published in ApJ
PROBING THE STRUCTURE OF GAMMA-RAY BURST
JETS WITH THE STEEP DECAY PHASE OF THEIR EARLY
X-RAY AFTERGLOWS
Kentaro Takami1, Ryo Yamazaki1, Takanori Sakamoto2, and Goro Sato2
ABSTRACT
We show that the jet structure of gamma-ray bursts(GRBs) can be inves-
tigated with the tail emission of the prompt GRB. The tail emission that we
consider is identified as a steep decay component of the early X-ray afterglow
observed by the X-Ray Telescope on board Swift. Using a Monte Carlo method,
we derive for the first time the distribution of the decay index of the GRB tail
emission for various jet models. The new definitions of the zero of time and the
time interval of a fitting region are proposed. These definitions for fitting the
light curve lead us to a unique definition of the decay index, which is useful to
investigate the structure of the GRB jet. We find that if the GRB jet has a
core-envelope structure, the predicted distribution of the decay index of the tail
has a wide scatter and multiple peaks, which cannot be seen for the case of the
uniform- and the Gaussian jet. Therefore, the decay index distribution gives us
information about the jet structure. Especially if we observe events whose decay
index is less than about 2, both the uniform- and the Gaussian jet models will
be disfavored, according to our simulation study.
Subject headings: gamma-rays: burst — gamma-rays: theory
1. Introduction
Gamma-ray burst (GRB) jet structure, that is, the energy distribution E(θ) in the ultra-
relativistic collimated outflow, is at present not yet fully understood (Zhang & Mészáros
1 Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan;
takami@theo.phys.sci.hiroshima-u.ac.jp, ryo@theo.phys.sci.hiroshima-u.ac.jp.
2 NASA Goddard Space Flight Center, Greenbelt, MD 20771.
http://arxiv.org/abs/0704.1055v3
– 2 –
2002b). There are many jet models proposed in addition to the simplest uniform-jet model:
the power-law jet model (Rossi et al. 2002; Zhang & Mészáros 2002a), the Gaussian jet
model (Zhang et al. 2004), the annular jet model (Eichler & Levinson 2004), the multiple
emitting subshell model (Kumar & Piran 2000; Nakamura 2000), the two-component jet
model (Berger et al. 2003), and so on. The jet structure may depend on the generation
process of the jet and therefore may provide us important information about the central
engine of the GRB. For example, in the collapsar model for long GRBs (e.g., Zhang et al.
2003, 2004), the jet penetrates into and breaks out of the progenitor star, resulting in the
E(θ) ∝ θ−2 profile (Lazzati & Begelman 2005). For the compact binary merger model for
short GRBs, hydrodynamic simulations have shown that the resulting jet tends to have a
flat core surrounded by the power-law-like envelope (Aloy et al. 2005).
In the pre-Swift era, there were many attempts to constrain the GRB jet structure.
Thanks to the HETE-2, statistical properties of long GRBs, X-ray-rich GRBs, and X-
ray flashes were obtained (Sakamoto et al. 2005), which were thought to constrain the
jet models (Lamb et al. 2004). These observational results constrain various jet models,
such as the uniform-jet model (Yamazaki et al. 2004a; Lamb et al. 2005; Donaghy 2006),
the multiple subshell model (Toma et al. 2005b), the Gaussian jet model (Dai & Zhang
2005), and so on. For BATSE long GRBs, Yonetoku et al. (2005) derived the distribu-
tion of the pseudo-opening angle, inferred from the Ghirlanda (Ghirlanda et al. 2004) and
Yonetoku (Yonetoku et al. 2004) relations, as f(θj)dθj ∝ θ
j dθj , which is compatible with
that predicted by the power-law jet model as discussed in Perna et al. (2003) (however, see
Nakar et al. 2004). Afterglow properties are also expected to constrain the jet structure
(e.g., Granot & Kumar 2003); however, energy redistribution effects prevent us from reach-
ing a definite conclusion. Polarization measurements of optical afterglows bring us important
information (Lazzati et al. 2004).
In the Swift era, rapid follow-up observation reveals prompt GRBs followed by a steep de-
cay phase in the X-ray early afterglow (Tagliaferri et al. 2005; Nousek et al. 2006; O’Brien et al.
2006a). In the most popular interpretations, the steep decay component is the tail emis-
sion of the prompt GRB (the so called high-latitude emission), i.e., the internal shock
origin (Zhang et al. 2006; Yamazaki et al. 2006; Liang et al. 2006; Dyks et al. 2005), al-
though there are some other possibilities (e.g., Kobayashi et al. 2007; Panaitescu et al. 2006;
Pe’er et al. 2006; Lazzati & Begelman 2006; Dado et al. 2006). Then, for the uniform-jet
case, the predicted decay index is α = 1 − β, where we use the convention Fν ∝ T
−αν1+β
(Kumar & Panaitescu 2000). For power-law jet case (E(θ) ∝ θ−q), the relation is modified
to α = 1−β+(q/2). However, these simple analytical relations cannot be directly compared
with observations, because they are for the case in which the observer’s line of sight is along
the jet axis and because changing the zero of time, which potentially lies anywhere within
– 3 –
the epoch where we see the bright pulses, substantially alters the early decay slope.
Recently, Yamazaki et al. (2006) (Y06) investigated the tail emission of the prompt
GRB, finding that the jet structure can be described and that the global decay slope is
not so much affected by the local angular inhomogeneity as it is affected by the global
energy distribution. They also argued that the structured jet model is preferable, because
steepening GRB tail breaks appeared in some events. In this paper, we calculate for the first
time the distribution of the decay index of the prompt tail emission for various jet models
and find that the derived distributions can be distinguished from each other, so that the jet
structure can be more directly constrained than previous arguments. This paper is organized
as follows. We describe our model in § 2. In § 3, we investigate the distribution of the decay
index of the prompt GRB emission. Section 4 is devoted to discussions.
2. Tail Part of the Prompt GRB Emission
We consider the same model as discussed in the previous works (Y06; Yamazaki et al.
2004b; Toma et al. 2005a,b). The whole GRB jet, whose opening half-angle is ∆θtot, consists
of Ntot emitting subshells. We introduce the spherical coordinate system (r, ϑ, ϕ, t) in the
central engine frame, where the origin is at the central engine and ϑ = 0 is the axis of the
whole jet. Each emitting subshell departs at time t
(0 < t
< tdur, where j = 1, · · · , Ntot,
and tdur is the active time of the central engine) from the central engine in the direction of
~n(j) = (ϑ(j), ϕ(j)), and emits high-energy photons, generating a single pulse as observed. The
direction of the observer is denoted by ~nobs = (ϑobs, ϕobs). The observed flux from the jth
subshell is calculated when the following parameters are determined: the viewing angle of the
subshell θ(j)v = cos
−1(~nobs ·~n
(j)), the angular radius of the emitting shell ∆θ
, the departure
time t
, the Lorentz factor γ(j) = (1− β2
)−1/2, the emitting radius r
0 , the low- and high-
energy photon indices α
B and β
B , the break frequency in the shell comoving frame ν
(Band et al. 1993), the normalization constant of the emissivity A(j), and the source redshift
z. The observer time T = 0 is chosen as the time of arrival at the observer of a photon
emitted at the origin r = 0 at t = 0. Then, at the observer, the starting and ending times
of the jth subshell emission are given by
start ∼ t
dep +
1 + γ2
, (1)
1 + γ2
, (2)
where θ
+ = θ
v +∆θ
= max{0, θ(j)v −∆θ
}, and we use the formulas β(j) ∼ 1−1/2γ
and cos θ ∼ 1 − θ2/2 for γ(j) ≫ 1 and θ ≪ 1, respectively. The whole light curve from the
– 4 –
GRB jet is produced by the superposition of the subshell emission.
