2 Application to GRB 970508

In the best-sampled GRB afterglow light curve yet available (the GRB 970508 R band data), the optical spectrum changed slope at $t_{\oplus}\sim 1.4 \, \hbox{day}$, suggesting the passage of the cooling break through the optical band (Galama et al. 1998). We explore the range of acceptable beaming angles for this burst by fitting the afterglow light curve for $1.3 \, \hbox{day}\le t_{\oplus}\le 95 \, \hbox{day}$assuming that $\nu_{\oplus,{\rm cool}}< c/0.7~\mu$m.

The range of acceptable energy distribution slopes p for swept-up electrons is taken from the optical colors. Precise measurements for give $F_\nu \propto \nu^{-\beta}$ with $\beta = 1.10 \pm 0.08$ (Zharikov et al. 1998), so that $p = 2.20 \pm 0.16$. We take this value to hold throughout the range $1.3 \, \hbox{day}\le t_{\oplus}\le 95 \, \hbox{day}$, thus assuming that p does not change as the afterglow evolves. We subtract the host galaxy flux ($R_{\rm H}= 25.55 \pm 0.19$; Zharikov et al. 1998) from all data points before fitting.

  

Table: Fitted break times $t_{\oplus,b}$ and magnitudes $R_{\rm c}(t_0)$ (at fiducial observed time t₀ = 3.231 day) for beamed GRB afterglow models for three pairs of acceptable host galaxy magnitude $R_{\rm H}$and electron power law index p. The fit included all 43 data points with $1.3 \, \hbox{day}\le t_{\oplus}\le 95 \, \hbox{day}$ in the compilation by Garcia et al. (1998)

  

Figure 1: Upper panel: The Cousins R band light curve for GRB 970508 with the three fits shown in Table 1. Lower panel: Residuals for the data and for models 2 and 3 (in order of increasing curvature) relative to model 1. A host galaxy flux corresponding to $R_{\rm H}=25.55$ has been subtracted from all data points

We fixed values of $R_{\rm H}$ and p, and then executed a grid search on the break time $t_{\oplus,b}$ and normalization of the model light curve. Results are summarized in Table 2 and Fig. 1. The final $\chi^2$ per degree of freedom is $\sim 4$.

These large $\chi^2$ values make meaningful error estimates on parameters difficult. Let us suppose $\chi^2$ is large because details omitted from the models (clumps in the ambient medium or blast wave instabilities) affect the light curve, and so attach an uncertainty of $0.1 \, \hbox{mag}$ to each predicted flux. Adding this in quadrature to observational uncertainties when computing $\chi^2$, we obtain $\chi^2 / \hbox{d.o.f.} \sim 1$. Error estimates based on changes in $\chi^2$ then rule out $\log(t_{\oplus,b}/\, \hbox{day}) < 3.5$ at about the 90% confidence level even for our "maximum beaming'' case (p=2.04, $R_{\rm H}=25.36$).

To convert a supposed break time $t_{\oplus,b}$ into a beaming angle $\zeta_{\rm m}$,we need estimates of the burst energy per steradian and the ambient density. Wijers & Galama (1998) infer $E_0 / \Omega = 3.7 \ 10^{52}\ \hbox{erg}/ (4 \pi \, \hbox{Sr})$ and $\rho = 5.8\ 10^{-26} \, \hbox{g}/\hbox{cm}^3$. Combining these values with gives . $E_0 / \Omega$ and $\rho$are substantially uncertain, but because $\zeta_{\rm m}\propto (\rho / E_0)^{1/8}$, the error budget for $\zeta_{\rm m}$ is dominated by uncertainties in p rather than in E₀ or $\rho$.

This beaming limit implies $\Omega \ge 0.75 \, \hbox{Sr}$, which is $6\%$of the sky. GRB 970508 was at $z \ge 0.835$ (Metzger et al. 1997). We then find gamma ray energy $E_\gamma = 2.8 \ 10^{50}\ \hbox{erg}\times (\Omega / 0.75 \, \hbox{Sr}) (d_{_{\rm L}}/ 4.82 \, \hbox{Gpc})^2 \left( 1.835 / [1+z] \right)$. If the afterglow is primarily powered by different ejecta from the initial GRB, as when a "slow'' wind ($\Gamma_0 \sim 10$) dominates the ejecta energy, then our beaming limit applies only to the afterglow emission. The optical fluence implies $E_{\rm opt} = 3.4 \ 10^{49}\ \hbox{erg}\times (\Omega / 0.75 \, \hbox{Sr}) (d_{_{\rm L}}/ 4.82 \, \hbox{Gpc})^2 \left( 1.835 / [1+z] \right)$. The irreducible minimum energy is thus $3.4 \ 10^{49}\ \hbox{erg}$, using the smallest possible redshift and beaming angle. We have reduced the beaming uncertainty, from the factor $\sim \Gamma_0^2 \sim 300^2 \sim 10^5$ allowed by $\gamma$-ray observations alone to a factor $(4 \pi \, \hbox{Sr})/(0.75 \, \hbox{Sr}) \sim 20$, and thus obtain the most rigorous lower limit on GRB energy requirements yet.

Acknowledgements

I thank Re'em Sari for useful comments, and KNPO for financial support.