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

11 pages, 8 figures, accepted for publication in MNRAS