Characterising the observational properties of Sct stars in the ...Characterising Kepler Sct stars 3...

MNRAS 000 1ndash18 (2018) Preprint 16 February 2018 Compiled using MNRAS LATEX style file v30

Characterising the observational properties of δ Sct starsin the era of space photometry from the Kepler mission

Dominic M Bowman12 and Donald W Kurtz21 Instituut voor Sterrenkunde KU Leuven Celestijnenlaan 200D 3001 Leuven Belgium2 Jeremiah Horrocks Institute University of Central Lancashire Preston PR1 2HE UK

Accepted 2018 February 14 Received 2018 February 13 in original form 2017 December 22

ABSTRACTThe δ Sct stars are a diverse group of intermediate-mass pulsating stars located on andnear the main sequence within the classical instability strip in the HertzsprungndashRusselldiagram Many of these stars are hybrid stars pulsating simultaneously with pressureand gravity modes that probe the physics at different depths within a starrsquos interiorUsing two large ensembles of δ Sct stars observed by the Kepler Space Telescopethe instrumental biases inherent to Kepler mission data and the statistical propertiesof these stars are investigated An important focus of this work is an analysis of therelationships between the pulsational and stellar parameters and their distributionwithin the classical instability strip It is found that a non-negligible fraction of mainsequence δ Sct stars exist outside theoretical predictions of the classical instabilityboundaries which indicates the necessity of a mass-dependent mixing length parame-ter to simultaneously explain low- and high-radial order pressure modes in δ Sct starswithin the HertzsprungndashRussell diagram Furthermore a search for regularities in theamplitude spectra of these stars is also presented specifically the frequency differencebetween pressure modes of consecutive radial order In this work it is demonstratedthat an ensemble-based approach using space photometry from the Kepler mission isnot only plausible for δ Sct stars but that it is a valuable method for identifying themost promising stars for mode identification and asteroseismic modelling The full sci-entific potential of studying δ Sct stars is as yet unrealised The ensembles discussed inthis paper represent a high-quality data set for future studies of rotation and angularmomentum transport inside A and F stars using asteroseismology

Key words asteroseismology ndash stars oscillations (including pulsations) ndash starsvariables δ Scuti ndash techniques photometry

1 INTRODUCTION

The δ Sct stars are a diverse group of pulsating stars foundat the intersection of the main sequence and the classicalinstability strip in the HertzsprungndashRussell (HR) diagramwhich corresponds to spectral types between A2 V and F2 Vfor main sequence Population I stars and effective temper-atures approximately between 6400 le Teff le 8600 (Breger2000b Rodrıguez amp Breger 2001 Aerts et al 2010 Uytter-hoeven et al 2011) This places δ Sct stars within an inter-esting transition region in the HR diagram between low-massstars with radiative cores and thick convective envelopes(M 1 M) and high-mass stars with large convective coresand radiative envelopes (M amp 2 M) mdash see Bowman (2017)for a recent review of δ Sct stars The observations of the

E-mail dominicbowmankuleuvenbe (DMB)

physical properties of δ Sct stars provide useful constraintsfor stellar structure and evolutionary models since the phys-ical properties such as interior rotation and the size andshape of the convective core have a significant impact on theevolution of intermediate and massive stars (Maeder 2009Meynet et al 2013)

The observation and modelling of stellar pulsations ndashknown as asteroseismology ndash provides insight of the inte-rior physics of stars for a wide range of stellar masses andevolutionary stages across the HR diagram (Aerts et al2010) The pulsation modes excited within a star are depen-dent on stellar structure thus pulsation mode frequenciesprovide direct insight of physics within a star where tradi-tional methods in astronomy such as photometry and spec-troscopy are unable to probe directly This makes astero-seismology unique with its strong observational constraintsof stellar physics used for improving theoretical models of

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stellar structure and evolution The dominant pulsations inδ Sct stars are self excited by the opacity (κ) mechanism op-erating in the He ii ionisation zone (Cox 1963 Breger 2000bAerts et al 2010) which produces pressure (p) modes withperiods as long as eight hours and short as 15 min (Uytter-hoeven et al 2011 Holdsworth et al 2014) These p modestypically have low radial orders and are most sensitive tothe surface layers of a star Another form of coherent modeexcitation in δ Sct stars is from turbulent pressure which isable to excite p modes with radial orders between 7 le n le 10with higher pulsation mode frequencies for stars within theclassical instability strip (Houdek 2000 Antoci et al 2014Xiong et al 2016)

The instability regions of the δ Sct and the lower-massγ Dor stars are predicted to overlap in the HR diagram(Dupret et al 2004 2005 Houdek amp Dupret 2015) withstars that simultaneously pulsate in p modes excited by theκ mechanism and gravity (g) modes excited by the convec-tive flux blocking (modulation) mechanism commonly re-ferred to as hybrid stars The exact nature the excitationmechanism of g modes and its physical relationship with theκ-mechanism in intermediate-mass stars is not fully under-stood (see eg Xiong et al 2016) with a significant fractionof δ Sct stars hotter than the γ Dor instability region ob-served to pulsate with g modes (Uytterhoeven et al 2011Bowman 2016 2017) Few hybrid stars were known priorto space photometry (see eg Handler 2009) The unprece-dented photometric precision duty cycle and total length ofthe Kepler mission (Borucki et al 2010 Koch et al 2010)revealed that many δ Sct stars are hybrid stars (Grigahceneet al 2010 Uytterhoeven et al 2011 Balona amp Dziembowski2011 Balona 2014) On the other hand it is also demonstra-bly true that pure δ Sct stars with no significant frequenciesbelow 5 dminus1 albeit rare exist (Bowman 2017)

An analysis of 750 A and F stars was carried out byUytterhoeven et al (2011) using 1 yr (Q0 ndash Q4) of Keplerdata and it was concluded that approximately 25 per centwere hybrid pulsators Later analyses of A and F stars us-ing longer time spans of Kepler data by Balona (2014) andBalona et al (2015) have found much higher fractions ofδ Sct stars with low frequencies in their amplitude spectrabut also demonstrate that less than fifty per cent of starswithin the classical instability strip pulsate This measuredincidence of pulsation amongst δ Sct stars should be consid-ered a lower limit with 50 per cent of A and F stars observedto pulsate using the detection threshold Kepler photometrywhich is of order a few micromag in the best of cases For manyA and F pulsators it is not established if observed low-frequency peaks are caused by pulsation modes rotationalmodulation combination frequencies or have some other as-trophysical cause (Bowman 2017) Recent studies by VanReeth et al (2016) and Saio et al (2018) have discoveredthat global normal modes of Rossby waves (r modes) whichcause temperature perturbations in a rotating star are vis-ible in A and F stars in the Kepler data set Specifically rmodes in rapidly rotating γ Dor stars exhibit a power excessat low frequency such that r modes with azimuthal order mproduce a group of frequencies slightly below m times a starrsquosrotation frequency (Saio et al 2018)

For chemically normal stars within the classical insta-bility strip it is not surprising that they are observed topulsate from the nature of the driving mechanism in δ Sct

stars (Murphy et al 2015) although it is also known thatapproximately half of stars within the classical instabilitystrip are not pulsating (Balona amp Dziembowski 2011) at thephotometric precision of order a few micromag in Kepler dataFurther work is needed to study the interplay between theflux blocking mechanism and the κ mechanism in A and Fstars with a focus on understanding the synergy between thepulsation excitation mechanisms and the observed minorityof δ Sct stars pulsating in purely p modes This providesmotivation for characterising the pulsation properties of alarge ensemble of δ Sct stars observed by Kepler presentedin this work

However one of the main difficulties encountered whenobserving δ Sct stars is the issue of identifying pulsationmodes in terms of their radial order n angular degree `and azimuthal order m The typical low-order p modes inδ Sct stars often do not show any regular spacing in theamplitude spectrum unlike the asymptotic high radial or-der p modes in low-mass solar-type stars mdash see Chaplin ampMiglio (2013) and Hekker amp Christensen-Dalsgaard (2017)for reviews of solar-like pulsators The rapid rotation modevisibility and our incomplete understanding of mode excita-tion in δ Sct stars make them one of the more challenginggroups of pulsating stars to study using asteroseismology(eg Reese et al 2017) Despite these complexities ongo-ing efforts to apply ensemble asteroseismology techniques toδ Sct stars are being made using similar methods to thoseused to study low-mass stars (Garcıa Hernandez et al 20092013 2015 Paparo et al 2016 Michel et al 2017)

Significant progress in understanding the physics ofδ Sct stars has been made since the space photometry revo-lution facilitated by the CoRoT (Auvergne et al 2009) andKepler (Borucki et al 2010) space missions The rich pulsa-tion spectra of δ Sct stars offer the potential to probe physicsat different depths using asteroseismology thus these starscan provide useful constraints of stellar structure and evo-lution theory in an important transition region on the mainsequence (Breger 2000b Aerts et al 2010) The full scientificpotential of studying pulsation modes in δ Sct stars with as-teroseismology in the era of space photometry has yet to beexploited

