MNRAS 484, 3935–3940 (2019) doi:10.1093/mnras/stz122Advance Access publication 2019 January 12

A wavelet analysis of photometric variability in Kepler white dwarf stars

S. R. de Lira ,1‹ J. P. Bravo,1,2 I. C. Leao,1 A. D. da Costa,1,3 B. L. Canto Martins,1‹

D. B. de Freitas4 and J. R. De Medeiros1‹

1Universidade Federal do Rio Grande do Norte, Departamento de Fısica, 59072-970 Natal, RN, Brazil2Instituto Federal de Educacao, Ciencia e Tecnologia do Rio Grande do Norte, 59115-000 Natal, RN, Brazil3Universidade da Integracao Internacional Lusofonia Afro-Brasileira, 62790-000 Redencao, Ceara, Brazil .4Universidade Federal do Ceara, Departamento de Fısica, Caixa Postal 6030, Campus do Pici, 60455-900 Fortaleza, Ceara, Brazil

Accepted 2019 January 9. Received 2019 January 04; in original form 2018 March 9

ABSTRACTThis work describes a wavelet analysis of 14 Kepler white dwarf stars, in order to confirmtheir photometric variability behaviour and to search for periodicities in these targets. From theobserved Kepler light curves we obtained the wavelet local and global power spectra. Throughthis procedure, one can perform an analysis in the time–frequency domain rich in detail, andso obtain a new perspective on the time evolution of the periodicities present in these stars.We identified a photometric variability behaviour in 10 white dwarfs, corresponding to periodvariations of ∼2 h to 18 d: among these stars, three are new candidates and seven, earlieridentified from other studies, are confirmed.

Key words: stars: white dwarfs – methods: data analysis – techniques: photometric.

1 IN T RO D U C T I O N

The observations of the space missions Kepler (Koch et al. 2010) andCoRoT (Baglin et al. 2007) are revolutionizing our understandingof stellar variability and generating new insights into the correlationbetween photometric variability and rotation of main-sequence stars(e.g.; De Medeiros et al. 2013; Nielsen et al. 2013; Walkowicz &Basri 2013; McQuillan, Mazeh & Aigrain 2014; Leao et al. 2015;Paz-Chinchon et al. 2015; Davenport 2017) and beyond the mainsequence (e.g. Mosser et al. 2012; Van Saders & Pinsonneault 2013;De Medeiros et al. 2013; Costa et al. 2015). These works haveshown that rotation is a major constraint on the study of the netangular momentum evolution as well as on the angular momentumtransport from core to surface and expanding envelope. In addition,the normality of the Sun’s rotation with respect to the main-sequencestars with surface physical parameters close to solar values (DeFreitas et al. 2013; Leao et al. 2015) and bimodality in the rotationperiod distribution for main-sequence stars (McQuillan et al. 2014;Davenport 2017) have also emerged from data acquired by thesespace missions.

The observation of photometric modulation also has a uniquepotential to probe for rotation in white dwarfs (hereafter WDs), oneof the few remaining traces of the physics of the formation processof WDs (Kawaler 2004, 2015). Such a procedure is more easilyapplied to magnetic white dwarfs, which comprise approximately20 per cent of white dwarf stars presently reported in the literature(e.g. Kawka et al. 2007; Kepler et al. 2013). Typically, the observed

� E-mail: suzierly@ufrn.edu.br (SRL); brunocanto@fisica.ufrn.br (BLCM);renan@fisica.ufrn.br (JRM)

rotation period for these stars ranges from less than one hour to afew days (e.g. Kawka et al. 2007; Brinkworth et al. 2013; Kawaler2015). Nevertheless, photometric modulation in normal whitedwarfs can also be revealed by surface dynamo activity, producinglocalized star-spots (e.g. Kawaler 2015). Finally, asteroseismologyanalysis, based on high photometric precision, can be used for themeasurements of the rotation period of white dwarfs, showing theappropriate range of effective temperature in which they undergonon-radial g-mode pulsations.

