A Minimal Timescale for the Continuum in 4U 1608-52 and Aql X-1
K. Mohamed, E. Sonbas, K. S. Dhuga, E. Gö?ü?, A. Tuncer, N. N. Abd Allah, A. Ibrahim
MMNRAS , 1–7 (2020) Preprint 13 January 2021 Compiled using MNRAS L A TEX style file v3.0
A Minimal Timescale for the Continuum in 4U 1608-52 andAql X-1
K. Mohamed, , E. Sonbas, , (cid:63) K. S. Dhuga, E. Göğüş, A. Tuncer, N. N. Abd Allah, and A. Ibrahim Department of Physics, Sohag University, Egypt Adiyaman University, Department of Physics, 02040 Adiyaman, Turkey Department of Physics, The George Washington University, Washington, DC 20052, USA Sabancı University, Orhanlı - Tuzla, Istanbul 34956, Turkey Istanbul University Science Faculty, Department of Astronomy and Space Sciences, 34119, University-Istanbul, Turkey Department of Physics, Faculty of Science, Cairo University, Egypt
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Similar to black-hole X-ray binary (BHXRB) transients, hysteresis-like state transitions are also seen in some neutron-star X-ray binaries (NSXRBs). Using a method based on wavelets and lightcurves constructed from archival RXTEobservations, we extract a minimal timescale over the complete range of transitions for 4U 1608-52 during the 2002 and2007 outbursts and the 1999 and 2000 outbursts for Aql X-1. We present evidence for a strong positive correlationbetween this minimal timescale and a similar timescale extracted from the corresponding power spectra of thesesources.
Key words: methods: data analysis, stars: neutron, X-rays: binaries
It is well known that transient neutron star (NS) X-raybinaries (NSXRBs) undergo repeating outbursts drivenby accretion and experience lengthy periods of quiescencewhen the accretion is considerably reduced. These changesin state provide a unique opportunity to probe both theproperties of the compact objects and the accretion process.Several spectral states can be associated with NSXRBsincluding the hard, intermediate and soft state, that exhibitsome overlap in spectral and timing properties with XRBshosting black holes (BHs) (van der Klis (2006); Fender etal. (2004); Belloni et al. (2005)). Furthermore, NSXRBsare divided into two sub-classes as Z sources (with L x (cid:38) L Edd ) and atoll sources (with 0.01 L Edd (cid:46) L x (cid:46) L Edd ). The original classification, however, was based onthe evolution of the softness/hardness of the source on acolor-color diagram (Hasinger & van der Klis 1989), whereone group traces a z-shaped track as the luminosity changes,and the other group renders a c-shaped path also referredto as the ’banana’ state (see also (Gierlinski & Done 2002)).The different patterns are thought to reflect a variation inthe accretion rate.A systematic study, using a large sample of NS low-mass X-ray binaries (LMXBs) monitored by the RXTE,was performed by Munoz-Darias et al. (2014) in which they (cid:63)
E-mail: [email protected] showed that the NS LMXBs exhibit hysteresis-like patterns(similar to those observed in BH LMXBs (Miyamoto et al.1995)) between the hard state and the soft state. The hardstate is thought to be dominated by Compton scattering offenergetic electrons in the corona; the soft state is associatedwith thermal emission from the accretion disk. They alsofound that the hysteresis patterns are not seen in NSXRBsat higher accretion rates where the sources remain in thethermal-dominated, low-variability state.Traditionally, the states have been identified throughvarious spectral and timing studies with a majority ofthem involving variability studies focusing primarily on theextraction of the fractional RMS and hardness ratios totrack the transitions as the sources undergo changes in theobserved count rates. For example, the hard state spectrum(in the 2 -20 keV band) can be described by a single powerlaw, with a photon index Γ of ∼ . in addition to a minorthermal contribution from the disk. The power spectraldensity (PSD) usually shows significant continuum noisewith fractional RMS in the range ∼ − . On the otherhand, the spectrum for the soft state ( ≤ keV) is domi-nated by the disk thermal component, and the power-lawcomponent (with Γ ∼ − ) is typically very weak. Thefractional RMS is reduced in strength down to ∼ . ForBH transients the essence of these features is encapsulatedin the hardness-intensity diagram (HID; Homan et al.