Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 6 August 2021 (MN LATEX style file v2.2)
Spectral analysis of Swift long GRBs with known redshift
J. I. Cabrera1⋆, C. Firmani1,2, V. Avila–Reese1, G. Ghirlanda2, G. Ghisellini2 and
L. Nava2,3
1Instituto de Astronomı́a, Universidad Nacional Autónoma de México, A.P. 70–264, 04510, México, D.F.
2Osservatorio Astronomico di Brera, via E.Bianchi 46, I–23807 Merate, Italy
3Universitá degli studi dell’Insubria, Dipartimento di Fisica e Matematica, via Valleggio 11, I–22100 Como, Italy
6 August 2021
ABSTRACT
We study the spectral and energetics properties of 47 long–duration gamma–ray bursts
(GRBs) with known redshift, all of them detected by the Swift satellite. Due to the
narrow energy range (15–150 keV) of the Swift–BAT detector, the spectral fitting is
reliable only for fitting models with 2 or 3 parameters. As high uncertainty and cor-
relation among the errors is expected, a careful analysis of the errors is necessary.
We fit both the power law (PL, 2 parameters) and cut–off power law (CPL, 3 pa-
rameters) models to the time–integrated spectra of the 47 bursts, and we present the
corresponding parameters, their uncertainties, and the correlations among the uncer-
tainties. The CPL model is reliable only for 29 bursts for which we estimate the νfν
peak energy Epk. For these GRBs, we calculate the energy fluence and the rest–frame
isotropic–equivalent radiated energy, Eγ,iso, as well as the propagated uncertainties
and correlations among them. We explore the distribution of our homogeneous sample
of GRBs on the rest–frame diagram E′pk vs Eγ,iso. We confirm a significant correlation
between these two quantities (the “Amati” relation) and we verify that, within the un-
certainty limits, no outliers are present. We also fit the spectra to a Band model with
the high energy power law index frozen to −2.3, obtaining a rather good agreement
with the “Amati” relation of non–Swift GRBs.
Key words: gamma rays: bursts — gamma rays: sources
1 INTRODUCTION
The Swift Mission (Gehrels et al. 2004) was designed mainly
to rapidly detect, locate, and observe gamma–ray bursts
(GRBs). After more than two years of operation, the Swift
satellite has observed approximately 180 GRBs, for which
more than 50 events have known redshifts reported in the
Mission homepage1. In this paper we present results from
an homogeneous spectral analysis that we have carried out
to the Swift long–duration GRBs with known z with the
main aim of estimating the time–averaged spectral parame-
ters of the prompt emission of these bursts, as well as related
global quantities like the prompt energy fluence, Sγ , and the
rest–frame isotropic–equivalent radiated energy, Eγ,iso.
Unfortunately, the spectral coverage of the Swift γ–ray
detector, the Burst Alert Telescope (BAT, Barthelmy et al.
2005), is very narrow. The recommendation is to limit the
spectral analysis to channels between 15 keV and 195 keV,
though the Swift–BAT’s team suggests a more conservative
⋆ E–mail: jcabrera@astroscu.unam.mx
1 http://swift.gsfc.nasa.gov/
upper limit of 150 keV because, above this energy, the cal-
ibration is not yet sufficiently reliable. Thus, the BAT en-
ergy range is not broad enough to allow the spectra to be
fitted unambiguously by the usual GRB broken power–law
function of 4 parameters. Therefore, the inference of γ–ray
spectral parameters and global quantities related to these
parameters for the Swift GRBs is not an easy task.
A main concern of this task is the determination of the
uncertainties of the spectral parameters and their propaga-
tion into the composite quantities. Due to the narrow energy
range of Swift BAT we know a priori that the errors in the
fitted spectral parameters will be large and correlated among
them. Therefore, the appropriate handling of errors is cru-
cial in order to make useful the spectral data generated by
BAT. The effort is justified by the great value that an homo-
geneous sample has, where the spectral information for all
the events is obtained with the same detector and analyzed
with the same techniques and methods. This is the case of
the growing–in–number sample of Swift GRBs.
The characterization of the time–averaged photon
spectra of GRBs with known redshift is important for
several reasons. On one hand, this characterization offers
c© 0000 RAS
http://arxiv.org/abs/0704.0791v3
2 Cabrera et al.
clues for understanding the radiation and particle accel-
eration mechanisms at work during the prompt phase of
the bursts (see for recent reviews e.g., Zhang & Mészáros
2004; Piran 2005; Mészáros 2006; Zhang 2007). On
the other hand, the spectral parameters and the global
quantities inferred from them (Sγ , Eγ,iso, etc.) are
among the main properties that characterize GRBs. The
study of the rest–frame correlations among these and
other properties (e.g,. Amati et al. 2002; Atteia 2003;
Ghirlanda, Ghisellini & Lazzati 2004a; Yonetoku et al.
2004; Liang & Zhang 2005; Firmani et al. 2006a) is cur-
rently allowing to learn key aspects of the nature of GRBs
(e.g., Thompson 2006; Thompson, Rees & Mészáros 2007;
Ramirez–Ruiz & Granot 2006). Furthermore, those corre-
lations that are tight enough can be used to standardize
the energetics of GRBs, making possible to apply GRBs
for constructing the Hubble diagram up to unprecedentedly
high redshifts (Ghirlanda et al. 2004b, 2006; Dai et al. 2004;
Firmani et al. 2005, 2006b, 2007; Liang & Zhang 2005;
Xu, Dai & Liang 2005; Schaefer 2007). These tight corre-
lations, for a given cosmological model, can be used also
to estimate the (pseudo)redshifts of GRBs with not mea-
sured redshifts, mainly from the extensive CGRO–BATSE
database2 (e.g., Lloyd-Ronning, Fryer & Ramirez-Ruiz
2002; Atteia 2003). Thus, by combining the distribution
in redshift of a large sample of GRBs with the observed
flux distribution, inferences on the luminosity function and
formation rate of GRBs can be obtained (Firmani et al.
2004; Guetta et al. 2005).
The outline of this paper is as follows. In §2 we ex-
plain the main steps of the spectral analysis carried out by
us: selection of the sample (§§2.1), spectral deconvolution
(fit) procedure and fitting models (§§2.2), and error analysis
(§§2.3). The results of the spectral analysis are given in §3,
along with the estimates of Sγ and Eγ,iso as well as some
correlations among the obtained quantities. The summary
and conclusions of the paper are presented in §4. In the Ap-
pendix the deconvolved spectra of the 47 GRBs studied here
are plotted.
We adopt the concordance ΛCDM cosmology with
ΩM = 0.3, ΩΛ = 0.7, and H0 = 70 km s
−1Mpc−1.
2 SPECTRAL ANALYSIS
The first step in our spectral analysis is the selection of the
Swift sub-sample of GRBs to be studied (§§2.1). Then we
proceed to the spectral deconvolution with XSPEC and de-
termination of the GRB spectral parameters for two– and
three–parameter photon models (§§2.2). By using these pa-
rameters, the energy fluence and isotropic–equivalent energy
are calculated. Finally, we construct the error confidence el-
lipses for the spectral parameters and Eγ,iso (§§2.3).
The BAT is a large aperture γ−ray telescope. It consists
of a (2.4×1.2)m coded aperture mask supported one meter
2 In a recent paper, Li (2007a) showed that due both to a red-
shift degeneracy in the so–called Amati correlation (Amati et al.
2002) and to its high scatter, is not possible to infer reliable
(pseudo)redshifts from this correlation.
above the 5200 cm2 area detector plane. The detector con-
tains 32,768 individual cadmium-zinc-telluride detector ele-
ments, each one of (4×4×2)mm size. The effective efficiency
of BAT starts at approximately 15 keV, attains a broad
maximum at approximately 30–100 keV and then falls off,
attaining at 195 keV the same level as at 15 keV. Due to the
1 mm thickness of the lead tiles, the BAT coded mask starts
to be transparent around 150 keV. We reduce the BAT’s
data using the analysis techniques provided by the BAT’s
instrument team3.
