The Energy-Dependence of GRB Minimum Variability Timescales

1.9 MB·English

by  V. Zach Golkhou

Accepted to ApJ
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THE ENERGY-DEPENDENCE OF GRB MINIMUM VARIABILITY TIMESCALES
V. Zach Golkhou1,2 , Nathaniel R. Butler1,2 & Owen M. Littlejohns1,2
Accepted to ApJ
ABSTRACT
We constrain the minimum variability timescales for 938 GRBs observed by the Fermi/GBM in-
strument prior to July 11, 2012. The tightest constraints on progenitor radii derived from these
timescalesareobtainedfromlightcurvesinthehardestenergychannel. Inthesofterbands–orfrom
measurements of the same GRBs in the hard X-rays from Swift – we show that variability timescales
5 tendtobeafactor2–3longer. Applyingasurvivalanalysistoaccountfordetectionsandupperlimits,
1
we find median minimum timescale in the rest frame for long-duration and short-duration GRBs of
0
45 ms and 10 ms, respectively. Fewer than 10% of GRBs show evidence for variability on timescales
2
below 2 ms. These shortest timescales require Lorentz factors >400 and imply typical emission radii
∼
n R ≈ 1×1014 cm for long-duration GRBs and R ≈ 3×1013 cm for short-duration GRBs. We discuss
u implications for the GRB fireball model and investigate whether GRB minimum timescales evolve
J
with cosmic time.

8 Subject headings: gamma rays: bursts — methods: statistical — Gamma-rays: general

]
E
1. INTRODUCTION function of timescales are required.
H A variety of time series analyses have previously been
Gamma-RayBursts(GRBs)arethemostluminousex-
. plosions in the Universe, originating at cosmological dis- used to explore the rich properties of prompt GRB light
h tances and releasing ∼1051 ergs over timescales of sec- curves. These include structure function (SF) analyses
p
(Trevese et al. 1994; Hook et al. 1994; Cristiani et al.
- onds to tens of seconds. The gargantuan energy release
o is accompanied by a very rapid and stochastic tempo- 1996; Aretxaga et al. 1997), autocorrelation function
r ral variability in the gamma-ray emission. The Swift (ACF) analyses (Link et al. 1993; Fenimore et al. 1995;
t in’t Zand & Fenimore 1996; Borgonovo 2004; Chatterjee
s (Gehrels et al. 2004) and Fermi Space Telescopes (Mee-
a gan et al. 2009) have deepened immensely our under- et al. 2012), and Fourier power spectral density (PSD)
[ analyses (Beloborodov et al. 2000; Chang 2001; Abdo
standing of these cosmological beacons (e.g., Gehrels &
et al. 2010; Guidorzi et al. 2012; Dichiara et al. 2013).
2 Razzaque 2013).
Compared to power-spectral analyses, the SF approach
v The pulses observed in prompt GRB light curves of-
is less dependent on the time sampling (Paltani 1999).
8 ten have a Fast Rising Exponential Decay (FRED) pro-
In, Golkhou & Butler (2014), Paper I hereafter, we de-
4 file (Norris et al. 1996). The time profiles can have a
veloped and applied a fast (i.e. linear) and robust SF
9 broadmorphologicaldiversityinboththenumberofand
estimator, based on non-decimated Haar wavelets, to
5 duration of these pulses. In the external shock model
0 for GRBs, shells of material produced by the GRB im- measure the minimum variability timescale, ∆tmin, of
1. pact material in the circumburst medium (e.g., Rees Swift GRBs. We used the first-order SF of light curves
as measured by the Swift Burst Alert Telescope (BAT;
0 & Meszaros 1992). Unless the circumburst medium is
Barthelmy et al. 2005) to infer the shortest timescale at
5 highly-clumped(Fenimoreetal.1999),thisprocesstends
which a GRB exhibit uncorrelated temporal variability.
1 to produce a smooth GRB light curve in contrast to
One limitation of the work presented in Paper I is
: the rapid temporal variability observed in many GRBs.
v thatweonlyconsiderthevariabilitytimescaleusinglight
Under the internal shock mechanism (Rees & Meszaros
i curves measured over the narrow 15–350 keV energy
X 1994), a variable central engine emits a relativistic out-
band of Swift/BAT. A fixed and narrow energy band
flow comprised of multiple shells with different Lorentz
r in the observer frame would probe different regions of
a factors, Γ. As faster shells collide with slower shells, ki-
the intrinsic GRB spectra, because GRBs are known to
netic energy is converted to radiation, and multiple shell
occur over a wide range of redshifts (see e.g. Salvaterra
collisions can lead to a complex GRB light curve (e.g.,
et al. 2009; Tanvir et al. 2009; Cucchiara et al. 2011;
Rees & Meszaros 1994).
Jakobssonetal.2012). Previousstudieshaveshownthat
Traditional duration measures such as T (Kouve-
90 GRBpulsesvaryindurationasafunctionofenergy,with
liotouetal.1993),whichdescribesthetimeduringwhich
harder energy channels having a lower observed dura-
the central 90% of prompt gamma-ray counts are re-
tion (Fenimore et al. 1995; Norris et al. 1996). Working
ceived, only describe bulk emission properties of the
at higher energies – where pulses are narrower – also
burst. Such a duration does not capture information
has the potential to provide tighter limits on variability
concerning individual collisions between shells. Instead,
timescales.
detailed temporal analyses that probe variability over a
We wish to use the broad Fermi Gamma-ray Burst
1School of Earth and Space Exploration, Arizona State Uni- Monitor (GBM;Meegan et al. 2009) energy coverage to
versity,Tempe,AZ85287,USA overcome this limitation and to effectively standardize a
2Cosmology Initiative, Arizona State University, Tempe, AZ measure of the minimum variability timescale by study-
85287,USA
2 Golkhou, Butler & Littlejohns
ingtheenergyevolutionand/orevaluatingtheminimum complex than linear.
timescale in a fixed rest frame bandpass. Broad en- Weanalyzethebackground-subtractedburstcountsin
ergy coverage can potentially also allow us to disentan- the full T region following the procedure outlined in
100
gle the role the ejecta velocity plays in relating radius PaperI.Onechangeismadetothealgorithmtooptimize
to minimum timescale and to understand how minimum forthedetectionofsignalvariationsonshorttimescales:
timescales measured for different instruments should be instead of re-binning the 200 µs light curve to a fixed
compared (see, e.g., Sonbas et al. 2014). Also, it is im- S/N per bin, we weight the unbinned light curve by the
portant to note that the GBM provides very fine time denoised (following, Kolaczyk 1997) signal. This zeros-
resolution (2µs) event mode data for the full GRB and out portions of the light curve containing no signal and
not just the first 1–2 s as was the case for BATSE (e.g., permits use of the full T region without adversely af-
100
Walker et al. 2000). fecting our ability to identify variations on much shorter
In the discussion below, we begin with a brief appli- timescales.
