GRB Energetics

One of the most impressive properties of the bursts is the huge amount of energy released (fig. ), typically ranging between few $10^{51}$ and few $10^{54}$ ergs (isotropic emission).

Figure: Energy Distribution of GRBs with known redshift in two cases: isotropic emissions (top panel), and geometry-corrected for jet-like emissions (bottom panel). The non-isotropic case tightly clusters the distribution around $5\times 10^{50}$ erg. (From [Frail et al., 2001]).

Assuming isotropic emission, the case of GRB990123 is really impressive: only in the gamma-ray domain, it released $3\times 10^{54}$ ergs, i.e. about twice the rest mass of the Sun. If one supposes a jet-like emission, the energy requirements are reduced by a factor of $\Omega/4\pi \sim \theta^2_0/4$ with respect to the spherical emission ($\Omega$ is the solid angle subtended by the emission cone). On this subject, searches for the so-called ``orphan afterglows'', corresponding to GRBs beamed away from our line of sight, have been performed, only providing limits on the beaming factor.

Figure: The two different cases of spherical and jet-like emission: the visibility cone, defined by the time-evolving bulk Lorentz factor, should distinguish at late times between the two cases. (From [Ghisellini, 2001]).

Fig.  nicely shows what kind of observable effects should be measured in the afterglow light curve for the two different cases of spherical and jet-like emission: the relativistic beaming produces a visibility cone with an opening angle of $1/\Gamma$. In the early phases, $\Gamma \sim 100$ (see Fireball Model section) and the two cases cannot be distinguished; as the afterglow evolves, $\Gamma$ decreases; when it is $\Gamma < \theta_j^{-1}$ ($\theta_j$ is the opening angle of the jet), the afterglow light curve in the jet-like geometry should break and decline more rapidly, since an edge effect occurs. Indeed, the break in the light curve is due also to another effect: the lateral spreading of the jet, since the ejecta, encountering more surrounding matter, decelerate faster than it happens for the spherical case. The clearest evidence for a jet can be found in the sharp achromatic break in the light curve of GRB990510. Also the detection of polarization for this burst ([Covino et al., 1999]) seems to agree with the jet-like emission. Recently, [Frail et al., 2001], have tried to estimate the opening angles $\theta_j$ for a set of bursts with known redshifts, whose break times $t_j$ (or limits) have been estimated; the set of $\theta_j$ coming out includes the range $3\rm ^{\circ}- 25\rm ^{\circ}$ with a strong concentration around $4\rm ^{\circ}$. The energy distribution derived is surprisingly more clustered with respect to the spherical case, around $E \sim 5\times 10^{50}$ ergs.