2.1 The sample
We analyse the time–integrated spectra of all available
Swift GRBs with known redshift z. The list of events was
taken from the daily updated compilation of J. Greiner4
for the period January 2005 to January 2007. The corre-
sponding file spectra were loaded from the legacy NASA
database5. The total number of events in this list was
55. We exclude from the list the short–duration bursts
(T90 < 2s; GRB050509, GRB050724, and GRB061217) as
well as those events with incomplete observational informa-
tion (GRB050730, GRB060124, GRB060218, GRB060505
and GRB060512), leaving us with a preliminary list of 47
long GRBs with measured z.
The list of 47 events is given in Table 1. The first four
columns of Table 1 give respectively the burst name, the
class (see below), the redshift z, and the time Tintthat we
use to integrate the spectrum. In principle T90 could be used
as an integration time. Nevertheless two problems arise in
this case. The first one has to do with the systematic under-
estimate of the fluence, hence of Eγ,iso. The second problem
may be more important; the elimination of the (early and
late) wings in the light curve can influence the estimates of
the spectral parameters. For example the elimination of the
late wing may harden artificially the spectrum. For these
reasons we have fixed the integration time by a visual in-
spection of the light curve as the period where the signal
is clearly identifiable in the background noise. A modest
overestimate of Tint does not influence on the values of
the integrated parameters, though increases the noise. We
would like to remark that Tint is not a good estimate of
the burst duration because its uncertainty is related to the
identification of faint (some time extended) wings.
2.2 Spectral fits
It is well known that the GRB photon spectra are in general
well described, in the energy range of ∼ 10 keV to a few
MeV, by the so–called Band function (Band et al. 1993),
which is a two smoothly connected power laws:
N(E) = N0
100 keV
E 6 E0(α− β) (1)
E0(α− β)
100 keV
](α−β)
exp(β − α)
100 keV
3 http://heasarc.gsfc.nasa.gov/docs/swift/analysis/threads
4 http://www.mpe.mpg.de/ jcg/grbgen.html
5 ftp://legacy.gsfc.nasa.gov/swift/data/obs
c© 0000 RAS, MNRAS 000, 000–000
Spectral analysis of Swift GRBs with known z 3
GRB Class z Tint LgN0[sd] α[sd] e1 n1 e2 n2 χ
sec 50 keV
(1) (2) (3) (4) (5) (6) (7) (8) (9)
050126 A 1.29 40.0 -2.656[0.026] -1.317[0.101] 0.026(-1.00, 0.01) 0.106(-0.01,-1.00) 1.399
050223 A 0.5915 30.0 -2.652[0.028] -1.837[0.106] 0.025(-0.99, 0.13) 0.109(-0.13,-0.99) 0.902
050315 D 1.949 129.0 -2.567[0.013] -2.101[0.055] 0.012(-0.99, 0.11) 0.056(-0.11,-0.99) 0.889
050318 A 1.44 34.5 -2.375[0.014] -1.968[0.057] 0.012(-0.99, 0.13) 0.060(-0.13,-0.99) 0.936
050319 D 3.24 190.0 -3.100[0.034] -2.045[0.118] 0.026(-0.98, 0.17) 0.119(-0.17,-0.98) 0.687
050401 B 2.9 43.0 -1.667[0.012] -1.472[0.046] 0.011(-1.00,-0.04) 0.047( 0.04,-1.00) 1.060
050416 D 0.6535 5.0 -2.225[0.055] -3.119[0.155] 0.026(-0.95, 0.30) 0.182(-0.30,-0.95) 0.971
050505 A 4.27 70.0 -2.418[0.018] -1.415[0.071] 0.018(-1.00,-0.03) 0.073( 0.03,-1.00) 0.789
050525 A 0.606 15.0 -0.932[0.003] -1.769[0.015] 0.003(-1.00,-0.09) 0.015( 0.09,-1.00) 2.934
050603 B 2.821 16.0 -1.347[0.010] -1.159[0.043] 0.010(-1.00,-0.05) 0.043( 0.05,-1.00) 0.913
050803 A 0.422 165.0 -2.868[0.022] -1.526[0.085] 0.021(-1.00, 0.04) 0.088(-0.04,-1.00) 0.882
050814 A 5.3 95.0 -2.750[0.030] -1.798[0.108] 0.027(-0.99, 0.12) 0.117(-0.12,-0.99) 1.036
050820 C 2.61 32.0 -2.461[0.016] -1.689[0.067] 0.015(-1.00, 0.09) 0.067(-0.09,-1.00) 0.988
050824 D 0.83 38.0 -3.102[0.081] -2.743[0.219] 0.036(-0.95, 0.30) 0.241(-0.30,-0.95) 1.0157
050904 C 6.29 207.0 -2.603[0.009] -1.205[0.041] 0.009(-1.00,-0.04) 0.041( 0.04,-1.00) 1.070
050908 A 3.344 24.0 -2.685[0.031] -1.947[0.107] 0.025(-0.99, 0.17) 0.114(-0.17,-0.99) 0.969
050922C B 2.198 10.0 -1.761[0.009] -1.371[0.038] 0.009(-1.00,-0.02) 0.038( 0.02,-1.00) 0.900
051016B D 0.9364 6.0 -2.509[0.048] -2.331[0.145] 0.030(-0.97, 0.24) 0.158(-0.24,-0.97) 1.344
051109 A 2.346 43.0 -2.052[0.033] -1.626[0.124] 0.032(-1.00, 0.05) 0.129(-0.05,-1.00) 0.819
051111 B 1.55 45.0 -2.080[0.008] -1.305[0.034] 0.008(-1.00,-0.04) 0.035( 0.04,-1.00) 0.764
051221 C 0.547 3.1 -1.422[0.008] -1.397[0.037] 0.008(-1.00,-0.03) 0.037( 0.03,-1.00) 0.996
060115 A 3.53 128.0 -2.851[0.018] -1.771[0.071] 0.016(-0.99, 0.11) 0.073(-0.11,-0.99) 0.875
060206 A 4.048 14.0 -2.184[0.011] -1.693[0.047] 0.011(-1.00, 0.06) 0.049(-0.06,-1.00) 1.061
060210 A 3.91 278.0 -2.579[0.015] -1.544[0.059] 0.015(-1.00,-0.01) 0.060( 0.01,-1.00) 0.928
060223 A 4.41 14.8 -2.320[0.020] -1.775[0.078] 0.019(-0.99, 0.11) 0.082(-0.11,-0.99) 1.009
060418 C 1.489 80.0 -1.993[0.006] -1.634[0.030] 0.006(-1.00,-0.01) 0.030( 0.01,-1.00) 0.815
060502A B 1.51 34.0 -2.157[0.011] -1.431[0.045] 0.011(-1.00,-0.01) 0.046( 0.01,-1.00) 1.031
060510B A 4.9 289.0 -2.839[0.011] -1.778[0.048] 0.010(-1.00, 0.07) 0.049(-0.07,-1.00) 0.643
060522 A 5.11 80.0 -2.829[0.024] -1.578[0.093] 0.023(-1.00, 0.08) 0.098(-0.08,-1.00) 1.188
060526 C 3.21 320 -3.387[0.035] -1.924[0.140] 0.028(-0.99, 0.17) 0.132(-0.17,-0.99) 0.967
060604 D 2.68 67.0 -3.356[0.085] -2.040[0.283] 0.060(-0.98, 0.21) 0.290(-0.21,-0.98) 0.804
060605 A 3.711 37.0 -2.810[0.026] -1.474[0.105] 0.025(-1.00, 0.07) 0.109(-0.07,-1.00) 0.912
060607 A 3.082 127.0 -2.686[0.011] -1.436[0.046] 0.010( 1.00, 0.02) 0.047(-0.02,-1.00) 0.947
060614 C 0.125 154.0 -1.877[0.005] -1.936[0.024] 0.004(-1.00, 0.01) 0.024( 0.01,-1.00) 1.072
060707 A 3.425 81.0 -2.652[0.022] -1.667[0.082] 0.021(-1.00, 0.06) 0.087(-0.06,-1.00) 1.116
060714 A 2.71 135.0 -2.614[0.016] -1.935[0.063] 0.013(-0.99, 0.12) 0.064(-0.12, 0.99) 1.161
060729 C 0.54 134.0 -2.658[0.021] -1.870[0.083] 0.019(-0.99, 0.12) 0.084(-0.12,-0.99) 0.863
060904B A 0.703 187.0 -3.059[0.022] -1.628[0.090] 0.021(-1.00, 0.10) 0.090(-0.10,-1.00) 0.855
060906 A 3.686 63.0 -2.386[0.020] -2.006[0.075] 0.017(-0.99, 0.14) 0.078(-0.14,-0.99) 1.114
060908 A 2.43 30.0 -2.040[0.009] -1.339[0.039] 0.009(-1.00,-0.01) 0.040( 0.01,-1.00) 0.864
060926 D 3.2 13.0 -2.724[0.044] -2.460[0.146] 0.026(-0.97, 0.24) 0.151(-0.24,-0.97) 0.972
060927 A 5.6 32.0 -2.388[0.013] -1.654[0.052] 0.012(-1.00, 0.07) 0.055(-0.07,-1.00) 1.158
061007 C 1.261 85.0 -1.334[0.004] -1.002[0.016] 0.003(-0.99,-0.13) 0.016( 0.13,-0.99) 0.523
061110B C 3.44 147.0 -3.055[0.021] -1.069[0.091] 0.021(-1.00,-0.05) 0.091( 0.05,-1.00) 1.175
061121 C 1.314 125.0 -1.936[0.004] -1.410[0.018] 0.004(-1.00,-0.08) 0.018( 0.08,-1.00) 0.450
061222B A 3.355 57.0 -2.367[0.023] -1.962[0.084] 0.020(-0.99, 0.14) 0.089(-0.14,-0.99) 1.280
070110 C 2.352 103.0 -2.806[0.016] -1.581[0.702] 0.016(-1.00, 0.06) 0.070(-0.06,-1.00) 1.007
Table 1. The sample of Swift long GRBs with known z analysed here (47 events). The z and duration (Tint) of the burst as well as the
results from the PL photon model fit are reported. Second column refers to the class group assigned to the burst according to our spectral
analysis (see text). The best fit values of the parameters and their one–dimensional standard deviations (sd) are given in columns 5 and
6; e1,2 and n1,2 in columns 7 and 8 are the semi-axis lengths and principal axes of the error ellipse, respectively. The reduced χ
reported in the last column (dof=58).