cation and summary of the method outlined in detail in For 109 bursts in the second Fermi/GBM GRB cat-
Paper I. We then investigate how ∆t depends on en- alog which also have Swift high-energy prompt cover-
min
ergy for a large sample of Fermi/GBM GRBs (Section age, the Swift/BAT data were obtained from the Swift
3.1). Wecompare∆tminestimatesfromSwift andFermi Archive4. Using calibration files from the 2008-12-17
for bursts detected in common to demonstrate stability BATdatarelease,weconstruct100µslightcurves,inthe
and accuracy of error estimates (Section 3.2). We then full15–350keVBATenergyrange. Weusethestandard
use spectral hardness to standardize the ∆tminestimate Swift software tools: bateconvert, batmaskwtevt
(Section3.4)andconcludebyderivingconstraintsonthe and batbinevt. Further details regarding the extrac-
sample Lorentz factors and emission radii (Section 3.5) tionoftheSwift/BATlightcurvescanbefoundinPaper
and by investigating potential evolution of ∆tminwith I.
cosmic time (Section 3.6).
3. DISCUSSIONANDRESULTS
2. DATA In Paper I, we demonstrate the power of a novel,
We consider 949 GRBs published in the second wavelet-based method – the Haar-Structure Function
Fermi/GBM GRB catalog (von Kienlin et al. 2014), (denoted σ ) – to robustly extract the shortest vari-
X,∆t
spanning the first four years of the Fermi mission (be- abilitytimescaleofGRBsdetectedbySwift/BAT.Inthis
tween 2008 July 14th and 2012 July 11th, inclusive). work, we implement our technique on GRBs detected
Eventlistsfor942oftheseburstsweredownloadedfrom by the Fermi/GBM instrument, which is sensitive to a
the online Fermi/GBM burst catalog3. much broader range of energies. We obtain constraints
We analyze the Fermi/GBM Time-Tagged Event on the minimum variability timescales for 938 of 949
(TTE) data for each of the 12 sodium iodide scintilla- GRBs reported in the second Fermi/GBM GRB cata-
tors. We only consider those detectors in which each log (von Kienlin et al. 2014). Of these, we are able to
GRB was brightest, as listed in column 2 of Table 7 in confirm the presence of a linear rise phase (see Section
von Kienlin et al. (2014). Typically, this entails using 3.1) in the Haar-Structure Function on short timescales
event lists for three detectors for each GRB. Following for 528 GRBs. We quote upper-limit values for the re-
MacLachlanetal.(2013),weextract200µsbinnedlight mainder. Most(421)oftheburstsinthissub-sampleare
curves in the full (8 keV – 1 MeV) energy range. We long-duration (T90 > 3 s) GRBs. In this sub-sample, 24
also extract light curves in four energy channels of an GRBs have measured redshift, z. The temporal speci-
equal logarithmic width (8–26, 26–89, 89–299 and 299– fications of all 938 GRBs discussed here are determined
1000 keV). These channels are referred to as channels 1, usingfully-automaticsoftwareandarepresentedinTable
2, 3, and 4 below. 2.
To remove background counts from the Fermi/GBM
3.1. Studying the Energy-Dependence of ∆t
we employ a two-pass procedure. Using the estimates of min
T fromTable7ofvonKienlinetal.(2014),webineach Ithasbeenrecognizedfordecades(e.g.,Fenimoreetal.
90
lightcurveataresolutionofT /100andfitalinearback- 1995; Norris et al. 1996) that a defining feature of GRB
90
ground model. The background is initially determined emission is a narrowing of pulse profiles observed in in-
considering two regions of each light curve, both T in creasingly higher energy bands. As a result, durations
90
length, occurring immediately before and after the iden- measured by different instruments can be different (e.g.,
tified period of burst emission. Using the background Virgili et al. 2012). Durations also appear to depend on
subtracted light curve, we then estimated T by accu- redshift, perhaps as a result of the dependence on band-
100
mulatingafurther5%oftheT intervalcountsoutward pass: recently, Zhang et al. (2013) have found evidence
90
from both the beginning and end of T90. The second that T90 duration – when z is known and used to evalu-
pass at fitting a linear background is then conducted, ate the GRB duration in a fixed rest frame energy band
masking out all bins included in the total T region. –maycorrelatelinearlywithredshiftasisexpectedfrom
100
Thissecondbackgroundfitisthenscaledtosubtractthe cosmological time dilation. This result is quite sensitive
predicted background counts in the fine-time-resolution to the particular choice of binning in the analysis (see,
light curve. Our analysis – which identifies variations on Littlejohns & Butler 2014). Here, we seek to understand
timescales short compared to the overall burst durations whether our measure of shortest duration in GRBs is
–doesnotrequirethefittingofbackgroundmodelsmore also highly-dependent upon the observed energy band,
and on the instrument detecting the GRB, in particular.
3 http://heasarc.gsfc.nasa.gov/W3Browse/fermi/
fermigbrst.html 4 ftp://legacy.gsfc.nasa.gov/swift/data
GRB Minimum Timescales 3
The prompt GBM Gamma-ray light curve for
GRB 110721A, split in 4 energy bands, and our derived 5000
σ curve for each channel are shown in Figure 1.
X,∆t
There is a clear evolution in ∆tminwith bandpass, de- ) 3500
creasingfromthesoftesttothehardestenergyband. To c 4000
e
guide the eye, several lines of constant σX,∆t ∝ ∆t are s
/
also plotted. The expected Poisson level (i.e., measure- 1
ment error) has been subtracted away, leaving only the ( 3000 2000
e
flux variation expected for each channel. t
a
Briefly, we review here how our ∆t is identified. A R
general feature observed in our GRBmsincaleograms, pro- t 2000
n
vided there is sufficient signal-to-noise ratio (S/N), is a u
o 0
linear rise phase relative to the Poisson noise. Poisson C
1000
noise sets a floor on the shortest measurable timescale
0.0.0.0.0.