Y06 discussed some kinematical properties of prompt GRBs in our model and found
that each emitting subshell with θ(j)v ≫ ∆θ
produces a single, smooth, long-duration, dim,
and soft pulse, and that such pulses overlap with each other and make the tail emission of
the prompt GRB. Local inhomogeneities in the model are almost averaged during the tail
emission phase, and the decay index of the tail is determined by the global jet structure,
that is the mean angular distribution of the emitting subshell because in this paper all
subshells are assumed to have the same properties unless otherwise stated. Therefore, we
are essentially studying the tail emission from the usual continuous jets at once, i.e., from
uniform- or power-law jets with no local inhomogeneity. In the following, we study various
energy distributions of the GRB jet through the change of the angular distribution of the
emitting subshell.
3. Decay Index of the Prompt Tail Emission
In this section, we perform Monte Carlo simulations in order to investigate the jet
structure by calculating the statistical properties of the decay index of the tail emission.
For a fixed-jet model, we randomly generate 104 observers with their line of sights (LOSs)
~nobs = (ϑobs, ϕobs). For each LOS, the light curve, F (T ) of the prompt GRB tail in the
15–25 keV band is calculated, and the decay index is determined. The adopted observation
band is the low-energy end of the Burst Alert Telescope(BAT) detector and near the high-
energy end of the X-Ray Telescope(XRT) on Swift. Hence, one can observationally obtain
continuous light curves, beginning with the prompt GRB phase to the subsequent early
afterglow phase (Sakamoto et al. 2007), so that it is convenient for us to compare theoretical
results with observations. However, our actual calculations have shown that our conclusion is
not qualitatively altered, even if the observation band is changed, for example, to 0.5–10 keV,
as usually considered for other references.
For each light curve, the decay index is calculated by fitting F (T ) with a single power-
law form, ∝ (T − T∗)
−α, as in the following (see Fig. 1). The decay index α depends on the
choice of T∗ (Zhang et al. 2006; Yamazaki et al. 2006)
1. Let Ts and Te be the start and end
time, respectively, of the prompt GRB, i.e.,
Ts = min{T
start} (3)
1 Recently, Kobayashi & Zhang (2007) have discussed the way to choose the time zero. According to
their arguments, the time zero is near the rising epoch of the last bright pulse in the prompt GRB phase.
– 5 –
Te = max{T
} . (4)
Then, we take T∗ as the time until 99% of the total fluence, which is defined by Stotal =
F (T ′) dT ′, is radiated, that is,
F (T ′) dT ′ = 0.99Stotal . (5)
Then, the prompt GRB is in the main emission phase for T < T∗, while it is in the tail emis-
sion phase for T > T∗. The time interval [Ta, Tb], in which the decay index α is determined
assuming the form F (T ) ∝ (T − T∗)
−α, is taken to satisfy
F (Ta,b) = qa,bF (T∗) , (6)
where we adopt qa = 1 × 10
−2 and qb = 1 × 10
−3, unless otherwise stated. We find that in
this epoch the assumed fitting form gives a well approximation.
At first, we consider the uniform-jet case, in which the number of subshells per unit
solid angle is approximately given by dN/dΩ = Ntot/(π∆θ
tot) for ϑ < ∆θtot, where ∆θtot =
0.25 rad is adopted. The departure time of each subshell t
is assumed to be homogeneously
random between t = 0 and t = tdur = 20 sec. The central engine is assumed to produce
Ntot = 1000 subshells. In this section, we assume that all subshells have the same values of
the following fiducial parameters: ∆θsub = 0.02 rad, γ = 100, r0 = 6.0×10
14 cm, αB = −1.0,
βB = −2.3, hν
0 = 5 keV, and A = constant. Our assumption of constant A is justified
as follows. Note that the case in which N subshells that have the same brightness A are
launched into the same direction, but a different departure time, is equivalent to the case of
one subshell emission with the brightness of NA. This is because in the tail emission phase,
the second terms in the r.h.s. of Eqs. (1) and (2) dominate the first terms, so that the time
difference effect, which arises from the difference of tdep for each subshell, can be obscured.
Hence, giving the angular distribution of the emission energy is equivalent to giving the
angular distribution of the subshells with constant A. Also, Y06 showed that to obtain a
smooth, monotonic tail emission as observed by Swift, the subshell properties, hν ′0
and/or
A(j), cannot have wide scatter in the GRB jet. Therefore, we can expect, at least as the
zeroth-order approximation, that the subshells have the same properties.
The left panel of Fig. 2 shows the decay index α as a function of ϑobs. For ϑobs . ∆θtot
(on-axis case), α clusters around ∼ 3. On the other hand, when ϑobs & ∆θtot (off-axis case),
α rapidly increases with ϑobs. The reason is as follows. If all subshells are seen sideways
(that is, θ(j)v ≫ ∆θ
for all j), the bright pulses in the main emission phase followed by the
tail emission disappear because of the relativistic beaming effect, resulting in a smaller flux
contrast between the main emission phase and the tail emission phase compared with the on-
axis case. Then T∗ becomes larger. Furthermore, in the off-axis case, the tail emission decays
– 6 –
more slowly (|dF/dT | is smaller) than in the on-axis case. Then both Ta−T∗ and Tb−T∗ are
larger for the off-axis case than for the on-axis case. As can be seen in Fig. 3 of Zhang et al.
(2006), the emission seems to decay rapidly, so that the decay index α becomes large. The
left panel of Fig. 3 shows α as a function of the total fluence Stotal which is the sum of the
fluxes in the time interval, [Ts, Te]. In Fig. 3, both α and Stotal are determined observationally,
so that our theoretical calculation can be directly compared with the observation.
A more realistic model is the Gaussian jet model, in which the number of subshells per
unit solid angle is approximately given by dN/dΩ = C exp(−ϑ2/2ϑ2c) for 0 ≦ ϑ ≦ ∆θtot,
where C = Ntot/2πϑ
c [1 − exp(−∆θ
tot/2ϑ
c)] is the normalization constant. We find only a
slight difference between the results for the uniform- and the Gaussian jet models. Therefore,
we do not show the results for the Gaussian jet case in this paper.
Next, we consider the power-law distribution. In this case, the number of subshells per
unit solid angle is approximately given by dN/dΩ = C[1 + (ϑ/ϑc)
2]−1 for 0 ≦ ϑ ≦ ∆θtot,
i.e., dN/dΩ ≈ C for 0 ≦ ϑ ≪ ϑc and dN/dΩ ≈ C(ϑ/ϑc)
−2 for ϑc ≪ ϑ ≦ ∆θtot, where C =
(Ntot/πϑ
c)[ln(1 + (∆θtot/ϑc)
2)]−1 is the normalization constant and we adopt ϑc = 0.02 rad
and ∆θtot = 0.25 rad. The other parameters are the same as for the uniform-jet case.