E > E0(α− β)
where α and β are the photon indices of the low and high
energy power laws, respectively, and E0 is the e−folding
(break) energy related to the peak energy in the νfν [or
E2f(E)] spectrum by Epk = E0× (2+α). Note that Epk is
well defined for α > −2 and β < −2. The normalization N0
is in photons s−1 cm−2 keV−1.
The spectral range of BAT is narrower than that one
of BATSE and other previous detectors. In the most recent
update of the BAT–team homepage6 it is written that due
to varying threshold levels in individual detectors, channels
below 15 keV should not be used for spectral analysis. Like-
6 http://swift.gsfc.nasa.gov/docs/swift/analysis/bat-digest.html
c© 0000 RAS, MNRAS 000, 000–000
4 Cabrera et al.
wise, channels above 150 keV are unreliable due to a lack of
calibration data at those energies. Then, we are limited to
the narrow energy band of 15–150 keV for the spectral anal-
ysis of Swift bursts. This implies that model fitting with the
4–parameter Band function will not be reliable for most of
these burst. Therefore, we proceed to fit the observed pho-
ton spectra to both the power law (PL) and cutoff power
law (CPL) models. The former has only two parameters, the
normalization, N0, and the photon index, α, and is given by:
N(E) = N0
50 keV
The CPL model implies one more parameter, an energy
e−folding spectral break, E0, related to Epk in the same
way as in the Band model (see above). The CPL model is
defined as:
N(E) = N0
50 keV
It is easy to verify that the CPL model (named also the
Compton –COMP– model) is the Band model with β →
As we will see later, the complex nature of the errors
due to the small energy range of the spectra makes it more
convenient to use the logarithms of N0 and E0 (or Epk)
rather than the linear quantities. This choice reduces the
asymmetry of the errors and of the confidence levels (CL)
contour shapes. Therefore, we have introduced into XSPEC
new models corresponding to Eqs. (2) and (3) (and also Eq.
(1); see §§3.3) to carry out the fits with LogN0 and LogEpk.
The spectral analysis was carried out by using the hea-
soft6.1.2 public software7. The first step in our analysis is to
define the background–subtracted light curve from the cor-
responding data file. The photon counts at each channel are
taken in the time interval Tint where the main light curve
is clearly above the background noise. The spectral file was
corrected by position and systematic errors. Then, the pho-
ton counts at each channel is convolved with a response ma-
trix, build up for a given event with the calibration matrix8,
in order to obtain the time–integrated energy spectrum. As
mentioned above, the BAT detector is well calibrated only
in the 15–150 keV energy range. Therefore, we used the 60
BAT channels in this range, with the default binning. The
spectral fit to the different photon models mentioned above
was performed with the heasoft package XPSEC (version
11.3.1).
For the GRBs for which the (CPL) Epk may be esti-
mated, we calculate the fluence, Sγ , corresponding to the
observed spectral band (15–150 keV) only with the aim of
completeness. In addition the rest isotropic–equivalent en-
ergy, Eγ,iso, is calculated from the observer bolometric flu-
ence SγB corresponding to the spectral range 1–10000 keV
at rest:
Eγ,iso =
4πSγBdL(z)
(1 + z)
, (4)
where dL(z) is the cosmology–dependent luminosity dis-
tance.
7 http://heasarc.gsfc.nasa.gov/docs/software/lheasoft/download.html
8 We use the January 2007 matrix calibration provided by the
Swift–BAT team.
2.3 Errors from the fit and error propagation
A main concern of the spectral deconvolution of BAT spec-
tra is the expected large uncertainty and correlations among
the fitted spectral parameters. Consequently, a careful inter-
pretation of the errors is crucial to make usable and reliable
the spectral analysis of Swift GRBs.
First, as mentioned above, we have carried out the fits
using LogN0 (PL model), and LogN0 and LogEpk (CPL
model), instead of the corresponding linear values. The er-
rors in the linear parameters are very asymmetric and corre-
lated among them in a very complex way; the corresponding
regions at a given CL, as plotted with the XSPEC com-
mand steppar in the plane of two parameters, have shapes
strongly deviating from ellipses. Instead, the errors and the
CL regions when the fits are carried out with logarithmic
quantities, tend to be more symmetric and ellipse–shaped,
respectively. This facilitates the handling of the errors to
calculate composite quantities and correlations among the
GRB parameters.
The fit command of XSPEC, generates ellipsoidal con-
fidence intervals that approximate the complex regions of
joint variation of the parameters. These ellipsoids are char-
acterized by the semi–axis lengths, ei (they are the square
root of the variances), where i is for each axis, and the cor-
responding principal axes (unitary vectors ni). We use these
data to calculate the covariance matrix, Cij , and the ellip-
tical boundary of the desired CL region in the subspace of
interest (see e.g., Press et al. 1999). The diagonal elements
of the matrix, Cii, are the variances of each model param-
eter (two for the PL model and three for the CPL model),
while the off–diagonal elements (covariances) show the cor-
relations among the parameters.
The error propagation from the spectral quantities
(LogN0, α, LogEpk) to the composite quantities LogE
(being E′pk the peak energy at rest E
pk = Epk× [1+z)])
and LogEγ,iso is performed in two steps. First, we have ex-
tracted sets of spectral quantities by a Monte Carlo method
according to their elliptical CL. Then, for each set we have
calculated the corresponding composite quantities as well
as their covariance matrix averaging on the overall extrac-
tions. From this last covariance matrix the elliptical CL of
the composite quantities is obtained.
3 RESULTS
In Table 1 we present the results from the spectral fit to the
PL model for the whole sample of 47 long GRBs. Columns
1 to 4 give the burst name, the class group of the burst, z,
and Tint. Columns 5 and 6 give the best fit values of the
normalization, LogN0, and the photon index, α, with the
corresponding one–dimensional symmetric standard devia-
tions in brackets, as obtained with the XSPEC error com-
mand. In columns 7 and 8 we report the square root of the
variances, ei, together with the unitary vectors ni (principal
axes), which provide respectively the semi–axis lenghts and
directions of each axis (i = 1, 2) of the joint error ellipse.
