The low-luminosity tail of the GRB distribution: the case of GRB 980425

Abstract

(abridged) The association of GRB 980425 with SN 1998bw at z=0.0085 implies the existence of a population of GRBs with an isotropic-equivalent luminosity which is about 10^4 times smaller than in the standard cosmological case. We investigate two scenarios to explain a weak GRB : a normal (intrinsically bright) GRB seen off-axis or an intrinsically weak GRB seen on-axis. For each of these two scenarios, we first derive the conditions to produce a GRB 980425-like event and we then discuss the consequences for the event rate. If we exclude the possibility that GRB 980425 is an occurence of an extremely rare event observed by chance during the first eight years of the 'afterglow era', the first scenario implies that (i) the local rate of standard bright GRBs is much higher than what is usually expected; (ii) the typical opening angle is much narrower than what is derived from observations of a break in the afterglow lightcurve. In addition, we show that the afterglow of GRB 980425 in this scenario should have been very bright and easily detected. For these reasons the second scenario appears more realistic. We show that the parameter space of the internal shock model indeed allows GRB 980425-like events, in cases where the outflow is only mildly-relativistic and mildly-energetic. The rate of such weak events in the Universe has to be much higher than the rate of standard bright GRBs to allow the discovery of GRB 980425 during a short period of a few years. However it is still compatible with the observations as the intrinsic weakness of these GRB 980425-like bursts does not allow detection at cosmological redshift. We finally briefly discuss the consequences of such a high local rate of GRB 980425-like events for the future prospects of detecting non-electromagnetic radiation, especially gravitational waves.

Full text

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arXiv:0707.0931v1 [astro-ph] 6 Jul 2007
Astronomy & Astrophysics manuscript no. weakgrb
February 1, 2008
c ? ESO 2008
The low-luminosity tail of the GRB distribution:
the case of GRB 980425.
F. Daigne1,2and R. Mochkovitch1
1Institut d’Astrophysique de Paris, 98 bis boulevard Arago, 75014 Paris France
2Universit´ e Pierre et Marie Curie-Paris VI, 4 place Jussieu, 75005 Paris, France
Received 21 July 2006 / Accepted 23 November 2006
ABSTRACT
Context. The association of GRB 980425 with the nearby supernova SN 1998bw at z=0.0085 implies the existence of a
population of gamma-ray bursts with an isotropic-equivalent luminosity which is about 104times smaller than in the standard
cosmological case. Apart from its weak luminosity, GRB 980425 appears as a “normal” burst regarding all its other properties
(variability, duration, spectrum), with however a rather low peak energy Ep ≃ 30 − 100 keV.
Aims. We investigate two scenarios to explain a weak gamma-ray burst such as GRB 980425 : a normal (intrinsically bright)
gamma-ray burst seen off-axis or an intrinsically weak gamma-ray burst seen on-axis.
Methods. For each of these two scenarios, we first derive the conditions to produce a GRB 980425-like event and we then discuss
the consequences for the event rate. In the second scenario, this study is done in the framework of the internal shock model.
Results. If we exclude the possibility that GRB 980425 is an occurence of an extremely rare event observed by chance during
the first eight years of the “afterglow era”, the first scenario implies that (i) the local rate of “standard” bright gamma-ray
bursts is much higher than what is usually expected; (ii) the typical opening angle in gamma-ray bursts ejecta is much narrower
than what is derived from observations of a break in the afterglow lightcurve. In addition to this statistical problem, we show
that the afterglow of GRB 980425 in this scenario should have been very bright and easily detected. For these reasons the
second scenario appears more realistic. We show that the parameter space of the internal shock model indeed allows GRB
980425-like events, in cases where the outflow is only mildly-relativistic and mildly-energetic. The rate of such weak events
in the Universe has to be much higher than the rate of “standard” bright gamma-ray bursts to allow the discovery of GRB
980425 during a short period of a few years. However it is still compatible with the observations as the intrinsic weakness
of these GRB 980425-like bursts does not allow detection at cosmological redshift with present gamma-ray instruments. We
finally briefly discuss the consequences of such a high local rate of GRB 980425-like events for the future prospects of detecting
non-electromagnetic radiation, especially gravitational waves.