As can be seen in the right panels of Figs. 2 and 3, both the ϑobs–α and Stotal–α diagrams
are complicated compared with the uniform-jet case. When ϑobs . ϑc, the observer’s LOS is
near the whole jet axis. Compared with the uniform-jet case, α is larger, because the power-
law jet is dimmer in the outer region, i.e., emitting subshells are sparsely distributed near
the periphery of the whole jet (see also the solid lines of Figs. 1 and 3 of Y06). If ϑobs ≫ ϑc,
the scatter of α is large. Some bursts have an especially small α of around 2. This comes
from the fact that the power-law jet has a core region (0 < ϑ . ϑc), where emitting subshells
densely distributed compared with the outer region. The core generates the light-curve break
in the tail emission phase, as can be seen in Fig. 4 (Y06). In the epoch before the photons
emitted by the core arrive at the observer (e.g., T − Ts . 7.5 × 10
2 s for the solid line in
Fig. 4), the number of subshells that contribute to the flux at time T , Nsub(T ), increases
with T more rapidly than for the uniform-jet case. Then, the light curve shows a gradual
decay. If the fitting region [Ta, Tb] lies in this epoch, the decay index α is around 2. In the
epoch after the photons arising from the core are observed (e.g., T −Ts & 7.5× 10
2 s for the
solid line in Fig. 4), the subshell emission with θ(j)v & ϑobs + ϑc is observed. Then Nsub(T )
rapidly decreases with T , and the observed flux suddenly drops. If the interval [Ta, Tb] lies
in this epoch, the decay index becomes larger than 4.
To compare the two cases considered above more clearly, we derive the distribution of
the decay index α. Here we consider the events whose peak fluxes are larger than 10−4 times
of the largest one in all simulated events, because the events with small peak fluxes are not
– 7 –
observed. Fig. 5 shows the result. For the uniform-jet case (solid line), α clusters around 3,
while for the power-law jet case (dotted line), the distribution is broad (1 . α . 7) and has
multiple peaks.
So far, we have considered the fiducial parameters. In the following, we discuss the
dependence on parameters, r0, γ, βB, tdur, and ∆θtot (It is found that the α-distribution
hardly depends on the value of αB, ∆θsub, and ν
0 within reasonable parameter ranges). At
first, we consider the case in which r0 = 1.0 × 10
14 cm is adopted, with other parameters
being fiducial. Fig. 6 shows the result. The shape of the α-distribution is almost the
same as that for the fiducial parameters, in both the uniform- and the power-law jet cases.
This comes from the fact that in a tail emission phase, the light curve for a given r0 is
approximately written as F (T ; r0) ≈ g(cT/r0), where a function g determines the light-curve
shape of the tail emission for other given parameters. Then, the light curves in the case of
r0 = r0,1 and r0 = r0,2, namely, F (T ; r0,1) and F (T ; r0,2), satisfy the relation F (T ; r0,2) ≈
F ((r0,1/r0,2)T ; r0,1). This can be seen, for example, by comparing the solid line with the
dotted one in Fig. 4. Hence, T∗, Ta, and Tb are approximately proportional to r0; in this
simple scaling, one can easily find that α remains unchanged for different values of r0.
Second, we consider the case of γ = 200 and r0 = 2.4× 10
15 cm, with other parameters
being fiducial. In this case, the angular spreading timescale (∝ r0/γ
2) is the same as in the
fiducial case, so that the tail emissions still show smooth light curves, although the whole
emission ends later, according to the scaling Te ∝ r0 (see the dot-dashed line in Fig. 4).
Fig. 7 shows the result. For large γ, the relativistic beaming effect is more significant, so
that the events in ϑobs & ∆θtot, which cause large α, are dim compared with the small-γ
case. Such events cannot be observed. For the power-law jet case, therefore, the number of
large-α events becomes small, although the distribution is still broad (1 . α . 4) and has
two peaks. On the other hand, for the uniform-jet case, the distribution of the decay index
α is almost the same as for the fiducial parameter set, because the value of the decay index
α in the case of ϑobs . ∆θtot is almost the same as that in the case of ϑobs & ∆θtot.
Third, we change the value of the high-energy photon index βB from −2.3 to −5.0, with
other parameters being fiducial. Fig. 8 shows the result. For the uniform-jet case, the mean
value is 〈α〉 ∼ 4, while 〈α〉 ∼ 3 for the fiducial parameters, so that the decay index defined
in this paper does not obey the well-known formula α = 1−βB (Kumar & Panaitescu 2000).
For the power-law jet case, the whole distribution shifts toward the higher value, and the
ratio of the two peaks changes. In the tail emission phase, the spectral peak energy Epeak is
below 15 keV (see also Y06), so that the steeper the spectral slope of the high-energy side
of the Band function, the more rapidly the emission decays, resulting in the dimmer tail
emission (see the dashed line in Fig. 4). Then, the fitting region [Ta, Tb] shifts toward earlier
– 8 –
epochs, because T∗ becomes small. Therefore, the number of events with small α increases,
and the number of events with large α decreases. Furthermore, we comment on the case in
which βB is varied for each event in order to more directly compare with the observation.
Here we randomly distribute βB according to the Gaussian distribution with a mean of −2.3
and a variance of 0.4. It is found that the results are not qualitatively changed.
Next, we change the value of the duration time tdur from 20 sec to 200 sec, with other
parameters being fiducial. The epoch of the bright pulses in the main emission phase becomes
longer than that in tdur = 20 sec. However, the behavior of the tail emission does not depend
on tdur very much (see Fig. 9). Therefore, the distribution of the decay index α is almost
the same as that for the fiducial parameters for both the uniform-jet case and the power-
law jet case. Even if we consider the case in which tdur is randomly distributed for each
event according to the lognormal distribution with an average of log(20 s) and a logarithmic
variance of 0.6, the results are not significantly changed.
Finally, we discuss the dependence on ∆θtot. Only the uniform-jet case is considered,
because the structured jet is usually quasi-universal and because we focus our attention
on the behavior of the uniform-jet model. The dotted line in Fig. 10 shows the result for
constant ∆θtot = 0.1 rad with other parameters being fiducial. We can see many events with
large α. The large α is observed because for small ∆θtot, although the off-axis events (i.e.,
∆θtot . ϑobs) are still dim because of the relativistic beaming effect, a fraction of such events
survives the flux threshold condition and are observable. Such events have large α & 5 (see
the 4th paragraph of this section, which explains the left panel of Fig. 2). This does not
occur in the large-∆θtot case. However, we still find in this case that there are no events
with α . 2. We consider another case in which ∆θtot is variable. Here we generate events
whose ∆θtot distributes as f∆θtotd(∆θtot) ∝ ∆θtot
−2d(∆θtot) (0.05 . ∆θtot . 0.4). Then for
a given ∆θtot, the quantities ν
0 and A are determined by hν
0 = (∆θtot/0.13)
−3.6 keV and
A ∝ (∆θtot)
−7.3, respectively. Other parameters are fiducial. If the model parameters are
chosen in this way, the Amati and Ghirlanda relations (Amati et al. 2002; Ghirlanda et al.
2004) are satisfied, and the event rates of long GRBs, X-ray-rich GRBs and X-ray flashes
become similar (Donaghy 2006). The solid line in Fig. 10 shows the result. Again we find
that there are no events with α . 2.
In summary, when we adopt model parameters within reasonable ranges, the decay
index becomes larger than ∼ 2 for the uniform- and the Gaussian jet cases, while a significant
fraction of events with α . 2 is expected for the power-law jet case. Therefore, if a non-
negligible number of events with α . 2 are observed, both the uniform- and the Gaussian
jet models will be disfavored. Furthermore, if we observationally derive the α-distribution,
the structure of GRB jets will be more precisely determined.