The reduced χ2 is reported in column 9.
As it may be appreciated from Table 1, the best–fit α is
smaller than −2 for 7 bursts out of the 47 (GRB 060906 is
marginal and it will be recovered by the CPL fitting). Most
c© 0000 RAS, MNRAS 000, 000–000
Spectral analysis of Swift GRBs with known z 5
Figure 1. Error CL contours projected in the plane of the pair of variables LogEpk–(-α) for each one of the bursts in samples A and
B (CPL model). The irregular contours (black lines in the electronic version) were calculated with the XSPEC command steppar, while
the elliptical contours (red lines) were constructed from the variances and principal axes given by the XSPEC command fit. Inner (thick
line) and outer (thin line) contours in each case are for ∆χ2 = 1 and 2.3, respectively (see text). The dotted line indicates the level of
150 keV.
c© 0000 RAS, MNRAS 000, 000–000
6 Cabrera et al.
GRB LgN0[sd] α[sd] LgEpk[sd] e1 n1 e2 n2 e3 n3 χ
50 keV keV
(1) (2) (3) (4) (5) (6) (7) (8)
050126 -2.38[0.21] -0.75[0.44] 2.07[0.19] 0.022 (-0.83,-0.17,-0.53) 0.519 ( 0.39,-0.85,-0.34) 0.092 ( 0.39,0.49,-0.78) 1.38
050223 -2.47[0.22] -1.50[0.42] 1.83[0.21] 0.022 (-0.89,-0.36,-0.29) 0.506 (-0.44, 0.83, 0.34) 0.140 (-0.12,-0.43, 0.90) 0.90
050318 -2.02[0.14] -1.32[0.26] 1.67[0.05] 0.010 (-0.77,-0.40,-0.49) 0.289 ( 0.46,-0.89,-0.02) 0.051 ( 0.43, 0.24,-0.87) 0.80
050401 -1.56[0.08] -1.23[0.19] 2.27[0.23] 0.011 (-0.96,-0.11,-0.25) 0.306 (-0.27, 0.61, 0.74) 0.056 (-0.07,-0.78, 0.62) 1.05
050505 -2.20[0.14] -0.95[0.31] 2.10[0.16] 0.016 (-0.87,-0.16,-0.46) 0.366 ( 0.37,-0.84,-0.41) 0.070 ( 0.32, 0.53,-0.79) 0.75
050525 -0.62[0.03] -0.98[0.07] 1.91[0.01] 0.002 (-0.73,-0.20,-0.65) 0.012 ( 0.59, 0.30,-0.75) 0.077 (-0.35, 0.93, 0.10) 0.34
050603 -1.27[0.07] -0.97[0.17] 2.51[0.31] 0.010 (-0.98,-0.04,-0.21) 0.358 (-0.20, 0.46, 0.86) 0.049 (-0.06,-0.89, 0.46) 0.90
050803 -2.60[0.18] -0.99[0.37] 1.99[0.15] 0.018 (-0.85,-0.23,-0.48) 0.425 ( 0.41,-0.86,-0.32) 0.081 ( 0.34, 0.47,-0.82) 0.85
050814 -2.10[0.30] -0.58[0.56] 1.73[0.59] 0.017 (-0.60,-0.27,-0.75) 0.0.75 ( 0.65, 0.38,-0.66) 0.628 (-0.46, 0.88, 0.05) 0.93
050908 -2.29[0.27] -1.26[0.48] 1.65[0.08] 0.019 (-0.76,-0.41,-0.51) 0.550 ( 0.49,-0.87,-0.02) 0.093 ( 0.43, 0.26,-0.86) 0.94
050922C -1.66[0.07] -1.16[0.15] 2.35[0.22] 0.009 (-0.97,-0.09,-0.22) 0.274 (-0.24, 0.54, 0.81) 0.045 (-0.04,-0.84, 0.54) 0.87
051109 -1.75[0.27] -1.04[0.55] 1.92[0.17] 0.026 (-0.82,-0.26,-0.52) 0.628 ( 0.42,-0.88,-0.22) 0.112 ( 0.40, 0.40,-0.83) 0.81
051111 -2.02[0.06] -1.17[0.14] 2.54[0.34] 0.008 (-0.99,-0.04,-0.15) 0.374 (-0.15, 0.35, 0.92) 0.040 (-0.01,-0.93, 0.36) 0.76
060115 -2.51[0.16] -1.13[0.32] 1.80[0.08] 0.013 (-0.80,-0.33,-0.51) 0.361 ( 0.45,-0.88,-0.15) 0.063 ( 0.40, 0.35,-0.85) 0.80
060206 -1.90[0.10] -1.11[0.21] 1.89[0.06] 0.009 (-0.83,-0.28,-0.48) 0.237 ( 0.42,-0.88,-0.22) 0.044 ( 0.37, 0.39,-0.85) 0.91
060210 -2.38[0.11] -1.12[0.26] 2.07[0.14] 0.013 (-0.89,-0.19,-0.41) 0.307 (-0.37, 0.82, 0.43) 0.060 (-0.25,-0.53, 0.81) 1.06
060223 -2.04[0.18] -1.16[0.35] 1.80[0.09] 0.015 (-0.80,-0.33,-0.50) 0.400 ( 0.45,-0.88,-0.15) 0.072 ( 0.39, 0.34,-0.85) 0.96
060502A -2.05[0.08] -1.19[0.18] 2.27[0.22] 0.010 (-0.96,-0.12,-0.25) 0.296 (-0.27, 0.60, 0.75) 0.054 (-0.06,-0.79, 0.61) 1.01
060510B -2.72[0.09] -1.53[0.19] 1.99[0.17] 0.009 (-0.94,-0.27,-0.22) 0.267 (-0.35, 0.70, 0.62) 0.069 ( 0.01,-0.66, 0.75) 0.62
060522 -2.37[0.23] -0.70[0.44] 1.85[0.08] 0.016 (-0.71,-0.26,-0.66) 0.071 ( 0.55, 0.39,-0.74) 0.501 (-0.45, 0.88, 0.13) 1.13
060605 -2.57[0.22] -1.00[0.44] 2.01[0.23] 0.022 (-0.87,-0.24,-0.43) 0.539 (-0.41, 0.82, 0.39) 0.103 (-0.26,-0.51, 0.82) 0.90
060607 -2.52[0.09] -1.09[0.19] 2.15[0.15] 0.010 (-0.93,-0.17,-0.34) 0.251 (-0.35, 0.74, 0.57) 0.050 (-0.15,-0.65, 0.75) 0.89
060707 -2.18[0.19] -0.73[0.40] 1.84[0.06] 0.015 (-0.68,-0.25,-0.69) 0.062 ( 0.59, 0.37,-0.72) 0.443 (-0.43, 0.90, 0.10) 1.00
060714 -2.52[0.12] -1.77[0.24] 1.80[0.20] 0.012 (-0.91,-0.38,-0.15) 0.304 (-0.40, 0.74, 0.54) 0.138 ( 0.09,-0.55, 0.83) 1.17
060904B -2.77[0.19] -1.07[0.37] 1.90[0.13] 0.017 (-0.84,-0.29,-0.47) 0.432 (-0.44, 0.86, 0.26) 0.081 (-0.33,-0.43, 0.84) 0.82
060906 -2.16[0.16] -1.60[0.31] 1.65[0.09] 0.014 (-0.83,-0.45,-0.33) 0.346 (-0.46, 0.89,-0.05) 0.094 (-0.31,-0.11, 0.94) 1.09
060908 -1.84[0.08] -0.90[0.17] 2.15[0.10] 0.008 (-0.89,-0.15,-0.42) 0.208 ( 0.36,-0.81,-0.47) 0.039 ( 0.27, 0.57,-0.78) 0.74
060927 -2.02[0.12] -0.93[0.24] 1.86[0.06] 0.009 (-0.77,-0.27,-0.58) 0.043 ( 0.47, 0.37,-0.80) 0.272 (-0.44, 0.89, 0.16) 0.98
061222B -2.03[0.20] -1.30[0.37] 1.67[0.06] 0.015 (-0.76,-0.40,-0.51) 0.422 (-0.47, 0.88, 0.01) 0.075 (-0.44,-0.24, 0.86) 1.30
Table 2. Spectral fit results using the CPL photon model for the class A and B GRBs. The best fit values of the parameters and
their one–dimensional standard deviations (sd) are given in columns 2–4; e1,2,3 and n1,2,3 in columns 5–7 are the semi-axis lengths and
principal axes of the error ellipsoid, respectively. The reduced χ2 is reported in the last column (dof=57).
likely, the νfν spectra of these GRBs peak at energies below
15 keV. They could be related to the X–ray rich events and
X–ray Flashes discovered by BeppoSAX and HETE–2 (Kip-
pen et al. 2001; Sakamoto et al. 2005).