Maoz et al. (2015) have conducted a fast Fourier transformanalysis of Kepler light curves (hereafter LCs) in 14 WDs, searchingfor photometric variability. Those authors detected periodic signalsranging from two hours to 10 days for seven among the 14 targetsthat they analysed. These signals were detected in both short- andlong-cadence-data LCs from the Kepler mission. In the presentwork, we perform a time–frequency analysis of the same KeplerWD datasets studied by Maoz et al. (2015) in the interest of pointingout the spectral components and following their time evolution,which leads to short- and long-lived signature formation that couldbe linked to WD rotation or activity. For this purpose, since thecontinuous wavelet transform (WT) is a powerful mathematicaltool for non-stationary signals, we used a sixth-order Morlet wavelet(Torrence & Compo 1998) to obtain the energy distribution of eachLC computed as a local wavelet spectrum, mapping short- and long-term features.

This paper is organized as follows. Section 2 presents ourstellar sample and the methods used for this analysis. Section 3brings the primary results, in which the periods detected from thewavelet method are analysed and compared with those obtainedby Maoz et al. (2015). Finally, our conclusions are provided inSection 4.

C© 2019 The Author(s)Published by Oxford University Press on behalf of the Royal Astronomical Society

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Figure 1. Light curves with additional treatment, wavelet local and globalspectra of KIC 5769827 (upper panel) and KIC 11604781 (lower panel).Contour levels are 90 per cent, 80 per cent, ..., 20 per cent and 10 per centof the map maximum. The variability periods are illustrated by the reddashed lines in the global spectra. A plausible aliasing period is displayedin blue.

2 SAMPLE SELECTION AND LC ANALYSIS

The NASA Kepler mission provides LCs in 18 runs (also knownas quarters) with standard treatment, such as simple aperturephotometry (SAP) data, and with a more refined treatment, inwhich instrumental effects were removed, such as pre-search dataconditioning (PDC) (Stumpe et al. 2012; Smith et al. 2012). For ourpurposes, we have chosen 14 WDs with long (30 min) and short (1min) cadence LCs retrieved from the Kepler Data Archive.1 Thesetargets were previously studied by Østensen et al. (2011, 2010),searching for pulsation among compact objects, and by Maoz et al.(2015), focusing on photometric variations using the FFT.

As underlined by Maoz et al. (2015), their analysis made useof Public Releases 14–21 of the long-cadence data and PublicRelease 21 of the short cadence, downloaded from the Keplermission archive. However, it seems that no additional treatmenton the LCs was performed by the authors. The detection thresholdfor periodicity varies from star to star, depending on its brightness,the number of quarters of observations, and the final cleaning of theLCs. To avoid possible distortions in the spectral wavelet maps, andconsequently changes in the power spectra, an additional treatmentof outlier removal and instrumental trend correction was performedfor the PDC LCs (e.g. Paz-Chinchon et al. 2015). The single long-term LC was obtained by assembling the available quarters for eachobject, based on the procedure of Banyai et al. (2013). The mapsignatures are then well defined, inducing more reliable variabilityperiodicities.

1http://archive.stsci.edu/kepler/

Figure 2. Light curves with additional treatment, wavelet local and globalspectra of Kepler WDs that were classified as periodic by Maoz et al.(2015), and their results reproduced via wavelet (from top to bottom) for KIC6669882, KIC 6862653, and KIC 11337598. Contour levels are 90 per cent,80 per cent, ..., 20 per cent and 10 per cent of the map maximum. Thevariability periods are illustrated by the red dashed lines in the global spectra.A plausible aliasing period is displayed in blue.

