Distribution of Gamma-ray Burst Ejecta Energy with Lorentz Factor Page: 3 of 8
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We find that the LF typically drops by a factor of ~ 2-4 during the shallow decline phase. This
is consistent with the basic picture suggested above, where a finite LF distribution for the ejecta
causes a more gradual decline of the forward-shock LF, which gives rise to a shallow light-curve,
and is an intermediate transition stage before the onset of the adiabatic Blandford-McKee solution.
2. Dependence of Burst Kinetic Energy on Lorentz Factor
The emission from an external shock can be described in terms of the shock front LF (F)
and the density profile of the circum-stellar medium (CSM). For a uniform CSM the synchrotron
characteristic frequency (v), the cooling frequency (v) and the flux at the peak of the spectrum
(F,max), in the observer frame, are proportional to F4, F-'t-2 and t3F respectively, where t is
the observed time. The flux at a frequency between the Un and v, is proportional to t3F6+2P and
for the observed band above Un and v, the flux scales as t2F4+2p. The observed flux is strongly
dependent on F and therefore even a small deviation from the F oc t-3/8 scaling has a very large
effect on the observed light-curve. The observed flux has a weaker dependence on F for a wind
like density stratification of the CSM; the flux in the two regimes considered above scales roughly
as F1+pt(-p>/2 and F2+Pt(p-2/2, respectively.
More generally, for a power law external density profile, pext =Ark, we have F,max oc FBR3-k O
2R3-3k/2 oc F8-3kt3-3k/2, Um oc FBjy oc F4R-k/2 o F4-kt-k/2, <Y o 1/FB2t and ve Oc FB27, oc F-1B-3t-2 O
R3k/2-4t-2 oc f3k-4t-2+3k/2 Therefore,
Fmax(v/Uc)-1/2 Oc F6-3k/2t2-3k/4 Uc < U < Ur
F I ~ Fmax(vU/ )(p->/2 OC Ft3-k(+5)/4 vm < v < v, (1)
Fv,max(vc/Um)(P1)/2(U/c-P/2 oc F4-k+p(4-k)/2t2-k(2+p)/4 U > max(Um, v)
Assuming that the LF distribution for the ejecta is E(> F) oc F-', we find g - -d log F/dlogt
is smaller by an amount 8 compared to the standard value of (3 -k)/2(4 -k), i.e. 3/8 (1/4) for a
uniform (wind) CSM, where
(3-k)a [3a/[8(8+a)] k=0,
2(4 -k)[2(4 -k)+a] a +j
The deviation to the LC temporal power-law index (Aa) from the standard case of Blandford-
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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/m1/3/: accessed April 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.