The Millisecond Magnetar Central Engine in short GRBs

1.5 MB·English

by  Hou-Jun Lü

PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THE MILLISECOND MAGNETAR CENTRAL ENGINE IN SHORT GRBS
Hou-Jun Lu¨1, Bing Zhang1, Wei-Hua Lei2, Ye Li1, Paul D Lasky3,4
ABSTRACT
One favored progenitor model for short duration gamma-ray bursts (GRBs) is the coalescence of
5 two neutron stars (NS−NS). One possible outcome of such a merger would be a rapidly spinning,
1 strongly magnetized neutron star (known as a millisecond magnetar). These magnetars may be
0 “supra-massive,” implying that would collapse to black holes after losing centrifugal support due to
2 magnetic dipole spin down. By systematically analyzing the Burst Alert Telescope (BAT)-XRT light
curvesofallshortGRBsdetectedbySwift,wetesthowconsistentthedataarewiththiscentralengine
y
model of short GRBs. We find that the so-called “extended emission” feature observed with BAT in
a
some short GRBs is fundamentally the same component as the “internal X-ray plateau” as observed
M
in many short GRBs, which is defined as a plateau in the light curve followed by a very rapid decay.
Based on how likely a short GRB is to host a magnetar, we characterize the entire Swift short GRB
6
sample into three categories: the “internal plateau” sample, the “external plateau” sample, and the
2
“no plateau” sample. Basedon the dipole spin down model, we derive the physicalparameters of the

putative magnetars and check whether these parameters are consistent with expectations from the
]
E magnetar central engine model. The derived magnetar surface magnetic field B and the initial spin
p
H period P0 fall into a reasonable range. No GRBs in the internal plateau sample have a total energy
exceeding the maximum energy budget of a millisecond magnetar. Assuming that the beginning of
.
h therapidfallphaseattheendoftheinternalplateauisthecollapsetimeofasupra-massivemagnetar
p to a black hole, and applying the measured mass distribution of NS−NS systems in our Galaxy, we
- constrainthe neutronstar equationofstate (EOS).The data suggestthatthe NSEOSis closeto the
o
r GM1 model, which has a maximum non-rotating NS mass of MTOV ∼2.37M⊙.
t Subject headings: gamma rays: general- methods: statistical- radiation mechanisms: non-thermal
s
a
[
1. INTRODUCTION withnearbyearly-typegalaxieswithlittlestarformation
3 (Barthelmyetal. 2005;Bergeretal. 2005;Gehrelsetal.
Gamma-ray bursts (GRBs) are classified into cate-
v 2005;Bloomet al. 2006),to have a large offset from the
gories of “long soft” and “shorthard” (SGRB) based on
9 center of the host galaxy (e.g. Fox et al. 2005; Fong et
the observed duration (T ) and hardness ratio of their
8 90 al. 2010), and to have no evidence of an associated SNe
5 prompt gamma-ray emission (Kouveliotou et al. 1993).
(Kann et al. 2011, Berger 2014 and references therein).
2 LongGRBsarefoundtobeassociatedwithcore-collapse
The evidence points toward an origin that does not in-
0 supernovae (SNe; e.g. Galama et al. 1998; Hjorth et al.
volve a massive star. The leading scenarios include the
. 2003; Stanek et al. 2003; Campana et al. 2006; Xu et
1 merger of two neutron stars (NS−NS, Paczy´nski 1986;
al. 2013), and typically occur in irregular galaxies with
0 Eichler et al. 1989) or the merger of a neutron star and
intense star formation (Fruchter et al. 2006). They are
5 a black hole (Paczy´nski1991). For NS−NS mergers, the
likelyrelatedtothedeathsofmassivestars,andthe“col-
1 traditional view is that a BH is formed promptly or fol-
lapsar” model has been widely accepted as the standard
: lowing a short delay of up to hundreds of milliseconds
v paradigm for long GRBs (Woosley 1993; MacFadyen &
i Woosley 1999). The leading central engine model is a (e.g. Rosswog et al. 2003; Rezzolla et al. 2011; Liu et
X al. 2012). Observations of short GRBs with Swift, on
hyper-accretingblack hole (e.g. Pophametal. 1999;Lei
r et al. 2013). Alternatively, a rapidly spinning, strongly the other hand, indicated the existence of extended cen-
a tral engine activity following at least some short GRBs
magnetized neutron star (millisecond magnetar) may be
in the form of extended emission (EE; Norris & Bonnel
formed during core collapse. In this scenario, magnetic
2006), X-ray flares (Barthelmy et al. 2005; Campana
fields extract the rotation energy of the magnetar to
et al. 2006), and, more importantly, “internal plateaus”
power the GRB outflow (Usov 1992; Thompson 1994;
with rapid decay at the end of the plateaus (Rowlinson
Dai & Lu 1998; Wheeler et al. 2000; Zhang & M´esza´ros
et al. 2010, 2013). These observations are difficult to
2001;Metzger et al. 2008,2011;Lyons et al. 2010; Buc-
interpret within the framework of a black hole central
ciantini et al. 2012; Lu¨ & Zhang 2014).
engine, but are consistent with a rapidly spinning mil-
In contrast, short GRBs are found to be associated
lisecond magnetar as the central engine (e.g. Dai et al.
2006; Gao & Fan 2006; Metzger et al. 2008; Rowlinson
1DepartmentofPhysicsandAstronomy,UniversityofNevada
et al. 2010, 2013; Gompertz et al. 2013, 2014).
Las Vegas, Las Vegas, NV 89154, USA; [email protected],
[email protected] About 20% of short GRBs detected with Swift have
2SchoolofPhysics,HuazhongUniversityofScienceandTech- EE (Sakamoto et al. 2011) following their initial short,
nology, Wuhan,430074, China
3MonashCentre forAstrophysics, School of Physics and As- hard spike. Such EE typically has a lower flux than the
tronomy,MonashUniversity,VIC3800, Australia initial spike,but canlast for tens of seconds (e.g. Perley
4School of Physics, University of Melbourne, Parkville, VIC etal. 2009). ThefirstshortGRBwithEEdetectedwith
3010, Australia
2 Lu¨ et al
Swift was GRB 050724,which had a hardspike of T ∼ statistical properties are presented. The implications on
90
3s followed by a soft tail with a duration of ∼150 s in theNSEOSarediscussed. Theconclusionsaredrawnin
the Swift Burst Alert Telescope (BAT; Barthelmy et al. §5 with some discussion. Throughout the paper, a con-
2005) band. The afterglow of this GRB lies at the out- cordance cosmology with parameters H = 71 km s−1
0
skirt of an early-type galaxy at a redshift of z=0.258. It Mpc −1, Ω =0.30, and Ω =0.70 is adopted.
