General kinematic considerations impose constraints on the temporal structure produced when the energy of a relativistic shell is converted to radiation. The enormous variability of the temporal profiles of GRBs from one burst to another in contrast to the relatively regular spectral characteristics, was probably the reason that until recently this aspect of GRBs was largely ignored. However, it turns out that the observed temporal structure sets a strong constraint on the energy conversion models [20, 230]. GRBs are highly variable (see section 2.2) and some configurations cannot produce such temporal profiles.
Special relativistic effects determine the observed duration of the burst from a relativistic shell (see Fig. 14).
Figure 14. Different time scales in terms of the arrival time of four photons: t_{A},t_{B}, t_{C}, and t_{D}. T_{radial} = t_{C} - t_{A}; T_{angular} = t_{D} - t_{A}, / c = t_{B} - t_{A}. |
The Radial Time Scale: T_{radial}: Consider an infinitely thin relativistic shell with a Lorentz factor _{E} (the subscript E is for the emitting region). Let R_{E} be a typical radius characterizing the emitting region (in the observer frame) such that most of the emission takes place between R_{E} and 2R_{E}. The observed duration between the first photon (emitted at R_{E}) and last one (emitted at 2R_{E}) is [205, 18]:
(29) |
The Angular Time Scale: T_{ang}: Because of relativistic beaming an observer sees up to solid angle of _{E}^{-1} from the line of sight. Two photons emitted at the same time and radius R_{E}, one on the line of sight and the other at an angle of _{E}^{-1} away travel different distances to the observer. The difference lead to a delay in the arrival time by [205, 18, 230] :
(30) |
Clearly this delay is relevant only if the angular width of the emitting region, is larger than _{E}^{-1}.
In addition there are two other time scales that are determined by the flow of the relativistic particles. These are:
Intrinsic Duration: T: The duration of the flow. This is simply the time in which the source that produces the relativistic flow is active. T = / c, where is the width of the relativistic wind (measured in the observer's rest frame). For an explosive source R_{i}. However, could be much larger for a wind. The observed duration of the burst must be longer or equal to / c
Intrinsic Variability T: The time scale on which the inner source varies and produces a subsequent variability with a length scale = cT in the flow. Naturally, T sets a lower limit to the variability time scale observed in any burst.
Clearly and must satisfy:
(31) |
Finally we have to consider the cooling time scale.
(32) |
Note that this differs from the usual time dilation which gives _{E}e / P.
For synchrotron cooling there is a unique energy dependence of the cooling time scale on frequency: T_{cool}() ^{-1/2} [103] (see Eq. 59). If T_{cool} determines the variability we will have T() ^{-1/2}. This is remarkably close to the observed relation: T ^{-0.4} [102]. Quite generally T_{cool} is shorter than the hydrodynamics time scales [103, 232, 69]. However, during the late stages of an afterglow, T_{cool} becomes the longest time scale in the system.