– 9 –
4. Discussion
We have calculated the distribution of the decay index, α, for the uniform-, Gaussian,
and the power-law jet cases. For the uniform-jet case, α becomes larger than ∼ 2, and its
distribution has a single peak. The Gaussian jet model predicts almost the same results as
the uniform-jet model. On the other hand, for the power-law jet case, α ranges between
∼ 1 and ∼ 7, and its distribution has multiple peaks. Therefore, we can determine the jet
structure of GRBs by analyzing a lot of early X-ray data showing a steep decay component
that is identified as a prompt GRB tail emission. However, one of the big challenges in the
Swift data for calculating the decay index in our definition is to derive the composite light
curve of BAT and XRT. Since the observed energy bands of BAT and XRT do not overlap,
we are forced to extrapolate one of the data sets to plot the light curve in a given energy
band. To derive the composite light curve unambiguously for a prompt and an early X-ray
emission, we need an observation of a prompt emission by current instruments, which overlap
the energy range of XRT.
The tail behavior with α . 2 does not appear in the uniform- and the Gaussian jet
models; hence, it is important to constrain the jet structure. However, in practical observa-
tions, such gradually decaying prompt tail emission might be misidentified with the external
shock component, as expected in the pre-Swift era. Actually, some events have shown such
a gradual decay, without the steep and the shallow decay phases, and their temporal and
spectral indices are consistent with a classical afterglow interpretation (O’Brien et al. 2006b).
Hence, in order to distinguish the prompt tail emission from the external shock component
at a time interval [Ta, Tb], one should study the spectral evolution and/or the continuity
and smoothness of the light curve (i.e., whether breaks appear or not) over the entire burst
emission.
In this paper, we adopt qa = 1×10
−2 and qb = 1×10
−3 when the fitting epoch [Ta, Tb] is
determined [see Eq. (6)]. Then, the prompt tail emission in this time interval is so dim that
it may often be obscured by the external shock component, causing a subsequent shallow
decay phase of the X-ray afterglow. One possible way to resolve this problem is to adopt
larger values of qa and qb, e.g., qa = 1/30 and qb = 1 × 10
−2, in which the interval [Ta, Tb]
shifts toward earlier epochs, so that the flux then is almost always dominated by the prompt
tail emission. We have calculated the decay index distribution for this case (qa = 1/30
and qb = 1 × 10
−2) and have found that the differences between uniform- and power-law
jets still arises as can be seen in the case of qa = 1 × 10
−2 and qb = 1 × 10
−3, so that our
conclusion remains unchanged. However, the duration of the interval, Tb−Ta, becomes short,
which might prevent us from observationally fixing the decay index at high significance. If
qa & 1/30, the emission at [Ta, Tb] is dominated by the last brightest pulse. Then, the light-
– 10 –
curve shape at [Ta, Tb] does not reflect the global jet structure, but reflects the properties of
the emitting subshell causing the last brightest pulse. Another way to resolve the problem
is to remove the shallow decay component. For this purpose, the origin of the shallow decay
phase should be clarified in order to extract the dim prompt tail emission exactly. The other
problem is contamination of X-ray flares, whose contribution has to be removed in order to
investigate the tail emission component. In any case, if the GRB occurs in an extremely
low-density region (a so-called naked GRB), where the external shock emission is expected
to be undetectable, our method may be a powerful tool to investigate the GRB jet structure.
This work was supported in part by Grants-in-Aid for Scientific Research of the Japanese
Ministry of Education, Culture, Sports, Science, and Technology 18740153 (R. Y.). T.S. was
supported by an appointment of the NASA Postdoctoral Program at the Goddard Space
Flight Center, administered by Oak Ridge Associated Universities through a contract with
NASA.
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This preprint was prepared with the AAS LATEX macros v5.2.
– 13 –
Ts T Te
 F(Tb)
 F(Ta)
 F(T  )
Ta Tb
1/100 1/1000
T [sec]*
fitting region
Fig. 1.— Example of how the decay index α is determined by the calculated light curve
F (T ). The start and end time of the burst are denoted by Ts and Te, respectively. The time
T∗ is determined by Eq. (5). The decay index α is determined by fitting F (T ) ∝ (T − T∗)
in the time interval [Ta, Tb].
Fig. 2.— Decay index α as a function of ϑobs, the angle between the whole jet axis and the
observers’ lines of sight. Red and green points represent events whose peak fluxes are larger
and smaller than 10−4 times the largest one in all simulated events, respectively. Left and
right panels are for the uniform- and power-law jet cases, respectively.
– 14 –
Fig. 3.— Decay index α as a function of the total fluence Stotal, the sum of the fluxes in
the time interval [Ts, Te]. Red and green points represent events whose peak fluxes are larger
and smaller than 10−4 times the largest one in all simulated events, respectively. Left and
right panels are for the uniform- and power-law jet cases, respectively.
– 15 –
10-12
10-10
102 103 104
T-Ts [sec]
fiducial
r0=1.0x10
γ=200 , r0=2.4x10
βB=-5.0
Fig. 4.— Examples of light curves of the prompt tail emission in the 15–25 keV band
for the power-law jet case and ϑobs > ϑc (ϑobs = 0.27 rad and ϑc = 0.02 rad). The solid
line shows the fiducial parameters. A bump caused by the core emission can be seen at
T − Ts ∼ 7.5 × 10
2 s. The dotted, dot-dashed, and dashed lines are for r0 = 1.0× 10
14 cm;
r0 = 2.4 × 10
15 cm and γ = 200; and βB = −5, respectively, with other parameters being
fiducial. Time intervals [Ta, Tb] for each case are denoted by the thick solid lines. The flux
is normalized by the peak value.
– 16 –
 0.05
 0.15
 0.25
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 5.— Distributions of the decay index α for uniform-jet (dN/dΩ = const.;solid line) and
power-law jet (dN/dΩ ∝ [1 + (ϑ/ϑc)
2]−1;dotted line) models, respectively. We assume that
all subshells have the same values of the following fiducial parameters: ∆θsub = 0.02 rad,
γ = 100, r0 = 6.0 × 10
14 cm, αB = −1.0, βB = −2.3, hν
0 = 5 keV, and A = constant. We
consider events whose peak fluxes are larger than 10−4 times the largest one in all simulated
events (red points in Fig. 2).
– 17 –
 0.05
 0.15
 0.25
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 6.— Same as Fig. 5, but for r0 = 1.0× 10
14 cm.
– 18 –
 0.05
 0.15
 0.25
 0.35
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 7.— Same as Fig. 5, but for γ = 200 and r0 = 2.4× 10
15 cm.
– 19 –
 0.05
 0.15
 0.25
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 8.— Same as Fig. 5, but for βB = −5.0.
– 20 –
 0.05
 0.15
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 9.— Same as Fig. 5, but for tdur = 200 sec.
– 21 –
 0.05
 0.15
 0.25
 0.35
 0  5  10  15  20
α ( decay index ) 
Fig. 10.— Distribution of the decay index α for the uniform-jet profile. The dotted line is for
∆θtot = 0.1 rad with other fiducial parameters. The solid line is for the variable-∆θtot case,
in which we generate events whose ∆θtot distributes as f∆θtotd(∆θtot) ∝ ∆θtot
−2d(∆θtot)
(0.05 . ∆θtot . 0.4), and for given ∆θtot, the quantities ν
0 and A are determined by
hν ′0 = (∆θtot/0.13)
−3.6 keV and A ∝ (∆θtot)
−7.3, respectively. Other parameters are fiducial.