M Λ
is therefore a “smoking-gun” burst of the compact star
merger population (Barthelmy et al. 2005; Berger et al. 2. DATAREDUCTIONANDSAMPLESELECTION
2005). GRB 060614 is a special case with a light curve CRITERIA
characterized by a short/hard spike (with a duration ∼ The Swift BAT and XRT data are downloaded from
5s) followed by a series of soft gamma-ray pulses lasting the Swift data archive.6 We systematically process the
∼100s. Observationally,it belongs to a long GRB with- BAT and XRT GRB data to extract light curves and
out an associated SNe (with very deep upper limits of time-resolvedspectra. WedevelopedanIDLscripttoau-
the SN light, e.g. Della Valle et al. 2006; Fynbo et al. tomatically downloadandmaintain allof the Swift BAT
2006; Gal-Yam et al. 2006). Some of its prompt emis- data. TheHEAsoftpackageversion 6.10,includingbate-
sion properties, on the other hand, are very similar to a convert, batbinevt Xspec, Xselect, Ximage, and the Swift
short GRB (e.g. Gehrels et al. 2006). Through simu- data analysis tools are used for data reduction. The de-
lations, Zhang et al. (2007b) showed that if this burst tailsofthe dataanalysismethod canbe foundin several
were a factor of 8 less luminous, it would resemble GRB previous papers (Liang et al. 2007; Zhang et al. 2007c;
050724 and appear as a short GRB with EE. Norris & Lu¨etal. 2014)inourgroup,andSakamotoetal. (2008).
Bonnell (2006) found a small fraction of short GRBs in We analyze 84 short GRBs observed with Swift be-
theBATSEcatalogqualitativelysimilartoGRB060614. tween 2005 January and 2014 August. Among these, 44
It is interesting to ask the following two questions. Are short GRBs are either too faint to be detected in the
short GRBs with EE different from those without EE? X-ray band, or do not have enough photons to extracta
What is the physical origin of the EE? reasonable X-ray light curve. Our sample therefore only
Swift observations of the X-ray afterglow of short comprises 40 short GRBs, including 8 with EE.
GRBs,ontheotherhand,providesomeinterestingclues. WeextrapolatetheBAT(15-150keV)datatotheXRT
A good fraction of Swift short GRBs exhibit an X-ray band(0.3-10KeV)byassuminga singlepower-lawspec-
plateau followed by a very sharp drop with a temporal trum(seealsoO’Brienetal. 2006;Willingaleetal. 2007;
decay slope of more than 3. The first case was GRB Evans et al. 2009). We then perform a temporal fit to
090515 (Rowlinson et al. 2010). It showed a nearly flat the light curve with a smooth broken power law in the
plateau extending to over 180 s before rapidly falling off rest frame,7
with a decay slope of α∼13.5 Such rapid decay cannot
be accommodated by any external shock model, and so t ωα1 t ωα2 −1/ω
F =F + (1)
theentireX-rayplateauemissionhastobeattributedto 0(cid:20)(cid:18)t (cid:19) (cid:18)t (cid:19) (cid:21)
b b
the internal dissipation of a central engine wind. Such
an “internal plateau” has previously been observed in to identify a possible plateau in the light curve. Here,
some long GRBs (e.g. Troja et al. 2007; Lyons et al. t is the break time, F = F ·2−1/ω is the flux at the
b b 0
2010), but they are also commonly observed in short breaktime t , α andα aredecayindicesbeforeandaf-
b 1 2
GRBs (Rowlinson et al. 2013). These plateaus can be terthebreak,respectively,andωdescribesthesharpness
interpreted as they internalemission of a spinning-down of the break. The larger the ω parameter, the sharper
magnetar which collapses into a black hole at the end of the break. An IDL routine named “mpfitfun.pro” is em-
the plateau (Troja et al. 2007; Rowlinson et al. 2010; ployed for our fitting (Markwardt 2009). This routine
Zhang 2014). performs a Levenberg-Marquardt least-square fit to the
IfmagnetarsareindeedoperatinginsomeshortGRBs, dataforagivenmodeltooptimizethemodelparameters.
then several questions emerge. What fraction of short Since the magnetar signature typically invokes a
GRBshaveamillisecondmagnetarcentralengine? What plateau phase followed by a steeper decay (Zhang &
are the differences between short GRBs with EE and M´esza´ros 2001), we search for such a signature to de-
those that have an internal plateau but no EE? Is the cide how likely a GRB is to be powered by a magnetar.
total energy of the magnetar candidates consistent with Similarto ourearlierwork(Lu¨ &Zhang2014),weintro-
themaximumrotationenergyofthemagnetarsaccording duce three grades to define the likelihood of a magnetar
to the theory? What are the physical parameters of the engine.
magnetar candidates derived from observational data?
How can one use the data to constrain the equation of • Theinternalplateau(Internal)sample: thissample
state (EOS) of neutron stars? is defined by those bursts that exhibit a plateau
This paper aims to address these interesting questions
throughasystematicanalysisofbothSwift/BATandX- 6 http://www.swift.ac.uk/archive/obs.php?burst=1
ray Telescope (XRT) data. The data reduction details 7 Another empirical model to fit GRB X-ray afterglow light
curves is that introduced by Willingale et al. (2007, 2010). The
and the criteria for sample selection are presented in §2.
function was found to be a good fit of the external plateaus of
In §3, the observational properties of short GRBs and longGRBs (e.g. Dainotti etal. 2010), but cannot fit the internal
their afterglows are presented. In §4, the physical pa- plateauswhicharelikelyduetothemagnetarorigin(e.g. Lyonset
rametersoftheputativemagnetarsarederivedandtheir al. 2010). We have tried to use the Willingale function to fit the
datainoursample,butthefits arenotgood. Thisisbecause our
short GRB sample includes a large fraction of internal plateaus.
5 The convention Fν ∝t−αν−β is adopted throughout the pa- We therefore do not use the Willingale function to fit the light
per. curvesinthispaper.