(2001); Belloni et al. (2005)) on which the various statestrace out tracks (or hysteresis-like loops) as the sources © 2020 The Authors a r X i v : . [ a s t r o - ph . H E ] J a n K. Mohamed et al. undergo transitions during outbursts. The HIDs serve astestbeds for not only the identification of the states andtheir transitions but are also useful in exploring the effects ofaccretion rate, and the evolution of the relative contributionsof the thermal and power-law components to the emissionprocess. Unfortunately, the tracks vary significantly fromsource to source and even within individual sources fromoutburst to outburst. These variations show up as localfluctuations in the intensity for any given state and thusresult in considerable dispersion in HIDs as the tracks formband-like structures (similar to hysteresis loops) rather thanwell-defined reproducible contours.Our motive for this study is straightforward: we wishto track the spectral changes by using a temporal scaleinstead of a hardness ratio. The temporal parameter ofinterest is the minimal timescale (MTS) that was recentlydeployed by Sonbas et al. (2020) in their study of GX339-4,a well studied BH transient. We focus our attention ontwo NSXRBs (well-studied transient atolls (Gierlinski &Done 2002), that have undergone a number of outburststhat have excellent RXTE coverage) with the primary goalof determining whether the MTS exhibits similar behaviorto that observed in GX339-4: We expect the timescaleto be useful in tracking the transitions in a way that iscomplementary to the the traditional HIDs. We followSonbas et al. (2020) and interpret the proposed timescaleto imply the smallest temporal feature in the lightcurvethat is consistent with a fluctuation above the Poisson noiselevel. In the frequency domain, this would be equivalent tothe highest frequency component in the signal at or justabove the noise threshold. Assuming that the (intrinsic)signal-noise threshold is different for different states, theproposed timescale implies a temporal tag for each stateand associated transitions, thus providing a tool that canbe used to track their dynamic evolution. Of course, asimilar timescale should be accessible from PSDs; we assumethis to be the cutoff frequency where the red-noise (signal)intersects the white noise (a combination of intrinsic noiseassociated with the source itself and an extraneous noiseindependent of the target, a component that is usuallyminimized/removed during the construction of the PSD).For cases where the broad-band (continuum) spectrumexhibits a simple 1/f β behavior, a simple phenomenologicalmodel such as a powerlaw suffices to extract this cutofffrequency. However, the spectra for different spectral statescan vary quite significantly in profile and complexity, thusnecessitating the use of models with increasing degrees offreedom. We, on the other hand, have chosen to estimate thistimescale through a wavelet decomposition (of lightcurves),where we focus on the variance of the signal; this tendsto minimize the effect of the complex features observedin some PSDs. If the MTS indeed can be taken to be thehighest frequency component in the signal with the parallelassumption that the (intrinsic) signal-noise threshold islikely different for different states, then the MTS couldpotentially serve as a tool for tracking the evolution of theunderlying continuum of those states and their transitions.There are several timescales that are present in accret-ing binary systems: typical ones of interest include thedynamical, viscous and thermal timescales. The dynamical timescale, t dyn , is related to the Keplerian frequency Ω ,which is given by (cid:112) ( GM/r ) , where M is the mass of thecompact object, r gives the size of the orbit, and G is thegravitational constant. Assuming typical numbers for agiven system, one obtains a timescale in the range ∼ tensto hundreds of ms. The viscous (accretion) timescale, t vis ,computed as 1/( α ( H/r ) Ω ), assuming the standard diskwith the α prescription for the viscosity (Shakura & Sunyaev(1973)), with typical values for α and H/r (representingthe vertical and radial extensions of the accretion disk)i.e., 0.1 and 0.01 respectively, yields a significantly longertime scale i.e., extending well over hundreds of secondscompared with the dynamical scale. The other importantscale worth a mention is the thermal timescale, t thm , whichprovides a measure of the heating and cooling rate of thedisk and, as noted by Lasota et al. (2016), is intermediate tothe other two scales, and is related to t dyn , by a factor of 1/ α .Our targets of interest are 4U 1608-52 and Aql