We have also fitted the 47 GRB spectra to the CPL
model and found that for 18 bursts the fit produces too
large uncertainties and/or unphysical solutions. For these
18 cases the PL model is definitively more reliable than the
CPL one. According to the results from our spectral fits
(judged by the joint 68% CL contour errors), we classify the
47 Swift GRBs analysed here in four classes:
Class A – The CPL–model fit is acceptable and Epk is
within the 15–150 keV range.
Class B – The CPL–model fit is acceptable, but Epk >
150 keV.
Class C – the CPL–model fit gives unreliable results,
while the PL–model fit is acceptable and gives α> −2, sug-
gesting that Epk is at energies > 150 keV.
Class D – The CPL–model fit gives unreliable results,
while the PL–model fit is acceptable and gives α< −2, sug-
gesting that Epk is smaller than 15 keV.
The class of each GRB is indicated in the second col-
umn of Table 1. This classification can be easily understood
after a visual inspection of the plots shown in Appendix and
figures 1 and 2. In Figs. A1, A2, and A3 of the Appendix we
present the deconvolved time–integrated νfν [or E
2f(E)]
observed spectra of the 47 Swift GRBs analysed here. The
CPL model was used for the spectral deconvolution. The
error bars show the observations while the continuous lines
show the fitted curves. The spectra of all the events that
we classify in groups A and B show some curvature, evi-
dence of a peak in the E2f(E) distribution. Nevertheless,
in most of the cases the formal F−test analysis says that
the spectral fits from the PL model to the CPL one (which
introduces one more parameter, Epk) do not improve sig-
nificantly. Therefore, one expects large uncertainties in the
determination of Epk for most of the events analyzed here.
This is why it is important to carry out a carefully statistical
analysis of the errors and to adequately propagate them for
calculating composite quantities and inferring correlations.
Figures 1 and 2 show different contours of the CLs pro-
jected in the plane of the pair of variables LogEpk vs. (−α)
for each burst in groups A and B (CPL model). The irregu-
lar CL contours (black lines in the electronic version) were
calculated with the XSPEC command steppar, while the el-
liptical CLs (red lines) were constructed from the joint vari-
ances and principal axes given by the XSPEC command fit
(see §§2.3; columns 5, 6, and 7 in Table 2 below). Thick lines
show the CLs for ∆χ2 = 1, while thin lines show the CLs for
∆χ2 = 2.3. For ∆χ2 = 1, the projections of the CL regions
in each axis enclose ≈ 68% of the one–dimensional probabil-
ity for the given parameter (one standard deviation), while
c© 0000 RAS, MNRAS 000, 000–000
Spectral analysis of Swift GRBs with known z 7
Figure 2. Continues Fig. 1
for ∆χ2 = 2.3, the plotted region encompasses ≈ 68% of
the probability for the joint variation of the two parameters
(one standard deviation) (e.g., Press et al. 1999). As can be
appreciated, the ellipses are a reasonable description of the
CL regions in most cases, with the great advantage that they
are handled analytically.
We see from Figs. 1 and 2 that for many of the GRBs,
the LogEpk and α spectral parameters are correlated (in-
clined, thin ellipses). A similar situation arises for the other
combination of parameters. Therefore, for this kind of data,
a simple statistical quantification of the errors, such as the
one–dimensional standard deviation, is not enough. This is
why we have worked out the full covariance matrix calcu-
lated from the error ellipse. We use it to propagate errors in
the calculation of E′pk and Eγ,iso (§§2.3). This is a crucial
c© 0000 RAS, MNRAS 000, 000–000
8 Cabrera et al.
GRB Sγ [s.d] LgE
[sd] LgE′
[sd] e1 n1 e2 n2
10−7erg cm2 1050erg s−1 keV
(1) (2) (3) (4) (5) (6)
050126 8.55[1.82] 1.70[0.18] 2.42[0.18] 0.062 (0.73,-0.69) 0.247 ( 0.69, 0.73)
050223 6.14[0.83] 1.00[0.20] 2.04[0.21] 0.113 (0.72,-0.69) 0.266 ( 0.69, 0.72)
050318 12.90[0.54] 2.07[0.11] 2.06[0.04] 0.107 (1.00, 0.03) 0.044 (-0.03, 1.00)
050401 89.41[7.24] 3.51[0.14] 2.86[0.23] 0.042 (0.86,-0.51) 0.261 ( 0.51, 0.86)
050505 25.79[3.06] 3.13[0.12] 2.82[0.16] 0.040 (0.79,-0.62) 0.195 ( 0.62, 0.79)
050525 158.20[1.59] 2.34[0.01] 2.12[0.01] 0.018 (0.79, 0.61) 0.008 (-0.61, 0.79)
050603 74.60[8.18] 3.55[0.22] 3.10[0.30] 0.061 (0.81,-0.59) 0.372 ( 0.59, 0.81)
050803 20.83[2.57] 1.13[0.14] 2.14[0.15] 0.044 (0.73,-0.68) 0.199 ( 0.68, 0.73)
050814 14.60[1.16] 2.94[0.11] 2.53[0.06] 0.114 (0.93, 0.36) 0.047 (-0.36, 0.93)
050908 4.36[0.46] 2.25[0.16] 2.29[0.08] 0.156 (1.00, 0.06) 0.079 (-0.06, 1.00)
050922C 16.97[0.94] 2.63[0.14] 2.86[0.22] 0.034 (0.85,-0.53) 0.257 ( 0.53, 0.85)
051109 35.27[5.85] 2.80[0.19] 2.44[0.16] 0.238 (0.77, 0.64) 0.077 (-0.64, 0.77)
051111 37.32[3.59] 2.80[0.22] 2.96[0.34] 0.056 (0.84,-0.54) 0.399 ( 0.54, 0.84)
060115 16.04[1.07] 2.78[0.10] 2.46[0.07] 0.121 (0.84, 0.54) 0.044 (-0.54, 0.84)
060206 8.29[0.35] 2.59[0.06] 2.59[0.06] 0.086 (0.72, 0.70) 0.024 (-0.70, 0.72)
060210 69.24[3.74] 3.52[0.10] 2.76[0.14] 0.030 (0.81,-0.59) 0.170 ( 0.59, 0.81)
060223 6.54[0.52] 2.56[0.12] 2.53[0.08] 0.139 (0.86, 0.50) 0.052 (-0.50, 0.86)
060502A 22.87[1.66] 2.42[0.14] 2.67[0.22] 0.039 (0.85,-0.53) 0.259 ( 0.53, 0.85)
060510B 38.64[2.88] 3.53[0.13] 2.76[0.17] 0.046 (0.80,-0.59) 0.207 ( 0.59, 0.80)
060522 10.49[1.04] 2.78[0.09] 2.63[0.08] 0.120 (0.74, 0.67) 0.036 (-0.67, 0.74)
060605 5.33[1.12] 2.31[0.20] 2.69[0.22] 0.074 (0.74,-0.67) 0.289 ( 0.67, 0.74)
060607 24.91[1.58] 2.94[0.10] 2.76[0.15] 0.026 (0.83,-0.55) 0.173 ( 0.55, 0.83)
060707 16.26[1.13] 2.70[0.08] 2.48[0.06] 0.097 (0.81, 0.59) 0.036 (-0.59, 0.81)
060714 30.58[3.96] 3.11[0.15] 2.37[0.20] 0.111 (0.85,-0.53) 0.219 ( 0.53, 0.85)
060904B 14.78[1.91] 1.44[0.13] 2.13[0.13] 0.174 (0.71, 0.71) 0.047 (-0.71, 0.71)
060906 23.82[1.92] 3.16[0.14] 2.32[0.09] 0.142 (0.98,-0.20) 0.087 ( 0.20, 0.98)
060908 26.56[1.11] 2.78[0.07] 2.68[0.10] 0.016 (0.81,-0.59) 0.123 ( 0.59, 0.81)
060927 11.66[0.46] 2.92[0.06] 2.68[0.06] 0.077 (0.72, 0.70) 0.022 (-0.70, 0.72)
061222B 22.35[1.23] 2.97[0.14] 2.31[0.06] 0.137 (1.00, 0.01) 0.064 (-0.01, 1.00)
Table 3. Calculated fluence and rest–frame isotropic and peak energies for our GRB groups A and B. The corresponding one–dimensional
standard deviations are given between square brackets. The last 2 columns give the estimated semi–axis lengths (e1,2) and principal
axes (n1,2) of the error ellipse associated to the pair of quantities LogE
and LogEγ,iso. Note that the errors in most of the cases are
correlated significantly (see also the plotted ellipses in Figs. 4 and 5).
