Distribution of Gamma-ray Burst Ejecta Energy with Lorentz Factor Page: 2 of 8
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the central source, or a short lived central explosion that produces ejecta with some distribution of
Lorentz factor (LF). In either of these scenarios the deceleration of the afterglow shock is reduced
due to the energy being added to it, and this in turn produces a slowly declining light curve.
A long lived activity of the central source is not very appealing since it would require the
source to be active up to several hours after the GRB, with a very smooth temporal behavior, where
most of the energy is in the outflow that is ejected around treaA2 - 104 s; this makes the problem
of the observed high efficiency for converting kinetic energy to gamma-ray radiation much worse
(Nousek et al. 2005). Another interesting way to produce an early flat phase in the afterglow
light curve (Eichler & Granot 2005) is by a line of sight that is slightly outside the (sharp) edge
of a roughly uniform jet (Granot et al. 2002; Granot, Ramirez-Ruiz & Perna 2005). This would,
however, naturally be accompanied by a weaker and softer prompt emission, perhaps resulting in
an X-ray flash or X-ray rich GRB rather than a classical GRB; the more pronounced this effect is
the flatter and longer lived the slow X-ray afterglow decay phase should be. Initial inspection of
the data does not show such a correlation, suggesting that viewing angle effects are probably not
the predominant cause of the early slow decay phase in the X-ray afterglows, at least under the
It is natural to expect that matter ejected in any explosion will have a range of velocities or
LFs. After a while (on a time scale, in the observer frame, of order a few times the duration of the
central engine activity) the ejecta will rearrange themselves such that the fastest moving plasma is
at the head of the outflow and the slowest at the tail end. This can occur either through internal
shocks within the outflow, or by a smooth decrease in the LF of the outflow toward the end of the
central source activity. If the ejecta have a finite range of LFs, the slower ejecta would gradually
catch up with the shocked external medium, injecting energy into the forward shock. If the slower
ejecta carry more energy than the faster ejecta, then this added energy would gradually increase
the energy of the afterglow shock, causing it to decelerate more gradually. Once the energy in
the lower LF ejecta becomes small compared to the energy already in the afterglow shock, the
blast wave evolution becomes impulsive (i.e. the subsequent small amount of energy injection
hardly effects the evolution of the forward shock), and if radiative losses are unimportant then it
approaches the adiabatic Blandford & McKee (1976) self-similar solution. This occurs when the
LF of the afterglow shock drops slightly below Fpa, the LF where dE/dlnF peaks and where
most of the energy in the outflow resides.
In this paper we use the Swift data to determine the time dependence of the blast wave LF.
aEichler & Granot (2005) point out that viewing angle effects might still be the dominant cause of the flat early
decay of the afterglow light curves if along some lines of sight the kinetic energy in the afterglow shock is very low
while the energy in gamma-rays remains high.
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Granot, Jonathan & Kumar, Pawan. Distribution of Gamma-ray Burst Ejecta Energy with Lorentz Factor, article, October 7, 2005; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc876998/m1/2/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.