Temporal evolution of long-timescale periodicities in ULX NGC 5408 X-1
aa r X i v : . [ a s t r o - ph . H E ] N ov Astronomy & Astrophysics manuscript no. 26182_am c (cid:13)
ESO 2018August 26, 2018
Temporal evolution of long-timescale periodicities in ULXNGC 5408 X-1
Tao An , , , Xiang-Long Lu , , and Jun-Yi Wang Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, Chinae-mail: [email protected] Key Laboratory of Cognitive Radio & Information Processing, the Ministry of Education, Guilin University ofElectronic Technology, Guilin 541004, China e-mail: [email protected] Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, ChinaReceived xx xxx 2015; accepted xx xxx 2015
ABSTRACT
Context.
NGC 5408 X-1 is one of the few ultraluminous X-ray sources (ULXs) with an extensive monitoring programin X-rays (a temporal baseline of 4.2 yr), making it one of the most suitable candidates to study the long-timescalequasi-periodic oscillations (QPOs).
Aims.
Previous timing analysis of the
Swift data of NGC 5408 X-1 led to detection of multiple periodicities ranging from2.6 d to 230 d. In this paper, we focus on the statistical significance and the temporal evolution of these periodicities.
Methods.
A time-series analysis technique in the time-frequency domain, the weighted wavelet Z -transform (WWZ),was employed to identify the periodicities and trace their variations with time. Results.
Three periodic components were detected from the WWZ periodogram, corresponding to periods of . ± . d, . ± . d and . ± . d. All three have statistical significance higher than 99.74% ( & σ ). The 2.65-dperiodicity is quite stable in the majority of the light curve, whereas it changes to a shorter 2.50 d period in the lastthird of the time interval covered by the light curve. The 115-d periodicity is the most prominent but appears variable.It starts at an initial value of ∼ Conclusions.
The long-timescale periodicities in NGC 5408 X-1 are most likely of super-orbital origin, and are probablyassociated with the precession of a warped accretion disc. The disc may have been broken into two distinct planes withdifferent precessing periods, i.e. the 189-d and 115-d periodicities corresponding to the outer and inner disc, respectively.
Key words. accretion, accretion discs – X-rays: binaries – X-rays: individual: NGC 5408 X-1 – methods: data analysis
1. Introduction
Ultraluminous X-ray sources (ULXs) comprise a popula-tion of bright non-nuclear X-ray point sources in externalgalaxies with luminosity > × erg s − , in excess of theEddington limit of a stellar-mass black hole (BH). Severaltheoretical models are invoked to explain the high X-rayluminosity, such as super-Eddington accretion by a stellar-mass black hole ( M BH < M ⊙ ; Motch et al. 2014) or sub-Eddington accretion by an intermediate-mass black hole(IMBH, M BH & M ⊙ ; e.g., Feng & Soria 2011; Pasham& Strohmayer 2013; Grisé et al. 2013). The central blackhole mass is crucial for distinguishing between the models.In particular, if quasi-periodic oscillations (QPOs) in theX-ray luminosity were tied to the binary orbital motion,the QPOs could be used to constrain the BH mass (e.g.Pasham et al. 2014).Periodic X-ray modulations with timescales of millisec-onds and seconds in X-ray binaries (XRBs) have been stud-ied for over 30 years by now. Long-timescale periods of tensto hundreds of days have also been observed in both Galac-tic and extragalactic XRBs, although the long-period bi-naries are relatively fewer. NGC 5408 X-1 is one of the brightest ULXs, with an X-ray luminosity of L X ≈ × erg s − in the 0.3 – 10 keV band (Strohmayer 2009a). Itis one of the few ULXs that have been monitored for ex-tended time intervals. Strohmayer (2009a) discovered 115-dperiodic modulations in the X-ray flux from ∼
500 days of
