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

Abstract

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 to 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 which 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 $R_{\rm jet}$, half-opening angle $\theta_{\rm jet}$, a comoving-frame emissivity law and an observer viewing angle $\theta_{\rm obs}$, and then study the effects for the conditions $\theta_{\rm obs} < \theta_{\rm jet}$ and $\theta_{\rm obs} >\theta_{\rm jet}$. 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 $\theta_{\rm obs} < \theta_{\rm jet}$; on the contrary, when $\theta_{\rm obs} > \theta_{\rm jet}$ 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.