	Introduction
	Tail Part of the Prompt GRB Emission
	Decay Index of the Prompt Tail Emission
	Discussion

  We show that the jet structure of gamma-ray bursts (GRBs) can be investigated
with the tail emission of the prompt GRB. The tail emission which we consider
is identified as a steep-decay component of the early X-ray afterglow observed
by the X-ray Telescope onboard Swift. Using a Monte Carlo method, we derive,
for the first time, the distribution of the decay index of the GRB tail
emission for various jet models. The new definitions of the zero of time and
the time interval of a fitting region are proposed. These definitions for
fitting the light curve lead us an unique definition of the decay index, which
is useful to investigate the structure of the GRB jet. We find that if the GRB
jet has a core-envelope structure, the predicted distribution of the decay
index of the tail has a wide scatter and has multiple peaks, which cannot be
seen for the case of the uniform and the Gaussian jet. Therefore, the decay
index distribution tells us the information on the jet structure. Especially,
if we observe events whose decay index is less than about 2, both the uniform
and the Gaussian jet models will be disfavored according to our simulation
study.

Introduction
Gamma-ray burst (GRB) jet structure, that is, the energy distribution E(θ) in the ultra-
relativistic collimated outflow, is at present not yet fully understood (Zhang & Mészáros
1 Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan;
takami@theo.phys.sci.hiroshima-u.ac.jp, ryo@theo.phys.sci.hiroshima-u.ac.jp.
2 NASA Goddard Space Flight Center, Greenbelt, MD 20771.
http://arxiv.org/abs/0704.1055v3
– 2 –
2002b). There are many jet models proposed in addition to the simplest uniform-jet model:
the power-law jet model (Rossi et al. 2002; Zhang & Mészáros 2002a), the Gaussian jet
model (Zhang et al. 2004), the annular jet model (Eichler & Levinson 2004), the multiple
emitting subshell model (Kumar & Piran 2000; Nakamura 2000), the two-component jet
model (Berger et al. 2003), and so on. The jet structure may depend on the generation
process of the jet and therefore may provide us important information about the central
engine of the GRB. For example, in the collapsar model for long GRBs (e.g., Zhang et al.
2003, 2004), the jet penetrates into and breaks out of the progenitor star, resulting in the
E(θ) ∝ θ−2 profile (Lazzati & Begelman 2005). For the compact binary merger model for
short GRBs, hydrodynamic simulations have shown that the resulting jet tends to have a
flat core surrounded by the power-law-like envelope (Aloy et al. 2005).
In the pre-Swift era, there were many attempts to constrain the GRB jet structure.
Thanks to the HETE-2, statistical properties of long GRBs, X-ray-rich GRBs, and X-
ray flashes were obtained (Sakamoto et al. 2005), which were thought to constrain the
jet models (Lamb et al. 2004). These observational results constrain various jet models,
such as the uniform-jet model (Yamazaki et al. 2004a; Lamb et al. 2005; Donaghy 2006),
the multiple subshell model (Toma et al. 2005b), the Gaussian jet model (Dai & Zhang
2005), and so on. For BATSE long GRBs, Yonetoku et al. (2005) derived the distribu-
tion of the pseudo-opening angle, inferred from the Ghirlanda (Ghirlanda et al. 2004) and
Yonetoku (Yonetoku et al. 2004) relations, as f(θj)dθj ∝ θ
j dθj , which is compatible with
that predicted by the power-law jet model as discussed in Perna et al. (2003) (however, see
Nakar et al. 2004). Afterglow properties are also expected to constrain the jet structure
(e.g., Granot & Kumar 2003); however, energy redistribution effects prevent us from reach-
ing a definite conclusion. Polarization measurements of optical afterglows bring us important
information (Lazzati et al. 2004).
In the Swift era, rapid follow-up observation reveals prompt GRBs followed by a steep de-
cay phase in the X-ray early afterglow (Tagliaferri et al. 2005; Nousek et al. 2006; O’Brien et al.
2006a). In the most popular interpretations, the steep decay component is the tail emis-
sion of the prompt GRB (the so called high-latitude emission), i.e., the internal shock
origin (Zhang et al. 2006; Yamazaki et al. 2006; Liang et al. 2006; Dyks et al. 2005), al-
though there are some other possibilities (e.g., Kobayashi et al. 2007; Panaitescu et al. 2006;
Pe’er et al. 2006; Lazzati & Begelman 2006; Dado et al. 2006). Then, for the uniform-jet
case, the predicted decay index is α = 1 − β, where we use the convention Fν ∝ T
−αν1+β
(Kumar & Panaitescu 2000). For power-law jet case (E(θ) ∝ θ−q), the relation is modified
to α = 1−β+(q/2). However, these simple analytical relations cannot be directly compared
with observations, because they are for the case in which the observer’s line of sight is along
the jet axis and because changing the zero of time, which potentially lies anywhere within
– 3 –
the epoch where we see the bright pulses, substantially alters the early decay slope.
Recently, Yamazaki et al. (2006) (Y06) investigated the tail emission of the prompt
GRB, finding that the jet structure can be described and that the global decay slope is
not so much affected by the local angular inhomogeneity as it is affected by the global
energy distribution. They also argued that the structured jet model is preferable, because
steepening GRB tail breaks appeared in some events. In this paper, we calculate for the first
time the distribution of the decay index of the prompt tail emission for various jet models
and find that the derived distributions can be distinguished from each other, so that the jet
structure can be more directly constrained than previous arguments. This paper is organized
as follows. We describe our model in § 2. In § 3, we investigate the distribution of the decay
index of the prompt GRB emission. Section 4 is devoted to discussions.
2. Tail Part of the Prompt GRB Emission
We consider the same model as discussed in the previous works (Y06; Yamazaki et al.
2004b; Toma et al. 2005a,b). The whole GRB jet, whose opening half-angle is ∆θtot, consists
of Ntot emitting subshells. We introduce the spherical coordinate system (r, ϑ, ϕ, t) in the
central engine frame, where the origin is at the central engine and ϑ = 0 is the axis of the
whole jet. Each emitting subshell departs at time t
(0 < t
< tdur, where j = 1, · · · , Ntot,
and tdur is the active time of the central engine) from the central engine in the direction of
~n(j) = (ϑ(j), ϕ(j)), and emits high-energy photons, generating a single pulse as observed. The
direction of the observer is denoted by ~nobs = (ϑobs, ϕobs). The observed flux from the jth
subshell is calculated when the following parameters are determined: the viewing angle of the
subshell θ(j)v = cos
−1(~nobs ·~n
(j)), the angular radius of the emitting shell ∆θ
, the departure
time t
, the Lorentz factor γ(j) = (1− β2
)−1/2, the emitting radius r
0 , the low- and high-
energy photon indices α
B and β
B , the break frequency in the shell comoving frame ν
(Band et al. 1993), the normalization constant of the emissivity A(j), and the source redshift
z. The observer time T = 0 is chosen as the time of arrival at the observer of a photon
emitted at the origin r = 0 at t = 0. Then, at the observer, the starting and ending times
of the jth subshell emission are given by
start ∼ t
dep +
1 + γ2
, (1)
1 + γ2
, (2)
where θ
+ = θ
v +∆θ
= max{0, θ(j)v −∆θ
}, and we use the formulas β(j) ∼ 1−1/2γ
and cos θ ∼ 1 − θ2/2 for γ(j) ≫ 1 and θ ≪ 1, respectively. The whole light curve from the
– 4 –
GRB jet is produced by the superposition of the subshell emission.