11 Regularities in the amplitude spectra of δ Sctstars

It has been the goal of various studies to search for regu-larities in the amplitude spectra of δ Sct stars (eg Gar-cıa Hernandez et al 2009 2013 2015 Paparo et al 2016Michel et al 2017) with a particular focus of identifyingthe asymptotic frequency separation between modes of con-secutive high radial order known as the large frequencyseparation ∆ν However δ Sct stars often have high modedensities in their amplitude spectra caused by dozens andsometimes hundreds of pulsation mode frequencies makingit difficult to distinguish intrinsic pulsation modes This iscomplicated by non-linearity of pulsation modes in δ Sctstars in the form of harmonics nνi and combination frequen-cies nνi plusmn mνj where n and m take integer values (Papics2012 Kurtz et al 2015 Bowman 2017) These non-linearcombination frequencies can have high Fourier amplitudesbut do not represent intrinsic pulsation modes since theydescribe the non-sinusoidal shape of a starrsquos light curve in

MNRAS 000 1ndash18 (2018)

Characterising Kepler δ Sct stars 3

the Fourier domain and can also create regularities in anamplitude spectrum (see chapter 6 from Bowman 2017)

Recently Michel et al (2017) searched for regularityin the amplitude spectra of 1860 δ Sct stars observed byCoRoT using two observables which they termed fmin andfmax corresponding to the lower and upper limits for sig-nificant frequencies in a starrsquos amplitude spectrum Michelet al (2017) define significant frequencies as those that havean amplitude larger than ten times the mean amplitude inan amplitude spectrum A subsample of approximately 250of these stars were classified as young δ Sct stars since theyhad high-frequency pulsation modes with fmax ge 400 microHz(amp 35 dminus1) such that they were consistent with theoreticalmodes of stars near the zero-age main sequence (ZAMS)Michel et al (2017) found an agreement between the ob-served frequency distribution and the corresponding distri-bution caused by island modes from theoretical models usinga non-perturbative treatment of fast rotation (Reese et al2009) Specifically regularities in the amplitude spectra ofthese stars produced ridge-like structures with spacings oforder a few tens of microHz (of order a few dminus1) consistent withconsecutive radial order pulsation modes Thus the observ-ables fmin and fmax were concluded to be potentially usefuldiagnostics to constrain the mass and evolutionary stage ofa δ Sct star (Michel et al 2017)

The ensemble approach of studying pulsating stars cer-tainly has its advantages especially for such diverse pul-sators as δ Sct stars (Grigahcene et al 2010 Uytterhoevenet al 2011 Balona amp Dziembowski 2011 Balona 2014 Bow-man et al 2016 Bowman 2017 Michel et al 2017) In thispaper the statistical properties of a large number of δ Sctstars observed by the Kepler Space Telescope are investi-gated with a discussion of the target selection methods pro-vided in section 2 an evaluation of the edges of the classicalinstability strip presented in section 3 an analysis of thecorrelations between the pulsational and stellar parametersgiven in section 4 and the results for a search for regulari-ties in the amplitude spectra given in section 5 Finally theconclusions are presented in section 6

2 COMPILING ENSEMBLES OF Kepler δ SctSTARS

The Kepler Space Telescope was launched on 2009 March 7and has an Earth-trailing orbit with a period of 3725 d(Borucki et al 2010) Within its 115 deg2 field of view inthe constellations of Cygnus and Lyra it observed approxi-mately 200 000 stars at photometric precision of order a fewmicromag (Koch et al 2010) To maximise its goal of findingtransiting Earth-like planets orbiting Sun-like stars Keplerhad two observing modes a long cadence (LC) of 2945 minand a short cadence (SC) of 585 s (Gilliland et al 2010)From the limited onboard data storage capacity only 512stars could be observed in SC at a given epoch The Ke-pler LC data were divided into quarters (from Q0 ndash Q17)covering a maximum length of 14705 d whereas SC datawere divided into 30-d segments which was motivated by therolling of the Kepler spacecraft and data downlink epochsapproximately every 90 and 30 d respectively Kepler dataare available from the Mikulski Archive for Space Telescopes

(MAST1) in pre- and post-pipeline formats mdash see Smithet al (2012) and Stumpe et al (2012) for details of the Ke-pler science data pipeline In this work the post-pipelinedata known as msMAP PDC data are used

Previously a large ensemble of 983 δ Sct stars with LCKepler data was compiled by Bowman et al (2016) to studyamplitude modulation of pulsation modes Later Bowman(2016 2017) used the same stars to investigate the relation-ship between the pulsation and stellar parameters in δ Sctstars This ensemble of δ Sct stars was comprised of Keplerstars characterised by 6400 le Teff le 10 000 K in the KeplerInput Catalogue (KIC Brown et al 2011) were observedcontinuously in LC by Kepler for 4 yr and had significantpeaks in the p mode frequency range (ν ge 4 dminus1) with am-plitudes above 010 mmag Although an amplitude cut-offof 010 mmag is higher than the typical noise level observedfor Kepler data it was chosen by Bowman et al (2016) toensure that all extracted peaks had reasonable phase uncer-tainties as these are dependent on the amplitude signal-to-noise (SN) ratio (Montgomery amp OrsquoDonoghue 1999) TheKIC ID numbers and stellar parameters such as effectivetemperature and surface gravity of these stars were providedas supplementary online data in Bowman et al (2016) witha machine-readable table also available through CDS

However as discussed by Bowman (2016 2017) thisensemble of δ Sct stars using LC Kepler data suffers frominstrumental biases Firstly the Kepler wavelength pass-band of 420 minus 900 nm (Koch et al 2010) was chosen tomaximise the goal of finding transiting exoplanets aroundSun-like stars This leads to the pulsation mode amplitudesof the early-type stars observed by the Kepler mission in-cluding δ Sct stars having lower pulsation mode ampli-tudes when observed in the white-light Kepler passbandcompared to bluer passbands such as Johnson B (Bowmanet al 2015 Holdsworth et al 2018) Furthermore the am-plitude visibility function (which is termed apodization byHekker amp Christensen-Dalsgaard 2017) is a strong func-tion of frequency such that pulsation mode amplitudes areheavily suppressed near integer multiples of the instrumen-tal sampling frequency which corresponds to 489 dminus1 and14769 dminus1 for LC and SC Kepler data respectively (Bow-man 2016 2017) This amplitude visibility function is givenby

A = A0 sinc( π

n

)= A0 sinc

(πν

νsamp

) (1)

where A is the observed amplitude A0 is the true amplituden is the number of data points per pulsation cycle ν is thepulsation mode frequency and νsamp is the instrumental sam-pling frequency Thus for example a hypothetical δ Sct starthat has a pulsation mode frequency of ν 40 dminus1 and an in-trinsic amplitude of 1 mmag would be observed with an am-plitude of 02 mmag using LC Kepler data This amplitudesuppression represents a noticeable bias towards extractinglow-frequency pulsation modes over high-frequency pulsa-tion modes in any iterative pre-whitening procedure and byextension a bias towards studying cool andor evolved δ Sctstars when using LC Kepler mission data (Bowman 2016

1 MAST website httparchivestsciedukepler



2017) These selection effects are significant when compar-ing results obtained with LC and SC Kepler data especiallyfor asteroseismology of δ Sct stars as they pulsate with modefrequencies that can span the Kepler LC sampling frequency

To evaluate the impact of these biases in the LC Keplerdata for the analysis of δ Sct stars specifically the ampli-tude visibility function suppressing high-frequency pulsationmodes a second ensemble of δ Sct stars compiled using SCKepler data is also analysed independently in this work Theδ Sct stars in the SC ensemble of stars were selected usingthe same criteria as the LC ensemble except no restrictionwas placed on the length of the SC data The ensembles ofSC and LC δ Sct stars have been filtered for known bina-rity using the Villanova binary catalogue2 (Prsa et al 2011Abdul-Masih et al 2016) It is important to note that δ Sctstars in eclipsing binaries and ellipsoidal variables have beenexcluded from the ensembles presented in this work becauseof the effects of contamination which could significantly af-fect the Teff determination using photometry such as thoseincluded in the KIC (Brown et al 2011)