Swift data, and suggested that the 115-d periodicity cor-responds to the orbital period of the black hole binary,which contains an IMBH with a mass exceeding M ⊙ (Strohmayer 2009b). However, the black hole mass andphysical origin of this 115-d QPO in NGC 5408 X-1 are stillissues of debate. Other authors argue against the existenceof an IMBH in NGC 5408 X-1 (e.g. Middleton M.J. & F.E.2011). Based on the period search in the He II 4686 lineshift, Cseh et al. (2013) constrained the BH mass in NGC5408 X-1 to be less than ∼ M ⊙ . Moreover, Foster et al.(2010) suggested that the 115-d periodicity could be super-orbital in nature. Recently, Pasham & Strohmayer (2013)and Grisé et al. (2013) analysed additional Swift data andperformed the periodicity analysis over a much longer timebaseline ( ∼ Article number, page 1 of 7 &A proofs: manuscript no. 26182_am were detected by Grisé et al. (2013). These extra periodshave comparable or even higher power than the 115-d pe-riod. In addition, both Pasham & Strohmayer (2013) andGrisé et al. (2013) noticed repeating dips in the light curveon an average interval of 250 days.In this paper, we focus on the statistical confidenceof the detected periodicities and their temporal evolu-tion. Section 2 presents the periodicity analysis using theLomb–Scargle (LS) periodogram and weighted wavelet Z -transform (WWZ). Section 3 discusses the temporal evolu-tion of individual periodic components. A summary is givenin Section 4.
2. Periodicity analysis
The X-ray light curve of NGC 5408 X-1 used in the presentanalysis was obtained from the published
Swift data byGrisé et al. (2013). Details of the observations and data re-duction process are given in Pasham & Strohmayer (2013)and Grisé et al. (2013). The monitoring lasted for about4.2 yr (1532 d) from 2008 April 9 to 2012 June 19. A to-tal of 354 observations have been made by the
Swift /XRT.The typical time interval between subsequent observationsis about 3.5 d during the sessions in 2008 and 2009, and6.4 d from 2010 to 2011 May. From 2011 May, there weretwo periods of quasi-daily observations (2011 May 1 – 2011August 24, and 2012 January 29 – 2012 February 26).Previous periodicity studies of NGC 5408 X-1 aremostly based on Fourier transform techniques (e.g. Fos-ter et al. 2010; Pasham & Strohmayer 2013; Grisé et al.2013). Conventional Fourier-based periodicity analyses per-form least-squares fits to the entire time series with a fixedcombination of trigonometric functions. These analyses canonly provide the frequency information, and the tempo-ral variation information of the periodic component is lostin this transformation. Moreover, any short-lived period-icity would be weakened in the overall integrated powerspectrum since it only appears significantly in a certaintime span. In the following, we will not only present thefrequency-domain LS periodicity analysis, but also give thewavelet transform power spectrum in the time-frequencydomain.
The LS periodogram (Lomb 1976; Scargle 1981) is a widelyused analysis technique in searching for periodic compo-nents from unevenly sampled time series. It works wellwhen the periodicity parameters (period, amplitude, andphase) remain constant over the whole time span. Becauseof bad observing conditions and complicated underlying as-trophysical processes, astronomical time series are often ir-regular and contaminated by random noise. As a result,instead of being interpreted as a periodic signal, an alter-native explanation for peaks in the power spectrum is thatthey result from noise. Random measurement uncertaintiescould generate Poisson noise, which behaves like white noisein the power spectrum. Another class of noise is related tofrequency-dependent physical processes, commonly knownas red noise (so-called flicker noise: Press 1978). These twotypes of noise are different in the power spectral density(PSD). The PSD of both white and red noises can be de-scribed by a power law function. The white noise spectrum -3 -2 -1 -3 -2 -1 Frequency(1/day) P o w e r P3 P2 P1LSTheo. AR(1)MC99.74%
Fig. 1.