Magnetar central engine in short GRBs 3
followed by a decay with t−2 or steeper than 3. 3. DERIVEDPHYSICALPARAMETERSANDSTATISTICS
The t−2 decay is expected by the magnetar dipole
In this section, we derive physical parameters of the
spin down model (Zhang & M´esza´ros 2001), while
shortGRBsinvarioussamples,andperformsomestatis-
a slope steeper thanthree is anindicationthat the
tics to compare among different samples.
emission is powered by internal dissipation of the
magnetar wind, since essentially no external shock 3.1. EE and internal plateau
model can account for such a steep decay. This Our first task is to investigate whether short GRBs
sample is similar to the “Gold” sample defined by with EE are fundamentally different from those without
Lu¨ & Zhang (2014)8, but with the inclusion of EE. The EE has been interpreted within the magnetar
two GRBs with a t−2 decay following the plateau. model as the epoch of tapping the spin energy of the
ThesetwoGRBs(GRB061201andGRB070714B) magnetar (Metzger et al. 2008;Bucciantini et al. 2012).
alsohaveaplateauindexcloseto0asdemandedby On the other hand, a good fractionof short GRBs with-
the magnetar spin down model, and therefore are out EE have an internal plateau that lasts for hundreds
strong candidates for magnetar internal emission. of seconds, which can also be interpreted as the inter-
For those cases with a post-plateau decay index nal emission of a magnetar during the spin-down phase
steeper than three, the rapid decay at the end of (Troja et al. 2007; Yu et al. 2010; Rowlinson et al.
plateau may mark the implosion of the magnetar 2013; Zhang 2014). It would be interesting to investi-
into a black hole (Troja et al. 2007; Zhang 2014). gate whether or not there is a connection between the
Altogether there are 20 short GRBs identified as two groups of bursts.
having such a behavior, 13 of which have redshift Analyzing the whole sample, we find that the short
measurementsand7ofwhichareshortGRBswith GRBs with EE do not show a plateau in the XRT band
EE.FortheselatterGRBs,the extrapolatedX-ray (except GRB 060614, which shows an external plateau
light curves from the BAT band in the EE phase at a later epoch). Extrapolating the BAT data to the
resemble the internal plateaus directly detected in XRT band, the EE appears as an internal plateau (Fig-
the XRT band in other GRBs. The light curves of ure.1). Fittingthejointlightcurvewithabrokenpower-
these 22 GRBs are presented in Fig.1, along with law model, one finds that there is no significant differ-
the smooth broken power-law fits. The fitting pa- ence in the distribution and cumulative distribution of
rameters are summarized in Table 1. the plateau durations for the samples with and with-
out EE (Figure 4(a)). The probability (p) that the two
• The external plateau (External) sample: this sam-
samples are consistent with one another, as calculated
ple includes those GRBs with a plateau phase fol-
using a student’s t-test, is 0.65.10 Figure 4(b) shows the
lowedbyanormaldecaysegmentwithapost-decay
redshift distributions of those short GRBs in our sam-
indexcloseto-1. Thepre-andpost-breaktemporal
ple which have redshift measurements. Separating the
and spectral properties are consistent with the ex-
sample into EE and non-EE sub-samples does not re-
ternalforwardshockmodelwiththe plateauphase
veal a noticeable difference. In Figure 4(c), we show
due to continuous energy injection into the blast-
the flux distribution of the plateau at the break time.
wave. Thissampleis similarto the SilverandAlu-
It is shown that the distribution for the EE sub-sample
minum samples in Lu¨ & Zhang (2014). We identi- (mean flux logF = −8.74±0.12 ergs s−1 cm−2) is sys-
fied10GRBsinthisgroup.9 TheXRTlightcurves b
tematically higher than that for the non-EE sub-sample
are presented in Figure 2 along with the smooth (mean flux logF =−9.84±0.07 ergs s−1 cm−2). How-
broken power-law fits. The fitting results are pre- b
ever, the combined sample (Figure 4(d)) shows a single-
sented in Table 1.
component log-normal distribution with a mean flux of
• Noplateau(Non)sample: weidentify8GRBsthat logFb = −9.34±0.07 ergs s−1 cm−2, with a student’s
t-test probability of p = 0.76 of belonging to the same
do not have a significant plateau behavior. They
parent sample. This suggests that the EE GRBs are
either have a single power-law decay, or have er-
simplythosewithbrighterplateaus,andthedetectionof
ratic flares that do not present a clear magnetar
EE is an instrumental selection effect. We also calculate
signature.
the luminosity of the internal plateau at the break time
Figure 3 collects all of the light curves of the GRBs in for both the GRBs with and without EE. If no redshift
our samples. The Internal sample with or without EE is is measured, then we adopt z =0.58, the center value
collected in Figures 3(a) and 3(b); the External sample for the measuredredshift distribution (Figure 4(b)). We
(without EE) is collected in Figure 3(c); and the Non find that the plateau luminosity of the EE (logL =
0
sample are collected in Figure 3(d). 49.41±0.07 ergs s−1) is systematically higher than the
no-EEsample(logL =48.68±0.04ergs s−1),seeFigure
8 Lu¨ & Zhang (2014) studied the magnetar engine candidates 0
4(e). However,thejointsampleisagainconsistentwitha
for long GRBs. The grades defined in that paper were based on
thefollowingcriteria. Goldsample: thoseGRBswhichdispalyan singlecomponent(logL0 =48.91±0.07ergs s−1, Figure
“internal plateau”; Silver sample: those GRBs which display an 4(f)), with astudent’s t-test probabilityofp=0.74. For
“external plateau,” whose energy injection parameter q is consis-
thesampleswiththemeasuredredshiftsonly,ourresults
tent with being 0, as predicted by the dipole spin down model of
(shown in the inset of Figure 4(e) and 4(f)), the results
GRBs; Aluminum sample: those GRBs whichdisplay anexternal
plateau,butthederivedqparameterisnotconsistentwith0;Non- are similar.
magnetar sample: thoseGRBs whichdonotshow aclear plateau
feature. 10 The hypothesis that the two distributions are from a same
9 TheSN-lesslongGRB 060614 isincludedinthis category. It parentsampleisstatisticallyrejectedifp<0.05. Thetwosamples
hasEEandanadditional externalplateauatlatetimes. arebelievedtohavenosignificantdifferenceifp>0.05.
4 Lu¨ et al
Thedistributionsoftheplateauduration,flux,andlu- (for details, see Lu¨ & Zhang, 2014). If no redshift is
minositysuggestthattheEEandX-rayinternalplateaus measured, then we use z =0.58 (see Table 2).
are intrinsically the same phenomenon. The different To calculate E , we apply the method described in
K,iso
plateau luminosity distribution, along with the similar Zhang et al. (2007a). Since no stellar wind environment
plateau duration distribution, suggest that the fraction is expected for shortGRBs, we apply a constantdensity
ofshortGRBswithEEshouldincreasewithsofter,more model. One important step is to identify the external
sensitive detectors. The so-called “EE” detected in the shock component. If an external plateau is identified,
BAT band is simply the internal plateau emission when thenitis straightforwardtouse the afterglowflux to de-
the emission is bright and hard enough. rive E . The derived E is constant during the
K,iso K,iso
normal decay phase, but it depends on the time during
3.2. The host offset and local environment of Internal theshallowdecayphase(Zhangetal. 2007a). Wethere-
and External samples fore use the flux in the normal decay phase to calculate
E . FortheNonsample,noplateauisderivedandwe
One curious question is why most (22) short GRBs K,iso
use any epoch during the normal decay phase to derive
haveaninternalplateau,whereassomeothers(10)show
E . For GRBs in the Internal sample, there are two
an external plateau. One naive expectation is that the K,iso
possibilities. (1) In some cases, a normal decay phase is
External sample may have a higher circumburst density
detected after the internal plateau, e.g. GRBs 050724,
thantheInternalsample,andsotheexternalshockemis-
062006,070724A,071227,101219A,and111121AinFig-
sion will be greatly enhanced. It has been found that
ure 1. For these bursts, we use the flux at the first data
short GRBs typically have a large offset from their host
point during the normal decay phase to derive E .
