Testing Gamma-Ray Burst Jet Structure with the Distribution of Gamma-Ray Energy Release

Abstract

We present a general method for testing gamma-ray burst (GRB) jet structure and carry out a comprehensive analysis about the prevalent jet structure models. According to the jet angular energy distribution, we can not only derive the expected distribution of the GRB isotropic-equivalent energy release for any possible jet structure, but also obtain a two-dimensional distribution including redshift z. By using the Kolmogorov-Smirnov test we compare the predicted distribution with the observed sample, and find that the power-law structured jet model is most consistent with the current sample and that the uniform jet model is also plausible. However, this conclusion is tentative because of the small size and the inhomogeneity of this sample. Future observations (e.g., Swift) will provide a larger and less biased sample for us to make a robust conclusion by using the procedure proposed in this paper. Comment: 36 pages including 16 figures, accepted for publication in ApJ

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arXiv:astro-ph/0508324v1 15 Aug 2005
Testing Gamma-Ray Burst Jet Structure with the Distribution of
Gamma-Ray Energy Release
L. Xu, X. F. Wu, and Z. G. Dai
Department of Astronomy, Nanjing University, Nanjing 210093, China
ABSTRACT
We present a general method for testing gamma-ray burst (GRB) jet struc-
ture and carry out a comprehensive analysis about the prevalent jet structure
models. According to the jet angular energy distribution, we can not only derive
the expected distribution of the GRB isotropic-equivalent energy release for any
possible jet structure, but also obtain a two-dimensional distribution including
redshift z. By using the Kolmogorov-Smirnov test we compare the predicted
distribution with the observed sample, and find that the power-law structured
jet model is most consistent with the current sample and that the uniform jet
model is also plausible. However, this conclusion is tentative because of the small
size and the inhomogeneity of this sample. Future observations (e.g., Swift) will
provide a larger and less biased sample for us to make a robust conclusion by
using the procedure proposed in this paper.
Subject headings: gamma rays: bursts — gamma rays: observations — ISM: jets
and outflows — methods: statistical
1.INTRODUCTION
Growing evidence such as the achromatic breaks in afterglow light curves of ∼ 20
gamma-ray bursts (GRBs) and the observation of polarized emission (Covino et al. 1999;
Wijers et al. 1999) suggests that GRBs are produced by collimated jets. Numerous models
for the jet configuration have been proposed due to its importance in understanding the
burst progenitor and GRB event rates. One leading model is the uniform jet one (e.g.,
Rhoads 1997; Sari, Piran, & Halpern 1999; Frail et al. 2001; Dai & Cheng 2001 ), in which
the angular energy distribution within the jet is uniform while the nearly constant total
energy is collimated into different opening angles. Another is the structured jet model, in
which the jet energy and structure are approximately identical for all GRBs, but the energy
per solid angle ǫ(θ) varies as a function of angle from the jet axis within the structured jet