Y06 discussed some kinematical properties of prompt GRBs in our model and found
that each emitting subshell with θ(j)v ≫ ∆θ
produces a single, smooth, long-duration, dim,
and soft pulse, and that such pulses overlap with each other and make the tail emission of
the prompt GRB. Local inhomogeneities in the model are almost averaged during the tail
emission phase, and the decay index of the tail is determined by the global jet structure,
that is the mean angular distribution of the emitting subshell because in this paper all
subshells are assumed to have the same properties unless otherwise stated. Therefore, we
are essentially studying the tail emission from the usual continuous jets at once, i.e., from
uniform- or power-law jets with no local inhomogeneity. In the following, we study various
energy distributions of the GRB jet through the change of the angular distribution of the
emitting subshell.
3. Decay Index of the Prompt Tail Emission
In this section, we perform Monte Carlo simulations in order to investigate the jet
structure by calculating the statistical properties of the decay index of the tail emission.
For a fixed-jet model, we randomly generate 104 observers with their line of sights (LOSs)
~nobs = (ϑobs, ϕobs). For each LOS, the light curve, F (T ) of the prompt GRB tail in the
15–25 keV band is calculated, and the decay index is determined. The adopted observation
band is the low-energy end of the Burst Alert Telescope(BAT) detector and near the high-
energy end of the X-Ray Telescope(XRT) on Swift. Hence, one can observationally obtain
continuous light curves, beginning with the prompt GRB phase to the subsequent early
afterglow phase (Sakamoto et al. 2007), so that it is convenient for us to compare theoretical
results with observations. However, our actual calculations have shown that our conclusion is
not qualitatively altered, even if the observation band is changed, for example, to 0.5–10 keV,
as usually considered for other references.
For each light curve, the decay index is calculated by fitting F (T ) with a single power-
law form, ∝ (T − T∗)
−α, as in the following (see Fig. 1). The decay index α depends on the
choice of T∗ (Zhang et al. 2006; Yamazaki et al. 2006)
1. Let Ts and Te be the start and end
time, respectively, of the prompt GRB, i.e.,
Ts = min{T
start} (3)
1 Recently, Kobayashi & Zhang (2007) have discussed the way to choose the time zero. According to
their arguments, the time zero is near the rising epoch of the last bright pulse in the prompt GRB phase.
– 5 –
Te = max{T
} . (4)
Then, we take T∗ as the time until 99% of the total fluence, which is defined by Stotal =
F (T ′) dT ′, is radiated, that is,
F (T ′) dT ′ = 0.99Stotal . (5)
Then, the prompt GRB is in the main emission phase for T < T∗, while it is in the tail emis-
sion phase for T > T∗. The time interval [Ta, Tb], in which the decay index α is determined
assuming the form F (T ) ∝ (T − T∗)
−α, is taken to satisfy
F (Ta,b) = qa,bF (T∗) , (6)
where we adopt qa = 1 × 10
−2 and qb = 1 × 10
−3, unless otherwise stated. We find that in
this epoch the assumed fitting form gives a well approximation.
At first, we consider the uniform-jet case, in which the number of subshells per unit
solid angle is approximately given by dN/dΩ = Ntot/(π∆θ
tot) for ϑ < ∆θtot, where ∆θtot =
0.25 rad is adopted. The departure time of each subshell t
is assumed to be homogeneously
random between t = 0 and t = tdur = 20 sec. The central engine is assumed to produce
Ntot = 1000 subshells. In this section, we assume that all subshells have the same values of
the following fiducial parameters: ∆θsub = 0.02 rad, γ = 100, r0 = 6.0×10
14 cm, αB = −1.0,
βB = −2.3, hν
0 = 5 keV, and A = constant. Our assumption of constant A is justified
as follows. Note that the case in which N subshells that have the same brightness A are
launched into the same direction, but a different departure time, is equivalent to the case of
one subshell emission with the brightness of NA. This is because in the tail emission phase,
the second terms in the r.h.s. of Eqs. (1) and (2) dominate the first terms, so that the time
difference effect, which arises from the difference of tdep for each subshell, can be obscured.
Hence, giving the angular distribution of the emission energy is equivalent to giving the
angular distribution of the subshells with constant A. Also, Y06 showed that to obtain a
smooth, monotonic tail emission as observed by Swift, the subshell properties, hν ′0
and/or
A(j), cannot have wide scatter in the GRB jet. Therefore, we can expect, at least as the
zeroth-order approximation, that the subshells have the same properties.
The left panel of Fig. 2 shows the decay index α as a function of ϑobs. For ϑobs . ∆θtot
(on-axis case), α clusters around ∼ 3. On the other hand, when ϑobs & ∆θtot (off-axis case),
α rapidly increases with ϑobs. The reason is as follows. If all subshells are seen sideways
(that is, θ(j)v ≫ ∆θ
for all j), the bright pulses in the main emission phase followed by the
tail emission disappear because of the relativistic beaming effect, resulting in a smaller flux
contrast between the main emission phase and the tail emission phase compared with the on-
axis case. Then T∗ becomes larger. Furthermore, in the off-axis case, the tail emission decays
– 6 –
more slowly (|dF/dT | is smaller) than in the on-axis case. Then both Ta−T∗ and Tb−T∗ are
larger for the off-axis case than for the on-axis case. As can be seen in Fig. 3 of Zhang et al.
(2006), the emission seems to decay rapidly, so that the decay index α becomes large. The
left panel of Fig. 3 shows α as a function of the total fluence Stotal which is the sum of the
fluxes in the time interval, [Ts, Te]. In Fig. 3, both α and Stotal are determined observationally,
so that our theoretical calculation can be directly compared with the observation.
A more realistic model is the Gaussian jet model, in which the number of subshells per
unit solid angle is approximately given by dN/dΩ = C exp(−ϑ2/2ϑ2c) for 0 ≦ ϑ ≦ ∆θtot,
where C = Ntot/2πϑ
c [1 − exp(−∆θ
tot/2ϑ
c)] is the normalization constant. We find only a
slight difference between the results for the uniform- and the Gaussian jet models. Therefore,
we do not show the results for the Gaussian jet case in this paper.
Next, we consider the power-law distribution. In this case, the number of subshells per
unit solid angle is approximately given by dN/dΩ = C[1 + (ϑ/ϑc)
2]−1 for 0 ≦ ϑ ≦ ∆θtot,
i.e., dN/dΩ ≈ C for 0 ≦ ϑ ≪ ϑc and dN/dΩ ≈ C(ϑ/ϑc)
−2 for ϑc ≪ ϑ ≦ ∆θtot, where C =
(Ntot/πϑ
c)[ln(1 + (∆θtot/ϑc)
2)]−1 is the normalization constant and we adopt ϑc = 0.02 rad
and ∆θtot = 0.25 rad. The other parameters are the same as for the uniform-jet case.
