A numerical jet model for the prompt emission of gamma-ray bursts

Gamma-ray bursts (GRBs) are known to be highly collimated events, and are mostly detectable when they are seen on-axis or very nearly on-axis. However, GRBs can be seen from off-axis angles, and the recent detection of a short GRB associated with a gravitational wave event has conclusively shown such a scenario. The observer viewing angle plays an important role in the observable spectral shape and the energetic of such events. We present a numerical model that is based on the single-pulse approximation with emission from a top-hat jet and has been developed to investigate the effects of the observer viewing angle. We assume a conical jet parametrized by a radius Rjet, half-opening angle θjet, a comoving-frame emissivity law and an observer viewing angle θobs, and then study the effects for the conditions θobs < θjet and θobs > θjet. We present results considering a smoothly broken power-law emissivity law in jet comoving frame, albeit the model implementation easily allows to consider other emissivity laws. We find that the relation E^{ m i}_{ m p}propto E_{ m iso}^{0.5} (Amati relation) is naturally obtained from pure relativistic kinematic when Gamma gtrsim 10 and θobs < θjet; on the contrary, when θobs > θjet it results E^{ m i}_{ m p}propto E_{ m iso}^{0.25}. Using data from literature for a class of well-know sub-energetic GRBs, we show that their position in the E^{ m i}_{ m p}!-!E_{ m iso} plane is consistent with event observed off-axis. The presented model is developed as a module to be integrated in spectral fitting software package xspec and can be used by the scientific community.