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(M´ esz´ aros, Rees, & Wijers 1998; Dai & Gou 2001; Rossi, Lazzati, & Rees 2002; Zhang &
M´ esz´ aros 2002). Furthermore, although the specific distribution of jet opening angles in the
uniform jet model is uncertain, several kinds of distribution such as the power-law, Gaussian
and exponential function have been suggested. In this paper we discuss these possible models
specifically.
Since a few leading models of jet structure can all explain the observed achromatic
breaks in afterglow light curves, some authors attempted to test the geometrical configu-
ration by using other observations. For example, the jet structure could be constrained
by considering the luminosity function (e.g., Firmani et al. 2004; Lloyd-Ronning, Dai, &
Zhang 2004; Guetta, Piran, & Waxman 2005). The logN − logP plot was also used to
constrain the jet opening angle distribution for the uniform jet and the star forming rate
(Lin et al. 2004; Guetta, Piran, & Waxman 2005). Perna, Sari, & Frail (2003) predicted the
observed one-dimensional (1D) distribution of viewing angles, n(θ) = dn/dθ, in the power-
law structured jet model. They found that the predicted distribution could fit the current
sample of 16 bursts with modelled angles and known redshifts. Considering the same model,
Nakar, Granot, & Guetta (2004) carried out an analysis about the two-dimensional (2D)
distribution n(z,θ) = d2n/dzdθ. However, they found that the 2D prediction shows a very
poor agreement with the same sample, and the hypothesis that the data are drawn from
this model is rejected at the 99% confidence level by a 2D Kolmogorov-Smirnov (K-S) test.
Based on the standard energy reservoir of GRB jets and the relationship between the cosmic
rest frame GRB spectral peak energy and the isotropic gamma-ray energy, i.e., Epeak∝ E1/2
Liang, Wu, & Dai (2004) simulated a sample including 106GRBs to test the GRB proba-
bility distribution as a function of viewing angle and redshift, and supported the power-law
structured jet model. Dai & Zhang (2005) summarized the above tests and performed a
global test on the quasi-universal Gaussian-like structured jet. Together with their previous
tests with the observed jet break angle vs. isotropic energy (Eiso− θj) and observed peak
energy vs. fluence relations (Eobs
current GRB data are generally consistent with the Gaussian-like jet model. These tests
can only tentatively support certain possible configuration of the GRB jet, partly because
the size of the sample use in the test is too small (but see Cui, Liang, & Lu 2005, who
recently adopted the BATSE sample). In this paper, we propose a new method to test the
GRB jet structure by using the sample of bursts with measured fluences and redshifts. By
considering three jet structure models, i.e., the power-law structured jet model, the Gaussian
structured jet model, and the uniform jet model, we work out the theoretical distribution
of burst isotropic-equivalent energy release in both 1D distribution ˙N(E) = dN/dE and
2D distribution˙N(E,z) = d2N/dEdz (hereafter, we use E instead of Eisofor brevity), and
compare them with the observed distribution of E derived from the sample with measured
iso,
peakvs. fluence), Zhang et al. (2004) concluded that the

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fluences and redshifts.
This paper is organized as follows: In §2.1 we present several prevalent models for
jet structure and the probability distribution of the isotropic-equivalent energy under these
models, while in §2.2 we describe our methodology in detail. In §3 we discuss what each
model predicts for the observed distribution (in 1D and 2D) of GRB isotropic-equivalent
energy release and assess how our numerical results are compared with the current sample
by using K-S test. We summarize our results and draw our main conclusions in §4.
2. THEORY
2.1. Models for jet structure
Since the energy release in an explosion plays an important role in determining the
progenitors of GRBs and the physics of their central engine, some authors have carried out
plenty of analyses about the gamma-ray energy release distribution. Within the uniform jet
model, Frail et al. (2001) found that the gamma-ray energy release after beaming correction
is narrowly clustered around 5 × 1050ergs by fitting a sample of 16 well observed GRBs.
Using a larger sample, Bloom, Frail, & Kulkarni (2003) obtained a similar result that the
gamma-ray energies are well clustered around a median value of 1.33×1051ergs. The gamma-
ray energy release should become larger if the gamma-ray radiation efficiency is considered
(Xu & Dai 2004). Panaitescu & Kumar (2001, 2002) modelled the broadband emission of
several GRB afterglows and obtained a similar value of the kinetic energy. Furthermore,
Wu, Dai, & Liang (2004) derived a jet break time-flux density relationship and constrained
some physical parameters of gamma-ray burst afterglows. The jet break relation supports
the “standard candle” hypothesis of the afterglows by an entirely different approach. We
therefore adopt a standard value for GRB jet energy Ej= 1051ergs throughout this paper.
According to the presently measured isotropic-equivalent energy, we adopt the upper limit
of the isotropic-equivalent energy as Emax= 1055ergs.
According to the angular energy distribution for any plausible jet model, one can obtain
the isotropic-equivalent energy as a function of θ. The isotropic-equivalent energy, E, is
defined as E(θ) = 4πǫ(θ), where θ is the angle from the jet axis and ǫ(θ) is the energy per
solid angle.
In the power-law structured jet (PLSJ) model, all GRBs are supposed to have a similar
jet profile, and their different observed properties are due to different observer’s viewing
angles. Under the power-law jet scenario, the angular distribution of the isotropic-equivalent

