 photons s^-1 cm^-2. Each burst must have at least one peak, as determined by a peak-finding algorithm (similar to [Li & Fenimore 1996]), in each third of its duration. The largest peak in each third was normalized to unity and a shifted in time, bringing the largest peaks of all bursts into common alignment. This method was applied in each third of the duration of the bursts. Thus, we obtained one curve of the averaged pulse shape for each different section of the bursts (as shown in Fig. 1). The average profile is notably identical in each 1/3 of T₉₀ (we estimate the differential spread, S, to be $\sim$ 1%).

We have shown the lack of temporal evolution of the peak width in the context of an average of many bursts. Now, we expand our analysis to individual bursts. An excellent analysis has been provided by [Norris et al. (1995)], where they examined the temporal profiles of bright GRBs by fitting those profiles with pulses. From the set of bursts that they analysed, we used the 28 bursts with the characteristic of having seven or more fitted pulses within their duration. To obtain the temporal dependency of the pulse width, we selected the five largest peaks in each burst and fit their FWHM to a function of the arrival time, $\Delta T=k \left(\frac{T-T_{\rm c}}{T_{90}}\right )^{\alpha}$. We chose this model because the expected dependency from the "external'' shock model scales as $\frac{T/T_{90}}{\Gamma(1+\beta)}$ (if a relativistic shell is responsible for the shape in the time histories, T₀ should be proportional to T₉₀; see [Fenimore et al. 1996]). The purpose of the $T_{\rm c}$ parameter is to correct the BATSE time to the time since the beginning of the explosion. Figure 2 shows the distribution of the power law indexes ($\alpha$) for the set of bursts analysed. The distribution shows that in individual bursts the pulse width does not increase throughout their duration as predicted by the external shock model, for which $\alpha$ is expected to be 1 (or larger if deceleration is occurring). In fact, most bursts show a diminution in pulse width.

  

Figure 2: Distribution of the $\alpha$ parameter from 28 bright BATSE bursts with durations longer than 1.5 s. The pulse width ($\Delta T$), as expected for the external model, should scale as T/T90. We fit $k \left (\frac{T-T_{\rm c}}{T_{90}}\right)^{\alpha}$ to find the temporal evolution of the pulse width in each burst. The dashed line is the expected index value, $\alpha$, for a single relativistic shell ($\alpha=1$). The distribution shows that the vast majority of the bursts present no time evolution (or negative) of the pulse width