Key words. Gamma rays: bursts – (Star:) Supernovae: individual: SN1998bw – Shock waves – Radiation mechanisms: non-
thermal
1. Introduction
Our knowledge of the gamma-ray burst (hereafter GRB)
population has dramatically increased in the recent
years. In addition to “standard” cosmological bright
GRBs peaking at 100 keV – 1 MeV have been dis-
coved X-ray Flashes (hereafter XRFs) and X-ray Rich
GRBs (hereafter XRRs) that peak at much lower en-
ergy (Heise et al. 2001; Kippen et al. 2001; Barraud et al.
2003), weak GRBs (such as GRB 031203, Sazonov et al.
2004; Soderberg et al. 2004) and even a few “local” events:
GRB 980425 (Galama et al. 1998) and GRB 060218
(Mirabal et al. 2006). It is usually believed that a long
GRB is produced by an ultra-relativistic outflow ejected
Send offprint requests to: F. Daigne, e-mail: daigne@iap.fr
from a black hole that has just formed in the collapse
of a very massive star (collapsars, Woosley (1993)). The
prompt GRB emission originates from the relativistic
ejecta itself, and is probably due to the formation of inter-
nal shocks (Rees & Meszaros 1994), whereas the afterglow
is emitted by the strong shock that propagates within the
ambient medium during the deceleration phase of the out-
flow (Meszaros & Rees 1997).
It is important to understand the physical origin of the ob-
served diversity in this theoretical framework. There are
two groups of explanations : either the diversity is only
apparent: all GRBs are intrinsically very similar, but the
observation conditions (viewing angle for instance) break
this similarity; or the diversity is intrinsic: all collapsars
do not produce the same relativistic outflow and therefore

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2 F. Daigne and R. Mochkovitch: The low-luminosity tail of the GRB distribution: the case of GRB 980425.
the same GRB.
In this paper, we focus on the closest GRB, i.e. GRB
980425. This burst has been detected by Beppo-SAX and
BATSE. It appears as a “standard” single-pulse burst re-
garding its gamma-ray properties : duration Tγ ∼ 31 s,
peak flux (40-700 KeV) P ∼ 2.4 × 10−7erg/cm2/s, Band
spectrum with low- and high-energy slopes α ∼ −0.8
and β ∼ −2.3 and peak energy Ep ∼ 68 ± 40 keV
(Frontera et al. 2000). With maybe the exception of a
somewhat lower peak energy, all these properties are
very close to those of standard GRBs. Note also that
Norris et al. (2000) derived an especially large time lag for
this burst between BATSE channels 1 and 3 : ∆13∼ 4.5 s.
A peculiar type Ic supernova, SN 1998bw, was dis-
covered simultaneously in the Beppo-SAX
(Galama et al. 1998). The probability to have two such
rare events occuring at the same time in the same direc-
tion is very low, so that GRB 980425 and SN 1998bw are
probably physically associated (see also Kouveliotou et al.
2004). This is reinforced by other associations that have
been found since 1998, the better case being probably the
association of GRB 030329 with SN2003dh (Stanek et al.
2003) at z = 0.168. If the association GRB 980425 / SN
1998bw is real, then this burst is far from “standard”. The
host galaxy of SN 1998bw is indeed located at z = 0.0085
and GRB 980425 is therefore the closest GRB ever de-
tected. As the peak flux of GRB 980425 is comparable to
typical peak fluxes of other Beppo-SAX GRBs, it means
that it is intrinsically much weaker (by more than four
orders of magnitude).
In section 2, we recall the various physical factors enter-
ing in the observed peak flux and peak energy of a cos-
mic GRB. We then study in section 3 a scenario where
GRB 980425 is a standard bright GRB seen off-axis, and
therefore apparently weak. In section 4 we detail an al-
ternative explanation, where GRB 980425 is intrinsically
weak. We show how the internal shock model can allow
for such weak bursts. Both in section 3 and 4 we discuss
the consequences of each scenario in terms of GRB rate.
Our results are summarized in section 5.
error-box
2. Peak flux and peak energy of a cosmic GRB
We assume that a GRB is produced by a relativistic out-
flow of Lorentz factor Γ and opening angle ∆θ ≫ 1/Γ
generated by a source at redshift z (see Figure 1). We de-
fine θ0as the angle between the line-of-sight and the axis
of the ejecta. The observed bolometric peak flux and peak
energy are given by
Pobs= KP(Γ;∆θ;θ0) ×Lrad,4π
4πD2
L
(1)
and
Eobs
p
= KE(Γ;∆θ;θ0) ×
Ep
1 + z,
(2)
where Lrad,4π and Ep are the isotropic equivalent lumi-
nosity and the peak energy measured in the GRB source
Fig.1. Geometry. A relativistic outflow of Lorentz fac-
tor Γ is emitted by the source (located at redshift z) in a
cone of opening angle ∆θ. The line-of-sight of the observer
makes an angle θ0with the axis of the cone. The obser-
vation is made on-axis (resp. off-axis) if θ0 ≤ ∆θ (resp.