LS periodogram spectrum (green line) of NGC 5408 X-1. The red dashed curve represents the theoretical AR(1) powerspectrum, and the blue dot-dashed curve marks the 99.74% (or σ ) statistical confidence level derived from Monte Carlo simu-lations performed 5000 times. Three peaks (P1, P2, and P3) areabove the 99.74% statistical significance level. P o w e r (a) LS of NGC 5408 X-1
P1 P2 P30100020003000 P o w e r P2 P3(b)
Integrated power of WWZPeriod(day) E op c h ( M J D - ) (c) P2 P3WWZ of NGC 5408 X-1
Fig. 2.
The WWZ power spectrum of NGC 5408 X-1 (bottompanel) in the test period range of 2 – 300 d. The WWZ spectrumis created using a decay parameter c = 0 . . For comparison,the LS periodogram is shown in the top panel, and the time-integrated WWZ power spectrum is shown in the middle panel. has a power law index α = 0 , i.e. P ( f ) ∝ f , and it of-ten shows suspicious peaks at high frequency. The PSD ofthe red noise follows a power law with α = 1 − . The rednoise spectrum shows relatively large stochastic peaks atlow frequencies.Hasselmann (1976) demonstrated that the climatic rednoise signature can be explained by the first-order autore-gressive (AR(1)) process. Robinson (1977) investigated theAR(1) process for discrete time series: x ( t i ) = ρ i x ( t i − ) + ǫ ( t i ) , where the autocorrelation coefficient ρ i = exp( − ( t i − t i − ) /τ ) , and ǫ ( t i ) is the Gaussian white noise with zeromean and variance σ ǫ ≡ − exp( − t i − t i − ) /τ ) . When Article number, page 2 of 7ao An et al.: Temporal evolution of periodicities in NGC 5408 X-1 P o w e r LS of NGC 5408 X-1 (a) P1 P o w e r P1(b)
Integrated power of WWZPeriod(day) E op c h ( M J D - ) P1 (c) WWZ of NGC 5408 X-1
Fig. 3.
The WWZ power spectrum of NGC 5408 X-1 (bottompanel) in the test period range of 2.4 – 2.8 d. The WWZ spec-trum is created using a decay parameter c = 0 . . The LSperiodogram and the time-integrated WWZ power spectrum arealso shown for comparison. the ρ i coefficients are zero, the AR(1) time series becomewhite noise type, i.e. x ( t i ) = ǫ ( t i ) . The parameter τ cor-responds to the characteristic timescale of the AR(1) pro-cess. This parameter is calculated from the average auto-correlation coefficient ρ and mean time interval ∆ t , basedon the relationship ρ ≡ exp( − ∆ t/τ ) , and ρ is estimatedfrom the least-squares algorithm of Mudelsee (2002). Basedon the above calculations, we can see that the variance σ ǫ is also dependent on τ ; this ensures that the AR(1) pro-cess is stationary and has unit variance. The power G ( f i ) corresponding to the stochastic AR(1) process is given as G ( f i ) = G (1 − ρ ) / (1+ ρ − ρ cos( πf i /f Nyq )) , where f i de-notes discrete frequency up to the Nyquist frequency f Nyq and G is the average spectral amplitude (Schulz & Stat-tegger 1997; Schulz & Mudelsee 2002).To discriminate the periodic signal from the noisy spec-trum, we compared the LS periodogram (green line in Fig.1) of NGC 5408 X-1 with the theoretical red noise spectrum(denoted by the red dashed line in Fig. 1), which is gener-ated from the AR(1) process described above. The spec-trum appears flat in the low-frequency segment, suggestingthat the red noise does not have significant effect on the pe-riodogram. The noisy spectrum in the high-frequency endis dominated by white noise. The blue dot-dashed line rep-resents 99.74% (or ∼ σ ) statistical confidence level, whichwas derived from 5000 Monte Carlo tests. More details onthe LS method and the production of the red noise spec-trum can be found in An et al. (2013). Three prominentpeaks are detected . ± . × − d − (P1 = . ± . d), . ± . × − d − (P2 = . ± . d), and . ± . × − d − (P3 = . ± . d). They areall evident above the 99.74% confidence level (Fig. 1). Z -transform In contrast to the traditional Fourier-based technique ofperiodicity analysis, the wavelet transform is remarkablefor its localization ability in both time and frequency do- P o w e r (a) LS of white noise P o w e r (b) Integrated power of WWZPeriod(day) E op c h ( M J D - ) (c) WWZ of white noise P o w e r LS of red noise(d) P o w e r (e) Integrated power of WWZPeriod(day) E op c h ( M J D - ) (f) WWZ of red noise Fig. 4.