X-1, apair of well studied NSXRBs. We deploy a wavelet-basedtechnique (MacLachlan et al. (2013a)) to extract a temporalscale for the targets over their respective range of statetransitions that were observed during the 2002 and 2007outbursts for 4U 1608-52 and the 1999 and 2000 outburstsfor Aql X-1, and relate that timescale to a scale extractedfrom PSDs using the traditional Fourier-based methods. Thelayout of the paper is as follows: in Section 2, we describethe selection and extraction of the available RXTE data,as well as, the extraction details of the MTS and the cutofffrequency. In Section 3, we present the main results of ouranalysis and the comparison of the extracted time scales.We conclude by summarizing our main findings in Section 4. We analyzed publicly available Rossi X-ray Timing Ex-plorer/ Proportional counter Array (RXTE/PCA: (Bradt,Rothschild & Swank 1993)) observations covering the 2002and 2007 outbursts of the atoll 4U1608-52, and the 1990-2000outbursts for Aql X-1 (Tables 1 and 2 respectively). We usedthe standard RXTE data analysis using the tools of HEA-SOFT V.6.26 to create lightcurves with a time resolutionof − s (i.e., ∼ µ s). We used PCA data modes fromdifferent channels with different time resolution includinghigh resolution Good-Xenon or Event data modes that coverthe full energy band (2 - 60 keV), otherwise we combinedSingle-Bit and Event data modes to cover the full energyband. The primary reason for using the full energy band wasto obtain a measure of the background noise which wouldappear in the highest available channels. The high resolutionlightcurves were created using standard screening criteria byapplying source elevation of greater than (10), SAA exclusiontime of (30) minutes, the pointing offset less than 0.02, andall PCUON. For each observation, we generated backgroundmodel for bright objects using the FTOOL (pcabackest) inthe IDL environment. This background model is used indetermining the PSD normalization.The PSDs were created (following the procedures ofNowak (2000), Pottschmidt et al. (2003) and Belloni etal. (2005)) for segments of 16s duration and for Nyquist MNRAS , 1–7 (2020) inimal Timescales frequency of 2048 Hz using Powspec 1.0 (Xronos5.22) in theIDL environment. In order to check for self consistency, anumber of PSDs were created for durations of 64s as well.Fast Fourier transforms were performed for each segment toextract a Leahy (Leahy et al. (1983)) normalized PSD foreach observation. The segments were averaged and rebinnedgeometrically to smooth the power spectrum.Wavelet transformations have been shown to be a nat-ural tool for multi-resolution analysis of non-stationarytime-series (Flandrin 1989, 1992; Mallat 1989). Indeed, anumber of applications in X-ray astronomy have appearedin the literature (Scargle et al 1993, Steiman-Cameron etal 1997, Lachowicz and Czerny 2005, and Lachowicz andDone 2010). In this study, we extract the MTS using themethodology (see Appendix A) described by MacLachlan etal. (2013a,b).In the following, we provide some background and notethe salient features of the methodology: The light curveis represented by a complete set of basis functions. In thecase of the Fourier series, the basis functions are sinusoidalfunctions. In the case of wavelets, we have a choice of manydiverse functions, common examples being the Haar wavelet,the Morlet function, and the Daubechies functions. Weuse the term ‘basis’ somewhat loosely since the functionsused are ‘scaled’ versions of an original function chosenfor the task (the Haar wavelet in our case). One of thebiggest differences in application of the Fourier series andthe use of wavelets is that the basis functions typically usedin wavelet analyses are localized in time i.e., have what’sknown as ‘compact’ support i.e., they are non-zero over afinite interval and vanish elsewhere. For example, the Haarwavelet, a ’box’-like function, is finite for the time interval0 < t <1 and is equal to 1 for t > 0, -1 for t > 0.5, andzero elsewhere. This is not true for the basis functions in theFourier series i.e., those functions provide global coverageand are technically finite over all time intervals. One of theprime reasons for using wavelets is to extract simultaneouslyboth time and frequency information for a given signal. Inthe case of the Fourier series one obtains only the frequencycontent but the location in time of the frequencies of interestis not available. While this is not an issue for signals that