θ0> ∆θ).
frame. The correction for the viewing angle are approxi-
matively given in the on-axis and the off-axis cases by
KP(Γ;∆θ;θ0) =
?
1
1
if θ0< ∆θ
3 if θ0> ∆θ
2(1+Γ2(θ0−∆θ)2)
(3)
and
KE(Γ;∆θ;θ0) =
?
1
1
if θ0< ∆θ
1+Γ2(θ0−∆θ)2 if θ0> ∆θ
. (4)
Therefore, the peculiar properties of GRB 980425 may
have two origins : (i) either GRB 980425 is a “stan-
dard” GRB, i.e. intrinsically bright with Lrad,4π
a few 1050erg.s−1, but appears as a weak GRB because
it is seen with a large viewing angle; (ii) or GRB 980425
is an intrinsically weak GRB (Lrad,4π≃ 3 × 1046erg.s−1)
seen on-axis. In this case GRB 980425 has been detected
only because of its very low redshift. In the following,
we successively investigate these two scenarios. The first
one has already been adressed by several authors (e.g.
Nakamura 1999; Salmonson 2001; Yamazaki et al. 2003;
Guetta et al. 2004) so we focus mainly on the second sce-
nario. For this case, the intrinsic properties of GRBs are
studied in the framework of the internal shock model,
where the prompt emission comes from a relativistic wind
which converts part of its kinetic energy into radiation via
the formation of shock waves within the wind itself. Such
internal shocks can occur if the wind is generated with a
highly variable Lorentz factor (Rees & Meszaros 1994).
>∼
3. GRB 980425 as an intrinsically bright GRB
seen off-axis
3.1. Conjugate effect of a low redshift and a large
viewing angle
In Figure 2, GRB 980425 has been plotted in a bolometric
peak flux vs peak energy diagram, together with the
GRBs detected by Beppo-SAX which have a known
redshift. A sensitivity of 10−7erg cm−2s−1represen-
tative of the BATSE and Beppo-SAX GRBM+WFC

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F. Daigne and R. Mochkovitch: The low-luminosity tail of the GRB distribution: the case of GRB 980425.3
Fig.2. Bolometric peak flux vs peak energy di-
agram: All Beppo-SAX GRBs with a known redshift
are indicated in this diagram (crosses). The peculiar
burst GRB 980425 is indicated by a circled cross. A
vertical dotted line stands for a threshold Pmin
10−7erg cm−2s−1representative of the BATSE and
Beppo-SAX WFC+GRBM
Effect of the redshift : we have plotted the evolution of
GRB 980425 when its redshift increases from z = 0.008
to z = 1 (cosmological distance). The final result is indi-
cated as GRB 980425∗. Effect of the viewing angle : we
have then plotted the evolution of GRB 980425∗when the
viewing angle decreases. We assume that GRB 980425∗is
seen off-axis with θ0= ∆θ+4/Γ (see text). The final result
for θ0= 0 (on-axis) is indicated as GRB 980425∗∗.
=
instruments (Band 2003).
instruments is also indicated as a dotted line (Band
2003). We use equations (1)–(4) to estimate under which
conditions on the redshift and the viewing angle GRB
980425 can be an intrinsically bright GRB despite its
low apparent intensity. To clarify the respective roles of
redshift and viewing angle, we perform the following two
transformations:
– Effect of the redshift : GRB 980425 has first been
moved from z = 0.008 to z = 1 (cosmological distance).
Both the peak energy and the peak flux decrease due
to the increase of redshift and luminosity distance. The
final result is named GRB 980425∗: this burst is clearly
much to weak to be observed as a GRB or even a XRF.
Its bolometric peak flux is indeed at least three orders
of magnitude below the sensitivity of past or current
detectors.