(denoted ∆tS/N, with ∆tS/N ≈ 0.1 s for channels 2 0015304560
and 3 in Figure 1, bottom). Unlike previous studies 0
by other authors (Bhat 2013; MacLachlan et al. 2013;
0 5 10 15 20 25 30
Walker et al. 2000), we do not implicate the shortest ob-
Time Since Trigger (sec)
servable timescale as ∆t . Instead, we recognize that
min
pulses can be temporally smooth on short timescales.
The departure from this smoothness creates a break in 101
the scaleogram, and this in turn defines our timescale Ch1 ∆t
∆t for temporally un-smooth variability. Naturally, Ch2 min
min
this timescale also corresponds to a length-scale, which Ch3
mustbereconciledwithGRBprogenitormodels(Section ∆t Ch4
X,
3.5). σ

We now focus on the softest energy band of n
o
GthRouBgh11t0h7e2r1eAi,sdeexncoetsisngsigtnhaellpigrhetsecnutrvoen atsimXes(cta).lesAals- ati 100
short as ∆t = 0.4 s (Figure 1 - channel 1), these ari
timescales correspond to a region of the plot where V

the first order SF rises linearly timescale, σ ≡ x
X,∆t u
(cid:10)|X(t+∆t)−X(t)|2(cid:11)1/2 ∝ ∆t. (Here, (cid:104).(cid:105) denotes Fl
t t
an average over time t.) We interpret this linear rise
as an indication that the GRB exhibits temporally-
smooth variations on these timescales (i.e., X(t+∆t)≈ 10-1
X(t)+X(cid:48)(t)∆t), while changing to exhibit temporally- 10-1 100 101
unsmooth variations on longer timescales. The σX,∆t ∆t (sec)
points deviate significantly from the σ ∝ ∆t curve
X,∆t
at∆tmin =0.56±0.09s. Thisisthetimescaleofinterest, Fig. 1.— Top panel: Fermi/GBM light curves of the
describingtheminimumvariabilitytimeforuncorrelated GRB 110721A split in 4 different energy bands. Bot-
variations in the GRB. This timescale is associated with tom panel: The Haar wavelet scaleogram σ , rescaled
X,∆t
theinitialriseoftheGRBinthischannel,ascanbeseen for plotting purposes, corresponding to each channel ver-
from the Figure 1 inset. sus timescale ∆t for GRB 110721A. We derive minimum
The value for ∆t is found by fitting a broken pow- timescales(markedwithgreencircles)–0.56±0.09s,0.28±
min
erlawtotheσ datapointsbelowthepeak,assuming 0.05 s, 0.24±0.04 s, and 0.22±0.04 s for the channels 1, 2,
X,∆t 3,and4,respectively–whichincreaseinlowerenergybands.
that σ initially rises linearly with ∆t (see, also, Pa-
X,∆t In the top panel, the inset displays the the pulse rise with
per I) until flattening at ∆t . Uncertainties quoted
min finertimebinning,withdashedlinesdroppedontothex-axis
hereandbelowfor∆t aredeterminedbydirectprop-
min to demark the derived ∆t values for each channel.
min
agation of errors and correspond to 1σ confidence. If
the lower-limit on ∆t falls below the lowest measur-
min
abletimescale(i.e., ∆tS/N), wereportonlythe1σ upper cerned here with those longer timescale structures, al-
limit for ∆tmin. though we do note that σX,∆t provides a rich, aggregate
Forthisparticularburst, ∆tminevolvesfromthehard- description of this temporal activity.
est energy band to the softest energy band as one might In order to characterize and measure the average
expect: the softest energy band of a burst has longer ∆t for the Fermi sample as a function of spectral en-
min
minimum variability timescale compared to the hard- ergy band, we utilize the Kaplan-Meier (KM; Kaplan
est energy band of that burst. On timescales longer & Meier 1958, see also Feigelson & Nelson 1985) sur-
than ∆tmin, σX,∆t is flatter than σX,∆t ∝∆t, indicating vival analysis. This is necessary because many bursts
the presence of temporally-variable structure on these only permit upper limit measurements of ∆t . Fig-
min
timescales. On a timescale of about 6 s, σX,∆t begins ure2summarizeshowtheminimumvariabilitytimescale
turningoveraswereachthetimescales(tensofseconds) varies withenergyband. TheKMcumulative plots–in-
describingtheoverallemissionenvelope. Wearenotcon- cluding the shaded 1σ error region – for each bandpass
4 Golkhou, Butler & Littlejohns
TABLE 1
The Kaplan-Meier median and 10th percentile
timescales for long-duration GRBs.
Band ∆t50% ∆t10% Number Number
min min
(keV) (msec) (msec) Detected UpperLimit
8–26 540±67 25±8 395 307
26–89 260±26 4+4 431 319
−2
89–299 150+23 2+2 413 335
−18 −1
299–1000 72+24 ... 156 278
−21
8–1000 130±18 2+5 421 334
−1
andthefull(allchannelscombined)Fermi/GBMenergy
range are shown in the top panel. The sample 50th per-
centiles (i.e., medians) and the lowest 10th percentiles
(shownwiththedotted-linesinthetoppanelofFigure2)
areplottedinthebottompanel. Table1summarizesthe
corresponding values. Since the KM cumulative estima-
tioncurveofchannel4doesnotcrossthe10%limitline,
there is no value reported in Table 1 for this case. The
reported values clearly show the tendency of increasing
∆t with decreasing energy band. Because we tend to
min
find a clear association between ∆t and the rise time
min
of the shortest GRB pulse (also, Paper I), this confirms
that GRB pulse structures are narrower at higher en-
ergy and that understanding this effect is important for
understanding any implications drawn from ∆t .
min
The KM median values of ∆t versus en-
min
ergy band are well-fitted by a line ∆t50% =
min
0.20(E/89keV)−0.53±0.06 s (with reduced χ2 = 0.64).
The derived power-law index here is in agreement with
the power-law index of the relationship found for the
average pulse width of peaks as a function of energy
(Fenimore et al. 1995 and also from Norris et al. 1996).
