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 R_jet, half-opening angle θ_jet, a comoving-frame emissivity law and an observer viewing angle θ_obs, and then study the effects for the conditions θ_obs < θ_jet and θ_obs > θ_jet. 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 $E^{\rm i}_{\rm p}\propto E_{\rm iso}^{0.5}$ (Amati relation) is naturally obtained from pure relativistic kinematic when $\Gamma \gtrsim 10$ and θ_obs < θ_jet; on the contrary, when θ_obs > θ_jet it results $E^{\rm i}_{\rm p}\propto E_{\rm iso}^{0.25}$ . Using data from literature for a class of well-know sub-energetic GRBs, we show that their position in the $E^{\rm i}_{\rm p}\!-\!E_{\rm iso}$ 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.