galaxies (Fong et al. 2010; Fong & Berger 2013; Berger K,iso
(2) For those bursts whose normal decay segment is not
2014), so that the local interstellar medium (ISM) den-
observedafter the rapid decay of the internal plateau at
sity may be much lower than that of long GRBs (e.g.
later times (the rest of GRBs in Figure 1), we use the
Fan et al. 2005; Zhang et al. 2009; Kann et al. 2011).
lastdata pointto place anupper limit to the underlying
This is likely due to the asymmetric kicks during the
afterglow flux. An upper limit of E is then derived.
SNeexplosionsofthebinarysystemswhenthetwocom- K,iso
Weadopttwotypicalvaluesofthecircumburstdensity
pact objects (NS or BH) were born (e.g. Bloom et al. to calculate the afterglow flux, n = 1 cm−3 (a typical
1999, 2002). If the circumburst density is the key factor
densityoftheISMinsideagalaxy)andn=10−3cm−3(a
in creating a difference between the Internal and Exter-
typical density in the ISM/IGM with a large offset from
nal samples, then one would expect that the offset from
thegalaxycenter). Forthelateepochswearediscussing,
thehostgalaxyissystematicallysmallerfortheExternal
fast cooling is theoretically disfavored and we stick to
sample than the Internal sample.
the slow cooling (ν < ν ) regime. Using the spectral
With the data collected from the literature (Fong et m c
and temporal information from the X-ray data, we can
al 2010, Leibler & Berger 2010, Fong & Berger 2013,
diagnosethespectralregimeoftheafterglowbasedonthe
Berger 2014), we examine the environmental effect of
closurerelations(e.g. Zhang&M´esza´ros2004;seeGaoet
short GRBs within the Internal and External samples.
al. 2013a for a complete review). Most GRBs belong to
Themasses,ages,andspecificstarformationratesofthe
the ν > max(ν ,ν ) regime, and we use Equations(10)
host galaxies do not show statistical differences between m c
and (11) of Zhang et al. (2007a) to derive E . In
the two samples. The physical offsets and the normal- K,iso
some cases, the spectral regime ν < ν < ν is inferred
ized offsets11 of these two samples are shown in the left m c
and Equation (13) of Zhang et al. (2007a)is adopted to
and right panels of Figure 5. It appears that the ob-
derive E
jects in the External sample tend to have smaller offsets K,iso
Inorderto placeanupper limitofE forthe Inter-
than those in the Internal sample, both for the physical K,iso
nalsampleGRBswithoutadetectedexternalshockcom-
andnormalizedoffsets. Thisisconsistentwiththeabove
ponent,oneneeds to assumethe spectralregimeandde-
theoretical expectation. Nonetheless, the two samples
cayslopeofthenormaldecay. Todoso,weperformasta-
are not well separated in the offset distributions. Some
tistical analysis of the decay slope and spectral index in
GRBs in the External sample still have a large offset.
thenormaldecayphaseusingtheExternalandNonsam-
This may suggest a large local density in the ISM or
ples(Figure6). FittingthedistributionswithaGaussian
intergalactic medium (IGM) far away from the galactic
distribution,weobtaincentervaluesofα =1.21±0.04
center,orthatsomeinternalemissionofthenascentmag- 0,c
andβ =0.88±0.05. Weadoptthesevaluestoperform
netars may have observational signatures similar to the X,c
the calculations. Since 2α ≈ 3β is roughly satisfied,
external shock emission. 0 X
the spectral regime belongs to ν < ν < ν , and again
m c
Equation (13) of Zhang et al. (2007a) is again used to
3.3. Energetics and luminosity
derive the upper limit of E .
K,iso
Similar to Lu¨ & Zhang (2014), we derive the isotropic Inourcalculations,themicrophysicsparametersofthe
γ-ray energy (E ) and isotropic afterglow kinetic en- shocks are assigned to standard values derived from the
γ,iso
ergy (E ) of all of the short GRBs in our sample. To observations(e.g. Panaitescu& Kumar 2002;Yost etal.
K,iso
calculate E , we use the observed fluence in the de- 2003): ǫ =0.1 and ǫ = 0.01. The Compton parameter
γ,iso e B
tector’senergybandandextrapolateittotherest-frame is assigned to a typical value of Y = 1. The calculation
1−104 keV using spectral parameters with k-correction results are shown in Table 2.
After obtaining the break time t through light curve
b
11 Thenormalizedoffsetsaredefinedasthephysicaloffsetsnor- fitting, we derive the bolometric luminosity at the break
malizedtore,thecharacteristic sizeofagalaxy definedbyEqua-
tion(1)ofFongetal. (2010).
Magnetar central engine in short GRBs 5
time t : where I is the moment of inertia, R, P , and Ω are
b 0 0
the radius, initial period, and initial angular frequency
L =4πD2F ·k, (2)
b L b of the neutron star, and M is normalized to the sum
whereF is the X-rayflux att andk is the k-correction of the masses of the two NSs (2.46M⊙) in the observed
b b
factor. For the Internal sample, we derive the isotropic NS−NSbinaries inourGalaxy.12 Hereafter,the conven-
internalplateauenergy,EX,iso,usingthebreaktimeand tion Q = 10xQx is adopted in cgs units for all of the
break luminosity (Lu¨ & Zhang 2014), i.e. parameters except the mass. It is found that the to-
tal energy of the GRBs are below the E line if the
rot
E ≃L · tb (3) medium density is high (n=1 cm−3). This energy bud-
X,iso b 1+z get is consistent with the magnetar hypothesis, namely,
all of the emission energy ultimately comes from the
This energy is also the isotropic emission energy due to
spin energy of the magnetar. For a low-density medium
internal energy dissipation. (n=10−3cm−3),however,afractionofGRBsintheEx-
Comparisonsofthestatisticalpropertiesofvariousde-
ternalsample exceed the total energy budget. The main
rived parameters for the Internal and External samples
reason for this is that a larger E is needed to com-
are presented in Figure 7. Figure 7(a) and (b) show K,iso
pensate a small n in order to achieve a same afterglow
the distributions of the internal plateau luminosity and
flux. IftheseGRBsarepoweredbyamagnetar,thenthe
duration. For the External sample, no internal plateau
data demand a relatively high n. This is consistentwith
is detected and we place an upper limit on the internal
theargumentthattheExternalsamplehasalargenand
plateau luminosity using the observed luminosity of the
so the external shock component is more dominant.