The KM estimation of the lowest 10% of ∆t values
min
versustheenergybandcanalsobefittedbyapower-law,
with a steeper index, ∆t10% =0.01(E/48keV)−0.97±0.20
min
s (with reduced χ2 = 1.4). The steeper index indicates
that rare GRBs, which tend to be bright and spectrally Fig. 2.— Top panel: The KM cumulative estimation curve
hard GRBs, allow for tighter constraints on minimum of all long-duration GRBs in Fermi sample for each energy
timescales. This shifts the typical minimum timescales bandincludingshaded1σ regionaroundeachbandpass. The
tosmallervaluesascomparedtothosefoundforthebulk dottedlinesshowthe50th percentileandthelowest10th per-
of the population. We explore the minimum timescale centile for each bandpass. The location of the of measured
dependence on S/N and spectral hardness below for ∆tmin values (top tics) and upper-limits (bottom tics of the
individual GRBs. same color) are shown in the sub-panel. Bottom panel: The
KMmedianestimationof∆t versusenergybandandthe
min
3.2. Consistency in the Joint Fermi/GBM and lowest10th percentileof∆tminvaluesversusenergyband,in-
cluding error bars. Note: since the KM cumulative estima-
Swift/BAT Sample
tion curve of channel 4 does not cross the 10% line, we plot
In Paper I, we studied the robustness of our mini- an upper-limit.
mum timescales extracted for simulated bursts as the
S/N is varied. It was demonstrated that the shapes of study this behavior. In addition to allowing us to verify
the σ curves are highly stable as the S/N is strongly consistency in the ∆t estimates for bursts with sim-
X,∆t min
decreased (factor of ten), but the determination of the ilar S/N values, we can also directly observe (in many
true ∆t can be challenging. This is because GRBs cases) the reliability of ∆t for different S/N values.
min min
tend to show evidence for temporally-smooth variation Figure 3 captures the variety of scaleograms pro-
between timescales of non-smooth variability (e.g., pulse duced for bursts detected by both the Swift/BAT and
rise times) – which become harder to measure as S/N is Fermi/GBM instruments. Here we utilize the 15–350
decreased – and the longer timescales associated with keVenergyrangeforbothSwift/BATandFermi/GBM,
non-smooth variability (e.g., the full duration of the andwealignthelightcurvesandextractcountsoverthe
pulse). The sample of bursts detected jointly by both same time intervals for each burst. Although the instru-
Swift/BAT and Fermi/GBM provides a rich dataset to mentsdonothaveidenticaleffectiveareacurvesinthese
GRB Minimum Timescales 5
100 GRB110213A 100 GRB080916A 100 GRB120119A
10-1
on σX,∆t ∆t∆βtt=Sm/0iNn.=2=400 s..4103 ss ∆t∆βtt=Sm/0iNn.=8=410 s..0692 ss 10-1 ∆t∆βtt=Sm/0iNn.=1=800 s..2008 ss
ariati100 10-1 100 101 100 100 101 100 10-1 100 101 102
V
x
u
Fl
10-1
∆tβt=S/0N.4=20 s.26 s 10-1 ∆tβt=S/0N.6=10 s.37 s ∆tβt=S/0N.1=70 s.08 s
10-1 ∆tmin=0.43 s ∆tmin=0.84 s ∆tmin=0.20 s
10-1 100 101 10-1 100 101 10-1 100 101 102
∆t (sec)
3.5 0.8 2.5
ec) 3.0 Fermi 0.7 Fermi 2.0 Fermi
s 2.5 Swift 0.6 Swift Swift
1/ 0.5 1.5
( 2.0
e 0.4
t 1.5 1.0
a 0.3
R 1.0
t 0.2 0.5
un 0.5 0.1
Co 0.0 0.0 0.0
0.5 0.1 0.5
0 10 20 30 40 50 0 10 20 30 40 50 60 0 20 40 60 80
Time Since Trigger (sec)
Fig. 3.—AgalleryofHaarscaleogramsσ ,representingavarietyofpossiblestructurefunctionscalculatedforFermi/GBM
X,∆t
and Swift/BAT (both: 15–350 keV) with different level of sensitivity for detection of various GRBs. The left, middle, and
rightpanelscorrespondtoGRB110213A,GRB080916A,andGRB120119A,respectively. Thefirstandsecondrowsshowthe
structurefunctionsretrievedfromtheGRBslightcurvesdetectedbyFermi/GBMandSwift/BAT,respectively. Thethirdrow
showsthelightcurvesintheT durationregion. Ineachofthese,thereddashed-linesrepresentapassagefromthetemporally-
100
smooth ( σ ∝∆t) region to a flatter region and the red circle marks the extracted minimum variability timescale, ∆t ,
X,∆t min
after which the light curves transition to a temporally-unsmooth behavior. Triangles denote 3σ upper limits.
ranges, choosing the same energy range should minimize minimum timescale in a more robust (although not-
differences due to energy band (discussed in more detail perfect, as we discuss more below) fashion.
in Section 3.4 below). Figure 4 displays a scatter plot of ∆t determined
min
InthecaseofGRB110213A(leftpanels),Fermi/GBM for Swift/BAT versus Fermi/GBM. A line fit through
captured the higher sensitivity burst light curve. Op- the data points (blue curve with shaded gray 90% con-
positely in the case of GRB 080916A (middle panels), fidence region) is consistent with the dotted-line repre-
Swift/BAT captured a higher S/N light curve. The S/N senting equality. The best-fit line has a normalization
level can be gauged from the light curves and taken di- = 1.13 ± 0.13 and a slope = 0.99 ± 0.02. For this fit
rectly from the ∆t values, with high S/N translating the reduced χ2 =2.86 (for 42 degrees of freedom) and is
S/N
directly to lower ∆t . There are many bursts (e.g., dominatedbyasmallnumberofoutliers. Thefractionof
S/N
GRB 120119A, right panels) in the joint Fermi/GBM bursts not consistent with the fit, both below and above
and Swift/BAT sample which correspond to closely sim- the line are: 12% and 15%, respectively. The close con-
ilar S/N values and for which the resulting scaleograms sistency of this line with the unit line demonstrates that
are almost identical. We note that minimum timescales our method is robust and that our error bars, calculated
based simply on ∆t (e.g., Walker et al. 2000) di- by direct error propagation, are likely to be accurate.