Wavelet power spectra were analysed based on proceduresdescribed by Bravo et al. (2014). The local map is defined asthe signal energy distribution in time and frequency, while theglobal power spectrum is computed from the time integration ofthe wavelet map. Examples of these are displayed in Section 3. TheWT application and the variability analysis was performed for eachquarter of the whole sample, as well as for merged LCs in short- andlong-cadence modes. Since WD time series present some gaps, thewavelet map was obtained individually for each set of continuousdata. Thus, the global spectrum was produced by considering theweighted average by time span of these parts (Costa et al. 2015)for the targets KIC 4829241, KIC 5769827, KIC 6669882, KIC6862653, KIC 8682822, KIC 10420021, KIC 11604781 and KIC11822535. For these stars, the data treatment is related to the long-cadence dataset. The objects of our sample are listed in Table 1,

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A wavelet analysis of white dwarf stars 3937

Figure 3. Upper panel: Light curve with additional treatment, wavelet localand global spectra of KIC 8682822. The period of 4.66 d with low amplitude,illustrated by a red solid line, is present in a short time interval, which is notenough to confirm a periodic variability. Lower panel: wavelet local spectraof KIC 8682822 of different short datasets. After assemblage, the weightedaverage by time span points to a 5-d periodicity with the highest amplitude(red solid line). Contour levels are 90 per cent, 70 per cent, 50 per cent,30 per cent, and 10 per cent of the map maximum. The 11-, 14-, 28-, and26-d periodicities, displayed in blue in both maps, are possibly aliasingperiods.

including their Kepler magnitude, Kp; their spectral type, effectivetemperature, Teff, and surface gravity, log g, (e.g. Østensen et al.2011; Doyle et al. 2017; Hallakoun et al. 2018); the periodicitiescomputed by Maoz et al. (2015) using the FFT; and those obtainedusing our wavelet procedure.

Furthermore, we have applied the FFT to our data after additionaltreatments to confirm the periods reported in Table 1, as well as theLomb–Scargle method (Scargle 1982) to verify the variability na-ture of some targets, which seems to be quasi-periodic after a visualinspection of the wavelet map signature although classified as non-periodic by Maoz et al. (2015). In each Lomb–Scargle periodogram,main peaks were identified with a false alarm probability (FAP)(Horne & Baliunas 1986) less than 0.01, or a significance levelgreater than 99 per cent. This FAP estimation was used with cautionjust as a qualitative indication of reliability because it assumesrandom noise in the LCs, which actually have more complex noisesignatures.

3 R ESULTS AND DISCUSSION

Considering that the observed rotation periods in white dwarfs rangefrom less than one hour to a few days (Kawaler 2015), in the presentwork we are mostly searching for short periodicities. Our waveletprocedure confirms the periodic modulation found by Maoz et al.(2015) for the stars KIC 5769827, KIC 6669882, KIC 6862653,

KIC 8682822, KIC 11337598 and KIC 11604781, with periodicitiessimilar or close to the values reported by those authors, as illustratedby their wavelet maps in Figs 1–3. Nevertheless, for the WD KIC11514682 we found a period of 17.59 d, in contrast with Maozet al. (2015) who computed a 9.89 d period. Among the seven WDstars classified by Maoz et al. (2015) as non-periodic, our waveletanalysis reveals unambiguous periodicities for three of them, KIC4829241, KIC 10420021 and KIC 11822535, with 16.04 d, 11.70 dand 13.83 d, respectively. The computed wavelet periodicities, alongwith those from Maoz et al. (2015), are listed in Table 1.

The results obtained using the long-cadence data are in agreementwith those found making use of short-cadence data. However, forbetter viewing of the variability signature in the WD wavelet maps,and hence a clear identification of periodicities, we have givenpriority to the display of long-cadence datasets. Thus, the waveletperiods in Table 1, also displayed in the respective wavelet globalspectra, are related to the Kepler long-cadence mode. One exceptionis the WD KIC 11337598, for which variability is better defined inthe wavelet spectra using short-cadence data.

Let us underline that, except for KIC 8682822 and KIC 11822535,the level of persistence of the periodicity in the wavelet maps of theanalysed stars is larger than 50 per cent, a fairly significant fractionof the entire dataset. For KIC 8682822 and KIC 11822535 thefraction persistence is about 5 per cent and 40 per cent, respectively.In the following we discuss a few particular features emerging fromour wavelet analysis, in particular for those stars with photometricvariability revealed in the present study.