Power spectra of simulated white noise (top) and rednoise time series (bottom) with the period range 2 – 300 d. (a):LS periodogram of the average of 5000 simulated white noisetime series; (b): integrated WWZ power spectrum of 5000 sim-ulated white noise time series; (c): WWZ power spectrum ofthe average of 5000 simulated white noise time series; (d): LSperiodogram of the average of 5000 simulated red noise timeseries; (e): integrated WWZ power spectrum of 5000 simulatedred noise time series; (f): WWZ power spectrum of the averageof 5000 simulated red noise time series. mains. In particular, Foster (1996) developed an alternativeto the conventional wavelet technique, the weighted wavelet Z -transform (WWZ), which is more suitable for analysingirregularly-sampled time series. The WWZ is based on Mor-let wavelet (e.g. Grossmann et al. 1989) and has becomea useful tool in fields, such as fluid dynamics (e.g. Farge1992), climate change (e.g. Johnson 2010), pulsar timingnoise (e.g. Lyne et al. 2010), AGN variability (e.g. Hovattaet al. 2008; An et al. 2013; Wang et al. 2014), etc. TheWWZ projects the signals onto three trial functions: a con-stant function ( t ) = 1 , sin( ω ( t − τ )) , and cos( ω ( t − τ )) ,where ω is the scale factor (frequency) and τ is the timeshift (position in time sequence). Furthermore, statisticalweights w α = exp( − cω ( t α − τ ) ) were used for the pro-jection on the trial functions, where the constant c con- Article number, page 3 of 7 &A proofs: manuscript no. 26182_am P o w e r LS of white noise (a)0100020003000 P o w e r (b) Integrated power of WWZPeriod(day) E op c h ( M J D - ) (c) WWZ of white noise P o w e r LS of red noise (d)0100020003000 P o w e r (e) Integrated power of WWZPeriod(day) E op c h ( M J D - ) (f) WWZ of red noise
Fig. 5.
Power spectra of simulated white noise (top) and rednoise time series (bottom) with the period range 2.4 – 2.8 d. (a):LS periodogram of the average of 5000 simulated white noisetime series; (b): integrated WWZ power spectrum of 5000 sim-ulated white noise time series; (c): WWZ power spectrum ofthe average of 5000 simulated white noise time series; (d): LSperiodogram of the average of 5000 simulated red noise timeseries; (e): integrated WWZ power spectrum of 5000 simulatedred noise time series; (f): WWZ power spectrum of the averageof 5000 simulated red noise time series. trols how rapidly the wavelet decays. The WWZ power isdefined as
W W Z = ( N eff − V y / V x − V y ) , where N eff is a quantity representing the local number density of thedata points, and V x and V y are the weighted variations ofthe data and the model functions, respectively. By apply-ing this modification, WWZ follows an F -distribution with N eff -3 and 2 degrees of freedom. For more details on themethodology, see Foster (1996).According to Fig. 1, the characteristic frequencies arebetween × − d − and × − d − . We therefore limitedthe periodicity search window in WWZ from . × − d − to × − d − (corresponding to periods from 2 to300 d). As mentioned before, the constant decay factor c controls how rapidly the exponential term ( − cω ( t α − τ ) ) in the wavelet function decreases significantly in a single Peak power N u m be r Peak power
MC: white noise
Peak power
Peak power N u m be r MC: red noise
Peak power
Peak power
Fig. 6.