arestationary i.e., the frequencies are not varying in time but itis problematic when the signal is non-stationary (or weaklynon-stationary). In this case one would like to extract notonly the significant frequencies (that contribute to the powerin the signal) but also their location in time i.e., are they inthe early part of the signal or some later part. This is wherewavelets are most useful. The signal of interest of courseis the measured X-ray lightcurve. We know already thatthe emission process leading to the lightcurves most likelyinvolves more than one mechanism because of the variationsin the observed power spectra along with the occasionaloccurrence of quasiperiodic features. Moreover, we knowthat the accretion rate is unlikely to be constant over timeespecially since we know that the X-ray sources undergospectral transitions involving a large variation in intensity.Under these conditions it is likely that the underlying processis at least weakly non- stationary thus justifying the waveletapproach. The main utility of wavelets arises due to two basicoperations: scaling and translation. The primary wavelet(sometimes referred to as the mother wavelet) can bestretched and compressed in time altering the frequencycontent with each (scaling) operation. The larger thescale-factor the larger the spread of the wavelet i.e., thewavelet is stretched in time, which is equivalent to a waveof low-frequency. Conversely, the smaller the scale, the morecompressed (in time) the wavelet, which corresponds to ahigh frequency wave. The freedom to adjust the scale ofthe wavelet enables one to search for time structures ina given lightcurve that match the scaled wavelet. All oneneeds to do is to scan the entire lightcurve in a systematicway, at different scales, to probe for time structures thatmatch a given scale factor. The scanning of the lightcurveis achieved by the so-called translation factor which simplymoves the location of the wavelet function from time zeroby incremental time steps until the entire duration of thelightcurve has been covered. The result of each (translation)step is coded in terms of coefficients (essentially the productof the wavelet function and the lightcurve). These coefficients(in terms of the scale and translation factors) provide ameasure of the power in the signal. Clearly one only obtainsfinite power where there is significant overlap (convolution)of the wavelet function and some structure at the righttemporal scale in the lightcurve. The process of scanning thelightcurve for every scale is repeated and the appropriatecoefficients extracted.We used the Haar wavelet to represent the observedlightcurves: It is the simplest wavelet of the Daubechiesfamily, essentially a ’box’ function (i.e., no built-in oscilla-tory and/or exponential features) with the fewest vanishingmoments, the most compact support (Addison 2002), andis constant over its interval similar to the model assumedin the Bayesian block method (Scargle et al. 2013). Thevariance of the resulting detail coefficients (d jk ; AppendixA) is used to construct a logscale diagram, a plot of log ofvariance vs. frequency (octaves). For comparative purposes,we performed a number of extractions of the MTS usingthe Meyer wavelet available in the Wavelet Toolbox inMATLAB; the Meyer wavelet has oscillatory structurewith a decay feature and is significantly different in profilecompared to the ‘box’-like structure of the Haar wavelet. Wefound the results to be in excellent agreement over a timescale ranging from milliseconds to seconds.By using the logscale diagrams, we access timescales thatare presumably associated with different emission processes.White-noise processes appear as flat regions while theprocesses generating red-noise appear as sloped regions(see Figure 1). The transition between these regions occursat some characteristic timescale that we interpret as theminimal timescale i.e., MTS; A regression method is utilizedto determine the intersection of these two regions and hencethe MTS, j , in octaves . The octave scale is readily convertedto a real time scale by using the binning time of the lightcurve data (i.e., the MTS (s) ∼ j x timebin). MNRAS , 1–7 (2020)
K. Mohamed et al. l og V a r i an c e j j (octave)Frequency (Hz)Obs ID. 93408-01-25-09 Figure 1.
Logscale diagram for the observation 93408-01-25-09(4U1608-52): variance vs octave scale. Intersection of the red-noise(sloped line) and the white-noise (flat portion) provides a measureof the MTS.
