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 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 $E^{\rm i}_{\rm p}\propto E_{\rm iso}^{0.5}$ (Amati relation) is naturally obtained from pure relativistic kinematic when $\Gamma \gtrsim 10$ and θobs < θjet; on the contrary, when θobs > θjet it results $E^{\rm i}_{\rm p}\propto E_{\rm iso}^{0.25}$. 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.

中文翻译：

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伽马射线暴迅速发射的数值喷流模型

众所周知，伽马射线爆发 (GRB) 是高度准直的事件，当它们在轴上或非常接近轴上时，它们大多是可检测到的。然而，可以从离轴角度看到 GRB，最近检测到与引力波事件相关的短 GRB 最终证明了这种情况。观察者视角在可观察到的光谱形状和此类事件的能量中起着重要作用。我们提出了一个数值模型，该模型基于顶帽式射流发射的单脉冲近似值，并已开发用于研究观察者视角的影响。我们假设一个由半径 Rjet、半开角 θjet、同动框架发射率定律和观察者视角 θobs 参数化的锥形射流，然后研究条件 θobs < 的影响。θjet 和 θobs ＞θ喷射。我们提出了考虑射流联动框架中平滑破坏的幂律发射率定律的结果，尽管模型实现很容易允许考虑其他发射率定律。我们发现关系 $E^{\rm i}_{\rm p}\propto E_{\rm iso}^{0.5}$（Amati 关系）在 $\Gamma \gtrsim 10 时自然是从纯相对论运动学中得到的$ 和 θobs < θ 喷射；相反，当 θobs ＞θjet 得到 $E^{\rm i}_{\rm p}\propto E_{\rm iso}^{0.25}$。使用文献中一类著名的亚高能伽马暴的数据，我们表明它们在 $E^{\rm i}_{\rm p}-E_{\rm iso}$ 平面中的位置与事件一致离轴观察。提出的模型是作为一个模块开发的，可以集成到光谱拟合软件包 XSPEC 中，可供科学界使用。