Statistical significance test of WWZ-detected periodsin Fig. 2-c and Fig. 3-c. The x -axis represents the peak power ofsimulated periodograms at the periodicity under investigation.The y -axis gives the occurrence number within equally wide binsof 1. Top : histogram obtained from 5000 simulated white noisetime series.
Bottom : histogram obtained from 5000 simulatedred noise time series. The vertical red dashed lines denote thepeak powers of the periodic components identified in Figs. 2 and3. In the regions to the right of the line, the WWZ power of thesimulated fake light curves is higher than the observed value.The ratio of the accumulated count of these light curves to thetotal number used in the MC test (i.e. 5000) determines theprobability of the observed periodicity caused by the noise. cycle π/ω . In practice, c sets a constraint to the width ofthe frequency window. A large c is suitable for identifyinglow-frequency components, and a small c is better for high-frequency components. In each plot (Figs. 2 and 3), theWWZ power spectrum is shown in the bottom panel. Thepower of wavelet transforms at a certain test period ( x -axis)and location in time ( y -axis) is shown in a colour scale. Theintegrated power over the observing time span is shown inthe middle panel, and the LS periodogram is presented inthe top panel as a comparison.The WWZ power spectrum for NGC 5408 X-1 with de-cay parameter c = 0 . and search range 2 – 300 d areshown in Fig. 2-c. Two periodic components are evidentat . ± . d (P2) and . ± . d (P3). Theyare consistent with the periods obtained from the Lomb–Scargle periodogram. The relatively larger uncertainty inthe WWZ-derived periods mainly results from the tempo-ral variation of the periodicities. Because of the high decayparameter c , P1 is not detected in Fig. 2-c. The additionalWWZ power spectrum for NGC 5408 X-1, with the periodranging from 2.4 to 2.8 d and c = 0 . , is given inFig. 3. The presence of P1 ( . ± . d) is evident for themajority of the time span of the light curve and the periodremains stable (Fig. 3-c). The integrated power spectrumof the WWZ in Fig. 3-b shows an excellent agreement withthe LS periodogram in Figs. 1 and 3-a. The spectrum disap-pears in the last third of the light curve, however, anotherperiodic component appears at ∼ Article number, page 4 of 7ao An et al.: Temporal evolution of periodicities in NGC 5408 X-1
As in to Sect. 2.1, we also evaluated how the noise af-fected the WWZ power spectrum. The statistical confidencetests were performed using Monte Carlo (MC) simulations(Linnell Nemec & Nemec 1985). We first generated 5000white noise time series, which had the same number of sam-pling points, temporal baselines, and sampling intervals asthe observed light curve. We then calculated the WWZ pe-riodograms for the 5000 white noise time series using thesame parameters as for Figs. 2-c and 3-c. The averagedwhite noise power spectrum is shown in the top of Figs. 4and 5. The red noise power spectrum was also obtainedin the same way (bottom of Figs. 4 and 5). No significantpeaks were found at the positions of P1, P2, and P3, ineither of the white noise or red noise spectra. For a quan-titative evaluation of the statistical confidence, we countedthe number of simulated light curves, which show higherpower than the observed periodic components (P1, P2, andP3) at their peak positions. The ratio of the accumulatednumber to the total number of 5000 test light curves givesthe probability of the false detection (i.e. the detected peakcould be due to random noise). The percentage of the falsedetections in each case is below 0.1 (see Fig. 6). There-fore, the possibility that P1, P2, and P3 are generated bystochastic noise can easily be ruled out.
3. Discussion
The periodicity analysis in Sect. 2 results in the detection ofthree possible periods. The properties of individual periodiccomponents are discussed in the following:1.