As can be seen in the right panels of Figs. 2 and 3, both the ϑobs–α and Stotal–α diagrams
are complicated compared with the uniform-jet case. When ϑobs . ϑc, the observer’s LOS is
near the whole jet axis. Compared with the uniform-jet case, α is larger, because the power-
law jet is dimmer in the outer region, i.e., emitting subshells are sparsely distributed near
the periphery of the whole jet (see also the solid lines of Figs. 1 and 3 of Y06). If ϑobs ≫ ϑc,
the scatter of α is large. Some bursts have an especially small α of around 2. This comes
from the fact that the power-law jet has a core region (0 < ϑ . ϑc), where emitting subshells
densely distributed compared with the outer region. The core generates the light-curve break
in the tail emission phase, as can be seen in Fig. 4 (Y06). In the epoch before the photons
emitted by the core arrive at the observer (e.g., T − Ts . 7.5 × 10
2 s for the solid line in
Fig. 4), the number of subshells that contribute to the flux at time T , Nsub(T ), increases
with T more rapidly than for the uniform-jet case. Then, the light curve shows a gradual
decay. If the fitting region [Ta, Tb] lies in this epoch, the decay index α is around 2. In the
epoch after the photons arising from the core are observed (e.g., T −Ts & 7.5× 10
2 s for the
solid line in Fig. 4), the subshell emission with θ(j)v & ϑobs + ϑc is observed. Then Nsub(T )
rapidly decreases with T , and the observed flux suddenly drops. If the interval [Ta, Tb] lies
in this epoch, the decay index becomes larger than 4.
To compare the two cases considered above more clearly, we derive the distribution of
the decay index α. Here we consider the events whose peak fluxes are larger than 10−4 times
of the largest one in all simulated events, because the events with small peak fluxes are not
– 7 –
observed. Fig. 5 shows the result. For the uniform-jet case (solid line), α clusters around 3,
while for the power-law jet case (dotted line), the distribution is broad (1 . α . 7) and has
multiple peaks.
So far, we have considered the fiducial parameters. In the following, we discuss the
dependence on parameters, r0, γ, βB, tdur, and ∆θtot (It is found that the α-distribution
hardly depends on the value of αB, ∆θsub, and ν
0 within reasonable parameter ranges). At
first, we consider the case in which r0 = 1.0 × 10
14 cm is adopted, with other parameters
being fiducial. Fig. 6 shows the result. The shape of the α-distribution is almost the
same as that for the fiducial parameters, in both the uniform- and the power-law jet cases.
This comes from the fact that in a tail emission phase, the light curve for a given r0 is
approximately written as F (T ; r0) ≈ g(cT/r0), where a function g determines the light-curve
shape of the tail emission for other given parameters. Then, the light curves in the case of
r0 = r0,1 and r0 = r0,2, namely, F (T ; r0,1) and F (T ; r0,2), satisfy the relation F (T ; r0,2) ≈
F ((r0,1/r0,2)T ; r0,1). This can be seen, for example, by comparing the solid line with the
dotted one in Fig. 4. Hence, T∗, Ta, and Tb are approximately proportional to r0; in this
simple scaling, one can easily find that α remains unchanged for different values of r0.
Second, we consider the case of γ = 200 and r0 = 2.4× 10
15 cm, with other parameters
being fiducial. In this case, the angular spreading timescale (∝ r0/γ
2) is the same as in the
fiducial case, so that the tail emissions still show smooth light curves, although the whole
emission ends later, according to the scaling Te ∝ r0 (see the dot-dashed line in Fig. 4).
Fig. 7 shows the result. For large γ, the relativistic beaming effect is more significant, so
that the events in ϑobs & ∆θtot, which cause large α, are dim compared with the small-γ
case. Such events cannot be observed. For the power-law jet case, therefore, the number of
large-α events becomes small, although the distribution is still broad (1 . α . 4) and has
two peaks. On the other hand, for the uniform-jet case, the distribution of the decay index
α is almost the same as for the fiducial parameter set, because the value of the decay index
α in the case of ϑobs . ∆θtot is almost the same as that in the case of ϑobs & ∆θtot.
Third, we change the value of the high-energy photon index βB from −2.3 to −5.0, with
other parameters being fiducial. Fig. 8 shows the result. For the uniform-jet case, the mean
value is 〈α〉 ∼ 4, while 〈α〉 ∼ 3 for the fiducial parameters, so that the decay index defined
in this paper does not obey the well-known formula α = 1−βB (Kumar & Panaitescu 2000).
For the power-law jet case, the whole distribution shifts toward the higher value, and the
ratio of the two peaks changes. In the tail emission phase, the spectral peak energy Epeak is
below 15 keV (see also Y06), so that the steeper the spectral slope of the high-energy side
of the Band function, the more rapidly the emission decays, resulting in the dimmer tail
emission (see the dashed line in Fig. 4). Then, the fitting region [Ta, Tb] shifts toward earlier
– 8 –
epochs, because T∗ becomes small. Therefore, the number of events with small α increases,
and the number of events with large α decreases. Furthermore, we comment on the case in
which βB is varied for each event in order to more directly compare with the observation.
Here we randomly distribute βB according to the Gaussian distribution with a mean of −2.3
and a variance of 0.4. It is found that the results are not qualitatively changed.
Next, we change the value of the duration time tdur from 20 sec to 200 sec, with other
parameters being fiducial. The epoch of the bright pulses in the main emission phase becomes
longer than that in tdur = 20 sec. However, the behavior of the tail emission does not depend
on tdur very much (see Fig. 9). Therefore, the distribution of the decay index α is almost
the same as that for the fiducial parameters for both the uniform-jet case and the power-
law jet case. Even if we consider the case in which tdur is randomly distributed for each
event according to the lognormal distribution with an average of log(20 s) and a logarithmic
variance of 0.6, the results are not significantly changed.
Finally, we discuss the dependence on ∆θtot. Only the uniform-jet case is considered,
because the structured jet is usually quasi-universal and because we focus our attention
on the behavior of the uniform-jet model. The dotted line in Fig. 10 shows the result for
constant ∆θtot = 0.1 rad with other parameters being fiducial. We can see many events with
large α. The large α is observed because for small ∆θtot, although the off-axis events (i.e.,
∆θtot . ϑobs) are still dim because of the relativistic beaming effect, a fraction of such events
survives the flux threshold condition and are observable. Such events have large α & 5 (see
the 4th paragraph of this section, which explains the left panel of Fig. 2). This does not
occur in the large-∆θtot case. However, we still find in this case that there are no events
with α . 2. We consider another case in which ∆θtot is variable. Here we generate events
whose ∆θtot distributes as f∆θtotd(∆θtot) ∝ ∆θtot
−2d(∆θtot) (0.05 . ∆θtot . 0.4). Then for
a given ∆θtot, the quantities ν
0 and A are determined by hν
0 = (∆θtot/0.13)
−3.6 keV and
A ∝ (∆θtot)
−7.3, respectively. Other parameters are fiducial. If the model parameters are
chosen in this way, the Amati and Ghirlanda relations (Amati et al. 2002; Ghirlanda et al.
2004) are satisfied, and the event rates of long GRBs, X-ray-rich GRBs and X-ray flashes
become similar (Donaghy 2006). The solid line in Fig. 10 shows the result. Again we find
that there are no events with α . 2.
In summary, when we adopt model parameters within reasonable ranges, the decay
index becomes larger than ∼ 2 for the uniform- and the Gaussian jet cases, while a significant
fraction of events with α . 2 is expected for the power-law jet case. Therefore, if a non-
negligible number of events with α . 2 are observed, both the uniform- and the Gaussian
jet models will be disfavored. Furthermore, if we observationally derive the α-distribution,
the structure of GRB jets will be more precisely determined.