These selection criteria produced a LC ensemble of 963stars and a SC ensemble of 334 δ Sct stars All stars in theLC ensemble have continuous Kepler observations spanning4 yr but the δ Sct stars in the SC ensemble have total timespans that range between 10 and 1470 d The longest con-tinuous light curve for each SC δ Sct star ranges between 10and 1000 d with approximately 50 per cent of stars havingonly 30 d of Kepler SC data There are 218 stars that areduplicated in both ensembles with 116 SC stars that are notincluded in the LC ensemble It is easy to understand whyadditional SC δ Sct stars were not included in the LC ensem-ble in terms of amplitude suppression from the amplitudevisibility function given in Eqn (1) and because SC observ-ing slots were at a premium during the Kepler mission Theduplicate stars are not removed from either ensemble norare the ensembles combined as each ensemble suffers fromdifferent biases which should not be compounded into un-known biases

21 Extracting δ Sct stellar parameters

The vast majority of the 200 000 Kepler target stars werecharacterised with values of Teff and log g using griz and2MASS JHK broad-band photometry in the KIC (Brownet al 2011) This catalogue has since been revised by Huberet al (2014) with values of Teff being accurate in a statisticalsense for such a large number of stars

High-resolution and high SN spectroscopy using theHERMES spectrograph mounted on the Mercator telescope(Raskin et al 2011) was obtained and analysed by Niem-czura et al (2015) for 117 bright (V 10 mag) A and Fstars observed by Kepler Accurate values for fundamentalstellar parameters including Teff log g rotational and micro-turbulent velocities and their respective uncertainties weredetermined for each star (Niemczura et al 2015) An addi-tional 44 A and F stars were also observed using the FIESspectrograph mounted on the Nordic Optical Telescope byNiemczura et al (2017) Similar to the analysis by Huber

2 an updated list of known Kepler eclipsing binary stars can be

found at httpkeplerebsvillanovaedu

et al (2014) a 200 K systematic offset was found betweenthe KIC and spectroscopic values of Teff (Niemczura et al2015 2017) for A and F stars High-resolution and high SNspectroscopy of B A and F stars observed by Kepler wasalso obtained by Tkachenko et al (2012) and Tkachenkoet al (2013) using the 2-m telescope of the Thuringer Lan-dessternwarte Tautenburg in Germany

However not all the stars studied by Niemczura et al(2015 2017) and Tkachenko et al (2012 2013) are δ Sctstars but those that are represent a subset of δ Sct starsfor which accurate values of Teff and log g are known withuncertainties for an individual star typically between 100 leσ (Teff) le 200 K and 01 le σ (log g) le 02 (cgs) respectivelyOnly 58 and 70 stars studied by Tkachenko et al (20122013) and Niemczura et al (2015 2017) are included in theLC and SC ensembles respectively but these stars are in-cluded in this work to show compatibility with the KIC andHuber et al (2014) values

22 Extracting δ Sct pulsation parameters

To characterise the pulsational properties for the δ Sct starsin both ensembles the amplitude spectrum for each starwas calculated from 0 le ν le 98 dminus1 which has an upperfrequency limit corresponding to four times the LC Nyquistfrequency The highest amplitude pulsation mode was ex-tracted and optimised using a non-linear least-squares fit tothe light curve using the function ∆m = A cos(2πν(tminus t0)+φ)where A ν and φ are the pulsation mode amplitude fre-quency and phase respectively calculated using the zero-point of the time scale as t0 = 2 455 68877 BJD Each ex-tracted pulsation mode frequency was checked to satisfy thecommonly-used significance criterion of SN ge 4 in ampli-tude (Breger et al 1993) and that it did not correspond toa known instrumental spurious frequency3 The amplitudespectrum over a large frequency range for each star is usefulfor studying regularities in δ Sct stars which is discussedlater in section 5

By not correcting the amplitude spectra for the δ Sctstars in either ensemble for the amplitude visibility func-tion given in Eqn (1) prior to extracting pulsation modefrequencies it is ensured that the extracted peak is a pul-sation mode frequency and not a Nyquist alias frequencysince Nyquist aliases have lower observed amplitudes thanthe corresponding real peaks (Murphy et al 2013) Further-more this approach is to avoid dividing by zero for pulsa-tion mode frequencies very close to the LC Kepler samplingfrequency which would cause some δ Sct stars to have ex-tremely large and unphysical pulsation mode amplitudesInstead pulsation mode amplitudes are corrected for theamplitude visibility function after they have been extractedfrom an amplitude spectrum The disadvantage of this ap-proach is that the distribution of LC pulsation mode fre-quencies will be biased towards on average lower valuesbut this is a compromise that yields the most meaningful re-sults This also emphasises the need to keep both ensemblesof δ Sct stars separate in this analysis since the amplitude

3 Kepler data release notes httpsarchivestsciedu

keplerdata_releasehtml



visibility function has a negligible effect for stars in the SCKepler ensemble

3 REVISITING THE EDGES OF THECLASSICAL INSTABILITY STRIP WITHKepler DATA

In this section the observed boundaries of the classical in-stability strip are evaluated using the high-quality photom-etry provided by the Kepler Space Telescope using the twoensembles of δ Sct stars Prior to space photometry fromCoRoT and Kepler evolutionary and pulsation models ofδ Sct stars were calibrated to catalogues of δ Sct starscollated using ground-based observations for example Ro-drıguez amp Breger (2001) Compared to space data ground-based observations have higher noise levels and suffer fromlower duty cycles and significant aliasing which introducesstrong selection effects when studying pulsating stars Theseissues make it difficult to extract low-amplitude pulsationmodes and thus a δ Sct star may be classified as a non-pulsating star from the ground if its pulsation mode am-plitudes are lower than the noise level Typically the noiselevel in the amplitude spectrum of ground-based photome-try is a few tenths of a mmag for a single observing site (egBowman et al 2015 Holdsworth et al 2018 but can be aslow as tens of micromag in the best of circumstances (Kurtzet al 2005) With Kepler data we are afforded the luxuryof 4 yr of continuous high-precision light curves resultingin a white-noise level of order a few micromag in an amplitudespectrum

The distribution of the 963 LC and 334 SC δ Sct stars inTeff minus log g diagrams using values from the KIC are shown inthe top-left and top-right panels of Fig 1 respectively Thelocation of each δ Sct star is shown by a black cross with atypical error bar shown in bold in the bottom-right corner ofeach panel For comparison the same ensembles using therevised parameters from Huber et al (2014) are shown inbottom-left and bottom-right panels of Fig 1 as filled greysquares Stellar evolutionary tracks using a metallicity ofZ = 002 from Grigahcene et al (2005) which are calculatedfrom the ZAMS (indicated by a short-dashed purple line)to a post-MS effective temperature of Teff = 5900 K forstars of 14 18 22 and 27 M are also shown in Fig 1as solid purple lines The theoretical blue and red edges ofthe classical instability strip from Dupret et al (2005) forradial p modes between 1 le n le 6 are shown as blue and redlines with solid and dashed lines indicating the boundariesfor n = 1 and n = 6 radial modes respectively The subgroupof δ Sct stars for which accurate spectroscopic values of Teffand log g from Tkachenko et al (2012 2013) and Niemczuraet al (2015 2017) are also plotted as filled green circles ineach panel with their respective 1-σ uncertainties in Fig 1

The majority of δ Sct stars shown in Fig 1 are schemati-cally consistent with the theoretical blue and red edges of theclassical instability strip calculated by Dupret et al (2005)if the lower and upper bounds of the uncertainties on Teffvalues are considered for hot and cool δ Sct stars respec-tively However there is no evidence to indicate that temper-atures of hot and cool δ Sct stars would be over- and under-estimated in such a systematic fashion When comparing thedistribution of stars using the KIC and Huber et al (2014)

values in Fig 1 it is clear that the Huber et al (2014) val-ues are shifted to temperatures approximately 200 K higherHowever this systematic increase in temperature of the Hu-ber et al (2014) compared to the KIC values produces moreoutliers beyond the blue edge These outliers represent asmall yet significant fraction of δ Sct stars in both ensem-bles which includes stars with accurate spectroscopic Teffvalues obtained by Tkachenko et al (2012 2013) and Niem-czura et al (2015 2017) that are significantly hotter thanthe blue edge for the n = 6 radial overtone p mode calculatedby Dupret et al (2005)

Many of the δ Sct stars in Fig 1 have surface gravi-ties between 35 le log g le 40 except for those near to theZAMS and the red edge Thus the middle of the main se-quence and the Terminal Age Main Sequence (TAMS) arewell sampled compared to the ZAMS for hot δ Sct stars ob-served by the Kepler Space Telescope The high density ofδ Sct stars shown in the left panels of Fig 1 that are nearthe ZAMS and red edge of the theoretical instability stripcalculated by Dupret et al (2005) implies that ZAMS starscooler than Teff 6900 K are able to pulsate in p modesMany of the outliers at the red edge of the classical insta-bility strip can be explained as within 1σ of the classical in-stability strip but equally importantly many of these δ Sctstars are within 1σ of being cooler than the red edge Thelack of δ Sct stars with spectroscopic effective temperaturesand surface gravities from Tkachenko et al (2012 2013) andNiemczura et al (2015 2017) that are cooler than the rededge is likely caused by a selection effect in these authorrsquostarget lists