3.1 KIC 11337598

Østensen et al. (2011) detected a Balmer line broadening in thespectrum of KIC 11337598, suggesting the possibility that thisWD is a rapid rotator with vsin i = 1500 km s−1. The fit used bythose authors corresponds to a rotational period of about 40 s or0.000 463 d. Moreover, Maoz et al. (2015) suggested a period of0.09 d as the variability period for KIC 11337598. In Fig. 2 (lowerpanel), our wavelet analysis reveals three dominant periods, P1 =0.93 d, P2 = 2.13 d and P3 = 0.09 d. The period P3 is persistent overthe entire time span, corresponding to the main wavelet period,in agreement with Maoz et al. (2015). P1 is also persistent; it isprobably a P3 multiple period related to rotation. It should be notedthat P1 is exactly 2000 times the 40-s periodicity found by Østensenet al. (2011). Therefore, we consider here both P1 and P3 linked tothe WD rotation, while P2 is a plausible aliasing period.

3.2 KIC 11514682

For this star the wavelet analysis reveals two periodicities, incontrast with the 9.89 d found by Maoz et al. (2015). The waveletmap displayed in the upper panel of Fig. 4 shows a predominantperiod of 40.42 d and a secondary one at 17.59 d. These two periodsare confirmed in the Lomb–Scargle periodogram displayed in thelower panel of Fig. 4. The periods of 17.59 d and 40.42 d haveapproximately a 1:2 correspondence, respectively. This suggeststhat the first one is more likely a physical period and the latter oneis a possible alias.

3.3 KIC 4829241, KIC 10420021 and KIC 11822535

The stars KIC 4829241, KIC 10420021 and KIC 11822535 wereclassified as non-periodic by Maoz et al. (2015). Instead, our studysuggests a periodic or quasi-periodic modulation well describing a

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Table 1. White dwarf parameters, including photometric periodic variations computed via the wavelet technique.

KIC Spectral Kp Teff log g Perioda Quartersb Wavelet variability periodstype (mag) (kK) (cm s−2) (d) (d)

Periodic (Maoz et al. 2015) Our analysis5769827 DA 16.6 66 8.2 8.30 ± 0.04 4,6 8.026669882 DA 17.9 30.5 7.4 0.367 ± 0.009 2 0.366862653 subdwarf 18.2 40.3 6.4 0.594 ± 0.02 2 0.608682822 DA 15.8 23.1 8.5 4.7 ± 0.3 5–9 5.2911337598 DA 16.1 22.8 8.6 0.093 28 ± 0.000 03 3,10 0.0911514682 DA 15.7 32.2 7.5 9.89 ± 0.06 2–17 17.5911604781 DA 16.7 9.1 8.3 4.89 ± 0.02 3,6,7,10,11 4.73

Non-periodic (Maoz et al. 2015) Our analysis3427482 DA 17.3 – – – 1 –4829241 DA 15.8 19.5 8.0 – 2–13 16.047129927 composite DA + DA 16.6 9.5 8.3 – 3,5,6 –9139775 DA 17.9 24.6 8.6 – 2 –10198116 DA 16.4 13.5 8.0 – 4–6 –10420021 DA 16.2 12.8 7.8 – 2,5,6,8–10 11.7011822535 DA 14.8 36.0 7.9 – 2–13 13.83

Notes: a Periods obtained by Maoz, Mazeh & McQuillan (2015).b Quarters of long cadence available in the Kepler public data and analysed in this work.

Figure 4. Upper panel: light curve with additional treatment, wavelet localand global spectra of KIC 1151468, which was classified as periodic by Maozet al. (2015). Contour levels are 90 per cent, 80 per cent, 70 per cent, ...,20 per cent, and 10 per cent of the map maximum. The variability periodis illustrated by the red dashed line in the global spectra. A possible aliasingperiod is displayed in blue. Lower panel: Lomb–Scargle periodogram for thesame LC. The main peak labelled ‘A’ corresponds to the period of 18.86 d,whereas ‘B’ indicates the 39 d period.

variability signature in the wavelet representations. We attribute tothis variability periodicities ranging between 11 and 16 d, which isreasonable for WDs (e.g. Kawaler 2015; Valeev et al. 2017; Brakeret al. 2017). In fact, these periodicities are more persistent in KIC4829241 and KIC 10420021 than in KIC 11822535, even thoughalso noticeable in the latter star. We do not expect to find magneticcool-spot signatures for KIC 4829241 and KIC 11822535 consid-ering their high Teff and their log g values (Tremblay et al. 2015).However, a transient signature for KIC 10420021 is conceivable inview of its Teff and log g being close to the limit value (Tremblayet al. 2015), so the presence of a convection zone at the WD surfacecan still be envisaged.