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energy is
E(θ) =
Ec
[1 + (θ/θc)kElE]1/lE,(0 ≤ θ ≤ π/2),(1)
where lEdetermines the sharpness of the transition from the core to wing, and kErepresents
the profile of the jet wing. The opening angle of the core θc is introduced to avoid the
divergence of the total energy. Ecis defined as Ec = E(0), and equals to the maximum
isotropic-equivalent energy Emax. Below we assume lE = +∞ for simplicity. If all bursts
were observable, then in a structured jet model, the probability for a viewing angle θ is
P(θ)dθ ≈ θdθ. From the relation P(θ)dθ = P(E)dE, we obtain the normalized probability
density at a given isotropic-equivalent energy E,
P(E) ≈
8
π2
θ2
c
kEEc
?Ec
E
?1+2/kE
,((2θc/π)kEEc≤ E ≤ Ec).(2)
In the Gaussian structured jet (GSJ) model, the isotropic-equivalent energy distribution
that varies with the angle from the jet axis is (Zhang & M´ esz´ aros 2002)
E(θ) = Ece
−θ2
2θ2
c,(0 ≤ θ ≤ π/2),(3)
where Ec = E(0), and θc is the characteristic jet angular width. According to the same
derivation as in the power-law structured jet model, we have the normalized probability at
a given isotropic-equivalent energy E,
P(E) =
8
π2
θc2
E,
(Ece−π2/8θc2≤ E ≤ Ec).(4)
Similarly we assume Ec= 1055ergs.
In the uniform jet (UJ) model (e.g., Lamb, Donaghy, & Graziani 2005), all GRBs
produce jets with different jet opening angles θj. The structure of a uniform jet reads
E(θ) =
?E,0 ≤ θ ≤ θj,
θj< θ ≤ π/2.0,
(5)
Assuming f(E) is the probability that a GRB releases the isotropic energy between E and E+
dE, the probability that we observe a specified E is given by P(E) = f(E)(1−cosθj). Using
the Frail’s relation, Ej= E(1−cosθj) ≈ constant, we obtain the probability distribution of
the isotropic energy
P(E) ≈ f(E)Ej
E,
(Ej≤ E ≤ Emax),(6)

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Next we turn to discuss the specified form of f(E). As for the GRB opening angle
distribution, Lin et al. (2004) studied three cases in which the GRB opening angle θjfollows
a power-law, exponential, or Gaussian distribution. Here we also consider these three cases.
We assume that the first probability distribution of isotropic-equivalent energy follows a
power-law function with index ζ,
f(E) =
1 + ζ
max− E1+ζ
E1+ζ
j
Eζ,(Ej≤ E ≤ Emax).(7)
It is worth noting that f(E) should be normalized, i.e.
probability distribution for f(E) is an exponential function with parameter λ,
?Emax
Ej
f(E)dE = 1. The second
f(E) =
λeλEj
1 − e−λ(Emax−Ej)e−λE,(Ej≤ E ≤ Emax),(8)
which has also been normalized. Finally, if f(E) is a Gaussian distribution with the mean
energy¯E and standard scatter σ, we have
f(E) =
a
√2πσe−(E−¯
E)2
2σ2 ,(Ej≤ E ≤ Emax),(9)
where a is the normalization factor satisfying?Emax
Ej
f(E)dE = 1.
2.2.Detections
For a power-law photon spectrum, the photon luminosity in the triggering band νl−νu
is given by
E
Thνl
α − 1
where α is the photon spectrum index, and νl and νu are the lower and upper limiting
frequencies of a GRB detector respectively. For the BATSE triggering channel, νl= 50 keV
and νu= 300 keV. The E is the isotropic-equivalent γ-ray energy in a range of ν1= 0.1 keV
to ν2= 104keV at the GRB’s cosmological rest frame. The mean spectral peak energy for
GRBs is ?Ep? ∼ 250 keV (Preece et al. 2000). According to the spectral fit, we take α ≈ 1
for the frequency range νl< ν < νuin the following calculations (Band et al. 1993). Then
the photon luminosity can be simplified as
Lph,[νl,νu]=
α − 2
?νl
ν1
?2−α1 − (νu/νl)1−α
1 − (ν2/ν1)2−α,(10)
Lph,[νl,νu]=
E
Th(ν2− ν1)ln
?νu
νl
?
, (11)