– Effect of the viewing angle : the viewing angle of
GRB 980425∗is then decreased down to θ0= 0 (on-axis
observation), assuming that GRB 980425 is initially seen
off-axis with a large viewing angle θ0= ∆θ + k/Γ (here
Fig.3. The ratio of detected off-axis to on-axis
GRBs in the local Universe : this ratio within a dis-
tance D is plotted as a function of D for a power-law lumi-
nosity function of slope −1.7 between Lmin= 1050erg s−1
and Lmax= 1054erg s−1and three different distributions
of the opening angle: (1) a uniformly distributed open-
ing angle between 1 and 10◦(solid line); (2) a power-law
distribution p(∆θ) ∝ ∆θ−2between 1/Γ and π/2 (dashed
line); (3) an opening angle correlated with the isotropic
luminosity (dotted line). In this last case, the true lumi-
nosity L(1 − cos∆θ) is assumed to be constant and equal
to Lmin.
k = 4). This evolution is computed from the approximate
formulae (3) and (4). The final result is named GRB
980425∗∗. Now this burst is clearly back in the “standard”
GRB region, very close to GRB 971214.
Therefore,asit wasalready
Yamazaki et al. (2003), the
GRB 980425 are compatible with those of an intrinsically
bright GRB observed off-axis. In the case illustrated in
Figure 2, the decrease of the flux of GRB 980425 by
a factor ∼ 104due to a viewing angle θ0 = ∆θ + 4/Γ
iscompensatedbya smaller
(DL(z = 0.008)/DL(z = 1))2∼ 2 × 10−4.
pointed outby
ofpeculiar properties
luminosity distance
If it is clear from the previous analysis that a slightly
off-axis viewing angle (θ0− ∆θ ∼ 4/Γ ∼ 2.3◦if Γ ≃ 100)
can account for the properties of GRB 980425, one should
however keep in mind that the peak flux decreases very
rapidly with viewing angle (see equation (3)). For θ0 =
∆θ +k/Γ with k ≥ 6, the peak flux is divided by a factor
larger than ∼ 105and the decrease of luminosity distance
is not able anymore to compensate: such off-axis GRBs
cannot be detected. As Γ>∼100 in GRB outflows (Baring
1995; Lithwick & Sari 2001), the angle 6/Γ<∼3.4◦is small

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4 F. Daigne and R. Mochkovitch: The low-luminosity tail of the GRB distribution: the case of GRB 980425.
and the probability to observe GRB 980425-like events
should be low.
3.2. A statistical problem ?
For a detector having a threshold Pmin(hereafter we as-
sume Pmin = 10−7erg cm−2s−1, which is representa-
tive of instruments such as Beppo-SAX GRBM+WFC
or BATSE) the observed rate of on-axis GRBs within
D = 40 Mpc is given by
Ron=4π
3D3R0×
?π/2
0
d∆θ p(∆θ)(1 − cos∆θ) ,(5)
where R0is the local GRB volumic rate and p(∆θ) is the
probability distribution of the opening angle ∆θ. Here we
assume that the minimum GRB luminosity Lminis larger
than 4πD2Pmin= 2 × 1046erg s−1so that all nearby on-
axis GRBs are detected. On the other hand, the rate of
off-axis GRBs in the same volume is affected by the rapid
decrease of the GRB flux with the opening angle so that
Roff = R0
?D
?
0
dD 4πD2
?π/2
?
0
∆θ +kmax(D,L)
d∆θ p(∆θ)
?Lmax
??
Lmin
dL p(L)
×
cos∆θ − cos
Γ
,(6)
where p(L) is the GRB luminosity function and kmax is
given by
kmax(D,L) ≃
??1
2
L
4πD2Pmin
?1/3
− 1 .(7)
In both equations (6) and (7), the probability distribution
of the Lorentz factor Γ is neglected and Γ is supposed
to be constant. In the most simple case where both the
luminosity L and the opening angle ∆θ are constant, we
find that the ratio of detected off-axis over on-axis GRBs
can be expressed as a ratio of two solid angles :
Roff
Ron
For D = 40 Mpc and L = 1051erg s−1, we get kmax= 5.4,
so that kmax/Γ<∼3.1◦for Γ>∼100. This leads to a rate
of detected off-axis over on-axis GRBs Roff/Ron∼ 16 for
∆θ = 1◦, Roff/Ron∼ 1 for ∆θ = 7.5◦, and Roff/Ron∼
0.7 for ∆θ = 10◦. This implies that the rates of detected
off-axis and on-axis GRBs should be comparable for typ-
ical ∆θ of a few degrees and that it is only for ∆θ<∼1◦
that the number of detected off-axis GRBs is much larger
than the number of on-axis GRBs. The local apparent
rate of “standard” on-axis bright GRBs can be estimated
to be of the order of ∼ 1/(5000− 30000 yr) within 40
Mpc (Porciani & Madau 2001; Schmidt 2001; Perna et al.