P1 = 2.65 d . This is the most prominent periodic com-ponent in both the LS periodogram and the WWZpower spectrum, which confirms the first detection byGrisé et al. (2013) using the LS technique. The com-ponent P1 is remarkable in the first two-thirds of theinterval covered by the light curve. It starts with aslightly longer period of 2.67 d, then bifurcates into twobranches (2.65 d and 2.68 d). After MJD ∼ ∼ ∼ P2 = 115 d . This periodicity was first discovered byStrohmayer (2009a) based on a 485-day light curve andconfirmed by Han et al. (2012). Grisé et al. (2013) andPasham & Strohmayer (2013) revisited the periodicityanalysis of P2, using much longer light curves of 4.2-yr. They found that P2 has significantly weakened inthe second half of the monitoring interval. In our time-frequency domain power spectrum (Fig. 2), P2 is the most significant in the first half of the light curve. FromMJD ∼ ∼ ∼ P3 = 189 d . This periodicity has an amplitude compa-rable to that of P2 in the LS periodogram. It was alsodetected by Grisé et al. (2013). Compared with vari-able P1 and P2, P3 is rather persistent throughout thewhole light curve and steadily varies from 193 d to 181 d(depicted in cyan-coloured dashed line in Fig. 2-c). Thestatistical confidence of P3 is & σ . The WWZ powerspectra simulated with either white noise or red noise donot show any fake components around the period of 189d, suggesting that P3 is associated with some intrinsicphysical process.4. Other candidate periodicities . Both Pasham &Strohmayer (2013) and Grisé et al. (2013) foundX-ray dips in the light curve in five successive cycleswith an average interval of ∼
250 d. The 250-d repeti-tion is not detected in our LS or WWZ power spectra.Moreover, the recurrence of the 25-d periodicity was notconfirmed in targeted follow-up monitoring (Grisé et al.2013), therefore, it could not be a strict periodicityassociated with orbital motion of the binary system. Apossibility is that the 250-d repeating dips are due toperiodic obscuration of the central X-ray source by atilted, precessing accretion disc as suggested by Griséet al. (2013).In summary, three periodicities (P1 = 2.65 d, P2 = 115d, and P3 = 189 d) were detected with high statistical sig-nificance. For the majority of the observation interval, P1and P3 are persistent and prominent, while P2 appears vari-able. There is a debate in the literature about the physicalnature of these long-timescale periodicities. Any theoreti-cal model should satisfactorily explain the multiplicity andlong timescale of the periodicities in NGC 5408 X-1.The periodicity associated with the orbital motion of thebinary system should be persistent and stable. Strohmayer(2009a) argued that the 115-d period (P2) manifested theorbital period of the binary system. However, our periodic-ity analysis showed that the periodicity P2 (115 d) is vari-able in amplitude and period, leaving P1 (2.65 d) and P3(189 d) as the possible orbital periods. The period search inthe He II 4686 line resulted in no periodic signal between1 d and 300 d (Cseh et al. 2013). Moreover, the gradualdecrease of P3 is inconsistent with the stable orbital periodargument. All these evidences argue against long orbitalperiods in NGC 5408 X-1.