step when using the data, for instance, for establishing cor-
relations among the spectral or composite quantities (§§3.3).
The CPL–model fittings for GRB groups A and B are
presented in Table 2. Column 1 gives the burst names.
Columns 2, 3, and 4 report the best fit values of LogN0,
α, and LogEpk with their corresponding one–dimensional
standard deviations, as calculated with the XSPEC error
command. In columns 5, 6, and 7 we report the lengths
ei of each semi–axis together with the unitary vectors ni
(principal axes), which provide the directions of each axis
(i = 1, 2, 3). The reduced χ2 is reported in column 8 (dof =
Finally, in Table 3 we report the calculated values for
Sγ , LogEγ,iso, and LogE
pk , as well as the corresponding
propagated errors for the samples A and B, using the CPL–
model fit results (see §2.3 for the used error propagation
procedure). Column 1 gives the burst name. Columns 2 and
3 give respectively the values of Sγ (in the observed range
15–150 keV) and of Eγ,iso (extrapolated to the rest–frame
energy range 1–10000 keV) corresponding to the best fit
spectral parameters. Column 4 gives the value of the rest–
frame peak energy, E′pk. The one–dimensional standard de-
viationss of Sγ , Eγ,iso, and E
pk are given within square
brackets. However, due to the high correlation among the
errors the simple standard deviation is not enough to char-
acterize statistically the errors. Thus, in columns 5 and 6
we report the error ellipse elements (semi–axis lenghts, ei,
and principal axes ni, i = 1, 2) associated to the pair of
quantities LogE′pk and LogEγ,iso. It is important to remark
that the correlation between the uncertainties of LogE′pk
and LogEγ,iso is not entirely due to the correlation among
LogN0, α, and LogEpk. For instance an increase in LogEpk
alone is able to produce an increase in both LogE′pk and
LogEγ,iso. This effect will turn out to be helpful in the next
section, where we will study how LogE′pk and LogEγ,iso do
correlate in the entire GRB sample.
The median and dispersion (quartiles) of the best–
fit CPL parameter α for the 24 + 5 classes A+B
GRBs are −1.12+0.15
−0.14, which is in agreement with the re-
sults from BATSE bright–GRB (Kaneko et al. 2006) and
HETE–2 GRB (Barraud et al. 2003; Sakamoto et al. 2005)
time–integrated spectra. Our α distribution is in particular
similar to the one from HETE–2, with only ∼ 25% of the
bursts having α . −1.2. The median and quartiles of the
LogEpk [keV] parameter are 1.91
+0.19
−0.11 . Unfortunately, due
to the low energy limit of the Swift–BAT, the sample anal-
ysed here covers only the ∼ 15% fraction of bursts in the
(CPL–model) Epk distribution inferred for BATSE bright
bursts (Kaneko et al. 2006).
The distribution of E′pk (at rest) is much broader than
c© 0000 RAS, MNRAS 000, 000–000
Spectral analysis of Swift GRBs with known z 9
Figure 3. Top panel: the observed Epk as derived by the CPL fit
as a function of the spectral index αCPL of the same fit. Bottom
panel: Epk (as derived by the CPL fit) as a function of the spectral
index αPL of the single power law fit. The correlation of Epk with
αPL is induced by Epk entering in the Swift energy band (see
text), and cannot be used to infer Epk when knowing only αPL.
the one of Epk (the median and quartiles of LogE
pk[keV] are
2.54+0.22
−0.21). This is expected due to the broad distribution in
redshfits of the Swift bursts analysed here. For our 24 + 5
classes A+B GRBs, the z distribution has a broad maximum
at z ∼ 3 − 4; the median and quartiles are 3.08+0.82
−1.5 . The
median and quartiles of LogEγ,iso [10
50 erg] for our 24 + 5
classes A+B GRBs are 2.78+0.30
−0.44 . For the 8 long–duration
Swift GRBs in common with the compilation presented in
Amati (2006b), the values of E′pk reported here and in that
paper are similar, but those of Eγ,iso are systematically
larger in Amati (2006b). The latter difference is probably
due to the fact that Amati (2006b) did not use for most of
those GRBs the spectral information from the Swift satel-
lite but from other space missions (Konus, HETE–2), which
allowed to find a fitting with the Band model; as will be
discussed in §§3.3, this implies typically a larger Eγ,iso with
respect to the case a CPL model is used.
3.1 Bursts detected also by other instruments
Among the bursts listed in Tables 2 and 3 there are four
GRBs which have been detected by other instruments be-
sides Swift. Three of them have been detected by Konus–
Wind (GRB 050401, 050603 and 051109A) and one by
HETE–2 (GRB 050922C).
For GRB 050401 the Konus–Wind results reported by
Golenetskii et al. (2005a) concern the spectral parameters
for the two peaks displayed by its light curve, fitted with
a Band model in the energy range [20–2000] keV. The first
peak had α = −1.15 ± 0.16, β = −2.65 ± 0.31 and Epk =
132 ± 16 keV, while the second peak is described by α =
−0.83 ± 0.21, β = −2.37 ± 0.14 and Epk = 119 ± 26 keV.
Our analysis with the Swift data (see Tab. 3) for the time
integrated spectrum and a CPL model yielded α = −1.23±
0.19 and Epk = 186
−76 keV. Within the (relatively large)
errors, the results are consistent.
The Konus–Wind data of GRB 050603 have been fitted
by Golenetskii et al. (2005b) with a Band model in the [20–
3000] keV energy range, with α = −0.79±0.06, β = −2.15±
0.09 and Epk = 349 ± 28 keV. Our analysis with a CPL
model gives α = −1.23± 0.19 and Epk = 323
−164 keV. The
derived low energy spectral index is somewhat softer, while
the value of Epk is consistent.
GRB 050922C was detected by HETE–2 with the FRE-
GATE instruments in the 7–30, 7–80, and 30–400 keV bands.
It was fitted with a CPL law model by Crew et al. (2005)
with α = −0.83+0.23
−0.26 and Epk = 130.5
+50.9
−26.8 keV, to be com-
pared with our values of α = −1.16±0.15 and Epk = 224
keV. Again, within the errors, there is consistency.
GRB 051109, detected by Konus–Wind and fitted with
a CPL model by Golenetskii et al. (2005c) in the [20–500]
keV energy range, showed α = −1.25+0.44
−0.59 and Epk =
161+224
−58 keV. This should be compared with our α = −1.04±
0.55 and Epk = 83
−27 keV. Even in this case there is con-
sistency, within the relatively large errors.