Combining the surface gravities and the effective temperatures,derived spectroscopically by Hallakoun et al. (2018), with evolu-tionary grids (Benvenuto & Althaus 1999), we are able to estimatethe mass of these three WDs. KIC 4829241 is a WD of mass∼0.45–0.80 M�. The Lomb–Scargle periodogram for this WDshows a period of 16.68 d with an amplitude of 286 ppm, asindicated by the letter A in the upper-right panel of Fig. 5. FollowingMaoz et al. (2015), given its mass, rotation combined with anaccretion of gas from the interstellar medium (ISM) and the materialcarried on the magnetic poles generating hotspots, with an accretionluminosity of �Lacc ∼ 1.4 × 1029 erg s−1, could explain the periodicmodulation. Fig. 5 (upper-left panel) displays the wavelet analysis,from which we noticed a long-lasting periodicity of 16.04 d overthe time observation. With a lower amplitude, we also found a 40-dperiod that could be considered as a possible aliasing period. The16-d period characterizes the WD photometric modulation sinceit is the main period obtained in both Fourier- and wavelet-basedtechniques. Based on a rough FAP estimation (see Section 2), theseperiods likely lie above the confidence level.

The middle left-hand panel of Fig. 5 displays the time seriesand wavelet local and global spectra of KIC 10420021. We canclearly track a long-lasting periodicity of 11.70 d and a second of5.46 d, although, the latter is only observed with high amplitude inquarters Q5–Q7. Its time evolution could reveal a likely emergenceof cool magnetic spots in the WD surface. Their short lifetimeis still viable considering the WD effective temperature of about12 000 K (Bergeron, Wesemael & Beauchamp 1995) for which

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A wavelet analysis of white dwarf stars 3939

Figure 5. Left: light curves with additional treatment, wavelet local and global spectra of Kepler WDs (from top to bottom) KIC 4829241, KIC 10420021,and KIC 11822535. Contour levels are 90 per cent, 80 per cent, 70 per cent, ..., 20 per cent, and 10 per cent of the map maximum. The variability periodsare illustrated by the red dashed lines in the global spectra. Possible aliasing periods are displayed in blue. Right: respective Lomb–Scargle periodogram forthe same WD. The main peak labelled ’A’ corresponds to the main period obtained in the wavelet spectrum.

the weak magnetic field still inhibits convective motions. Bothperiodicities are in accordance with the values obtained throughFourier techniques. The main peak of a period of 11.68 d in theLomb–Scargle periodogram is indicated by the letter A in the middleright-hand panel of Fig. 5. From the wavelet analysis, we considerthe 12-d period to be related to the variability modulation, probablydue to the WD rotation. Given its small mass of ∼0.45–0.60 M�estimated using the Teff calculated by Hallakoun et al. (2018),KIC 10420021 could be one of the WDs for which photometricvariability is due to rotation plus UV line absorption plus opticalfluorescence.

KIC 11822535 has a mass ranging approximately from 0.45–0.60 M�. The wavelet analysis, shown in the bottom left-handpanel of Fig. 5, is very similar to that of KIC 4829241. However,for this star, the main period of 13.83 d describes short quasi-periodic variations over the entire time span with lower amplitude.In accordance, a period of 13.56 d corresponds to the main peak

of the respective Lomb–Scargle periodogram (Fig. 5, bottom right-hand panel) with a small amplitude of 74.3 ppm. As the effectivetemperature of this WD is higher than 30 kK, the quasi-periodicvariability observed in the wavelet map could not be associatedwith magnetic spots, but could have a photometric modulation dueto the rotation. This modulation could result from WD rotation plusmagnetic dichroism or UV line absorption plus optical fluorescence.

