In the internal shock model for gamma-ray bursts (GRBs), the synchrotron spectrum from the fast cooling electrons in a homogeneous downstream magnetic field (MF) is too soft to produce the low-energy slope of GRB spectra. However the magnetic field m
ay decay downstream with distance from the shock front. Here we show that the synchrotron spectrum becomes harder if electrons undergo synchrotron and inverse-Compton cooling in a decaying MF. To reconcile this with the typical GRB spectrum with low energy slope $ u F_ upropto u$, it is required that the postshock MF decay time is comparable to the cooling time of the bulk electrons (corresponding to a MF decaying length typically of $sim10^5$ skin depths); that the inverse-Compton cooling should dominate synchrotron cooling after the MF decay time; and/or that the MF decays with comoving time roughly as $Bpropto t^{-1.5}$. An internal shock synchrotron model with a decaying MF can account for the majority of GRBs with low energy slopes not harder than $ u^{4/3}$.
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.
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.
