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B. Relativistic time effects

Consider first a source moving relativistically with a constant velocity along a line towards the observer and two photons emitted at R1 and R2. The first photon (emitted at R1) will reach the observer at time (R2 - R1) / v - (R2 - R1) / c before the second photon (emitted at R2). For Gamma >> 1 this equals approx (R2 - R1) / 2c Gamma2. This allows us to associate an "observer time" R / 2c Gamma2 with the distance R and for this reason I have associated a scale c deltat Gamma-2 with fluctuations on a time scale deltat in the optical depth equation earlier (see Section IVA). This last relation should be modified if the source moves a varying velocity (v=v(R)). Now

Equation 7 (7)

which reduces to

Equation 8 (8)

for motion with a constant velocity. The difference between a constant velocity source and a decelerating source introduces a numerical factor of order eight which is important during the afterglow phase [363].

Consider now a relativistically expanding spherical shell, or at least a shell that is locally spherical (on a scale larger than 1 / Gamma). Emission from parts of the shell moving at angle theta relative to the line of sight to the observer will arrive later with a time delay R(1 - costheta) / c. For small angles this time delay equals Rtheta2 / 2c. As the radiation is beamed with an effective beaming angle approx 1 / Gamma most of the radiation will arrive within a typical angular time scale:

Equation 9 (9)

The combination of time delay and blueshift implies that if the emitted spectrum is a power law spectrum with a spectral index alpha then the observed signal from the instantaneous emission of a thin shell will decay at late time as a power law with t-(2-alpha) [94, 272]. The observed pulse from an instantaneous flash from a thin shell is shown in Fig. 14.

Figure 14

Figure 14. The observed pulse from an instantaneous flash from a spherical relativistic thin shell moving relativistically and emitting emitting with a power low nu-0.6.

As I discuss later (see Section VIA) the similarity between the angular time scale and the radial time scale plays a crucial role in GRB models.

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