Previous studies have found that the width of gamma-ray burst (GRB) pulse is energy dependent and that it decreases as a power-law function with increasing photon energy. In this work we have investigated the relation between the energy dependence of
pulse and the so-called Band spectrum by using a sample including 51 well-separated fast rise and exponential decay long-duration GRB pulses observed by BATSE (Burst and Transient Source Experiment on the Compton Gamma Ray Observatory). We first decompose these pulses into rise, and decay phases and find the rise widths, and the decay widths also behavior as a power-law function with photon energy. Then we investigate statistically the relations between the three power-law indices of the rise, decay and total width of pulse (denoted as $delta_r$, $delta_d$ and $delta_w$, respectively) and the three Band spectral parameters, high-energy index ($alpha$), low-energy index ($beta$) and peak energy ($E_p$). It is found that (1)$alpha$ is strongly correlated with $delta_w$ and $delta_d$ but seems uncorrelated with $delta_r$; (2)$beta$ is weakly correlated with the three power-law indices and (3)$E_p$ does not show evident correlations with the three power-law indices. We further investigate the origin of $delta_d-alpha$ and $delta_w-alpha$. We show that the curvature effect and the intrinsic Band spectrum could naturally lead to the energy dependence of GRB pulse width and also the $delta_d-alpha$ and $delta_w-alpha$ correlations. Our results would hold so long as the shell emitting gamma rays has a curve surface and the intrinsic spectrum is a Band spectrum or broken power law. The strong $delta_d-alpha$ correlation and inapparent correlations between $delta_r$ and three Band spectral parameters also suggest that the rise and decay phases of GRB pulses have different origins.
In this paper we have analyzed the temporal and spectral behavior of 52 Fast Rise and Exponential Decay (FRED) pulses in 48 long-duration gamma-ray bursts (GRBs) observed by the CGRO/BATSE, using a pulse model with two shape parameters and the Band m
odel with three shape parameters, respectively. It is found that these FRED pulses are distinguished both temporally and spectrally from those in long-lag pulses. Different from these long-lag pulses only one parameter pair indicates an evident correlation among the five parameters, which suggests that at least $sim$4 parameters are needed to model burst temporal and spectral behavior. In addition, our studies reveal that these FRED pulses have correlated properties: (i) long-duration pulses have harder spectra and are less luminous than short-duration pulses; (ii) the more asymmetric the pulses are the steeper the evolutionary curves of the peak energy ($E_{p}$) in the $ u f_{ u}$ spectrum within pulse decay phase are. Our statistical results give some constrains on the current GRB models.
Employing two samples containing of 56 and 59 well-separated FRED (fast rise and exponential decay) gamma-ray burst (GRB) pulses whose spectra are fitted by the Band spectrum and Compton model, respectively, we have investigated the evolutionary slop
e of $E_{p}$ (where $E_{p}$ is the peak energy in the $ u F u$ spectrum) with time during the pulse decay phase. The bursts in the samples were observed by the Burst and Transient Source Experiment (BATSE) on the Compton Gamma-Ray Observatory. We first test the $E_{p}$ evolutionary slope during the pulse decay phase predicted by Lu et al. (2007) based on the model of highly symmetric expanding fireballs in which the curvature effect of the expanding fireball surface is the key factor concerned. It is found that the evolutionary slopes are normally distributed for both samples and concentrated around the values of 0.73 and 0.76 for Band and Compton model, respectively, which is in good agreement with the theoretical expectation of Lu et al. (2007). However, the inconsistence with their results is that the intrinsic spectra of most of bursts may bear the Comptonized or thermal synchrotron spectrum, rather than the Band spectrum. The relationships between the evolutionary slope and the spectral parameters are also checked. We show the slope is correlated with $E_{p}$ of time-integrated spectra as well as the photon flux but anticorrelated with the lower energy index $alpha$. In addition, a correlation between the slope and the intrinsic $E_{p}$ derived by using the pseudo-redshift is also identified. The mechanisms of these correlations are unclear currently and the theoretical interpretations are required.
Employing a sample presented by Kaneko et al. (2006) and Kocevski et al. (2003), we select 42 individual tracking pulses (here we defined tracking as the cases in which the hardness follows the same pattern as the flux or count rate time profile) wit
hin 36 Gamma-ray Bursts (GRBs) containing 527 time-resolved spectra and investigate the spectral hardness, $E_{peak}$ (where $E_{peak}$ is the maximum of the $ u F_{ u}$ spectrum), evolutionary characteristics. The evolution of these pulses follow soft-to-hard-to-soft (the phase of soft-to-hard and hard-to-soft are denoted by rise phase and decay phase, respectively) with time. It is found that the overall characteristics of $E_{peak}$ of our selected sample are: 1) the $E_{peak}$ evolution in the rise phase always start on the high state (the values of $E_{peak}$ are always higher than 50 keV); 2) the spectra of rise phase clearly start at higher energy (the median of $E_{peak}$ are about 300 keV), whereas the spectra of decay phase end at much lower energy (the median of $E_{peak}$ are about 200 keV); 3) the spectra of rise phase are harder than that of the decay phase and the duration of rise phase are much shorter than that of decay phase as well. In other words, for a complete pulse the initial $E_{peak}$ is higher than the final $E_{peak}$ and the duration of initial phase (rise phase) are much shorter than the final phase (decay phase). This results are in good agreement with the predictions of Lu et al. (2007) and current popular view on the production of GRBs. We argue that the spectral evolution of tracking pulses may be relate to both of kinematic and dynamic process even if we currently can not provide further evidences to distinguish which one is dominant. Moreover, our statistical results give some witnesses to constrain the current GRB model.