external plateau. The internal plateau luminosity of the
Figure 8(a) shows the observed X-ray luminosity at
Internal sample is L ∼ 1049 ergs s−1. The distribution
b t = 103 s (L ) as a function of the decay slope α .
of the upper limits of L of the External sample peaks t=103s 2
at a smaller value of L ∼b 1047.5 ergs s−1. This suggests Figures 8(b) and 8(c) show the respective distributions
b of L and α . The Internal and External samples
that the distribution of internal plateau luminosity L t=103s 2
b are marked in red and black, respectively. On average,
hasanintrinsicallyverybroaddistribution(Figure7(a)).
the Internalsample hasarelativelysmallerL than
The distribution of the duration of the plateaus for the t=103s
that of the External sample (Figure 8(b)). The fitting
Internal sample peaks around 100 s, which is systemat-
results from the distributions of various parameters are
ically smaller than the duration of the plateaus in the
collected in Table 3.
External sample, which peaks around 103.3 s. In Figure
7(a)and(b),wealsocomparetheplateauluminosityand
duration distributions of our sample with those of long 4. THEMILLISECONDMAGNETARCENTRALENGINE
MODELANDIMPLICATIONS
GRBs (Dainotti et al. 2015) and find that the Internal
sample is quite different from long GRBs, whereas the In this section, we place the short GRB data within
External sample resembles the distributions of the long theframeworkofthemillisecondmagnetarcentralengine
GRBs well. According to our interpretation, the dura- model and derive the relevant model parameters of the
tion of the internal plateaus is defined by the collapse magnetar, and discuss the physical implications of these
time of a supra-massive neutron star (Troja et al. 2007; results.
Zhang 2014). For the external plateaus, the duration of
the plateau is related to the minimum of the spin-down 4.1. The millisecond magnetar central engine model
time and the collapse time of the magnetar. Therefore,
We first briefly review the millisecond magnetar cen-
by definition, the External sample should have a higher
tral engine model of short GRBs. After the coalescence
central value for the plateau duration than the Internal
of the binary NSs, the evolutionary path of the central
sample. The observations are consistent with this hy-
post-merger product depends on the unknown EOS of
pothesis.
the neutron stars and the mass of the protomagnetar,
Figure7(c)and(d)showthedistributionofγ−rayen-
M . If M is smaller than the non-rotating Tolman-
ergy(E )andtheinternaldissipationenergy(E ). p p
γ,iso X,iso Oppenheimer-Volkoff maximum mass M , then the
The E of the Internal sample is a little bit less than TOV
γ,iso magnetarwill be stable in equilibrium state (Cook et al.
that of the External sample, but E is much larger
X,iso 1994;Giacomazzo&Perna2013,Ravi&Lasky2014). If
(for the External sample, only an upper limit of E
X,iso M isonlyslightlylargerthanM ,thenitmaysurvive
can be derived). This means that internal dissipation is p TOV
to form a supra-massive neutron star (e.g. Duez et al.
a dominant energy release channel for the Internal sam-
2006),whichwouldbe supportedby centrifugalforcefor
ple. Figures 7(e) and (f) show the distributions of the
an extended period of time, until the star is spun down
blastwave kinetic energy (E ) for different values of
K,iso enough so that centrifugal force can no longer support
the number density, n = 1 cm−3 and n = 10−3 cm−3.
the star. At this epoch, the neutron star would collapse
In both cases, E of the Internal sample is system-
K,iso into a black hole.
atically smaller than that for the External sample. The
Before the supra-massive neutron star collapses, it
results are presented in Tables 2 and 3.
would spin down due to various torques, the most dom-
In Figures 7(g) and (h) (for n=1,10−3 cm−3, respec-
inant of which may be the magnetic dipole spin down
tively), we compare the inferred total energy of GRBs
(E =E +E +E ) with the total rotation energy
total γ X K 12 Strictlyspeaking,M isnormalizedtothemeanofthesumof
E of the millisecond magnetar:
rot massesofbinaryNSsystems,takingintoaccounttheconservation
1 ofrestmass(Lasky et al. 2014), and ignoringthenegligiblemass
E = IΩ2 ≃3.5×1052 erg M R2P−2 , (4) lostduringthemergerprocess(e.g.,Hotokezaka etal. 2013).
rot 2 0 2.46 6 0,−3
6 Lu¨ et al
(Zhang & M´esza´ros 2001).13 The characteristic spin- the measured L and τ can be directly used to derive
0
down timescale τ and characteristic spin-down luminos- these two parameters. For the Internal sample, both P
0
ity L depend on Ω = 2π/P and the surface mag- andB canbederivedifα =2. Ifα >3,wecanderive
0 0 0 p 2 2
neticfieldatthepoleB ,whichread(Zhang&M´esza´ros theupper limitforP andB . Theresultsarepresented
p 0 p
2001) in Table 2 and Figure 9(a).14
Figure9(b)showsthedistributionofthecollapsetimes
3c3I 3c3IP2
τ= = 0 for our Internal sample. For GRB 061201 and GRB
Bp2R6Ω20 4π2Bp2R6 070714B,thedecayslopefollowingtheplateauisα2 ∼2,
=2.05×103 s (I B−2 P2 R−6), (5) which means that we never see the collapsing feature. A
45 p,15 0,−3 6 lowerlimitofthe collapsetime canbe setbythe lastob-
servationaltime, so that the starsshould be stable long-
IΩ2 lived magnetars. For the collapsing sample, the center
L = 0 =1.0×1049 erg s−1(B2 P−4 R6). (6) value of the t distribution is ∼ 100 s, but the half
0 2τ p,15 0,−3 6 col
width spans about one order of magnitude.