A value of αMLT between 18 and 20 was determinedto produce a reasonably good agreement between the the-oretical models and observations of δ Sct stars by Dupretet al (2004 2005) However it should be noted that thehighest mass model used by Dupret et al (2005) to inves-tigate αMLT was 18 M but higher-mass δ Sct stars existwithin the Kepler data set especially since stars with massesM gt 27 M are included in Fig 1 and have been confirmedby spectroscopy It was discussed in detail by Dupret et al(2005) that decreasing the value of αMLT to 10 has the ef-fect of shifting the instability strip to lower values of effectivetemperature and vice versa Thus a mass-dependent valueof αMLT is needed to reproduce both the observed blue andred edges of the classical instability strip It is importantto note that the observed boundaries of the classical insta-bility strip are purely empirical and are subsequently usedto constrain theoretical models Furthermore the observededges using ground-based observations of δ Sct stars whichspan 7400 le Teff le 8600 K at log g 44 (Rodrıguez ampBreger 2001 Bowman 2017) are narrower than the theo-retical boundaries shown in Fig 1 Clearly the theoreticalred edge of the classical instability strip is more difficult tomodel compared to the blue edge because of the complexinteraction of surface convection and the κ mechanism incool δ Sct stars with the convective envelope predicted toeffectively damp pulsations in stars cooler than the red edge(Pamyatnykh 1999 2000 Dupret et al 2004 2005 Houdekamp Dupret 2015) These results indicate that further theo-retical work is needed in understanding mode excitation inδ Sct stars specifically those pulsating in higher radial or-ders (n ge 6) and the large temperature range of observedδ Sct stars



Figure 1 The Teff minus log g diagrams for the ensemble of LC and SC δ Sct stars observed by Kepler are shown in the left and right panels

respectively The locations of δ Sct stars using parameters from the KIC are shown by black crosses in the top row and Huber et al

(2014) values are shown as grey squares in the bottom row with a typical error bar shown in bold in the bottom-right corner of eachpanel Stellar evolutionary tracks using a metallicity of Z = 002 from Grigahcene et al (2005) calculated between the ZAMS (shown as a

short-dashed purple line) and a post main sequence effective temperature of Teff = 5900 K are shown as solid purple lines The subgroupof δ Sct stars with spectroscopic values of Teff and log g from Tkachenko et al (2012 2013) and Niemczura et al (2015 2017) are plotted

as filled green circles The theoretical blue and red edges of the classical instability strip from Dupret et al (2005) for p modes with

radial order of n = 1 are shown as solid blue and red lines respectively and radial order of n = 6 shown as dashed blue and red linesrespectively

In summary it is concluded that a statistically non-negligible fraction of δ Sct stars are located cooler than thered edge and hotter than the blue edge of the classical insta-bility strip calculated by Dupret et al (2005) as shown inFig 1 A plausible explanation for the minority of δ Sct starslocated outside of the classical instability strip using the KICand Huber et al (2014) values are that they have incorrectTeff and log g values since these were determined using pho-tometry but outliers do exist with some stars having stellarparameters confirmed using spectroscopy Therefore theseresults demonstrate the need for a mass-dependent value ofαMLT to reproduce all of the δ Sct stars observed by the Ke-pler Space Telescope The δ Sct stars that lie outside theclassical instability strip will be interesting to study furtheras they lie in a region where the excitation of low-radial or-der p modes by the κ mechanism is not expected to occur(Pamyatnykh 1999 2000 Dupret et al 2004 2005)

4 CHARACTERISING THE PULSATIONPROPERTIES OF δ Sct STARS OBSERVEDBY Kepler

In this section correlations between the pulsation proper-ties and stellar parameters of δ Sct stars are investigatedusing the LC and SC ensembles The pulsation propertiesof δ Sct stars are diverse and upon first inspection littlecommonality is often found when comparing the amplitudespectra of individual δ Sct stars in terms of the number ofexcited pulsation modes their amplitudes and the observedfrequency range Since the number of significant pulsationmodes in δ Sct stars ranges from as small as a single mode4

to hundreds it is difficult to characterise and compare thetotal pulsation energy in a systematic way for an ensembleof stars (Bowman 2016) This is complicated by the fact thatthe majority of δ Sct stars have variable pulsation mode am-

4 Examples of mono-periodic δ Sct stars include KIC 5989856

KIC 8196006 KIC 8196381 and KIC 11509728 which were iden-tified by Bowman et al (2016) as having only a single significant

pulsation mode in the p mode frequency range



plitudes with the total observable pulsational budget beingunconserved on time scales of order a few years (Bowman ampKurtz 2014 Bowman et al 2016) Also in a classical pre-whitening procedure it is important to exclude harmonicsand combination frequencies when quoting the range of ob-served pulsation mode frequencies as these frequencies arecaused by the non-linear character of the pulsation modesand do not represent independent pulsation mode frequen-cies (Papics 2012 Kurtz et al 2015 Bowman 2017)

41 Pulsation mode amplitudes

Amongst the δ Sct stars a subgroup of high-amplitude δ Sct(HADS) stars exists which were originally defined by Mc-Namara (2000) as δ Sct stars with peak-to-peak light am-plitudes exceeding 03 mag HADS stars are also typicallyslowly-rotating (v sin i 40 km sminus1) and pulsate in the fun-damental andor first-overtone radial modes These stars areknown to be rare making up less than one per cent of pul-sating stars within the classical instability strip (Lee et al2008) but the physical cause of their differences to typi-cal δ Sct stars remains unestablished if any exists (Balona2016)

To demonstrate the rarity of HADS stars in the Ke-pler mission data and examine the typical pulsation modeamplitudes in δ Sct stars the (logarithmic) distributions ofthe amplitude of the highest amplitude pulsation mode forthe LC and SC ensembles are shown in the top and bottompanels of Fig 2 respectively In each panel of Fig 2 theamplitudes of the dominant pulsation modes have been ap-proximately separated into p modes (ν ge 4 dminus1) and g modes(ν lt 4 dminus1) and are showed as the filled grey and red areasrespectively It is possible for g-mode frequencies and combi-nation frequencies of g modes to exceed 4 dminus1 but typicallythey have lower amplitudes and do not represent the domi-nant pulsation mode in a star Each amplitude is correctedfor the amplitude visibility function given in Eqn 1 with thecorrected distributions for p and g modes shown as the black-and red-hatched regions respectively As expected the am-plitude visibility function has a negligible effect on the SCensemble By separating δ Sct stars into if their dominantpulsation mode corresponds to a g- or p-mode it is demon-strated that a non-negligible fraction more than 15 per centof δ Sct stars have their highest-amplitude pulsation modein the g-mode frequency range We also investigated usinga value of 7 dminus1 to separate g-mode and p-mode dominatedδ Sct stars which yielded similar results with a difference oforder a few per cent

From inspection of Fig 2 the amplitude of the dom-inant pulsation mode in a δ Sct star typically lies in be-tween 05 and 10 mmag In each panel a tail extendingfrom moderate (A 10 mmag) to high pulsation mode am-plitudes (A 100 mmag) can be seen which is caused by asmall number of HADS stars in each ensemble Since the SCensemble contains all δ Sct stars irrespective of the lengthof data available coupled with insignificant amplitude sup-pression described by the amplitude visibility function inEqn (1) this demonstrates the intrinsic scarcity of HADSstars in the Kepler data set (Lee et al 2008 Balona 2016Bowman 2017)

It is known that the majority of δ Sct stars exhibit am-plitude modulation over the 4-yr Kepler data set (Bowman

Figure 2 The distributions of the highest amplitude (ie domi-

nant) pulsation mode for the ensemble of 963 LC and 334 SC δ Sctstars are shown in the top and bottom panels respectively The

black and red regions represent the stars for which the dominantpulsation mode is in the p-mode (ν ge 4 dminus1) or g-mode (ν lt 4 dminus1)

frequency regimes respectively The filled regions represent the

extracted pulsation mode amplitudes with the hatched regionsof the same colour representing each distribution after being cor-

rected for the amplitude visibility function given in Eqn (1)

et al 2016 Bowman 2016) so it could be argued that a dif-ferent dominant pulsation mode for each δ Sct star could beextracted when using different subsets of Kepler data How-ever the amplitude distribution of δ Sct stars would largelyremain similar This can be understood by comparing thetop and bottom panels of Fig 2 calculated using LC and SCKepler data respectively The SC observations of δ Sct starsare randomly distributed throughout the 4-yr Kepler mis-sion but the distributions in Fig 2 have similar median andFWHM values This demonstrates that the pulsation prop-erties of an ensemble of δ Sct stars are less time-dependentthan the pulsation properties of an individual δ Sct star