2003; Guetta et al. 2004). This simple analysis would then
imply that with GRB 980425 we have observed by chance
a very rare event. Even if it has some rather different prop-
erties, GRB 060218/SN2006aj at z = 0.0331 (144 Mpc)
≃
cos∆θ − cos
?
∆θ +kmax(D,L)
Γ
?
1 − cos∆θ
. (8)
(Cusumano et al. 2006; Mirabal et al. 2006; Masetti et al.
2006; Soderberg et al. 2006a)) is another case of a nearby
burst. At this distance Roff/Ronshould be even smaller.
Statistical studies (Bosnjak et al. 2006) also indicate that
there are probably other GRB980425-like events in the
BATSE catalog. For all these reasons, we conclude that
the off-axis interpretation probably suffers a statistical
problem. In the more general case, the two solid angles
in equation (8) have to be averaged over the luminos-
ity function, the distance and the opening angle distri-
bution, according to equations (6) and (7). We did that
for a power-law luminosity function with slope −1.7 (this
slope giving a good fit to the logN − logP diagram, see
e.g Daigne et al. 2006) between Lmin= 1050erg s−1and
Lmax = 1054erg s−1. We tested three possible distribu-
tions for the opening angle: (1) a uniformly distributed
opening angle; (2) a power-law distributed opening angle;
(3) an opening angle correlated with the isotropic equiva-
lent luminosity so that the true luminosity is constant, as
suggested by observations of achromatic breaks in after-
glow lightcurves (Frail et al. 2001). We find (see Figure 3)
that except below ∼ 5 Mpc where the total (on+off axis)
event rate is very low due to a small volume, the expected
rate of off-axis GRBs is either comparable (uniformly dis-
tributed opening angle) or smaller than the on-axis GRB
rate. This is in full agreement with our simple estimate
(equation (8)). This equation also indicates that the only
way to have a much higher rate of off-axis GRBs is to
assume a very small opening angle (∆θ ≪ 1/Γ), in con-
tradiction with observations (Frail et al. 2001).
4. GRB 980425 as an intrinsically weak GRB
We now consider in this section an alternative scenario
where GRB 980425 is an intrinsically weak GRB seen on-
axis. The intrinsic GRB properties are considered in the
framework of the internal shock model (Rees & Meszaros
1994).
4.1. Internal shocks
The dynamics of the internal shocks as well as the tem-
poral and spectral properties of the emission have been
studied in details in Daigne & Mochkovitch (1998) us-
ing a simplified model where the outflow is made of
a large number of discrete relativistic shells that in-
teract by direct collision only. This approach has been
validated by 1D relativistic hydrodynamical simulations
(Daigne & Mochkovitch 2000). We adopt here an even
simpler version of the model, where we consider only a
typical internal shock due to the collision of two shells of
equal mass M and Lorentz factors Γ1and Γ2> Γ1. This
toy model is described in Barraud et al. (2005). It is found
that the collision between the two shells occurs at radius
Ris≃
8κ2
(κ − 1)(κ + 1)3Γ2cτ , (9)

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F. Daigne and R. Mochkovitch: The low-luminosity tail of the GRB distribution: the case of GRB 980425.5
that the isotropic equivalent radiated luminosity due to
internal shocks is given by
Lrad,4π≃ ǫe(√κ − 1)2
κ + 1
˙E (10)
and that the peak energy takes the form
Ep≃ K
˙Exφxy(κ)
τ2xΓ6x−1, (11)
where˙E is the isotropic equivalent kinetic energy injection
rate in the relativistic outflow, τ is the duration of the
relativistic ejection (i.e. here the time interval between the
two shell ejections), Γ is the mean Lorentz factor (Γ1+
Γ2)/2 and the contrast κ = Γ2/Γ1 is a measure of the
initial amplitude of the variations of the Lorentz factor in
the ejecta; ǫeis the fraction of the dissipated energy in the
shock that is injected into non-thermal electrons, and K
is a constant. The function φxy(κ) is given by
φxy(κ) =(√κ − 1)2y(κ − 1)2x(κ + 1)6x−1
κ2x+y−1
2
(12)
and is steadily increasing with κ. The x and y parameters
have been introduced by considering that the peak energy
in the comoving frame of the shocked material scales as
E′
p∝ ρxǫy,
where ρ is the comoving density and ǫ the dissipated
energy per unit mass.