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Others proposed that the long-term periodic X-ray mod-ulations could be super-orbital (e.g. Foster et al. 2010; Griséet al. 2013). In this case, the required BH mass in NGC 5408X-1 would drastically decrease. Cseh et al. (2013) set an up-per limit of ∼ M ⊙ for the black hole mass based on theradial velocity curve of the He II emission line. A variety ofphysical processes may be responsible for the super-orbitalperiodic modulations, such as precessing jet (e.g. in SS 433,Begelman et al. 2006), warped or precessing disc inducedby radiation pressure (e.g. Ogilvie & Dubus 2001; White-hurst & King 1991), or tidal interactions (e.g. Homer et al.2001), and other mechanisms summarized by Charles et al.(2008), Kotze & Charles (2012), and Grisé et al. (2013).Currently, there is only an extended radio nebula detectedin NGC 5408 X-1 (Cseh et al. 2012) and the existence ofa precessing jet still lacks robust evidence. Moreover thereare multiple X-ray periods. This is not consistent with aprecessing jet model either.In the framework of the warped disc model, the tempo-ral evolution of the detected periodicities in NGC 5408 X-1would indicate unstable or chaotic warping. Similar phe-nomena have been found in Galactic BH-XRBs. For exam-ple, Kotze & Charles (2012) investigated the time variationsof the QPOs in a sample of high-mass and low-mass BHbinaries using a dynamic power spectrum technique. Theauthors found that the majority of the sample shows vari-able, unstable QPOs, or even intermittent QPOs in somecases. The theoretical calculations of the radiation-drivenwarped disc suggest that stable, persistent precessing discsare relatively rare compared to the commonly seen unsta-ble discs (Ogilvie & Dubus 2001). High-mass XRBs, whichhave larger accretion discs, are more likely to be unstable(Kotze & Charles (2012); see also the discussion in Griséet al. (2013)). The variations of the periodicities in NGC5408 X-1 could also be attributed to the non-linear dynam-ical process of the warped disc driven by tidal or radiationpressure (Ogilvie & Dubus 2001).During the review process of this paper, a paper by Linet al. (2015) appeared in astro-ph. The paper analysed the Swift light curves of four ULXs including NGC 5408 X-1 using various time series analysis techniques to searchfor long-term periodicities. The authors detected severallow-frequency quasi-periods, and also argued that theselong-term quasi-periods are super-orbital and related to thestructure of accretion discs. The methods and conclusionsfor NGC 5408 X-1 in the Lin et al. paper are consistentwith the present paper.In addition, if the unstable warped disc model is invokedto interpret the multiple periodicities in NGC 5408 X-1, itrequires either variable disc warping with a variety of timescales or multiple discs at work. The former mechanismwas used to explain the multiple periods found in GalacticX-ray BH binaries, such as Cyg X-1, GRS 1747 − −
0, and X1730 −
333 (Kotze & Charles 2012), in whichthe periodicities may be produced by ˙ M variations withdifferent timescales. However, this seems less likely here be-cause the spectral properties of NGC 5408 X-1 suggest thatthe mass accretion rate is quite stable. An alternative pos-sibility is that both the 189-d and 115-d periodicities aresuper-orbital periods of the warped disc. The accretion discaround a spinning BH is warped by the Lense–Thirring ef-fect, and under some circumstances, non-linear fluid effectscan break the disc into inner and outer parts, which precesswith different periods (Nixon & King 2012). If this mech- anism works for NGC 5408 X-1, the persistent, relativelystable P3 (189 d) is associated with the outer disc and theunstable P2 is related to the inner disc. The 2.65-d peri-odicity P1 may have different origins from P2 and P3. Thepossibility that P1 is the orbital period of the binary systemstill stands. Further X-ray monitoring, with dense regularsampling and a time interval of less than one day, is neededto verify this speculation.
4. Summary
We performed a periodicity search in the X-ray lightcurve of NGC 5408 X-1, using the frequency domain,Lomb–Scargle periodogram, and the time-frequency do-main, wavelet weighted Z -transform techniques. Threeprominent peaks were detected, corresponding to charac-teristic periods of 2.65 d, 115 d, and 189 d. Monte Carlotests of the detected periodicities show that all three havehigh statistical significance ( > σ ). The 115-d periodicityis prominent in the first half of the light curve; the periodincreases to 136 d between epoch MJD ∼ ∼ ∼ ∼ Swift has been carried outfrom mid-December 2013 to the end of April 2014 with acadence of about two days, with the initial motivation tocheck the re-appearance of the X-ray dips. This follow-upmonitoring is also important to verify the persistency of P1(2.65 d).
Acknowledgements.
We thank the anonymous referee for his/her con-structive comments, which greatly improved the manuscript. Thiswork is supported by the China Ministry of Science and Technol-ogy 973 programme under grant No. 2013CB837900, and the China–Hungary Exchange Program of the Chinese Academy of Science andShanghai Rising Star programme. The authors are grateful to DávidCseh, Sándor Frey, and Prashanth Mohan for their helpful commentson the manuscript.
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