The MTS is interpreted as the smallest temporal feature inthe lightcurve that is consistent with a fluctuation abovePoissonian noise. Moreover, we consider this timescale to beassociated more with the continuum rather than any spectralfeature such as a QPO or some other resonant-like behaviorin the underlying spectrum. In the frequency domain theMTS implies equivalence to the highest frequency componentin the signal at or just above the noise threshold. In order toexplore this implied connection, we use the standard Fouriertechnique to transform the lightcurves into PSDs and use asimple (broken) powerlaw (BPL) model to extract the cutofffrequency for each PSD. The cutoff frequency correspondsto the signal-noise threshold (defined by the intersectionof the red and white noise components). An example of aPSD fit with the BPL is shown in Figure 2 (upper panel).Admittedly, the BPL is not an ideal model to describeevery possible PSD: Indeed, considerable care has to betaken in the fitting procedure as a fair fraction of the PSDs,especially for the soft-intermediate and soft states, conveycomplex profiles and frequently exhibit an evolving spectralslope that progressively becomes shallower at the higherfrequencies (see lower panel of Figure 2). Nonetheless, theBPL model is convenient to use and does have a limited setof parameters, which makes the model and its parameters(in principle) considerably simpler to interpret in terms ofmore physically motivated models.The extracted lightcurves for Aql X-1 and 4U 1608-52are displayed in Figure 3. In addition, we show in Figure4 the corresponding hysteresis loops as function of thehardness ratio (computed as the ratio of counts in thefollowing bands respectively: 10 - 16 keV and 6 - 10 keV):These results are in excellent agreement with those reportedby Munoz-Darias et al. (2014). Specifically, the Aql X-1lightcurves exhibit almost identical and rapid rise for bothoutbursts. The decays however display different profiles,with the 1999 (blue) burst falling in rate almost immediatelyafter reaching the peak count rate whereas the 2000 (red)burst shows a much more steady decay that proceeds intwo stages, a slow decay followed by a precipitous drop inrate similar to that seen in the 1999 burst. The slow/steadyportion of the decay presumably coincides with a period of sustained balance between the effects of accretion andcooling. In the case of the 4U 1608-52, the rise portion,especially for the 2007 burst (green), where more observa-tions are available, indicates somewhat episodic accretionas the lightcurve shows structure and an overall gradual in-crease in count rate. The decay portions of both bursts of 4U1608-52 show a two-stage decay, a slow steady decay over sig-nificant period of time followed by a rapid drop in count rate.Our main results are displayed in Figures 5 and 6 where wehave plotted the extracted cutoff frequencies (converted to atime scale by simple inversion) vs MTS, and the hardness vsMTS respectively for the combined data. We make severalobservations regarding these results: A strong positivecorrelation between MTS and the cutoff frequency is clearlyevident. Interestingly, the individual hysteresis loops for thetwo sources show significant variation as function of hardnessratios, whereas the MTS-frequency correlation appears tobe universal. The slope of the best-fit line (Figure 5) to thecombined data is 1.01 ± ∼ σ ).This provides strong evidence that the two time scales areequivalent. In this paper, we have analyzed the available archival RXTEdata for the 2002-2007 and 1999-2000 outbursts respectivelyfor the NSXRBs 4U 1608-52 and Aql X-1. Following the pro-cedures described by Nowak (2000), Pottschmidt et al. (2003)and Belloni et al. (2005), we have constructed PSDs and haveextracted the cutoff frequencies by fitting the PSDs with aBPL model. We used the technique developed by MacLachlanet al. (2013a), that employs wavelet basis states, to extracta minimal time scale for lightcurves constructed for all theobservations of 4U1608-52 and Aql X-1. We summarize ourmain findings as follows: • We find a strong correlation between the MTS and therespective cutoff frequencies of the corresponding PSDs forboth sources • Although the hysteresis loops for the sources exhibit sig-nificant variation as function of the hardness ratios, the cut-off frequency-MTS correlation appears to be ’universal’ as
MNRAS , 1–7 (2020) inimal Timescales − − − − kp1.dat P = − . , P = . , P = − . E − , P = . , W V = . N = . edasonbas 19 − Oct − Frequency [Hz] P o w e r Obs ID. 70058-01-17-01Cutoff Freq. = 1.50 +/- 0.30 Hz − − − − kp1.dat P = − . , P = . , P = − . , P = . , P = . E − P = . , W V = . , N = . edasonbas 1 − Dec − Frequency [Hz] P o w e r Obs. ID 50049-01-04-00Cutoff Freq = 47.60 +/- 1.40 Hz
Figure 2.
PSDs for the observations 70058-01-17-01 (4U1608-52;upper panel) and 50049-01-04-00 (Aql X-1; lower panel) respec-tively, fitted with the BPL. Notice the change of slope of the fit inthe lower panel.
