The importance of off-jet relativistic kinematics in gamma-ray burst jet models

GRBs are widely thought to originate from collimated jets of material moving at relativistic velocities. Emission from such a jet should be visible even when viewed from outside the angle of collimation. I summarize recent work on the special relativistic transformation of the burst quantities Eiso (isotropic-equivalent energy of the burst) and Epeak (peak of the burst spectrum in the power νFν) as a function of viewing angle. The resulting formulae serve as input for a Monte Carlo population synthesis method, with which I investigate the importance of off-jet relativistic kinematics as an explanation for a class of GRBs termed X-ray flashes (XRFs) in the context of several top-hat-shaped variable opening-angle jet models. For certain parameters, such models predict a large population of off-jet bursts that are observable and that lie away from the Epeak ∝ E relation. This predicted burst population is not seen in current data sets. I investigate the effect of the bulk γ value on the properties of this population of off-jet bursts, as well as the effect of including an Ω0-Eγ correlation to jointly fit the Epeak ∝ E and Epeak ∝ E relations, where Ω0 is the opening solid angle of the GRB jet. I find that the XRFs seen by HETE-2 and BeppoSAX cannot be easily explained as classical GRBs viewed off-jet. I also find that an inverse correlation between γ and Ω0 has the effect of greatly reducing the visibility of off-jet events. Therefore, unless γ > 300 for all bursts, or there is a strong inverse correlation between γ and Ω0, top-hat variable opening-angle jet models produce a significant population of bursts away from the Epeak ∝ E and Epeak ∝ E relations, in contradiction with current observations.