1 Introduction

The observed fading multi-wavelength afterglows of gamma-ray bursts are so far consistent with the simple relativistic blastwave model (Mészáros & Rees 1997). The emission mechanism resulting in the prompt gamma rays remains a mystery. Studies of gamma-ray burst (GRB) spectral evolution have uncovered many trends which may be used to confront models. The discovery that peak power energy $E_{\rm pk}$(which is the maximum of $\nu F_{\nu}$, where $\nu$ is photon energy and $F_{\nu}$ is the specific energy flux) often decays exponentially in bright, long, smooth BATSE GRB pulses as a function of photon fluence $\Phi$($=\int_{t'=0}^{t'=t} F_{\rm N}(t'){\rm d}t'$)(Liang & Kargatis 1996, hereafter LK96) provided a new constraint on emission mechanisms (Liang et al. 1997; Liang 1997; Daigne & Mochkovitch 1998). In their analysis, LK96 fit the function

$$E_{\rm pk}(t) = E_{\rm pk(0)} {\rm e}^{-\Phi(t) / \Phi_{0}^{\rm LK}}$$ (1)

to 37 GRB pulses in 34 bursts. To interpret this empirical trend, they differentiated Eq. (1) to find

$$-{\rm d}E_{\rm pk}/{\rm d}t = E_{\rm pk}\ {F}_{\rm N}/ \Phi_{0}^{\rm LK}\approx F_{\rm E}/ \Phi_{0}^{\rm LK}$$ (2)

where $F_{\rm E} = \int_{{\rm E} \approx 30\ {\rm keV}}^{E \approx 2000\ {\rm keV}} E~N(E)~{\rm d} E$ is the BATSE energy flux and $F_{\rm N}=\int_{E \approx 30\ {\rm keV}}^{E \approx 2000\ {\rm keV}} N(E)~{\rm d}E$ is the BATSE photon flux (see Eq. (1) of LK96). We wished to avoid the assumption that $E_{\rm pk}~F_{\rm N} \approx F_{\rm E}$ and instead directly tested the trend $-{\rm d}(E_{\rm pk})/{\rm d}t= F_{\rm E} / \Phi_{0}$ by integrating it to give

$$E_{\rm pk}(t) = E_{{\rm pk}(0)} - \mathcal{E}\mathrm{(t)} \mathrm{/ \Phi_{0}}$$ (3)

where $\mathcal{E}\mathrm{(t)}$ ($=\int_{t'=0}^{t'=t} F_{\rm E}(t') {\rm d}t'$) is the BATSE energy fluence. Our motivation for using the $E_{\rm pk}$-energy fluence relation (Eq. (3)) as opposed the $E_{\rm pk}$-photon fluence relation (Eq. (1)) is that we believe that the former represents a more physical quantity. The BATSE LAD energy window was designed to contain the peak of GRB energy spectra, not the peak of the photon spectra. By using energy fluence, we avoid the somewhat shaky assumption that the BATSE LAD photon flux is proportional to the bolometric photon flux.