4.2. Relativistic Beaming?
Radiation from relativistically moving matter is beamed in the direction of the motion to within an angle -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 M be the angular size of the relativistically moving matter that emits the burst. The beaming angle will be M if M > -1 and -1 otherwise. Thus if M = 4 - 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 -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 > -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. Radiation from a relativistic beam with a width . Each observer will detect radiation only from a very narrow beam with a width -1. The overall angular size of the observed phenomenon can vary, however, with -2 < 2 < 4.
Beaming requires, of course, an event rate larger by a ratio 4 / 2 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 -1 with given by the compactness limit of ~ 100.