3.2 Peak energy vs spectral index
In Fig. 3 we show Epk, derived with the CPL fit, as a func-
tion of the spectral index αCPL of the CPL fit (top panel)
and as a function of αPL, the index derived when fitting
with a single power law. As can be seen, although there is
no relation between Epk and αCPL, a correlation appears
when fitting with a single power law. This correlation, how-
ever, has no physical meaning, since it is the result of the
attempt of the single power law to account for the data
point above the peak, with smaller flux (in νfν ). As Epk
decreases, a larger fraction of data lies above the peak, in-
ducing the single power law to steepen. This also means
that this effect vanishes when Epk is outside the Swift en-
ergy range, therefore if we do not know Epk, we cannot use
this correlation to infer it. Furthermore, note the difference
in the derived values of the spectral indices. As expected for
these bursts for which we could derive Epk, the index αPL
is systematically softer than αCPL. Given the obvious result
of this subsection, we have considered unnecessary to show
the upper panel of Fig. 3 with the 68% CL error ellipses. A
complete treatment of the errors in this sense confirms the
previous result.
3.3 Correlation between E′pk and Eγ,iso
In Fig. 4, E′pk is plotted vs Eγ,iso for the class A (red–line
ellipses) and class B (green–line ellipses) Swift GRBs. The
ellipses correspond to the joint 68% CL error regions calcu-
lated as explained in §§2.3. The dotted straight lines encom-
pass the 3σ scatter of the updated Amati correlation pre-
sented in Ghirlanda (2007) for 49 GRBs; the central thin
solid straight line is the best fit for that sample. A visual
inspection shows that the Swift data analyzed in this paper
are indeed correlated in the E′pk–Eγ,iso diagram. Taking
into account the uncertainties, no GRB may be classified
as outlier. Eight GRBs show their central point above the
c© 0000 RAS, MNRAS 000, 000–000
10 Cabrera et al.
Figure 4. Correlation between E′
and Eγ,iso (rest frame) as obtained from spectral fits using the CPL model for the class A (red
ellipses) and B (green ellipses) bursts. The ellipses correspond to the joint 68% CL error regions of each data point. The dotted lines
encompass the 3σ scatter of the updated Amati correlation presented in Ghirlanda (2007) for 49 GRBs; the thin solid line is the best
fit for that sample. The thick solid line is the best fit that we find for the Swift data presented here taking into account the correlated
errors, and the dot–dashed line is for the 1σ uncertainty, computed in the barycentre of the data points.
3σ strip, but taking into account the 68% CL contours as
well as the fact that the spectral fitting was performed us-
ing a CPL model instead of the Band one (see below), make
such GRBs compatible with the 3σ strip. Notice that the
5 (marginal) class B events are not outliers. Therefore, we
may include them for a fit along with the 24 class A events.
The thick solid line in Fig. 4 is the best fit that we
find for the 29 class A and B GRBs. The linear fit has been
performed by an iterative minimum χ2 method, where the
coordinate origin of the data is shifted to the barycentre, and
the uncertainty component of each datapoint is calculated
from the corresponding error ellipse in the orthogonal direc-
tion of the straight fitted line. The dot–dashed line curves
describe the standard error of the mean values of the fit
parameters computed in the barycentre of the data points.
The best fit in logarithmic quantities is:
= (2.25±0.01)+(0.54±0.02)Log
Eγ,iso
1052.44erg
We point out the weight of GRB050525 in determining the
best fit due to its very–reduced spectral parameter uncer-
tainties. We have also carried out the linear fit for only the
24 class A GRBs:
= (2.23±0.01)+(0.54±0.02)Log
Eγ,iso
1052.42erg
As expected, the fit is very similar to the one that includes
the 5 class B GRBs.
Thus, a correlation between E′pk and Eγ,iso is confirmed
for 29 long GRBs observed by the Swift satellite and ho-
mogeneously analyzed by us. The values of the power–law
index (α ∼ 0.54) and normalization (K ∼ 100 keV for
Eγ,iso given in units of 10
52erg, E′pk = KEγ,iso
α) corre-
sponding to the best fit, lie in the ranges of values found
with different previous samples (see for a review of previ-
ous studies in Amati 2006a). It is encouraging that with
the Swift data, as analysed here, one obtains an E′pk–Eγ,iso
c© 0000 RAS, MNRAS 000, 000–000
Spectral analysis of Swift GRBs with known z 11
correlation fully consistent with the one obtained for GRBs
observed with other satellite detectors (the so–called Am-
ati relation; Amati et al. 2002, see also Amati 2006a and
Ghirlanda 2007 for the most recent updates), which had a
much broader energy (spectral) range than BAT. The er-
rors in the fit parameters reported in eqs. (5) and (6) are at
the 68% CL. These errors are somewhat larger than those
reported in Ghirlanda (2007) and Amati (2006a; note that
therein the errors are at the 90% CL).
In general, the scatter around the correlation obtained
here is within the range of different previous determinations
of the Amati correlation. The value of the reduced χ2 for
the fit with the 29 class A+B GRBs is χ2ν = 379/27 = 14
close to the values χ2ν = 493/41 = 12 and χ
ν = 530/47 =
11.3 obtained for larger samples (43 and 49 GRBs) by
Ghirlanda et al. (2006) and Ghirlanda (2007), respectively.
We have also estimated the variance of our fit, s2 (see e.g.,
Bevington & Robinson 1992, Chapter 11), by taking into
account the joint variance of the data, i.e. the elliptical CL
contours. For the fit using the 24+5 GRBs, we find s = 0.13
(dof=27). Note that for calculating the deviations, we use
orthogonal distances to the best–fitting line rather than dis-
tances projected in the LogE′pk (Y) axis. An estimate of the
vertical 1σ spread of the data around the best–fitting model
would be σ
LogE′pk
= 0.15, in agreement with previous esti-
mates (see e.g., Amati 2006a).
The virtue of the Swift sample analysed here is its homo-
geneity, in the sense of observations, reduction, and analysis.
Previous samples used to infer the E′pk–Eγ,iso correlation
were constructed from different detectors, with an informa-
tion collected from different authors, who used different data
processing and assumptions. This heterogeneity in the data
should introduce a spurious scatter. In spite that the data
used here have large observational errors, the dispersion of
our fit lies within the range of fits reported previously. This
is namely due to the homogeneity of the sample and be-
cause, having taken into account the whole error ellipses
associated to the data points, it results that the major–axis
orientation of most of these ellipses is close to the direction
of the correlation line.
Examining in more detail Fig. 4, one sees that the nor-
malization of the correlation found here [eq. (5)] is slightly
higher (by ≈ 0.1 dex at the sample median value of Eγ,iso)
than the previously established one. This is indeed expected.
Due to the narrow energy range of BAT, we had to force
the GRB spectra to be fitted by the CPL model, while most
spectra could have been better described by the Band model,
if higher energies had been detected. It is known that when
fitting the CPL model to a spectrum whose shape is ac-
tually described by the Band model, then (i) Epk is larger
than it would otherwise be, and (ii) the low–energy power–
law index α is more negative than it would otherwise be
(Band et al. 1993; Kaneko et al. 2006). On the other hand,
(iii) in the CPL model the νfν spectrum is cut–off exponen-
tially after the break energy E0, while in the Band model
the spectrum decreases as a power law. Item (i) implies di-
rectly that Epk is likely overestimated when the CPL model
is used in the spectral analysis, and items (ii) and (iii) im-
ply that Eγ,iso, which is calculated in the rest–frame energy
range of 1–10000 keV by extrapolating the fitted spectrum,
is likely underestimated.
In order to explore in a statistical sense the effect in
the E′pk–Eγ,iso correlation of using the CPL model instead
of the Band one for the spectral fitting, we carried out the
following experiment. The time–averaged spectra of the 29
long GRBs that had a reasonable CPL–model fit (groups A
and B) were fitted with XSPEC to a Band model, but with
the high–energy power law index β frozen to the typical
value of −2.3. As in the case of CPL, we have modified the
Band model in order to perform the fit for the logarithms
of the normalization and Epk. Then, we followed the same
procedures described in §2 for handling the errors, taking
into account the correlation among them. Fig. 5 shows the
final data points and their 68% CL ellipses in the Epk–Eγ,iso
diagram. As expected, the normalization of the E′pk–Eγ,iso
correlation decreases (with respect to Fig. 4), now being
almost equal to the one of the updated Amati relation given
in Ghirlanda (2007) for the range of Eγ,iso values studied
here. The best fit in logarithmic quantities is:
= (2.23±0.01)+(0.53±0.04)Log
Eγ,iso
1052.61erg
4 SUMMARY AND CONCLUSIONS
We have analysed homogeneously the time–integrated spec-
tra of a relatively large sample of long–duration GRBs with
known redshift obtained by a same detector, the Swift–BAT
instrument. We have carried out spectral fits to 47 GRBs
by using the two–parameter PL and three–parameter CPL
photon models. Due to the narrow spectral range of BAT,
15–150 keV, only two– or three–parameter models can give
reliable fits. However, even in these cases, large uncertain-
ties are expected in the fitted parameters, making the study
of the (highly correlated) errors important. We have per-
formed Monte Carlo simulations in order to propagate the
errors and calculate composite quantities such as E′pk and
Eγ,iso. The main results and conclusions are as follow:
• For 29 GRBs (classes A+B) out of the 47 bursts anal-
ysed here, the CPL model, which identifies a peak energy
Epk in the νfν spectrum, gives reliable fits, though in 5
cases (class B), the best fit value for Epk lies beyond the
upper limit (150 keV) of the BAT energy band. The un-
certainties in the fitted parameters are in most cases highly
correlated among them.