42 Pulsation and Effective Temperature

A semi-empirical relationship exists between the effectivetemperature and the observed pulsation mode frequenciesin a δ Sct star which is caused by the depth of the He iidriving region in the stellar envelope being a function of Teff(Christensen-Dalsgaard 2000 Aerts et al 2010) A ZAMSδ Sct star near the blue edge of the classical instability strip



is expected to pulsate in higher radial overtone modes lead-ing to higher observed pulsation mode frequencies comparedto a ZAMS δ Sct star near the red edge of the classical in-stability strip (Pamyatnykh 1999 2000 Dupret et al 20042005) This relationship between effective temperature andpulsation mode frequencies in δ Sct stars has been previ-ously been demonstrated using ground-based observations(see eg Breger amp Bregman 1975 Breger 2000b Rodrıguezamp Breger 2001) Observationally the blue and red edges areknown for ground-based data (Rodrıguez amp Breger 2001)but the situation becomes less clear when the criterion ofSN ge 4 is applied to a data set with a noise level of order afew micromag such as Kepler mission data as demonstrated insection 3

The frequency of the dominant pulsation mode in eachδ Sct star in the LC and SC ensembles are shown in the leftand right columns of Fig 3 respectively In the top rowthe distributions of the dominant pulsation mode frequencyare shown by the black region with the subgroup of starsstudied by Tkachenko et al (2012 2013) and Niemczuraet al (2015 2017) shown as the green-hatched region forcomparison Although the LC ensemble suffers from a biascaused by the amplitude visibility function a dearth of δ Sctstars with pulsation mode frequencies above 60 dminus1 exists inboth ensembles such that 0 lt ν le 60 dminus1 can be consideredthe typical frequency range of pulsation modes of δ Sct starsin the Kepler data set These results are not necessarilyrepresentative of all δ Sct stars since the TAMS is bettersampled compared to the ZAMS in the Kepler data set withmore-evolved δ Sct stars expected to have lower pulsationmode frequencies The frequency of the dominant pulsationmode does not exceed 60 dminus1 in any δ Sct star in eitherensemble which is in agreement with the expectations fromtheoretical models of main sequence δ Sct stars near theblue edge of the classical instability strip (Pamyatnykh 19992000)

The frequency of the highest amplitude pulsation modeis plotted against Teff for the ensemble of 963 LC and 334SC δ Sct stars in the middle row of Fig 3 in which theKIC parameters the revised Huber et al (2014) parametersand the subgroup of stars studied by Tkachenko et al (20122013) and Niemczura et al (2015 2017) are shown as blackcrosses grey squares and green circles respectively Statis-tically significant linear regressions are also shown as solidlines with the gradient m intercept c coefficient of corre-lation R and p-value obtained from a t-test for each dataset given in Table 1 Linear regressions of the frequency ofmaximum amplitude and effective temperature using eitherthe KIC or Huber et al (2014) values yield statistically-significant positive correlations with R 02 and R 03for the LC and SC ensembles respectively The amplitudesuppression of high frequency pulsation modes in LC Keplerdata explains the weaker correlation between pulsation andeffective temperature found in the LC ensemble compared tothe SC ensemble since few δ Sct stars with pulsation modefrequencies above 40 dminus1 are included On the other handstatistically-significant correlations between pulsation andeffective temperature for stars in the spectroscopic subgroupwere not found From the relatively large p-values calculatedfrom the linear regression in the spectroscopic subgroup wecannot claim that the positive correlation found between ef-fective temperature and frequency of the highest-amplitude

pulsation mode is statistically significant With few δ Sctstars available with high-resolution spectroscopy more areneeded to investigate this relationship further

However it is important to distinguish that the expec-tation of hotter δ Sct stars having higher pulsation modefrequencies is predicted for ZAMS stars with the relation-ship breaking down for TAMS and post-main sequence δ Sctstars As for any star the frequencies of its pulsation modesdecrease albeit slowly during the main sequence because ofthe increase in stellar radius This creates a degeneracy whenstudying an ensemble of stars with pulsation mode frequen-cies expected to be correlated with Teff but also correlatedwith log g (ie inversely with age on the main sequence)In practice the sensible approach is to test the relationshipbetween the frequency of the dominant pulsation mode andeffective temperature for different groups of stars based ontheir log g value This is shown in the bottom row of Fig 3for stars in the LC and SC ensembles that have groupedinto log g ge 40 (ZAMS) 35 le log g lt 40 (Mid-Age MainSequence MAMS) and log g lt 35 (TAMS) using their KICparameters

Linear regressions for each of these log g subgroups forthe KIC and Huber et al (2014) values are also included inTable 1 which demonstrate the need to differentiate ZAMSand TAMS stars when studying the pulsational propertiesof δ Sct stars For example as shown in Fig 3 and given inTable 1 ZAMS stars in the LC and SC ensembles show astatistically-significant positive correlation between effectivetemperature and frequency of the highest-amplitude pulsa-tion with values of R 03 and R 05 respectively Asdiscussed previously the lack of high-frequency pulsators inthe LC ensemble explains the difference in these two valuesConversely no statistically-significant correlation was foundfor TAMS δ Sct stars in either the LC or SC ensembles

Linear regressions for subgroups based on log g for theδ Sct stars with spectroscopic parameters from Tkachenkoet al (2012 2013) and Niemczura et al (2015 2017) donot provide reliable results because most of these stars havesurface gravities between 35 le log g le 40 and becauseno significant correlation was found for the subgroup as awhole Clearly the ZAMS δ Sct stars shown in blue in thebottom row of Fig 3 have the strongest statistical corre-lation between Teff and frequency of maximum amplitudewith R 05 which is in agreement with theoretical modelsand previous observations (Pamyatnykh 1999 2000 Breger2000b Rodrıguez amp Breger 2001 Dupret et al 2004 2005Houdek amp Dupret 2015)

43 Low-frequency variability in δ Sct stars

The incidence of g modes in stars that exceed Teff amp8000 K especially those with accurate stellar parametersobtained from high-resolution spectroscopy are interestingcases These stars are significantly hotter than typical γ Dorstars observed by Kepler with effective temperatures ofγ Dor stars determined using high-resolution spectroscopytypically between 6900 le Teff le 7400 K (Tkachenko et al2013 Van Reeth et al 2015b) The excitation of g modesin δ Sct stars that are located hotter than the blue edgeof the γ Dor instability region warrants further study withsuch stars found in both LC and SC ensembles These hotlow-frequency pulsators can be clearly seen in each panel



Figure 3 The top row shows the distribution of the frequency of maximum amplitude shown as the black region with the subgroup

of stars studied by Tkachenko et al (2012 2013) and Niemczura et al (2015 2017) plotted as the green-hatched area The middle rowshows the frequency of maximum amplitude versus the effective temperature using the KIC Huber et al (2014) and spectroscopic values

from Tkachenko et al (2012 2013) and Niemczura et al (2015 2017) as black crosses grey squares and green circles respectively with

statistically-significant linear regressions shown as solid lines The bottom shows the same relationship as the middle row but for onlythe KIC stars that have grouped based on their log g value into bins of log g ge 40 (ZAMS) 35 le log g lt 40 (MAMS) and log g lt 35(TAMS) For all rows the left and right hand plots correspond to the LC and SC ensembles respectively which have been plotted using

the same ordinate and abscissa scales for comparison

of Fig 3 in which a non-negligible fraction of stars (origi-nally chosen because they contain only p modes or p andg modes) have their dominant pulsation mode in the g modefrequency regime

A distinct bi-modality is present in the top-left panelof Fig 3 with a minimum in the distribution at a frequencyof approximately 7 dminus1 This minimum in the distributioncorresponds to the approximate upper limit of g-mode pul-sation frequencies in fast-rotating stars (Bouabid et al 2013Van Reeth et al 2016 Aerts et al 2017b) and an approxi-mate lower limit of p-mode pulsation frequencies althoughoverlap of g and p-modes and their combination frequen-cies is expected between 4 le ν le 7 dminus1 Therefore the starswith the dominant pulsation mode in the g-mode frequency-regime shown in Fig 3 represent hybrid stars in which the

dominant pulsation mode corresponds to a g mode Howeverit remains unclear how or why there should be a significantdifference if any in the respective amplitudes of the p andg modes in a δ Sct star with all stars in both ensemblesselected to contain at least p modes

The Kepler mission data have proven extremely usefulfor studying the interior physics of intermediate-mass starsspecifically concerning measurements of radial rotation andangular momentum transport (Aerts et al 2017b) The firstexamples of hybrid main sequence A and F stars with mea-sured interior rotation were by Kurtz et al (2014) and Saioet al (2015) and exemplify the power of asteroseismologywhen applied to hybrid stars such that model-independentmeasurements of rotation inside stars can be made To datea few dozen intermediate-mass main sequence stars have