For a millisecond magnetar, the open field line region Figure 10(a) presents an anti-correlation between L
0
opensa verywide solidangle,andsothe magnetarwind and t , i.e.
col
can be approximated as roughly isotropic.
logL =(−2.79±0.39)logt −(0.45±0.28)
Another relevant timescale is the collapse time of a 0,49 col,2
supra-massive magnetar, t . For the Internal sample, (12)
col
the observed break time t either corresponds to t or
b col with r = 0.87 and p < 0.0001. This suggests that a
τ, depending on the post-break decay slope α . If α ≃
2 2 longer collapse times tends to have a lower plateau lu-
2, then the post-break decay is consistent with a dipole
minosity. This is consistent with the expectation of the
spin-down model, so that t is defined by τ and one has
b magnetarcentralengine model. Thetotalspinenergyof
t > τ. On the other hand, if the post-decay slope is
col the millisecond magnetars may be roughly standard. A
steeper than 3, i.e. α >3, then one needs to invoke an
2 stronger dipole magnetic field tends to power a brighter
abrupt cessation of the GRB central engine to interpret
plateau, making the magnetar spin down more quickly,
the data (Trojaetal. 2007;Rowlinsonet al. 2010,2013;
and therefore giving rise to a shorter collapse time (see
Zhang 2014). The break time is then defined by the
also Rowlinson et al. 2014).
collapse time t , and one has t ≤τ. Overall,one can
col col Figure 10(b) presents an anti-correlation between
write
E and t .
total,iso col
=t /(1+z),α =2,
τ b 2 (7) logE =(−1.08±0.27)logt +(0.11±0.18)
(cid:26)≥tb/(1+z),α2 >3. total,iso,52 col,2
(13)
and
with r = 0.71 and p = 0.0009. This may be understood
t >tb/(1+z),α2 =2, (8) as the follows. A higher plateau luminosity corresponds
col(cid:26)=tb/(1+z),α2 >3. to a shorter spin-down timescale. It is possible that in
this case, the collapse time is closer to the spin-down
In both cases, the characteristic spin-down luminosity
timescale, and so most energy is already released before
is essentially the plateau luminosity, which may be esti-
the magnetar collapses to form a black hole. A lower
mated as
plateau luminosity corresponds to a longer spin-down
L ≃L (9) timescale, and it is possible that the collapse time can
0 b
be much shorter than the spin-down timescale, so that
4.2. Magnetar parameters and correlations only a fraction of the total energy is released before the
With the above model, one can derive magnetar pa- collapse.
rameters and perform their statistics. Two important Empirically, (Dainotti et al. 2008, 2010, 2013) dis-
magnetar parameters to define magnetar spin down, i.e. covered an anti-correlation between Lb and tb for long
the initial spin period P and the surface polar cap GRBs. In Figure 10(c) we plot our short GRB Internal
0
magnetic field B , can be solved from the characteristic +Externalsampleandderiveanempiricalcorrelationof
p
plateauluminosity L (Equation(6)) andthe spin-down
0 logL =(−1.41±0.14)logt −(0.46±0.37), (14)
b,49 b,3
timescale τ (Equation (5); (Zhang & M´esza´ros 2001),
i.e. with r = 0.88 and p < 0.001. The slope of the correla-
B =2.05 G(I R−3L−1/2τ−1), (10) tion is slightly steeper than that of the “Dainotti rela-
p,15 45 6 0,49 3 tion” (e.g. Dainotti et al. 2008, data see gray dots in
Fig.10(c)). This is probably related to different progen-
itor systems for long and short GRBs, in particular, the
1/2 −1/2 −1/2
P0,−3 =1.42 s (I45 L0,49 τ3 ). (11) dominanceofInternalplateausinoursample. Rowlinson
Since the magnetar wind is likely isotropic for short
14 Thederivedmagnetar parametersofmostGRBsareslightly
GRBs (in contrast to long GRBs; Lu¨ & Zhang 2014),
differentfromthosederivedbyRowlinsonetal. (2013). Onemain
discrepancyisthattheyusedMp=1.4M⊙ tocalculatetheproto-
13 Deviationsfromthesimpledipolespin-downformulamaybe magnetar’smomentofinertiaI,wheareasweusedMp=2.46M⊙,
expected (e.g. Metzger et al. 2011; Siegel et al. 2014), but the which is more relevant for post-merger products. The different
dipoleformulamaygiveareasonable first-orderapproximationof data selection criteria and fitting methods also contribute to the
thespin-downlawofthenascent magnetar. discrepanciesbetweenthetwoworks.
Magnetar central engine in short GRBs 7
et al. (2014)performeda jointanalysisof both longand Internal sample with redshift measurements. Five NS
short GRBs taking into account for the intrinsic slope equations of state, i.e. SLy (black, Douchin & Haensel.
ofthe luminosity-timecorrelation(Dainottietal. 2013). 2001),APR(red,Akmaletal. 1998),GM1(green,Glen-
We focus on short GRBs only but studied the Internal denning & Moszkowski. 1991), AB-N, and AB-L (blue
and External sub-samples separately. andcyan,Arnett&Bowers. 1997)areshownindifferent
vertical color bands. The gray shaded region is the pro-
4.3. Constraining the neutron star EOS tomagnetarmassdistribution,M ,discussedabove. The
p
horizontal dashed line is the observed collapse time for
The inferred collapsing time can be used to constrain
each short GRB. Our results show that the GM1 model
the neutron star EOS (Lasky et al. 2014; Ravi & Lasky
givesanM bandfallinthe2σregionoftheprotomagne-
2014). The basic formalism is as follows. p
tarmassdistribution, sothat the correctEOSshouldbe
The standarddipole spin-downformulagives(Shapiro
closetothismodel. Themaximummassfornon-rotating
& Teukolsky 1983)
NS in this model is MTOV =2.37M⊙.
4π2B2R6 Lasky et al. (2014) applied the observational collapse
P(t)=P0(1+ 3c3 IpP2 t)1/2 time of short GRBs to constrain the NS EOS (see also
0 a rough treatment by Fan et al. 2013a). Our results
=P (1+ t)1/2. (15) are consistent with those of Lasky et al. (2014), but
0 τ using a larger sample. Another improvement is that we
introduce a range of τ rather than one single τ to derive
ForagivenEOS,amaximumNSmassforanon-rotating
the range of plausible M , since the observed collapse
NS, i.e. M , can be derived. When an NS is supra- p
TOV time only gives the lower limit of τ. This gives a range
massive but rapidly rotating, a higher mass can be sus-
ofthe allowedM (ratherthanafine-tunedvalueforthe
tained. The maximum gravitational mass (M ) de- p
max single τ scenario) for each GRB for a given observed t .
pends on spin period, which can be approximated as b
(Lyford et al. 2003) 5. CONCLUSIONSANDDISCUSSION
M =M (1+αˆPβˆ) (16) In this paper, by systematically analyzing the BAT-
max TOV XRT light curves of short GRBs detected by Swift be-
whereαˆ andβˆdependontheEOS.Thenumericalvalues fore 2014 August, we systematically examine the mil-
lisecond magnetar central engine model of short GRBs.
of αˆ and βˆ for various EOSs have been worked out by
About 40 GRBs have bright X-ray afterglows detected
Lasky et al. (2014), and are presented in Table 4 along
with Swift/XRT, 8 0f which have EE detected with
with M , R, and I.