4 C O N C L U S I O N S

We have carried out a wavelet analysis of the variability behaviourof 14 stars observed by Kepler mission, previously analysed byMaoz et al. (2015) using the fast Fourier transform approach. Weconfirm the variability for all seven stars classified as periodic bythose authors. Nevertheless, for one WD, KIC 11514682, we havefound different periodicities. Let us underline that the variable starKIC 6862653, early classified as a WD, turned out to be a carbon-

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rich helium-dominated subdwarf (Hallakoun et al. 2018). We alsoidentified noticeable variability for three WD stars reported by Maozet al. (2015) as non-periodic. The stars KIC 4829241, KIC 10420021and KIC 11822535 present wavelet maps revealing a clear signatureof periodicities ranging approximately from 11 to 16 d. These arethen placed within the list of WDs with periodic modulations, whilefor KIC 3427482, KIC 7129927, KIC 9139775 and KIC 10198116we confirm their non-periodic behaviour.

The 10 stars studied here present different features in their time–frequency representations. For KIC 5769827 and KIC 116047812 aparticular semi-regular colour-filled contour, lasting all along theirtime span, leads us to assume that the rotation modulation is the mostprobable agent of such local map signature. As the KIC 116047812effective temperature is about 9 000 K, it is very likely that theobserved rotation modulation is combined with cool magnetic spots.Otherwise, the maps of KIC 6669882, KIC 6862653 and KIC11337598 reveal a short-term variability that is hardly unhiddenonly by visual inspection of the data. The wavelet procedure helpswith the identification of such short periodicities, which define avariability modulation characterized by their persistence, despitetheir low amplitude, along the time span. The 5-d periodicity foundfor the KIC 8682822 WD is considered to be less reliable becauseit only appears for almost 20 d of the total time span of 360 d. Itwas necessary to cut the entire light curve into small time intervalsand apply the wavelet treatment to each dataset to pick up shorttime-scales.

The time–frequency behaviours for KIC 11514682, 4829241and 11822535 are very similar. The main peaks in the waveletglobal spectra ranging from 13 to 18 d describe a semi-regulartrack probably associated with the WD variability. Other periodshigher than 11 d are considered in those cases as aliasing of the mostpredominant period. Finally, KIC 10420021, classified as a DA-typeWD by Østensen et al. (2011), presents the expected conditions forcool magnetic spots, in view of its effective temperature value closeto the threshold limit value for which the atmospheres are expectedto be fully radiative. This is noticeable in its wavelet analysis, forwhich we found two main periodicities of 11.70 and 5.46 d.

As underlined, the wavelet procedure offers a unique possibilityfor LC analysis because we can map short- and long-term featuresfrom the energy distribution of each LC computed as a localwavelet spectrum. In the present study, in addition to an enlargedtreatment of the LCs, performed to avoid possible distortions in thespectral wavelet maps and changes in the power spectra, our waveletapproach was able to reveal new variability traces in the analysedstellar sample.

AC K N OW L E D G E M E N T S

The research activities of the astronomy observational board at theFederal University of Rio Grande do Norte are supported by contin-uous grants from the national funding agencies CNPq, and FAPERNand by the INCT-INEspaco. SRdeL acknowledges CAPES grad-uate fellowships. JPB and ADC acknowledge a CAPES/PNPDfellowship. ICL acknowledges a postdoctoral fellowship from theBrazilian agency CNPq (Science Without Borders program, GrantNo. 207393/2014-1). DBdeF acknowledges financial support fromthe Brazilian agency CNPq-PQ2 (grant No. 306007/2015-0). Wealso thank the reviewer, Prof. S. O. Kepler, for very useful comments

to the original manuscript. This paper includes data collected by theKepler mission. Funding for the Kepler mission is provided by theNASA Science Mission Directorate. The data presented in this paperwere obtained from the Mikulski Archive for Space Telescopes(MAST). STScI is operated by the Association of Universities forResearch in Astronomy, Inc., under NASA contract NAS5-26555.

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