Table 1 Statistics for linear regressions of the frequency of maximum amplitude and effective temperature for the δ Sct stars in thispaper using KIC values (Brown et al 2011) the revised KIC values from Huber et al (2014) and spectroscopic values from Tkachenko

et al (2012 2013) and Niemczura et al (2015 2017) The column headers include the number of stars Nstars the gradient m the

ordinate-axis intercept of the linear fit c the coefficient of correlation R and the corresponding p-value (obtained from a t-test)Separate regressions for the log g subgroups are given for KIC and Huber et al (2014) values

Long Cadence (LC) ensemble Short Cadence (SC) ensemble

Nstars m c R p-value Nstars m c R p-value

(K dminus1) (K) (K dminus1) (K)

KIC

All 963 128 plusmn 20 7195 plusmn 17 020 lt 00001 334 141 plusmn 22 7229 plusmn 21 033 lt 00001ZAMS (log g ge 40) 421 156 plusmn 30 6972 plusmn 16 024 lt 00001 89 227 plusmn 40 6819 plusmn 21 052 lt 00001MAMS (35 le log g lt 40) 494 98 plusmn 25 7396 plusmn 17 017 00001 221 114 plusmn 25 7375 plusmn 22 029 lt 00001TAMS (log g lt 35) 48 minus77 plusmn 99 7404 plusmn 14 -011 04438 24 23 plusmn 115 7442 plusmn 14 004 08460

Revised KIC


Spectroscopy

All 58 114 plusmn 67 7515 plusmn 17 022 00947 70 62 plusmn 44 7625 plusmn 21 017 01584

been found to be almost rigidly rotating or have weak dif-ferential rotation (Degroote et al 2010 Kurtz et al 2014Papics et al 2014 Saio et al 2015 Van Reeth et al 2015aTriana et al 2015 Murphy et al 2016 Ouazzani et al 2017Papics et al 2017 Zwintz et al 2017 Van Reeth et al 2018)

The full asteroseismic potential of studying the hybridstars using the high-quality Kepler data has yet to be ex-ploited For example the observed power excess at low fre-quencies in γ Dor stars has recently been interpreted asRossby modes which further inform our understanding ofrotation (Van Reeth et al 2016 Saio et al 2018) Further-more expanding the systematic search for period spacingpatterns beyond the γ Dor stars to the hybrid stars has theprospect of extending the studies of rotation and angularmomentum transport to higher masses and fill the gap be-tween the published cases of B and F stars in the literatureThe hybrid stars in the LC ensemble have the necessary dataquality to extract identify and model g and p modes whichprovide valuable physical constraints of the near-core andnear-surface regions within a star respectively with moreobservational studies of intermediate- and high-mass starsneeded to address the large shortcomings in the theory ofangular momentum transport when comparing observationsof main sequence and evolved stars (Tayar amp Pinsonneault2013 Cantiello et al 2014 Eggenberger et al 2017 Aertset al 2017b)

The study of angular momentum transport in stars hasrevealed that Internal Gravity Waves (IGWs) stochastically-driven in stars with convective cores can explain the ob-served near-rigid rotation profiles in intermediate-mass stars(Rogers et al 2013 Rogers 2015) The detection of IGWscan be inferred by the power-excess at low frequencies ina starrsquos amplitude spectrum The few observational detec-tions of IGWs in the literature have been for massive starsof spectral type O and were made by matching the observedmorphology of the low-frequency power excess with that pre-dicted by state-of-the-art simulations of IGWs (Rogers et al2013 Rogers 2015 Aerts amp Rogers 2015 Aerts et al 2017a

2018 Simon-Dıaz et al 2017) Any star with a convectivecore is predicted to excite IGWs thus the ensemble of δ Sctstars presented in this work may contain observational sig-natures of IGWs (eg Tkachenko et al 2014) and warrantfurther study

The Kepler data set will remain unchallenged for theforeseeable future for studying δ Sct stars because of its totallength duty cycle and photometric precision Furthermoreit represents a homogeneous data set for detections of IGWsand the study of mixing and angular momentum transportin intermediate-mass A and F stars

44 Pulsation within the Classical Instability Strip

The relationships between pulsation mode frequencies andeffective temperature and evolutionary stage (ie log g) aredegenerate with pulsation mode frequencies predicted andobserved to be correlated with both effective temperatureand surface gravity Therefore using a Teff minus log g diagram isa sensible way to investigate these correlations further TheLC and SC ensembles of δ Sct stars are once again plotted inTeff minus log g diagrams in the left and right columns of Fig 4respectively using the KIC parameters for each star How-ever unlike the previous diagrams (shown in Fig 1) eachstar in Fig 4 is plotted as a filled circle that has been colour-coded by the frequency of its highest-amplitude pulsationmode in the top row and colour-coded by the amplitude ofits highest-amplitude pulsation mode in the bottom row

In each panel in Fig 4 contour lines corresponding tothe 10th 50th and 90th percentiles of the normalised proba-bility density after applying a Gaussian kernel to the distri-bution of stars are shown for three subgroups of stars thatfall into three parts of the colour bar scale For examplethe density maxima in the top-right panel of Fig 4 corre-spond to approximate Teff values of 7000 K 7500 K and8000 K for the 0 lt νmax lt 20 dminus1 20 le νmax lt 40 dminus1 andνmax ge 40 dminus1 subgroups respectively Note that since thereare so few high-frequency pulsators in the LC ensemble only



Figure 4 The left and right columns correspond to the LC and SC ensembles of δ Sct stars respectively The top row shows the

location of each δ Sct in a Teff minus log g diagram using the KIC parameters with each starrsquos location colour-coded by the frequency of the

highest-amplitude pulsation mode Similarly the bottom row is colour-coded by the amplitude of the highest-amplitude pulsation modeusing a logarithmic colour-bar scale The same stellar evolutionary tracks ZAMS line typical KIC error bar and theoretical edges of

the classical instability strip as in Fig 1 are also shown In each panel density contours are plotted for the stars contained within threeparts of the shown colour bar scale

two sets of contours are shown in the top-left panel of Fig 4For the maximum amplitude distributions shown in the bot-tom row of Fig 4 the three sets of contours correspond tolog10(Amax) le 0 0 lt log10(Amax) lt 1 and log10(Amax) ge 1The choice of these subgroups is somewhat arbitrary butthey demonstrate the average trends in each panel and helpto guide the eye compared to the distribution using individ-ual stars

As expected and demonstrated using a subsample ofZAMS stars (log g gt 40) in Fig 3 the δ Sct stars withhigher pulsation mode frequencies show higher effective tem-peratures and are found closer to the blue edge of the classi-cal instability strip The same relationship between effectivetemperature and pulsation mode frequencies can be seen inthe top row of Fig 4 with high-frequency δ Sct stars (shownin dark purple) being typically located nearer the blue edgeof the classical instability strip The situation is somewhatless clear when studying the relationship between the am-plitude of the dominant pulsation mode and location in theTeff minus log g which is shown in the bottom row of Fig 4 Inthe LC ensemble the highest-amplitude pulsators (shown indark purple) are more centrally located in the classical in-stability strip whereas the high-amplitude pulsators in theSC ensemble are located closer to the blue edge but this

discrepancy is likely caused by so few high-amplitude starsin each ensemble It has been previously demonstrated us-ing ground-based observations that HADS stars are oftenfound in the centre of the classical instability strip (McNa-mara 2000 Breger 2000b) but with so few HADS stars inthe Kepler data set and the uncertainties in Teff and log g

values it is difficult to make any meaningful conclusions

It should be noted that these results may not be com-pletely representative of all δ Sct stars as only a single pulsa-tion mode frequency was used to characterise each star Thisin turn may explain the lack of significant correlation foundbetween the frequency of maximum amplitude νmax andeffective temperature for all subgroups except ZAMS starsas described by the results in Table 1 Using space-basedobservations such as those from Kepler many δ Sct starshave a large range of pulsation mode frequencies with smallamplitudes The density and range of the observed pulsationmode frequencies are not accounted for when extracting onlya single pulsation mode mdash this is explored in more detail insection 5 Nonetheless the high-frequency δ Sct stars aretypically located nearer the blue edge of the instability stripas shown in Fig 4 which is in agreement with theoreticalpredictions and previous observations of δ Sct stars (Bregeramp Bregman 1975 Pamyatnykh 1999 2000 Breger 2000b



Rodrıguez amp Breger 2001) An interesting result from thisanalysis is that δ Sct stars across the instability strip areable to pulsate in low and high pulsation mode frequencieswith no obvious explanation for which pulsation modes areexcited in δ Sct stars Further observational and theoreticalstudy of this is needed as it will provide constrains of modeselection mechanism(s) at work in these stars