S/N
rectlytrackthenoisefloorlevel. Thisisalsothecasefor We do note, however, that the ∆tminvalues calculated
t , calculated according to the prescription of MacLach- for Swift versus Fermi do exhibit small, systematic dif-
β
lan et al. (2013). In the most extreme examples (i.e., ferences. On average, bursts detected by Swift (in the
GRBs090519Aand101011A),the∆t valuesdifferby sameenergyband)tendtohave13%longer∆tminvalues
S/N
approximatelyanorderofmagnitude,thet valuesdiffer as compared to Fermi. Histograms showing the spread
β
by approximately a factor of five, while the ∆t values in the overall populations are also drawn along the axes
min
are consistent (Table 2). Our method distinguishes be- in Figure 4.
tween the minimum detectable timescale and the true TostudytheoriginoftheoutlierstothefitinFigure4,
6 Golkhou, Butler & Littlejohns
we scale the relative size of the circles with the absolute 3.3. Distribution of ∆t Values for Fermi/GBM
min
valueofthelogoftheratiooffluxvariationattheshort-
Figure 5 (left) shows histograms for the Fermi GRBs
est observable variability timescale ∆t . This is in-
S/N permitting measurement of and also upper limits on
tended to provide an indication of whether each satellite
∆t . The two distributions have consistent mean val-
sampled the same (small circles) or very different (large min
ues. The middle and right panels of Figure 5 show the
circles) regions of the scaleogram at the inferred ∆t .
min KM cumulative histograms in the observer and source
The color bar can be used to identify which instrument
frames, respectively. The dotted-lines correspond to the
generated the higher σ .
X,∆t minimumtimescaleofthelowest10%and50%(median)
In general, we find that once the log(∆t ) ratios
S/N of short and long-duration bursts.
exceed 0.5 dex (corresponding to 0.5 dex in log(S/N) or Wefindamedianminimumtimescaleforlong-duration
roughly a factor 10 in flux) the more sensitive satellite (short-duration) GRBs in the observer frame of 134 ms
tends to yield a lower measurement of ∆tmin. This is (18ms). Inthesourceframe,wefindamedianminimum
consistent with our findings from Paper I. Given that timescale for long-duration (short-duration) GRBs of 45
suchvariationisnotknowna-prioriinthiscase(because ms (10 ms). It is interesting that these numbers are
the light curves are not based on a simulation), the a factor of 3–10 smaller than those we found for Swift
tendency to detect lower ∆tmin when possible suggests a in Paper I. The largest differences, in the case of short-
fractal nature of the phenomenon. Care must be taken durationGRBs,areattributabletotheincreasednumber
in interpreting GRB minimum timescales, because the of well-detected short-duration GRBs by Fermi. As we
phenomenology suggests these could always be limits discuss below (Section 3.4), ∆t also appears to vary
min
on the true minimum timescales. However, we do note byafactorof≈3dependingonthebursthardness. The
the important feature of the scaleograms: hidden (i.e., Fermi sample is studied using the full energy range, and
low S/N) minimum timescales will always correspond to thesampleappearstobespectrallyharderthantheSwift
smaller variations in the fractional flux levels. In this sample, overall.
sense, a perfect accounting of the minimum timescales We also report ∆t of the most exotic GRBs in
min
may not be necessary, because very short minimum Fermi sample – the lowest 10th percentile of bursts with
timescales tend to represent fractionally tiny (or alter- the shortest ∆t . The 10th percentile ∆t values
min min
natively very rare) episodes in the GRB emission. for long-duration (short-duration) GRBs in the observer
frame found to be 2.2 ms (1.9 ms). In the source frame,
we find 2.9 ms (2.4 ms). These numbers are consistent
with the findings in Paper I that millisecond variability
0 510
1.5 appears to be rare in GRBs.
From Figure 5, we find that the ∆t distribution
min
1.0 of long-duration GRBs is displaced from that of short-
duration GRBs (16σ, t-test 17σ, log-rank test (Mantel
ec) 0.5 1966)). The log-rank test includes the upper limits, un-
s like the t-test. This finding is consistent with the pre-
(
) sented results in Paper I for Swift. This discrepancy is
V) 0.0 >1
ke still present in the source frame (2.3σ, t-test and 3.4σ,
Swift(15350−min 10..05 σ<σ∆∆FSet1twmrmimiifnnti lcpmoelgaen-itrnoeafrnrssekahapostopertsne-tad)fruoeurrdnattltihiokoeenboeiGnbcsRoPenBravsspiesedtwreindtIthe.wgekThnneheorerewaSnctwyh-z.ieftidTssihsmltiekraielsblliyugstnatimiohfine--
t
∆
cant observer frame discrepancy is likely driven by the
(
g10 1.5 fact that short-duration GRBs tend to be detected only
o
l at low-redshift, unlike long-duration GRBs which span
2.0 a broad range of redshifts. Examining the dispersion
10 in log(∆t ) values, we see no strong evidence for dis-
min
5
similar values for the long and short-duration samples
2.5 0
2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 (< 1.3σ, F-test). This finding is also fully consistent
log10(∆tmFienrmi(15−350keV)) (sec) with the presented results in Paper I, where it was also
found (using a sample of Swift GRBs) that the two his-
Fig. 4.— ∆t for the sample of joint Fermi/GBM and
min tograms are quite broad and very similar in dispersion.
Swift/BAT bursts. The red and blue histograms correspond
to the short and long-duration GRBs, respectively. The ar- Figure 6 displays our minimum variability timescale,
rowsshowtheupperlimitburstcases. Theblackdotted-line ∆tmin, versus the GRB duration, T90. The short and
represents equality. The relative size of the circles is scaled long-duration GRBs are shown with diamond and circle
withtheabsolutevalueofthelogoftheratiooffluxvariation symbols, respectively. In this plot the relative size of
at the shortest observable variability timescale ∆t , pro- symbols is proportional to the ratio between minimum
S/N
viding a measure of whether each satellite samples the same variability and S/N timescale (∆t /∆t ). As de-
min S/N
(small circles) or very different (large circles) regions of the scribedabove,∆t representsthefirststatisticallysig-
scaleogram at the inferred ∆t . The color bar can be used S/N
min nificant timescale in the Haar wavelet scaleogram. The
to identify which instrument generated the higher σ .
