Most gamma-ray bursts ( GRBs ) observed by the Swift satellite show an early steep decay phase ( SDP ) in their X-ray lightcurve , which is usually a smooth continuation of the prompt gamma-ray emission , strongly suggesting that it is its tail .
However , the mechanism behind it is still not clear .
The most popular model for this SDP is High Latitude Emission ( HLE ) , in which after the prompt emission from a ( quasi- ) spherical shell stops photons from increasingly large angles relative to the line of sight still reach the observer , with a smaller Doppler factor .
This results in a simple relation between the temporal and spectral indexes , \alpha = 2 + \beta where F _ { \nu } \propto t ^ { - \alpha } \nu ^ { - \beta } .
While HLE is expected in many models for the prompt GRB emission , such as the popular internal shocks model , there are models in which it is not expected , such as sporadic magnetic reconnection events .
Therefore , testing whether the SDP is consistent with HLE can help distinguish between different prompt emission models .
In order to adequately address this question in a careful quantitative manner we develop a realistic self-consistent model for the prompt emission and its HLE tail , which can be used for combined temporal and spectral fits to GRB data that would provide strict tests for the HLE model .
We model the prompt emission as the sum of its individual pulses with their HLE tails , where each pulse arises from an ultra-relativistic uniform thin spherical shell that emits isotropically in its own rest frame over a finite range of radii .
Analytic expressions for the observed flux density are obtained for the internal shock case with a Band function emission spectrum .
We find that the observed instantaneous spectrum is also a Band function .
Our model naturally produces , at least qualitatively , the observed spectral softening and steepening of the flux decay as the peak photon energy sweeps across the observed energy range .
The observed flux during the SDP is initially dominated by the tail of the last pulse , but the tails of one or more earlier pulses can become dominant later on .
A simple criterion is given for the dominant pulse at late times .
The relation \alpha = 2 + \beta holds also as \beta and \alpha change in time .
Modeling several overlapping pulses as a single wider pulse would over-predict the emission tail .