3 Light curve, spectra & the burst parameters

The light curve and spectra of an afterglow are flat near the peak. The exact shape, and the value of the peak frequency and peak flux depend on the values of the physical parameters of the burst (see GPSa). In Fig. 2 we see the peak in the optical light curve of GRB 970508 (Sokolov et al. 1997; Metzger et al. 1997), with three theoretical light curves. These light curves are for p=2.57, which corresponds to the power law decay that follows the peak, and differ by the values of the remaining parameters (GPSa).

It is quite evident from Fig. 2 that the shape of the optical peak displayed by GRB 970508 is not accounted for by the theoretical light curve arising from the model we used, and a different explanation should be considered.

  

Figure: Optical observations of GRB 970508, made by Sokolov et al. (1997, circles) and Metzger et al. (1997, triangles). The three curves are three possible theoretical light curves

The observed flux density near the self absorption frequency (i.e., $\nu \ll \nu_{\rm m}$) can be approximated, with an accuracy better than 3%, by the following simple expression:  

$$F_{\nu}=F_{\rm \nu_a, ext}\psi^2\left( 1-\exp[-\psi^{-5/3}]\right), \quad \psi \equiv \nu/\nu_{\rm a}.$$ (1)

Where $F_{\nu_{\rm a,ext}}$ and $\nu_{\rm a}$ are defined in Fig. 3 and depend on the values of the physical parameters (GPSb).

In Fig. 3 we show a fit of our theoretical spectra near $\nu_{\rm a}$ to radio afterglow observations of GRB 970508 (Shepherd et al. 1998), which were made about one week after the burst. Extracting the values of $F_{\nu_{\rm a,ext}}$ and $\nu_{\rm a}$ from this fit and comparing them to the theoretical values results in two constraints on the parameters of the burst (GPSb).

Using the values of $\nu_{\rm a}, \nu_{\rm m}, F_{\nu_{\rm m}}$ and the cooling break frequency $\nu_{\rm c}$ extracted from the broad band spectra of GRB 970508, and comparing them to their theoretical values, Wijers & Galama (1998) calculated the parameters of this burst, using a simple broken power law theoretical spectrum (Sari et al. 1998). Making a similar calculation, using the more detailed description of the self absorption break and of the spectral peak, we obtained the following values for the physical parameters of GRB 970508:

$$\begin{eqnarray} E &= 5.3\ 10^{51} \ {\rm ergs} \quad \quad\ \ \ \epsilon_{\rm e... ...\rm cm^{-3}} \quad \quad \quad \quad \ \ \epsilon_{\rm B} &=0.014,\end{eqnarray}$$ (2)

where E is the total energy of the shell and n₀ is the ambient number density.

  

Figure 3: A fit of our calculated spectra to radio afterglow observations of GRB 970508, made about one week after the burst