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4.2. Relativistic Beaming?

Radiation from relativistically moving matter is beamed in the direction of the motion to within an angle gamma-1. In spite of this the radiation produced by relativistically moving matter can spread over a much wider angle. This depends on the geometry of the emitting region. Let thetaM be the angular size of the relativistically moving matter that emits the burst. The beaming angle theta will be thetaM if thetaM > gamma-1 and gamma-1 otherwise. Thus if thetaM = 4pi - that is if the emitting matter has been accelerated spherically outwards from a central source (as will be the case if the source is a spherical fireball) - the burst will be isotropic even though each observer will observe radiation coming only from a very small region (see Fig. 11). The radiation will be beamed into gamma-1 only if the matter has been accelerated along a very narrow beam. The opening angle can also have any intermediate value if it emerges from a beam with an opening angle theta > gamma-1, as will be the case if the source is an anisotropic fireball [222, 223] or an electromagnetic accelerator with a modest beam width.

Figure 11

Figure 11. Radiation from a relativistic beam with a width theta. Each observer will detect radiation only from a very narrow beam with a width Gamma-1. The overall angular size of the observed phenomenon can vary, however, with Gamma-2 < theta2 < 4pi.

Beaming requires, of course, an event rate larger by a ratio 4pi / theta2 compared to the observed rate. Observations of about one burst per 10-6 year per galaxy implies one event per hundred years per galaxy if theta approx gamma-1 with gamma given by the compactness limit of ~ 100.

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