X,∆t color of the points in Figure 6 corresponds to the flux
variation level, σ , at ∆t . A curved black line is
X,∆t min
also plotted to show a typical value for the minimum
GRB Minimum Timescales 7
Fig. 5.— Left panel: the histograms of ∆t with measurements (blue) and for GRBs allowing for upper limits only (red).
min
Middleandrightpanels: thecumulativehistogramsofburstsintheobserverandsourceframes,respectively. TheKMestimation
curve with 1σ error region around the curve is shown in these panels. The dotted lines correspond to the minimum timescale
of the lowest 10% and 50% of bursts, shown for the short and long-duration GRBs, separately. Sub-panels show the locations
of detections and upper-limits, as in Figure 2. For long-duration (short-duration) GRBs, we have 421 (107) measurements and
334 (76) upper limits in the observer frame and 24 (3) measurements and 18 (1) upper limits in the source frame.
observable time (∆t ) versus T . Values for T are 102
S/N 90 90
taken from Table 7 of von Kienlin et al. (2014). Long GRBs
We first note from the colors in Figure 6 that GRBs short GRBs
with ∆tmin close to T90 tend to have flux variations of 101 σX,∆t
min
order unity. These are bursts with simple, single-pulse
timeprofiles. Ascanbeseenfromtherangeofpointsizes 1.00
in Figure 6, most are not simply low S/N events where
finetimestructurecannotbeobserved. Also,weseethat c) 100 0.74
e
there are GRBs with both high and low S/N which have s
complex time-series (∆t (cid:28) T ). Based on the point ( 0.55
min 90 n
sizes, the short-timescale variation have higher ratio of mi
∆t /∆t for the short-duration GRBs of the simi- ∆t 10-1 0.40
min S/N
lar ∆t in comparison with that of the long-duration
min
0.30
GRBs. Short-durationGRBstendtohaveahigherσ
X,∆t
for the similar value of ∆tmincompared with the long- 10-2
duration GRBs.
These findings are all consistent with the similar results
explained in Paper I; although we have a better ratio of
short-duration GRBs to long-duration GRBs, here. 10-3
10-1 100 101 102 103
From a Kendall’s τ-test (Kendall 1938), we find only
marginal evidence that ∆tmin and T90 are correlated T90 (sec)
(τk =0.33, 11σ above zero). The ∆tmin values in Figure Fig. 6.—TheGRBminimumtimescale,∆tmin,plottedver-
6areboundfromabovebyT ,andtheydonotstrongly sustheGRBT duration. Circles(diamonds)representlong-
90 90
correlate with T within the allowed region of the plot. duration (short-duration) GRBs. The point colors represent
90
In Paper I, we studied this relation for the entire sample the flux variation level (σX,∆tmin) at ∆tmin. Also plotted as
of Swift GRBs and found only a marginal evidence that a curved line is the typical minimum observable timescale,
∆tminand T90 are correlated (τk = 0.38, 1.5σ). Even ∆tS/N, as a function of T90. The symbol sizes are propor-
tionaltotheratioof∆t /∆t foreachGRB.Thedashed
whenweutilizedtherobustdurationestimateT (Re- min S/N
R45 line shows the equality line.
ichartetal.2001)inplaceofT nosignificantcorrelation
90
was found (τ = 0.6, 2.4σ). If we perform a truncated
k
Kendall’s τ test which only compares GRBs above one-
another’s threshold (Lloyd-Ronning & Petrosian 2002), the hardness ratio (HR) as the total counts in the hard
the correlation strength drops precipitously (τ = 0.06, compositechannel(89–1000keV,ourcombinedchannels
k
1.4σ). We, therefore, believe there is no strong evidence 3 and 4) divided by the total counts in the soft com-
supporting a real correlation between ∆t and T . posite channel (8–89 keV, our channels 1 and 2). We
min 90
plot in Figure 7 (top panel) the ratio of ∆t for these
3.4. The Dependence of ∆t on Spectral Hardness min
min twocompositechannelsagainsttheHR ofthetwocorre-
We investigate here how a burst’s spectral hardness spondingbandpasses. GRBswithharderspectratendto
impacts its minimum variability timescale. We define have a lower ∆t ratio, by as much as a factor ≈3, for
min
8 Golkhou, Butler & Littlejohns
theline-of-sitehasaDopplerfactorΓ(1−β),propagating
101
withaspeedv =βcandLorentzfactorΓ,materialabove
V) or below the line of site at angle θ will have a Doppler
ke factorΓ(1−βcos(θ))≈(1+(Γθ)2)/2Γ,largerbyafactor
9
8−n 1+(Γθ)2. The off-axis emission will also arrive later, at
(8∆tmi 100 a time t−te =R/c(1−cos(θ)), where R is the emission
radius,afterthestartoftheemissionatt . Ifweassume
/ e
V) R=2Γ2cte,thentheDopplerfactorincreasesintime,in
e
k the observer frame, as t/t . As a result, the photon flux
0 e
100 10-1 observed at fixed energy E will decrease as higher and
9−n higher rest-frame-energy photons reach the bandpass, as
(8mi (t/t )α−2. Here, α is the photon index and the power of
t e
∆ slope= 0.34 0.04 2 arises from relativistic beaming.
− ±
Thus, we expect that impulsive releases of energy in
10-2
10-1 100 the rest frame will be smoothed over – in a fashion that
is stronger at low energy (α≈−1) as compared to high
V) energy(aboveE ,α<−2)–asviewedintheobserver
e peak ∼
k
00 frame. The degree of smoothing expected above Epeak
10 is a factor 2–3 less than the smoothing expected at ob-
−n
(8tmi 100 server frame energies below Epeak. This effect naturally
∆ explains the decreasing minimum timescale we observe
/ with increasing spectral bandpass, and it suggests that
V)
e thetightestconstraintsonminimumtimescaleshouldbe
k
9 obtainedfromthehighestavailableinstrumentbandpass.
8
(8−min It should also be sufficient to confirm that Epk is below,
∆t or perhaps within, a given bandpass.
slope=0.12 0.02
10-1 ± Figure7(middlepanel)showstheratioof∆tminforthe
soft composite channel over the full energy band against
101 10-1 100 theHR.Thisplotshowshow∆t isapproximatelythe
min
V) same in each bandpass until the hardness ratio goes be-
e
k yond roughly its median value. The bursts in this plot
0
0
10 well-fitted by a line with slope =0.12±0.02.
(8−min The ratio of ∆tminfor the hardest channel (#4) over
∆t 100 the full energy band against the HR is shown in Fig-
/ ure 7 (bottom panel). Here, the best-fit line (slope
eV) = −0.02 ± 0.09) is consistent with being flat: the
k
0 minimum timescales appear to be independent of this
0
10 hardness ratio for all but perhaps the hardest handful
−
99n of Fermi GRBs. We conclude that utilizing the full
(2∆tmi10-1 slope= 0.02 0.09 Fermi/GBM bandpass – which yields ∆tminconstraints
− ± consistent with those derived from the soft energy chan-
nel for soft GRBs and also ∆t constraints consistent
10-1 100 withthosederivedfromtheharmdinenergychannelforhard
HR=Cts(89−1000keV) GRBs – is an acceptable procedure for determining the
Cts(8 89keV) tightest constraints on ∆t .