TOV Swift/BAT.Basedontheexistenceofplateaus,their ob-
As the neutron star spins down, the maximum mass
servation properties, and how likely a GRB is powered
M gradually decreases. When M becomes equal
max max by a millisecond magnetar central engine, we character-
tothetotalgravitationalmassoftheprotomagnetar,M ,
p ized short GRBs into three samples: Internal (plateau),
the centrifugal force can no longer sustain the star, and
External (plateau), and Non (plateau). We compared
so the NS will collapse into a black hole. Using equa-
the statistical properties of our samples and derived or
tionEquation(15)andEquation(16), onecanderivethe
placed limits on the magnetar parameters P and B
collapse time: 0 p
from the data. Using the collapse time t of the proto-
col
tcol=4π32Bc32IR6[(MpαˆM−MTOV)2/βˆ−P02] Imnategrnneatlarsainmfperlere,dwferowmenttheonpltaotecaounsbtrreaaiknttihmeeNtbSiEnOthSe.
p TOV The following interesting results are obtained.
= τ [(Mp−MTOV)2/βˆ−P2]. (17) • At least for the Internal sample, the data seem to
P2 αˆM 0
0 TOV be consistent with the expectations of the mag-
As noted, one can infer B , P , and t from the obser- netar central engine model. Assuming isotropic
p 0 col
emission,thederivedmagnetarparametersB and
vations. Moreover,astheGalacticbinaryNSpopulation p
has a tight mass distribution (e.g., Valentim et al. 2011; P0 fall into a reasonable range. The total energy
Kiziltan et al. 2013), one can infer the expected distri- (sumofEγ,EX,andEK)iswithinthebudgetpro-
vided by the spin energy of the millisecond mag-
bution of protomagnetar masses, which is found to be
Mp = 2.460−.01.315M⊙ (for details see Lasky et al. 2014). ncoertraerla(tEioronti∼sg3e.n5er×al1ly05c2onersgis)t.enTthweitLh0t−hetchoylpaontthi--
The only remaining variables in Equation (16) are re-
esisthatthetotalspinenergyofthemagnetarmay
lated to the EOS, implying that the observations can be
be standard, and a higher dipolar magnetic field
usedto deriveconstraintsonthe EOSofnuclearmatter.
powers a brighter but shorter plateau.
For most GRBs in our Internal sample, only the lower
limit of τ is derived from tb (Equation (7)). One can • The so-calledEEfollowingsome shortGRBs is es-
alsoinferthe maximumτ bylimiting P tothebreak-up
0 sentially the brightest internal plateau commonly
limit. Considering the uncertainties related to gravita-
observed in short GRBs. A more sensitive and
tional wave radiation, we adopt a rough limit of 1 mil-
softer detector would detect more EE from short
lisecond. By doing so, one can then derive a range of τ,
GRBs.
and hence a range of M based on the data and a given
p
EOS. • The External sample may also be consistent with
Figure11presentsthecollapsetime(t )asafunction havingamagnetarcentralengine,eventhoughthe
col
of protomagnetar mass (M ) for each short GRB in the evidence is not as strong. If both the Internal
p
8 Lu¨ et al
and External samples are powered by a millisec- other interesting possibility is that a fast radio burst
ond magnetar central engine, then the difference (e.g. Lorimer et al. 2007; Thornton et al. 2013) may
betweenthetwosamplesmayberelatedtothecir- be released when the supra-massive magnetar collapses
cumburst medium density. The physical and host- into a black hole (Falcke & Rezzolla 2014; Zhang 2014).
normalizedoffsetsoftheafterglowlocationsforthe The discovery of an FRB following a GRB at the end
Internal sample is somewhat larger than those of of the internal plateau (see Bannister et al. 2012)would
the External sample, even though the separation naildowntheoriginofFRBs,althoughsuchobservations
between the two samples is not clear cut. In any require fast telescope response times given the expected
case, it is consistent with this expectation. The distributionofcollapsetimesfollowingSGRBs(seefigure
totalenergybudgetofthe GRBis withinthe mag- 9(b) and Ravi & Lasky 2014). The GRB-FRB associa-
netar energy budget for the External sample only tions, if proven true, would be invaluable for cosmology
iftheambientdensityisrelativelylarge,andhence studies (Deng & Zhang 2014; Gao et al. 2014; Zheng et
powers a strong external shock emission compo- al. 2014 Zhou et al. 2014).
nent. There is no significant difference between Recently,Rezzolla&Kumar(2014)andCiolfi&Siegel
thosetwogroupsforthestarformationrate,metal- (2014) proposed a different model to interpret the short
licity, and age of the host galaxy. GRB phenomenology. In their model, the post-merger
productisalsoasupra-massiveNS,butthecollapsetime
• Usingthecollapsetimeofsupra-massiveprotomag- is allocated as the epoch of the short GRB itself, rather
netar to form a black hole and the distribution of than the end of the Internal plateau. Our conclusions
the total mass of NS−NS binaries in the Galaxy, drawninthispaperdonotapplytothatmodel. Acrucial
one can constrain the NS EOS. The data point to- observationaltesttodifferentiatebetweenourmodeland
wardanEOSmodelclosetoGM1,whichhasanon- theirsiswhetherornotthereexistsstrongX-rayemission
spinning maximum NS mass MTOV ∼2.37M⊙. before the short GRB itself. This may be tested in the
future with a sensitive wide-field XRT.
The short GRB data are consistent with the hypoth-
esis that the post-merger product of an NS−NS merger
is a supra-massive neutron star. The existence of such We thank an anonymous referee for helpful sugges-
a long-lived post-merger product opens some interest- tions, Hui Sun, and Luciano Rezzolla for useful com-
ing prospects in the multi-messenger era. In particular, ments. We acknowledge the use of the public data
the dipole spin-downpowerofthe supra-massiveNS can from the Swift data archive and the UK Swift Science
power bright electromagnetic radiation even if the short Data Center. This work is supported by the NASA
GRB jet does not beam toward earth, and so some in- ADAPprogramundergrantNNX14AF85G[BZ],theNa-
teresting observationalsignatures are expected to be as- tionalNaturalScienceFoundationofChinaundergrants
sociatedwithgravitationalwavesignalsintheAdvanced U1431124, 11361140349(China-Israel jointed program)
LIGO/Virgo era (Fan et al. 2013b; Gao et al. 2013; Yu [WHL], and an Australian Research Council Discovery
et al. 2013; Zhang 2013; Metzger & Piro 2014). An- Project DP140102578[PDL].
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TABLE 1
Observed propertiesof shortGRBsin our samples.