45 Pulsation and rotation

The effects of rotation modify the pulsation mode frequen-cies of a star by lifting the degeneracy of non-radial modesinto their 2` + 1 components (Aerts et al 2010) whichare observed as multiplets in a starrsquos amplitude spectrumThe splitting of these component frequencies is significantlyasymmetric for moderate and fast rotating stars with theCoriolis force acting against the direction of rotation andaffects prograde and retrograde pulsation modes differently(Aerts et al 2010) For a star with numerous non-radial pul-sation modes as is the case for numerous δ Sct stars theeffects of rotation act to increase the observed range of fre-quencies in an amplitude spectrum (see eg Breger et al2012) However there is no expectation for the dominantpulsation mode frequency andor amplitude of a δ Sct starto be correlated with rotation

In Fig 5 the subgroups of LC and SC δ Sct stars thathave stellar parameters from Tkachenko et al (2012 2013)and Niemczura et al (2015 2017) are shown in Teff minus log g

diagrams in the left and right panels respectively The toprow shows the location of each star in a Teff minus log g dia-gram with each star shown as a filled circle colour-codedby v sin i determined from spectroscopy There is no clearcorrelation amongst effective temperature surface gravityand rotation with slow and fast rotators found across theclassical instability strip Of course since the spectroscopicvalues of v sin i are projected surface rotational velocitiesthey represent lower limits of the true rotation of the starsshown in Fig 5

In the bottom row of Fig 5 the relationship between thefrequency and amplitude of the dominant pulsation modeis shown with each star colour-coded by v sin i This figuresupports the findings of Breger (2000a) that δ Sct stars withhigh pulsation mode amplitudes are typically slow rotators(v sin i 50 km sminus1) but clearly not all slowly rotating δ Sctstars have high pulsation mode amplitudes However thelack of any significant correlation in the panels of Fig 5 isnot surprising with so few δ Sct stars having been observedwith high-resolution spectroscopy

5 REGULARITIES IN THE AMPLITUDESPECTRA OF δ Sct STAR STARS

In this section a similar methodology as that employed byMichel et al (2017) who searched for regularities in the am-plitude spectra of approximately 1800 δ Sct stars observedby CoRoT is applied to the SC ensemble of δ Sct stars pre-sented in this work It is not informative to perform thisanalysis using the LC ensemble of δ Sct stars because ofthe significantly longer integration times of LC Kepler datasuppressing high-frequency signals Consequently few δ Sct

stars exist in the LC ensemble that satisfy the required se-lection criteria of having high-frequency pulsation modes mdashie young δ Sct stars The SC Kepler data were not ob-tained using as short a cadence as the CoRoT data but it isstill significantly higher than the pulsation mode frequenciesin δ Sct stars

Following Michel et al (2017) an amplitude significancecriterion was chosen as ten times the mean amplitude ineach starrsquos amplitude spectrum with the frequency range ofνhigh and νlow calculated as the frequency of the highest- andlowest-amplitude peaks that satisfied the amplitude signifi-cance criteria for frequencies above ν ge 4 dminus1 (amp 50 microHz)This minimum in frequency is approximately twice the valuechosen by Michel et al (2017) but we use 4 dminus1 in this studyto reduce the chance that the extracted values of νlow rep-resent g-mode pulsation frequencies Each starrsquos amplitudespectrum was interpolated onto a frequency resolution of005 dminus1 using the maximum amplitude value of each binin the original spectrum This downgrades each amplitudespectrum to the approximate Rayleigh resolution criterionfor a data set of approximately 20 d in length but preservesthe amplitude content from the original high-resolution andoversampled Fourier transform This step is necessary toproduce a figure with structure that can be resolved by thehuman eye Also it has the added advantage of homogenis-ing the amplitude spectra in the SC ensemble irrespective ofdata set length The modified amplitude spectra shown be-tween 0 lt ν lt 90 dminus1 are stacked in order of increasing effec-tive temperature using KIC values and are shown in Fig 6In this figure solid red lines are used to indicate a movingaverage of the observables νlow and νhigh along the star num-ber ordinate axis The stacked amplitude spectra show anincrease in the frequencies of pulsation modes for increas-ing Teff but also show an increase in the range of observedpulsation mode frequencies for hotter δ Sct stars Thereforenot only does the frequency of maximum amplitude νmaxincrease with increasing Teff but so do the observables νlowand νhigh

The distribution of the νhigh and νlow values for eachstar in the SC ensemble is shown in Fig 7 with each starshown as a filled circle colour-coded by the amplitude of thedominant pulsation mode Using the same criteria as Michelet al (2017) a subsample of young δ Sct stars was created byincluding the δ Sct stars with νhigh ge 35 dminus1 (amp 400 microHz) andthe ratio of the highest and lowest frequency in each star sat-isfying (νhighνlow) lt 45 These selection criteria are showngraphically by the individual solid black lines in Fig 7 withthe enclosed grey region indicating the subsample of youngδ Sct stars This subgroup contains 60 δ Sct stars whichis noticeably fewer than the sim250 CoRoT stars presentedby Michel et al (2017) To include more young stars us-ing the SC Kepler data the νhigh restriction would have to

be relaxed substantially below 35 dminus1 but this boundary ismotivated by theoretical models of pulsation mode frequen-cies in young δ Sct stars being above 35 dminus1 and changingthis criterion removes the compatibility with the analysis byMichel et al (2017)

The stacked amplitude spectra of this subgroup ofyoung δ Sct stars using SC Kepler data are plotted in thetop panel of Fig 8 in order of ascending values of νhigh Thehigh-frequency ridges observed in the amplitude spectra ofyoung δ Sct stars studied by Michel et al (2017) were shown



Figure 5 The left and right columns correspond to the LC and SC ensembles of δ Sct stars respectively for which spectroscopic values

of Teff log g and v sin i are available The top row shows the location of each δ Sct in a Teff minus log g diagram using spectroscopic values

from Tkachenko et al (2012 2013) and Niemczura et al (2015 2017) with each starrsquos location colour-coded by v sin i The same stellarevolutionary tracks ZAMS line and theoretical edges of the classical instability strip as in Fig 1 are also shown in the top row The

bottom row shows the relationship between the frequency and amplitude of the dominant pulsation mode colour-coded by v sin i

Figure 6 The stacked amplitude spectra for the ensemble of SC δ Sct stars in order of increasing Teff using KIC values in an upwardsdirection with the red lines indicating a moving average of the νlow and νhigh values along the star number ordinate axis Indicativeeffective temperature values are also included on the ordinate axis to demonstrate how νlow and νhigh vary with Teff



Figure 7 The distribution of lowest and highest frequencies of

peaks with amplitudes greater than ten times the mean amplitudein the amplitude spectrum above ν ge 4 dminus1 for the ensemble of

SC δ Sct stars Each star has been colour-coded by the amplitude

of the highest-amplitude peak in the amplitude spectrum and thegrey region denotes the subsample of young δ Sct stars based

on νhigh gt 35 dminus1 and a ratio of (νhighνlow) lt 45 Since νlow and

νhigh are extracted in the frequency range of 4 le ν le 98 dminus1 theycan be significantly smaller or larger than the corresponding νmaxvalue for each star (cf section 4)

to be consistent with axisymmetric island modes using the-oretical models of fast rotating stars (Reese et al 2009)To examine any similar ridges in the amplitude spectra us-ing SC Kepler data the amplitude spectra were normalisedby νhigh and are plotted on a logarithmic frequency scalein the bottom panel of Fig 8 in which the rows above thewhite space show an average amplitude spectrum for the60 δ Sct stars in the panel The ridges and regularities dis-cussed by Michel et al (2017) are evident in a fraction butnot all of the amplitude spectra shown in Fig 8 Further-more the ridge-like structure is not always near the value ofνhigh in a starrsquos amplitude spectrum For example the laststar in the top panel of Fig 8 has several ridges separatedby approximately 3 dminus1 in its amplitude spectrum between35 le ν le 45 dminus1 which could correspond to the large fre-quency separation in this star but the value of νhigh 80 dminus1

using the SN ge 10 amplitude criterion described previouslyclearly corresponds to higher frequencies that are harmonicsand combination frequencies of these ridges and not intrinsicpulsation modes

Other stars similarly have ridges in their amplitudespectra that do not appear near their νhigh values whichexplains the lack of regularity in the average amplitude spec-trum shown in the rows above the white space in the bottompanel in Fig 8 Therefore this method of normalising ampli-tude spectrum by νhigh is strongly dependent on the selectioncriteria used in determining νhigh and is not consistent forall stars This is easily understood since all peaks in a starrsquosamplitude spectra above SN ge 10 in amplitude were in-cluded in the determination of νlow and νhigh However it ispossible for a δ Sct star to have harmonics and combinationfrequencies that have amplitudes larger than the chosen am-plitude criterion so a larger νhigh value can be used for sucha star which corresponds to a harmonic or a combinationfrequency rather than a pulsation mode frequency