− min
Fig. 7.— Top panel: The ratio of minimum variability
timescale for channels 3 + 4 and channels 1 + 2, plotted
againsthardnessratioforthecorrespondingcompositechan-
nels. Middleandbottompanels: Theratioof∆t forchan- 3.5. Constraints on the Size of the Central Engine
min
nels 1+2 and channel 4 over full energy band, separately
The minimum timescale provides an upper limit on
plotted against hardness ratio. The best fitted linear model
the size of the GRB emission region, in turn providing
through the bursts including shaded 1σ error region is also
hints on the nature of the GRB progenitor and poten-
shown in each panel.
tially shedding light on the nature of emission mecha-
nism. In Paper I, we summarized how an association of
both short and long-duration GRBs. This relationship a minimum timescale with a physical size is not unique,
canbecapturedusingabest-fittedlinearmodelthrough because the observed timescales depend strongly also on
all the bursts, shown in Figure 7 (top panel), with slope the emitting surface velocity.
=−0.34±0.04. The minimum Lorentz factor Γ can be estimated from
The change in minimum timescale with hardness can thecompactnessargument(Lithwick&Sari2001). Ifwe
be understood from the effects of relativistic beaming assumeaspectrumwithphotonindexα=−2(see,Ack-
on emission instantaneously emitted in the rest frame ermann et al. 2013, Figure 25) – typical for GRB spec-
by a moving shell (e.g., Fenimore et al. 1996; Ryde & tra above the pair-production limit and also appropriate
Petrosian 2002; Kocevski et al. 2003). If the material on for the range of energies which dominate the luminosity
GRB Minimum Timescales 9
(near the νF spectral peak) – we find – where the emission radius was argued to simply scale
ν
with the T duration – we find a broader overlap in the
(cid:18) L 1+z (cid:19)1/5 population9s0.
Γ>110 , (1)
∼ 1051erg/s ∆t /0.1sec
min
0 75150
whereListhegamma-rayluminosity. Ifweregard∆t 17
min Long Short
as corresponding to the bolometric emission, it is most
known z
natural to use the full Fermi/GBM bandpass for its es- 16 assigne−d z
timation rather than a fixed rest frame bandpass. It −
could be argued that corrections should also be made
15
to account for spectral hardness, based perhaps on the
assumption that GRBs have a single, fixed rest frame m)
c
hLaorrdenntezssfa–ctaonr.uHnoliwkeevlyerp,obsassiebdiliotnyt–hemaondaulylasitsedinoSnelcytibony (dius)14
a
3.4 above, any corrections would be small. R 13
Utilizing our ∆tminestimates and limits for the full og(10
Fermi/GBMbandpass,wefindthat50%ofFermi GRBs l
12
must have Γ > 190. In the case of the most energetic
events, 10% of Fermi GRBs require Γ > 410. To cal-
culate these fractions for short-duration bursts without 11
100
measured redshift, we follow D’Avanzo et al. (2014) in
50
assigning an average z = 0.85. For long-duration GRBs 10 0
lacking redshift, we assign the average z =2.18. 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5
Similarly, for some maximally allowed Γmax, compact- log10(1T+90z) (sec)
ness limits the emission radius to be greater than Fig. 8.—The characteristic emission radii R (Equation 4)
c
plottedversusrestframeT fortheFermi/GBMbursts. The
R (cid:39)2.8×1010 L (cid:18)Γmax(cid:19)−3 cm. (2) shaded region shows the in90terval between the minimum and
min 1051erg/s 1200 maximum emission radii allowed. The bursts with known-
z and assigned-z are denoted with filled and unfilled circles,
This minimum bound on the radius can be compared to respectively. Theshortandlong-durationGRBsaredenoted
the maximum bound on the radius established by the with red and blue colors, respectively.
temporal variability:
∆t
R =c min Γ2 ,
max 1+z max 3.6. Evolution of ∆t with z
min
∆t /0.1sec (cid:18)Γ (cid:19)2 Because GRBs are present over a very broad redshift
(cid:39)4.4×1015 min max cm. (3)
1+z 1200 range,thesignatureoftime-dilation–andperhapsofany
evolution in GRB time-structure with redshift – should
Here, we conservatively take Γmax ∼ 1200 from Racusin be present in GRB time-series. Finding the signature
et al. (2011). of time-dilation in GRBs has remained elusive (Norris
Ifemissionweretooccurattheminimumallowablera- et al. 1994; Kocevski & Petrosian 2013, but see, e.g.,
dius, Rmin, it would correspond to variability timescales Zhangetal.2013). Inourpreviousattemptdescribedin
as short as ∆t = Rmin/(2cΓ2max) ∼< 1µs. Because such PaperI,weutilizedSwift GRBsanddemonstratedacor-
timescales are not observed, a more realistic bound on relationbetween∆t andredshift,marginallystronger
min
the minimum emission radius is R =2cΓ2 ∆t /(1+ than expected simply from time-dilation. We discussed
c min min
z), or how this excess correlation strength was possibly due to
the utilization of a fixed observer frame bandpass in-
R (cid:39)7.3×1013 (cid:18) L (cid:19)2/5(cid:18)∆tmin/0.1sec(cid:19)3/5 cm. stead of a fixed rest frame bandpass in the analysis. For
c 1051erg/s 1+z Fermi/GBM,thebroadinstrumentenergyrangepermits
(4) analysis in a fixed rest frame bandpass.
Figure 8 shows the emission radius, R , for all the We identify 46 Fermi GRBs, including 4 short-
c
bursts with measured ∆t in Fermi/GBM sample ver- duration GRBs, with measured redshifts. Light curves
min
susrest-frameT . Theshadedregionshowstheinterval areextractedintherestframe89–299keVbandandan-
90
between the Rmin and Rmax. The interpretation of Rc alyzed. In Figure 9 we plot ∆tmin/(1+z) versus 1+z
as a characteristic minimum radius for the emission is for the long-duration GRBs. Redshift values are taken
motivated further in Section 4. from Butler et al. (2007, 2010, and references therein),
The short-duration GRBs have a KM mean R = Butler (2013), and this webpage5. The blue circles in
c
3.3×1013 cm. This is about four times smaller than the Figure9correspondtotheKMmeanvaluesof∆tminfor
KM mean R = 1.3×1014 cm for long-duration GRBs. sets of between 7 and 10 bursts, grouped by redshift
c
While this represents a statistically significant separa- intervals. The unbinned data are plotted in the back-
tion (18σ, t-test), it is substantially less than the factor ground for the entire sample and for those with mea-
of approximately twenty separation between the mean sured ∆tminusing unfilled and filled circles, respectively.
