5 Implied energies and luminosities

The maximum energy $({E_{\rm GRB}})_{\rm max}$ that has been observed for a GRB imposes an important requirement on GRB models, and is therefore of great interest to theorists. $({E_{\rm GRB}})_{\rm max}$has increased as the number of GRB redshift distances that have been determined has increased. Currently, the record holder is GRB 971214 at z=3.4, which implies $E_{\rm GRB} \sim 5 \ 10^{53}$ erg from its gamma-ray fluence, assuming isotropic emission and $\Omega_{\rm M} = 0.3$and $\Omega_\Lambda = 0.7$ (Kulkarni 1999).

The table below summarizes the redshifts and energies of the bursts for which these are currently known:

$$\begin{array} {ccc} \hline\ \ \ \ \ \ \ \ \ \ $Gamma-Ray\ \ ... ...6\ & 8 \ 10^{52}\ {\rm erg} \\ & & \\ [-10pt] \hline\end{array}$$

This kind of energy is difficult to accommodate in $\rm NS-NS$ or $\rm NS-BH$ binary merger models without invoking strong beaming. "Collapsar'' or "hypernova'' models have an easier time of it, and can perhaps reach $\sim 10^{54}$ erg without invoking strong beaming by assuming a high efficiency for the conversion of gravitational binding energy into gamma-rays (Woosley 1999).

Both classes of models can be "saved'' by invoking strong beaming ($f_{\rm beam} \sim 1/10 - 1/100$ (but see the lack of evidence of beaming discussed below). Even if GRBs are strongly beamed, they are still far and away the brightest electromagnetic phenomenon in the Universe, as the following comparison illustrates:

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$L_{\rm GRB} \sim 10^{51}\ (f_{\rm beam}/10^{-2})\ {\rm erg}\ {\rm s}^{-1}$.

The luminosities of GRB 970508 and GRB 971214 differ by a factor of about one hundred. Thus (if there was previously any doubt), determination of the redshift distances for just three GRBs has put to rest once and for all the idea that GRBs are "standard candles.'' The extensive studies by Loredo & Wasserman (1998a,b) and the study by Schmidt (1999) reported at this workshop show that the luminosity function for GRBs can be, and almost certainly is, exceedingly broad, with . The results of Loredo & Wasserman (1998a,b) show that the burst luminosity function could be far broader; and indeed, if GRB 980425 is associated with SN 1998bw, .

Even taking a luminosity range implies that , given the range in the distances of the three GRBs whose redshifts are known. This is far broader than the range of peak fluxes in the BASTE GRB sample, and implies that the flux distribution of the bursts extends well below the BATSE threshold.

The enormous breadth of the luminosity function of GRBs suggests that the differences (such as time stretching and spectral softening) between the apparently bright and the apparently dim bursts are due to intrinsic differences between intrinsically bright and faint bursts, rather than to cosmology.

Finally, a broad luminosity function is naturally expected in models with ultra-relativistic radial outflow and strong beaming (jet-like behavior). But then why is no large special relativistic Doppler redshift seen in GRB spectra; i.e., why is the spread in $E_{\rm peak}$so narrow?