10 100 1000 0 20 40 60 80 100 120 140 P CU C oun t R a t e Time (Day) Aql X-1/1999Aql X-1/20004U1608-52/20024U1608-52/2007
Figure 3.
Lightcurves of Aql X-1 and 4U 1608-52. Each pointrepresents the average count rate per observation (for PCU2: theenergy band was 2-15 keV.) indicated by essentially an identical best-fit to the separatedatasets • The best-fit slope is unity and the offset is consistentwith zero (within 3 σ ); this provides strong evidence that thetwo time scales are equivalent and justify the assumption thatthey represent the signal-noise threshold where the noise ispredominantly of an intrinsic nature • The extracted timescale ranges from ∼
10 ms to 10 sec-onds; this places the range in the thermal and viscous groupof time scales, indicating a strong role of the accretion diskin the observed emission.
10 100 1000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P CU C oun t R a t e Hardness RatioAql X-1/1999Aql X-1/20004U1608-52/20024U1608-52/2007
Figure 4.
PCU2 count rate vs Hardness ratio for several hysteresisloops: hard and soft states are well separated. C u t o ff F r eq . [ s e c ] MTS [sec] Aql-X1/1999Aql-X1/20004U1608-52/20024U1608-52/2007best fit = 1.01+/-0.01
Figure 5.
PSD cutoff frequencies (as a time scale) vs the corre-sponding MTS. The solid line is the best fit to the data. M T S [ s e c ] Hardness Ratio Aql X-1/1999Aql X-1/20004U1608-52/20024U1608-52/2007
Figure 6.
MTS vs Hardness ratio for several hysteresis loops: hardand soft states are well separated.
ACKNOWLEDGEMENTS
This research is supported by the Scientific and Technologi-cal Research Council of Turkey (TUBITAK) through projectnumber 117F334. In addition, KM acknowledges financialsupport provided by the Egyptian Cultural and EducationBureau in Washington DC, USA. The Referee’s feedback wasconstructive and helped to clarify a number of issues.
MNRAS , 1–7 (2020)
K. Mohamed et al.
Table 1.
Cutoff frequency and MTS for 2002 and 2007 outburstsin 4U1608-52. Full version of the table is available online.Obs. Id. Cutoff Freq. δ Cutoff Freq. τ δτ ± Hz Hz sec sec
Table 2.
Cutoff frequency and MTS for 1999 and 2000 outburstsin Aql X-1. Full version of the table is available online.Obs. Id. Cutoff Freq. δ Cutoff Freq. τ δτ ± Hz Hz sec sec
DATA AVAILABILITY
The RXTE (Rossi X-ray Timing Explorer) data analyzedin the course of this study are available in HEASARC, theNASA’s Archive of Data on Energetic Phenomena. The dataunderlying this article are available in the article and in itsonline supplementary material.
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APPENDIX A: WAVELET TRANSFORMS
The wavelet analysis employed in this study, as with the fastFourier transform, begins with a light curve with N elements, X i = { X . . . X N − } , (A1)where N is an integer power of two. The light curve is con-volved with a scaling function, φ j,k ( t i ) , and wavelet func-tion, ψ j,k ( t i ) which are rescaled and translated versions ofthe original scaling and wavelet functions φ ( t i ) = φ , , and ψ ( t i ) = ψ , . Translation is indexed by k and rescaling is in-dexed by j . The rescaling and translation relation is given by ψ j,k ( t ) = 2 − j/ ψ (2 − j t − k ) . (A2)The scaling function acts as a smoothing filter for the inputtime-series and the wavelet function probes the time-seriesfor detail information at some time scale, ∆ t , which is twicethat of the finest binning of the data, T bin . In the analysis,the time scale is doubled ∆ t → t and the transform isrepeated until ∆ t = NT bin . In this analysis we choose theHaar (Addison 2002) scaling/wavelet basis because it has thesmallest possible support, has one vanishing moment, and isequivalent to the Allan variance (Howe & Percival 1995), al-lowing for a straightforward interpretation. Convolving thelight curve, X , with the scaling functions yields approxima-tion coefficients, a j,k = (cid:104) φ j,k , X (cid:105) . (A3)Interrogating X with the wavelet basis functions yields scaleand position dependent detail coefficients: The coefficients ofthe transform are written as d j,k = (cid:104) ψ j,k , X (cid:105) . (A4)The values which j and k assume obey the dyadic partitioningscheme (Mallat 1989; Addison 2002; Percival 2002). That is,for a time series whose number of elements is given by N =2 m , ≤ j ≤ m − , and ≤ k ≤ j − . (A5) MNRAS , 1–7 (2020) inimal Timescales Applying the dyadic partitioning scheme removes any redun-dant encoding of information by the wavelet transform coeffi-cients and guarantees orthogonality among the wavelet basisfor any change in j or k , (cid:104) ψ j,k , ψ j (cid:48) ,k (cid:48) (cid:105) = δ j,j (cid:48) δ k,k (cid:48) . (A6)It is interesting to point out that for the trivial N = 2 casethe Haar wavelet transform and the Fourier transform areidentical. A1 Logscale Diagrams and Scaling