T durations (Figure 8, also Kouveliotou et al. 1993).
90
In contrast to the findings of Barnacka & Loeb (2014) 5 http://www.mpe.mpg.de/~jcg/grbgen.html
10 Golkhou, Butler & Littlejohns
We find that the binned data can be well-fitted by a line MinimumtimescaleestimatesusingthefullFermi/GBM
∆t /(1+z) ∼ 140((1+z)/2.8)0.5±1.0 ms, suggesting bandpass are a factor 2–3 times more constraining than
min
possibly increase in timescale with z but also consistent thosedeterminedfromFermi/GBMdatainaSwift/BAT
the prediction of simple time-dilation (dotted line). bandpass.
Considering measurements and limits, we find a me-
dianminimumvariabilitytimescaleintheobserverframe
101 of 134 ms (long-duration; 18 ms for short-duration
Fixed Rest Frame Energy Band Measured GRBs). In the source frame, for a smaller sample of
Upper limit 33 GRBs, we find a median timescale of 45 ms (long-
duration; 10 ms for short-duration GRBs). This finding
) validatesourpreviousresultsinPaperI,confirmingthat
c
e 100 millisecondvariabilityappearstoberareinGRBs. Inthe
s most extreme examples, 10% of the long-duration GRB
(
) sample yields evidence for 2.2 ms variability (1.9 ms for
z
+ short-durationGRBs). Inthesourceframe, wefindsim-
1 ilar numbers, 2.9 ms for long-duration GRBs and 2.4 ms
(
/ for short-duration GRBs. Even if we restrict to the 67
min10-1 ∆tmin∼1+z GRBs within minimum detectable timescales tS/N < 10
∆t ms, only 10% of the brightest and/or most impulsive
GRBs show evidence for variability on timescales below
4.2 ms in the observer frame.
∆1+tmzin∼ 140(12+.8z)0.5±1.0 ms 4.1. Constraints on the Fireball Model
10-2
100 101 In the “external shock” model (e.g., Rees & Meszaros
1992), gamma-rays are produced as the GRB sweeps up
1+z
and excites clouds in the external medium. The ex-
tracted ∆t can circumscribe the size scale of the im-
min
Fig. 9.— Minimum variability timescale in the rest frame pactedcloudalongthelineofsight. Forathinshell(e.g.,
89–299 keV energy band versus redshift, z. The blue circles
M´esz´aros2006),thegamma-rayradiationwillstartwhen
showtheKMmeanvaluesof∆t forgroupsof7–10bursts
min therelativisticshellhitstheinnerboundaryofthecloud
of similar redshift. The shaded region represents the 1σ con-
withthepeakfluxproducedastheshellreachesthedens-
fidence interval for the fitted red line. The dotted black line
est region or center of the cloud. The size scale of the
showstheexpectedevolutionduetosimplecosmologicaltime
impacted cloud is limited by 2Γ2c∆t since the shock
dilation, namely ∆tmin ∼ 1+z. The faint blue circles show min
allGRBswithmeasured∆t andknown-z andtheunfilled is moving near light speed (Fenimore et al. 1996). For
min
circles show GRBs with upper limit values for ∆tmin. thesmallest∆tminfound∼1ms,andassumingΓ<1000,
the cloud size must be smaller than 4 AU.
Iftheangularsizeofanimpactedcloudasviewedfrom
the GRB central engine is Θ, the minimum variability
4. CONCLUSIONS timescalesisconstrainedtobeδΘΓ<∆t /2T (Pa-
min rise
Using a technique based on Haar wavelets, previously per I). Here, T denotes the overall time to reach the
rise
developedinPaperI,westudiedthetemporalproperties maximum gamma-ray flux. The fraction of the emit-
of a large sample of GRB gamma-ray prompt-emission ting shell that becomes active is expected to be of order
light curves captured by the GBM instrument onboard 0.1∆t /2T (Fenimore et al. 1999). For the bursts
min rise
Fermi prior to July 11, 2012. We analyzed the time his- in the Fermi sample with typical minimum variability
tories in four energy bands. While the derived values for timescale ∆t ∼ T , there is no need to consider a
min rise
∆t arehighly-dependentuponbandpass,wefindthat highly-clumped external medium and the external shock
min
the use of the full energy band allows for the tightest scenario is viable.
constraints on the size of the emission region. In princi- However, there are many bursts (e.g., Figure 6) which
ple,thehighest-energybandpassshouldyieldthetightest do exhibit ∆t /T (cid:28) 1. If this variability results
min rise
constraint (Section 3.4). However, S/N in the highest- fromaclumpedexternalmedium,thenasignificantfrac-
energy channels is often low; the full energy bandpass tionoftheenergyfromtheGRBmustescapewithoutin-
allows for increased S/N while maintaining a consistent teractingandproducinggamma-rays. EarlyX-rayafter-
∆t estimate. glowobservations(e.g.,Nouseketal.2006),ontheother
min
Applying our technique to the joint Fermi/GBM and hand, demonstrate the need for a high (order unity) ef-
Swift/BATsample, wefindcloseconsistencyinthemin- ficiency in tapping the kinetic energy of the flow to pro-
imum timescales derived for each instrument. However, duce gamma-rays. Thus, external shocks likely cannot
assuggestedbysimulationsinPaperI–andobservedfor explain the finest-time-scale variability.
a handfull of bursts of widely varying S/N in Section 3.2 Inthe“internalshock”scenario(e.g.,Rees&Meszaros
–∆t valuesbelowthemeasurementlimit(∆t )can 1994), the relativistic expanding outflow released from a
min S/N
be present. It is thus important to consider our ∆t central engine is assumed to be variable, consisting of
min
valuesasdefinedgiventheobserveddata,withthepossi- multiple shells of different Γ. The dispersion in Γ is re-
bilityofimprovedlimitsgivenbetterdata. Weurgecau- lated to the observed variability of the light curve, as
tion, in particular, in interpreting minimum timescales ∆Γ/Γ ≈ 1/2(∆t /T ) (Paper I), with many of the
min rise
determined using hard X-ray data (e.g., Swift/BAT). Fermi light curves requiring ∆Γ ≈ Γ. Efficient pro-