• The spectra of the remaining 18 bursts (classes C+D)
are well fitted by the simple PL model. For 11 bursts (class
C) of the 47 GRBs, the best PL–fit value of the photon index
α is greater than −2, suggesting that Epk > 150 keV. The
other 7 bursts (class D) show a photon index α lower than
−2 which implies a decreasing power–law νfν spectrum in
the analysed energy range. In these cases, the peak energy
could lie in the X–ray range.
• The fluence, Sγ , and rest–frame isotropic equivalent
energy, Eγ,iso, as well as their uncertainties, were calculated
for the 29 (classes A+B) GRBs with reliable fits to the CPL
model. This is currently the largest homogeneous sample of
long GRBs with determined spectral parameters and Eγ,iso.
The mean values of the rest peak energyE′pk and Eγ,iso in
the sample are ∼ 350 keV and ∼ 4× 1052 erg, respectively.
• The Epk as inferred from the CPL fit and αPL as in-
ferred from the PL fit do correlate. However, this correlation
c© 0000 RAS, MNRAS 000, 000–000
12 Cabrera et al.
Figure 5. Same as Fig. 4 but for E′
and Eγ,iso inferred from spectral fits using the Band model with β frozen to −2.3. Note that
using the Band model the correlation shifts to the higher (lower) Eγ,iso (E
) side with respect to the case when the CPL model is
used.
is not physical, but is the result of the attempt of the sin-
gle PL to account for the data points above the peak, with
smaller flux. As Epk decreases, a larger fraction of data lies
above the peak, inducing the single power law to steepen.
Therefore, the correlation should not be used to infer Epk
when knowing only αPL. Indeed, we show that Epk and
αCPL do not correlate.
• A correlation between E′pk and Eγ,iso (the ’Amati’
relation) is confirmed for our sample [Eq. (5)]. This correla-
tion and its scatter are consistent with the ones established
previously for non–Swift bursts, showing that the E′pk–Eγ,iso
correlation is hardly an artifact of selection effects. The zero–
point of our correlation (in Log–Log) is larger by ∼ 0.1 dex
at the sample median value of Eγ,iso than the latest updated
“Amati” correlation (Ghirlanda 2007). This difference is ex-
pected due to the use of a CPL model for describing the
observed spectra instead of the Band model.
• When the Band model with the high–energy photon
index frozen to β = −2.3 is fitted to the spectra of our 29
(classes A+B) GRBs, the zero–point of the resultant E′pk–
Eγ,iso correlation becomes smaller than in the case of the
CPL model. For the Band model, the obtained E′pk–Eγ,iso
correlation is given by Eq. (7), which is virtually indistinct
from the ’Amati’ correlation established previously for un-
even observable data-sets from different satellites.
Although in this work we were able to process ho-
mogeneously the spectra of a significant fraction of the
Swift GRBs with known redshift, the sample is still lim-
ited for conclusive results regarding the physical meaning
of the correlations among the burst observable properties.
During the refereeing process of this work, appeared a pa-
per by Li (2007b), where the author finds evidence of sig-
nificant change in the ’Amati’ correlation parameters with
redshift for the heterogeneous sample of 48 GRBs compiled
by Amati (2006a). The number of GRBs with known red-
shifts and full spectral information obtained homogeneously
should increase in order to attain more conclusive results.
c© 0000 RAS, MNRAS 000, 000–000
Spectral analysis of Swift GRBs with known z 13
ACKNOWLEDGMENTS
JIC and VA-R gratefully acknowledge the hospitality ex-
tended by Osservatorio Astronomico di Brera in Merate dur-
ing research stays. JIC is supported by a CONACyT (Mex-
ico) fellowship. We are grateful to the anonymous Referee for
his/her comments, which helped to correct some bugs in the
processed data and to improve the manuscript. This work
was supported by the PAPIIT–UNAM grant IN107706-3 to
VA-R, and by an italian 2005 PRIN–INAF grant. The au-
thors acknowledge the use of the Swift publicly available
data as well as the public XSPEC software package. We
thank J. Benda for grammar corrections to the manuscript.
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APPENDIX A: THE OBSERVED SPECTRA
In this Appendix section we present the 47 Swift spectra
(see Table 1) analysed here. As described in §2, the 60 energy
channels in the range 15–150 keV of the BAT detector are
used to deconvolve the spectra. The spectral fluxes were
averaged over the integration time, Tint, determined for each
burst from its light curve (see §2). In Figs. A1, A2, and A3,
the CPL model was used to fit the observed spectra.
c© 0000 RAS, MNRAS 000, 000–000
http://arxiv.org/abs/astro-ph/0611189
http://arxiv.org/abs/astro-ph/0702212
http://arxiv.org/abs/astro-ph/0608379
http://arxiv.org/abs/astro-ph/0608282
14 Cabrera et al.
Figure A1. Time–averaged νfν spectra (red error bars) of the Swift GRBs with known redshift included in our sample. Continuous
(green) line: best fit curve with the CPL photon model. The spectral flux of each burst was averaged over its integration time, Tint.
c© 0000 RAS, MNRAS 000, 000–000
Spectral analysis of Swift GRBs with known z 15
Figure A2. Continues Fig. A1.
c© 0000 RAS, MNRAS 000, 000–000
16 Cabrera et al.
Figure A3. Continues Fig. A1.
c© 0000 RAS, MNRAS 000, 000–000
	Introduction
	Spectral analysis
	The sample
	Spectral fits
	Errors from the fit and error propagation
	Results
	Bursts detected also by other instruments
	Peak energy vs spectral index
	Correlation between Epk and E,iso
	Summary and Conclusions
	The observed spectra

  We study the spectral and energetics properties of 47 long-duration gamma-ray
bursts (GRBs) with known redshift, all of them detected by the Swift satellite.
Due to the narrow energy range (15-150 keV) of the Swift-BAT detector, the
spectral fitting is reliable only for fitting models with 2 or 3 parameters. As
high uncertainty and correlation among the errors is expected, a careful
analysis of the errors is necessary. We fit both the power law (PL, 2
parameters) and cut--off power law (CPL, 3 parameters) models to the
time-integrated spectra of the 47 bursts, and present the corresponding
parameters, their uncertainties, and the correlations among the uncertainties.
The CPL model is reliable only for 29 bursts for which we estimate the nuf_nu
peak energy Epk. For these GRBs, we calculate the energy fluence and the rest-
frame isotropic-equivalent radiated energy, Eiso, as well as the propagated
uncertainties and correlations among them. We explore the distribution of our
homogeneous sample of GRBs on the rest-frame diagram E'pk vs Eiso. We confirm a
significant correlation between these two quantities (the "Amati" relation) and
we verify that, within the uncertainty limits, no outliers are present. We also
fit the spectra to a Band model with the high energy power law index frozen to
-2.3, obtaining a rather good agreement with the "Amati" relation of non-Swift
GRBs.

Introduction
	Spectral analysis
	The sample
	Spectral fits
	Errors from the fit and error propagation
	Results
	Bursts detected also by other instruments
	Peak energy vs spectral index
	Correlation between Epk and E,iso
	Summary and Conclusions
	The observed spectra
