3 GRB phenomenology explained

Some $20-30\%$ of GRBs show the FRED (Fast Rise/ Exponential Decay) behavior. For a simple FRED-type GRB, or for well-defined peaks within a burst light curve, hard-to-soft evolution is usually measured (e.g., Norris et al. 1986). A hardness-intensity correlation is also generally observed for well-defined pulses within a GRB (Golenetskii et al. 1983). Both of these viewpoints are probably correct insofar as each well-defined pulse in a GRB time profile is accompanied by a spectral hardening and subsequent softening (Ford et al. 1995). When ${\epsilon}_{\rm peak}$is plotted against photon or energy fluence, then ${\epsilon}_{\rm peak}$decays as a power-law or a broken power law (Liang & Kargatis 1996; Crider et al. 1998). We (Dermer et al. 1999b) have recently shown how this behavior can be understood in terms of the analytic function described above.

Our model spectrum makes definite predictions if one assumes that the FRED-type profile of GRB peaks is a consequence of a smooth CBM that is reasonably well-described by a density $n(x) \propto n_0 x^{-\eta}$. At early times $t \ll t_{\rm d}$, where the deceleration time scale $t_{\rm d}\propto \Gamma_0^{-8/3} (E_0/n_0)^{1/3}$, the temporal profile of the rising flux ($\phi \propto t^{-\chi}$) of a decelerating blast wave exhibits temporal indices $\vert\chi\vert = 2-\eta+\eta\u/2$ at low energies $\epsilon \ll {\cal E}_0$ and $\vert\chi\vert = 2-\eta-\eta\delta/2$ at high energies $\epsilon \gg {\cal E}_0$. Here ${\cal E}_0 \equiv {\epsilon}_{\rm peak}(t_{\rm d})$. Because ${\cal E}_0$, $\u$, and $\delta$ can all be measured, fast timing over a broad energy band should give shifts in the temporal indices of the rising portion of the GRB time profiles approaching these limits. We also predict (Dermer et al. 1999b) that the time of the peak flux of a well-defined GRB pulse measured at different photon energies ($\varepsilon < {\cal E}_0$) should follow the relation

$$\begin{eqnarray} t_{\rm p}(\epsilon) = \frac{t_{\rm d}}{ 2g+1}\; \left[ 2g + \left({\epsilon\over {\cal E}_0}\right)^{-(2g+1)/(4g+\eta/2)}\right].\end{eqnarray}$$ (1)

Thus the peaks should align at high energies and lag at lower energies. The peak alignment in GRBs is quite well-documented at $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$\gt$}}}50$ keV energies, but the character of peak alignment and lagging is less well-studied at $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}20$ keV (see, however, Piro et al. 1998; Connors & Hueter 1998).

Perhaps the most crucial test for a model of GRBs is to explain the overall GRB duration distribution, including the strong evidence for bimodality (Kouveliotou et al. 1996). In our recent work (Böttcher & Dermer 1999) we show that the GRB duration distribution measured with BATSE and other instruments is a consequence of the emission properties of blast waves, the triggering criteria of GRB detectors, and the range of values of $\Gamma_0$ and E₀.