– 9 –
4. Discussion
We have calculated the distribution of the decay index, α, for the uniform-, Gaussian,
and the power-law jet cases. For the uniform-jet case, α becomes larger than ∼ 2, and its
distribution has a single peak. The Gaussian jet model predicts almost the same results as
the uniform-jet model. On the other hand, for the power-law jet case, α ranges between
∼ 1 and ∼ 7, and its distribution has multiple peaks. Therefore, we can determine the jet
structure of GRBs by analyzing a lot of early X-ray data showing a steep decay component
that is identified as a prompt GRB tail emission. However, one of the big challenges in the
Swift data for calculating the decay index in our definition is to derive the composite light
curve of BAT and XRT. Since the observed energy bands of BAT and XRT do not overlap,
we are forced to extrapolate one of the data sets to plot the light curve in a given energy
band. To derive the composite light curve unambiguously for a prompt and an early X-ray
emission, we need an observation of a prompt emission by current instruments, which overlap
the energy range of XRT.
The tail behavior with α . 2 does not appear in the uniform- and the Gaussian jet
models; hence, it is important to constrain the jet structure. However, in practical observa-
tions, such gradually decaying prompt tail emission might be misidentified with the external
shock component, as expected in the pre-Swift era. Actually, some events have shown such
a gradual decay, without the steep and the shallow decay phases, and their temporal and
spectral indices are consistent with a classical afterglow interpretation (O’Brien et al. 2006b).
Hence, in order to distinguish the prompt tail emission from the external shock component
at a time interval [Ta, Tb], one should study the spectral evolution and/or the continuity
and smoothness of the light curve (i.e., whether breaks appear or not) over the entire burst
emission.
In this paper, we adopt qa = 1×10
−2 and qb = 1×10
−3 when the fitting epoch [Ta, Tb] is
determined [see Eq. (6)]. Then, the prompt tail emission in this time interval is so dim that
it may often be obscured by the external shock component, causing a subsequent shallow
decay phase of the X-ray afterglow. One possible way to resolve this problem is to adopt
larger values of qa and qb, e.g., qa = 1/30 and qb = 1 × 10
−2, in which the interval [Ta, Tb]
shifts toward earlier epochs, so that the flux then is almost always dominated by the prompt
tail emission. We have calculated the decay index distribution for this case (qa = 1/30
and qb = 1 × 10
−2) and have found that the differences between uniform- and power-law
jets still arises as can be seen in the case of qa = 1 × 10
−2 and qb = 1 × 10
−3, so that our
conclusion remains unchanged. However, the duration of the interval, Tb−Ta, becomes short,
which might prevent us from observationally fixing the decay index at high significance. If
qa & 1/30, the emission at [Ta, Tb] is dominated by the last brightest pulse. Then, the light-
– 10 –
curve shape at [Ta, Tb] does not reflect the global jet structure, but reflects the properties of
the emitting subshell causing the last brightest pulse. Another way to resolve the problem
is to remove the shallow decay component. For this purpose, the origin of the shallow decay
phase should be clarified in order to extract the dim prompt tail emission exactly. The other
problem is contamination of X-ray flares, whose contribution has to be removed in order to
investigate the tail emission component. In any case, if the GRB occurs in an extremely
low-density region (a so-called naked GRB), where the external shock emission is expected
to be undetectable, our method may be a powerful tool to investigate the GRB jet structure.
This work was supported in part by Grants-in-Aid for Scientific Research of the Japanese
Ministry of Education, Culture, Sports, Science, and Technology 18740153 (R. Y.). T.S. was
supported by an appointment of the NASA Postdoctoral Program at the Goddard Space
Flight Center, administered by Oak Ridge Associated Universities through a contract with
NASA.
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This preprint was prepared with the AAS LATEX macros v5.2.
– 13 –
Ts T Te
 F(Tb)
 F(Ta)
 F(T  )
Ta Tb
1/100 1/1000
T [sec]*
fitting region
Fig. 1.— Example of how the decay index α is determined by the calculated light curve
F (T ). The start and end time of the burst are denoted by Ts and Te, respectively. The time
T∗ is determined by Eq. (5). The decay index α is determined by fitting F (T ) ∝ (T − T∗)
in the time interval [Ta, Tb].
Fig. 2.— Decay index α as a function of ϑobs, the angle between the whole jet axis and the
observers’ lines of sight. Red and green points represent events whose peak fluxes are larger
and smaller than 10−4 times the largest one in all simulated events, respectively. Left and
right panels are for the uniform- and power-law jet cases, respectively.
– 14 –
Fig. 3.— Decay index α as a function of the total fluence Stotal, the sum of the fluxes in
the time interval [Ts, Te]. Red and green points represent events whose peak fluxes are larger
and smaller than 10−4 times the largest one in all simulated events, respectively. Left and
right panels are for the uniform- and power-law jet cases, respectively.
– 15 –
10-12
10-10
102 103 104
T-Ts [sec]
fiducial
r0=1.0x10
γ=200 , r0=2.4x10
βB=-5.0
Fig. 4.— Examples of light curves of the prompt tail emission in the 15–25 keV band
for the power-law jet case and ϑobs > ϑc (ϑobs = 0.27 rad and ϑc = 0.02 rad). The solid
line shows the fiducial parameters. A bump caused by the core emission can be seen at
T − Ts ∼ 7.5 × 10
2 s. The dotted, dot-dashed, and dashed lines are for r0 = 1.0× 10
14 cm;
r0 = 2.4 × 10
15 cm and γ = 200; and βB = −5, respectively, with other parameters being
fiducial. Time intervals [Ta, Tb] for each case are denoted by the thick solid lines. The flux
is normalized by the peak value.
– 16 –
 0.05
 0.15
 0.25
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 5.— Distributions of the decay index α for uniform-jet (dN/dΩ = const.;solid line) and
power-law jet (dN/dΩ ∝ [1 + (ϑ/ϑc)
2]−1;dotted line) models, respectively. We assume that
all subshells have the same values of the following fiducial parameters: ∆θsub = 0.02 rad,
γ = 100, r0 = 6.0 × 10
14 cm, αB = −1.0, βB = −2.3, hν
0 = 5 keV, and A = constant. We
consider events whose peak fluxes are larger than 10−4 times the largest one in all simulated
events (red points in Fig. 2).
– 17 –
 0.05
 0.15
 0.25
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 6.— Same as Fig. 5, but for r0 = 1.0× 10
14 cm.
– 18 –
 0.05
 0.15
 0.25
 0.35
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 7.— Same as Fig. 5, but for γ = 200 and r0 = 2.4× 10
15 cm.
– 19 –
 0.05
 0.15
 0.25
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 8.— Same as Fig. 5, but for βB = −5.0.
– 20 –
 0.05
 0.15
 0  1  2  3  4  5  6  7  8
α ( decay index ) 
Fig. 9.— Same as Fig. 5, but for tdur = 200 sec.
– 21 –
 0.05
 0.15
 0.25
 0.35
 0  5  10  15  20
α ( decay index ) 
Fig. 10.— Distribution of the decay index α for the uniform-jet profile. The dotted line is for
∆θtot = 0.1 rad with other fiducial parameters. The solid line is for the variable-∆θtot case,
in which we generate events whose ∆θtot distributes as f∆θtotd(∆θtot) ∝ ∆θtot
−2d(∆θtot)
(0.05 . ∆θtot . 0.4), and for given ∆θtot, the quantities ν
0 and A are determined by
hν ′0 = (∆θtot/0.13)
−3.6 keV and A ∝ (∆θtot)
−7.3, respectively. Other parameters are fiducial.
	Introduction
	Tail Part of the Prompt GRB Emission
	Decay Index of the Prompt Tail Emission
	Discussion
