Radial time scale Edit

Consider a thin (of width $ \ll R/\Gamma^2 $) relativistic shell ($ \Gamma \gg 1 $) emitting as it is propagating from $ R_0 $ to $ R_f=R_0+\Delta R $. The arrival time of a photon emitted at $ R_0 $ is:

$ t_0=\frac{D-R_0}{c} $,

where $ D $ is the distance between the center of expansion and the observer and is typically (for instance, for GRBs) much larger than $ R_0 $. A photon emitted at $ R_f $ arrives at:

$ t_f=\frac{\Delta R}{\beta c}+\frac{D-R_f}{C} $,

where the first term is the time it takes the shell to propagate between $ R_0 $ and $ R_f $. The difference between arrival times of these photons is called the "radial time scale" and is given by:

$ t_R=t_f-t_0=\frac{\Delta R}{\beta c}-\frac{\Delta R}{c}\approx \frac{\Delta R}{2 \Gamma^2 c} $. This is approximately $ \Gamma^2 $ times shorter than the time it takes the shell to travel between $ R_0 $ and $ R_f $.

Angular time scale Edit

Another factor that has to be taken into account is that photons emitted at larger angles above the line of sight will take longer to reach the observer. A photon emitted at an angle $ \theta $ above the line of sight, will reach the observer later than a photon emitted at the same time along the line of sight. The arrival time difference will be:$ t_{\theta}=\frac{D-R cos \theta}{c}-\frac{D-R}{c}=\frac{R(1-cos \theta)}{c} \approx \frac{R\theta^2}{2c} $,

where the last transition assumes $ \theta $ is a small angle above the line of sight. Recall, that due to Relativistic Beaming, the observer can effectively see photons only up to an angle of $ \theta=1/\Gamma $. This implies an angular time scale of:$ t_{\theta}=\frac{R}{2 c \Gamma^2} $. Notice that this time also approximately equals the adiabatic expansion time (which is the timescale for particles to cool because of the expansion of the source).

Resulting time scale Edit

In general the resulting time scale of the signal seen by the observer is the sum of the radial and angular time scales. If $ \Delta R \gg R $, then the radial time-scale dominates the observed emission, whereas for $ \Delta R \ll R $ the angular scale dominates. Notice that in the latter regime, even if the emission is instantaneous in the source frame, it will still be smeared over the angular time scale in the observer's frame. In this sense, both the time-scales discussed here, set a minimum for the variability that can be observed from a given source.