(13)
In the standard synchrotron model with constant
equipartition parameters it is assumed that the mag-
netic field is amplified to reach a fraction ǫB of the
total dissipated energy in the shock, and that a frac-
tion ǫe of this dissipated energy is injected into a frac-
tion ζ of the electrons, which are therefore accelerated
to high Lorentz factors. This leads to x = 1/2 and
y = 5/2, as the synchrotron energy scales as E′
the magnetic field as B ∝ ǫ1/2
cal Lorentz factor of the electrons as Γe ∝ (ǫe/ζ)ǫ.
However Daigne & Mochkovitch (2003) have shown that
smaller values of x and y are required to reproduce
the hardness-intensity and hardness-fluence correlations
observed in many GRB pulses (Golenetskii et al. 1983;
Liang & Kargatis 1996). They consider x = y = 0.5 and
x = y = 0.25. Such exponents can for example be obtained
with the standard synchrotron process if the equipartition
parameters vary with shock intensity.
p∝ BΓ2
e,
B(ρǫ)1/2and the typi-
4.2. Producing GRB980425-like events
The value of Lrad,4πand Epgiven by equations (10) and
(11) are fixed by 6 physical quantities :˙E,¯Γ, κ, τ, ǫeand
K. The value of ǫe and K depend on the details of the
physical processes in the shocked material, which are not
studied here. We adopt ǫe= 1/3 and we fix the value of
K by demanding that a “typical” GRB at redshift z = 1
with τ = 5 s (observed duration of (1 + z)τ = 10 s),
κ = 4,˙E = 1.5 × 1052erg.s−1(Lrad,4π = 1051erg.s−1)
and¯Γ = 300 has an observed peak energy Eobs
Note that in this case, the efficiency of internal shocks is
Lrad,4π/˙E = 6.7%, as the product of ǫe = 1/3 with an
internal shock dynamical efficiency of 20% for κ = 4 (see
equation 10). The other parameters are limited by several
constraints (Daigne & Mochkovitch 2004):
p
= 200 keV.
(1) Transparency during the internal shock phase. The
relativistic ejecta has to be transparent during the internal
shock phase. The Thomson optical depth of the outflow is
given by (M´ esz´ aros & Rees 2000; Daigne & Mochkovitch
2002) :
τ(Ris) ≃
σT˙Eτ
4πΓRis(Ris+ 2Γ2cτ)mpc2. (14)
Then the condition τ(Ris) < 1 leads to
˙E
Γ5<4πmpc4
σT
f1(κ) τ ,(15)
with
f1(κ) =
8κ2
(κ − 1)(κ + 1)3
?
8κ2
(κ − 1)(κ + 1)3+ 2
?
.(16)
(2) No pair production. The relativistic ejecta has to
be transparent to pairs during the internal shock phase.
Pairs are produced by photon-photon annihilation in
the high-energy part of the GRB spectrum. At radius
Ris, the approximate optical depth for pair creation is
(M´ esz´ aros & Rees 2000; Lithwick & Sari 2001) :
τ±(Ris) ≃
?
α±σTLrad,4πτ
4πR2Γmec2
?
Ep
Γmec2
?β−2?2
,(17)
where β is the high-energy slope of the spectrum and α±
is a dimensionless number that depends on the spectral
shape. In the following, we adopt β = 2.5 (Preece et al.
2000). This leads to α± ≃ 0.06. Then the condition
τ±(Ris) < 1 leads to
˙E1+x(β−2)
Γ5+6x(β−2)<4πmec4
α±σT
f2(κ)
ǫe
?
K
mec2
?2−β
τ1+2x(β−2)(18)
with
f2(κ) =
64κ4
(√κ − 1)2(κ − 1)2(κ + 1)5Φβ−2
We also considered an additional constraint provided
by the lack of an observed high-energy cutoff due to
photon-photon annihilation in GRB spectra (see e.g.
Lithwick & Sari 2001). However, the constraint expressed
by equation (18) is dominant in most cases;
xy (κ)
(19)