GRB za T90/EEb Γγc βXd HostOffsete HostOffsete tbf α1f α2f χ2/dof Ref
name (s) (kpc) (re) (s)
Internal
050724 0.2576 3/154 1.89±0.22 0.58±0.19 2.76±0.024 — 139±9 0.20±0.1 4.16±0.05 980/835 (1,2)
051210 (0.58) 1.27/40 1.06±0.28 1.1±0.18 24.9±24.6 4.65±4.6 67±4 0.15±0.04 2.96±0.09 118/132 (1,2)
051227 (0.58) 3.5/110 1.45±0.24 1.1±0.4 — — 89±5 0.10±0.05 3.19±0.13 681/522 (3,4,5)
060801 1.13 0.49/N 1.27±0.16 0.43±0.12 — — 212±11 0.10±0.11 4.35±0.26 81/75 (1)
061006 0.4377 0.5/120 1.72±0.17 0.76±0.28 1.3±0.24 0.35±0.07 99±7 0.17±0.03 9.45±1.14 111/138 (1,2)
061201 0.111 0.76/N 0.81±0.15 1.2±0.22 32.47±0.06 14.91±0.03 2223±43 0.54±0.06 1.84±0.08 20/24 (1,6)
070714B 0.9224 3/100 1.36±0.19 1.01±0.16 12.21±0.53 5.55±0.24 82±2 0.10±0.07 1.91±0.03 672/581 (1,6)
070724A 0.46 0.4/N 1.81±0.33 0.5±0.3 5.46±0.14 1.5±0.04 77±6 0.01±0.1 6.45±0.46 256/222 (1,6)
071227 0.381 1.8/100 0.99±0.22 0.8±0.3 15.5±0.24 3.28±0.05 69±8 0.27±0.08 2.92±0.06 244/212 (1,6)
080702A (0.58) 0.5/N 1.34±0.42 1.03±0.35 — — 586±14 0.51±0.22 3.56±0.31 3/5 (7)
080905A 0.122 1/N 0.85±0.24 0.45±0.14 17.96±0.19 10.36±0.1 13±3 0.19±0.09 2.37±0.07 43/52 (6,7)
080919 (0.58) 0.6/N 1.11±0.26 1.09±0.36 — — 340±26 0.40±0.14 5.20±0.55 7/5 (7)
081024A (0.58) 1.8/N 1.23±0.21 0.85±0.3 — — 102±5 0.27±0.02 5.89±0.3 50/42 (7)
090510 0.903 0.3/N 0.98±0.21 0.75±0.12 10.37±2.89 1.99±0.39 1494±87 0.69±0.04 2.33±0.11 112/132 (6,7)
090515 (0.58) 0.036/N 1.61±0.22 0.75±0.12 75.03±0.15 15.53±0.03 178±3 0.10±0.08 12.62±0.5 42/38 (6,7)
100117A 0.92 0.3/N 0.88±0.22 1.1±0.26 1.32±0.33 0.57±0.13 252±9 0.55±0.03 4.59±0.13 84/92 (6,7,8)
100625A 0.425 0.33/N 0.91±0.11 1.3±0.3 — — 200±41 0.26±0.44 2.47±0.18 3/6 (7,9)
100702A (0.58) 0.16/N 1.54±0.15 0.88±0.11 — — 201±6 0.62±0.13 5.28±0.23 82/69 (7)
101219A 0.718 0.6/N 0.63±0.09 0.53±0.26 — — 197±10 0.13±0.19 20.52±8.01 3/5 (7)
111121A (0.58) 0.47/119 1.66±0.12 0.75±0.2 — — 56±9 0.10±0.13 2.26±0.04 274/289 (7)
120305A (0.58) 0.1/N 1.05±0.09 1.4±0.3 — — 188±14 0.73±0.14 6.49±0.63 14/18 (7)
120521A (0.58) 0.45/N 0.98±0.22 0.73±0.19 — — 270±55 0.30±0.27 10.74±4.76 3/7 (7)
External
051221A 0.55 1.4/N 1.39±0.06 1.07±0.13 1.92±0.18 0.88±0.08 25166±870 0.12±0.13 1.43±0.04 52/63 (1,2,7)
060313 (0.58) 0.71/N 0.71±0.07 1.06±0.15 2.28±0.5 1.23±0.23 2294±65 0.3±0.15 1.52±0.04 54/45 (1,2,7)
060614 0.1254 5/106 2.02±0.04 1.18±0.09 — — 49840±3620 0.18±0.06 1.9±0.07 70/54 (1,10)
070714A (0.58) 2/N 2.6±0.2 1.1±0.3 — — 892±34 0.11±0.09 0.95±0.06 15/18 (7)
070809 0.219 1.3/N 1.69±0.22 0.37±0.21 33.22±2.71 9.25±0.75 8272±221 0.18±0.06 1.31±0.17 17/22 (6,7)
080426 (0.58) 1.7/N 1.98±0.13 0.92±0.24 — — 566±97 0.11±0.16 1.29±0.05 28/21 (7)
090426 2.6 1.2/N 1.93±0.22 1.04±0.15 0.45±0.25 0.29±0.14 208±53 0.12±0.07 1.04±0.04 15/11 (6,7)
100724A 1.288 1.4/N 1.92±0.21 0.94±0.23 — — 5377±331 0.72±0.08 1.61±0.12 16/19 (11,12)
130603B 0.356 0.18/N 1.83±0.12 1.18±0.18 5.21±0.17 1.05±0.04 3108±356 0.4±0.02 1.69±0.04 126/109 (6,13,14)
130912A (0.58) 0.28/N 1.21±0.2 0.56±0.11 — — 231±54 0.04±0.39 1.34±0.04 28/21 (15)
References. — (1)Zhang et al.(2009),(2)Fong et al.(2010),(3)Hullinger et al.(2005),(4)Butler et al.(2007),(5)Gompertz et al.(2014),(6)Fong &
Berger(2013),(7)Rowlinson et al.(2013),(8)Fong et al.(2011),(9)Fong et al.(2013),(10)Lu¨ & Zhang.(2014),(11)Thoene et al.(2010),(12)Markwardt et
al.(2010),(13)Cucchiaraetal.(2013),(14)Barthelmyetal.(2013),(15)Krimmetal.(2013).
a
ThemeasuredredshiftarefromthepublishedpapersandGNCs.Whentheredshiftisnotknown,0.58isused.
b
ThedurationoftheGRBwithoutandwithextendedemission(ifEEexists). “N”denotesnoEE.
c
ThephotonindexintheBATband(15-150keV)fittedusingapowerlaw.
d
Thespectralindexoftheabsorbedpower-lawmodelforthenormalsegments.
e
Physicalandhost-normalizedoffsetsfortheshortGRBswithHubbleSpaceTelescope(HST)observations.
f
Thebreaktimeofthelightcurvesfromourfitting,α1andα2arethedecayslopesbeforeandafterthebreaktime.