Logscale diagrams are useful for identifying scaling and noiseregions. Construction of a logscale diagram for each GRBproceeds from the variance of detail coefficients (Flandrin1992), ρ j = 1 n j n j − (cid:88) k =0 | d j,k | , (A7)where the n j are the number of detail coefficients at a par-ticular scale, j . A plot of log variances versus scale, j , takesthe general form log ( ρ j ) = αj + constant , (A8)and is known as a logscale diagram. A linear regression ismade of each logscale diagram and the slope parameter, α ,(depicting a measure of scaling) is estimated. White-noiseprocesses appear in logscale diagrams as flat regions whilenon-stationary processes appear as sloped regions with thefollowing condition on the slope parameter, α > (Abry etal. 2003; Percival 2002; Flandrin 1989). A1.1 Statistical Uncertainties and Spurious Artifacts
We have considered the statistical uncertainties in the lightcurve by a typical bootstrap approach in which the squareroot of the number of counts per bin is used to generate anadditive poisson noise. A new poisson noise is considered foreach iteration through the bootstrap process. Spurious arti-facts due to incidental symmetries resulting from accidentalmisalignment (Percival 2002; Coifman 1995) of light curveswith wavelet basis functions are minimized by circularly shift-ing the light curve against the basis functions. Circular shift-ing is a form of translation invariant de-noising (Coifman1995). It is possible a shift will introduce additional arti-facts by moving a different symmetry into a susceptible lo-cation. Thus, our approach is to circulate the signal throughall possible values, or at least a representative sampling, andthen take an average over the cases which do not show spu-rious correlations. Both discrete Fourier and discrete wavelettransformations imply an overall periodicity equal to the fulltime-range of the input data. This can be interpreted to meanthat for a series of N elements, { X , X . . . X N − } then X is made a surrogate for X N and X is made a surrogate for X N +1 , and so forth. This assumption may lead to troubleif X is much different from X N − . In this case, artificiallylarge variances may be computed. Reverse-tail concatenationminimizes this problem by making a copy of the series whichis then reversed and concatenated onto the end of the original series resulting in a new series with a length twice that of theoriginal. Instead of matching boundary conditions like, X , X , . . . , X N − , X , (A9)we match boundaries as, X , X , . . . X N − , X N − , . . . , X , X . (A10)Note that the series length has thus artificially been in-creased to N by reversing and doubling of the originalseries. Consequently, the wavelet variances at the largestscale in a logscale diagram reflect this redundancy. Thisis the reason the wavelet variances at the largest scale areexcluded from least-squares fits of the scaling region.As noted, we use a bootstrap technique to estimatethe uncertainties on ρ (the variance of the signal) calcu-lated from the detailed coefficients. The basic algorithmis the following: from each original lightcurve, we select arandom sample of data points (equal to the dimension ofthe lightcurve). To this new set of data points, we add apoissonian contribution according to the count rate per eachbin — this new lightcurve is then run through our waveletanalyzer and the detailed coefficients are extracted whichenables the calculation of ρ for each scale factor j . Theprocess is repeated 100 times leading to a set of 100 ρ ’s foreach scale factor. The mean, minimum and maximum ρ ’s areextracted for each scale factor, and finally, we compute thestandard deviation for each scale factor in the standard way. This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000