The frequency spacing between consecutive radial orderp modes in a δ Sct star is of order a few dminus1 but the specificvalue depends on the stellar parameters and the radial ordersof the pulsation modes (Breger et al 2009 Garcıa Hernandezet al 2015 Van Reeth et al 2018) Furthermore althoughp modes in δ Sct stars are typically low-radial order pulsa-tion modes of higher radial order can be considered in thetransition region where the asymptotic relation for p modesapplies such that p modes are equally-spaced in frequencyThe mode-dependent transition in the validity of the asymp-totic relation for p modes would break the expectation offinding equally-spaced ridges in the logarithmic plot shownin the bottom panel of Fig 8 but the amplitude spectrain linear unnormalised frequencies would remain unaffectedfor high radial orders This is certainly true for the starsin Fig 1 that lie hotter than the blue edge of the classicalinstability strip and are inferred to be pulsating in p modeswith radial orders of n ge 6

Therefore the stars identified as young δ Sct stars inFig 8 which were selected because they have high pulsationmode frequencies (νhigh ge 35 dminus1) corresponding to youngstars and inferred to be close to the ZAMS based on predic-tions from theoretical models (Michel et al 2017) representa subgroup of δ Sct stars for which mode identification issimplified because of the presence of regularities in their am-plitude spectra Further observational and theoretical studyfor the stars showing regularities in their amplitude spectrais required as they provide insight of mode excitation andinterior physics such as rotation and mixing in δ Sct starswhere mode identification is possible cf Van Reeth et al(2018)

6 DISCUSSION AND CONCLUSIONS

Two ensembles of δ Sct stars using LC and SC Kepler dataconsisting of 963 and 334 stars respectively were compiledand used to characterise the ensemble pulsational proper-ties of intermediate-mass A and F stars The LC Keplersampling frequency of 489 dminus1 introduces a bias towardsextracting low-frequency pulsation modes in an iterativepre-whitening procedure with higher frequencies being sup-pressed in amplitude The amplitude visibility function givenin Eqn (1) explains the dearth of δ Sct stars with pulsationmode frequencies above ν ge 40 dminus1 in LC Kepler data Theamplitude visibility function is negligible in the SC ensemblesince the SC sampling frequency is 14769 dminus1 the limita-tion of these data is that fewer δ Sct stars were observed inSC than in LC and typically not for time spans longer than30 d which has a poor frequency resolution in comparison

The distributions of δ Sct stars in Teff minus log g diagramsshown in Fig 1 demonstrate that the theoretical edges ofthe classical instability strip for modes of radial orders be-tween 1 le n le 6 calculated by Dupret et al (2005) aremostly consistent with observations of δ Sct stars from theKepler Space Telescope However a minority of δ Sct starsincluding some with stellar parameters obtained using high-resolution spectroscopy by Tkachenko et al (2012 2013)and Niemczura et al (2015 2017) are hotter than the blueedge andor cooler than the red edge of the classical in-stability strip depending on the source of stellar parame-ters (eg Brown et al 2011 Huber et al 2014) Further-



Figure 8 The top panel shows the subsample of young δ Sct stars from the SC ensemble which were chosen as the stars that satisfied

νhigh gt 35 dminus1 and a ratio of (νhighνlow) lt 45 similar to Michel et al (2017) The solid red line indicates a moving average of the νhighvalues along the star number ordinate axis The bottom panel is the same subgroup of stars as the top panel but with each starrsquos

amplitude spectrum having been normalised by its νhigh value and shown on a logarithmic abscissa scale The solid red line indicates a

moving average of the νlow values along the star number ordinate axis In the bottom panel the three rows above the white space show anaverage of the 60 normalised amplitude spectra The colour scale in both panels has been purposefully chosen to have an upper limit of

02 mmag to reveal low-amplitude peaks which is similar to the arbitrary upper value of 02 parts-per-thousand (ppt) chosen by Michel

et al (2017)

more the boundaries of the classical instability strip calcu-lated by Dupret et al (2005) were calibrated using ground-observations of δ Sct stars which typically have high levelsof noise in their amplitude spectra thus are limited to highpulsation mode amplitudes The analysis of a large numberof δ Sct stars in this work supports the requirement for amass-dependent value of the αMLT parameter in theoreticalmodels to reproduce all δ Sct stars in a Teff minus log g diagramThis is particularly necessary to explain the hot δ Sct starsthat are beyond the blue edge of the n = 6 radial mode usingαMLT = 18 (Dupret et al 2004 2005)

The distributions of maximum pulsation amplitude forthe LC and SC ensembles were shown to be consistent witheach other as shown in Fig 2 with typical pulsation modeamplitudes between 05 mmag and 10 mmag found for mostδ Sct stars Similar high-amplitude tails in the amplitude dis-tributions of the LC and SC data are also evident which arecaused by the presence of HADS stars in the Kepler missiondata These stars have been previously demonstrated to berare (Lee et al 2008 Balona 2016 Bowman 2017) a finding

that is supported by the scarcity of these stars in both LCand SC ensembles in this work It remains unclear if HADSstars are physically distinct from their low-amplitude coun-terparts (Breger 2000b Balona 2016 Bowman 2016 2017)Further work is needed to address this

Despite the lack of amplitude suppression few δ Sctstars exist in the Kepler data set with pulsation mode fre-quencies above 60 dminus1 with no δ Sct stars having their dom-inant pulsation mode above 60 dminus1 as shown in Fig 3 Thisis in agreement with previous analyses of δ Sct stars us-ing subsets of Kepler data (Balona amp Dziembowski 2011)if such studies had been corrected for the amplitude visibil-ity function (Eqn 1) The dearth of high frequency δ Sctstars observed by Kepler can be explained by the lack ofhot ZAMS stars with the TAMS being better sampled incomparison (Tkachenko et al 2012 2013 Niemczura et al2015 2017) The correlations between the pulsation modefrequencies effective temperature and evolutionary stage(log g by proxy) were investigated and shown in Fig 3 Theensembles were separated into ZAMS (log g ge 40) MAMS



(35 le log g lt 40) and TAMS (log g lt 35) subgroups usingthe KIC values with separate linear regressions between thefrequency of the highest-amplitude pulsation mode and Tefffor each log g subgroup demonstrating a stronger correlationfor ZAMS stars in the SC data For a ZAMS δ Sct star nearthe blue edge of the classical instability strip a high effec-tive temperature facilitates the excitation of higher overtonep modes and higher observed pulsation mode frequencies be-cause the κ mechanism operating in the He ii ionisation zoneis closer to the surface of the star (Pamyatnykh 1999 2000Dupret et al 2004 2005) This relationship was also found inground-based observations of δ Sct stars (Breger amp Bregman1975 Breger 2000b Rodrıguez amp Breger 2001)

Following a similar methodology to that employed byMichel et al (2017) who classified and studied approxi-mately 250 young δ Sct stars observed by CoRoT regu-larities were searched for in the amplitude spectra of 60young δ Sct stars identified in the SC ensemble of Keplerobservations using the same requirement of high-frequency(ν ge 35 dminus1) pulsation modes in their amplitude spec-tra Similar ridges consistent with consecutive radial orderp modes can be seen at high frequency in the amplitude spec-tra of some but not all of these young δ Sct stars in theSC ensemble This can partially be explained by the ampli-tude significance criterion used to determine the observablesνlow and νhigh specifically how harmonics and combinationfrequencies need to be identified when searching for the fre-quency separation between radial order p modes However afraction of the stars shown in Fig 8 do show frequency ridgeswith spacings of order a few dminus1 which are consistent withaxisymmetric island modes predicted by theoretical modelsof fast rotating stars (Reese et al 2009 Michel et al 2017)

The lack of a complete theory for pulsation mode exci-tation specifically the non-linear effects that determine theamplitudes of pulsation modes means we are unable to re-alistically predict the amplitudes of pulsation modes usingcurrent theoretical models To test if one expects significantregularities caused by the separation between consecutiveradial modes in the amplitude spectra of δ Sct stars Reeseet al (2017) investigated multiplying the intrinsic ampli-tudes from theoretical amplitude spectra by random num-bers uniformly drawn from between 1 and 100 on a logarith-mic scale This proxy for a non-linear mode saturation mech-anism combined with fast rotation pulsation mode visibil-ity and observing a star at an unknown inclination angle ledReese et al (2017) to conclude that finding such regularitiesin δ Sct stars is unlikely The exceptions to this were foundin so-called favourable scenarios where 2Ω coincided with ∆νor ∆ν2 corresponding to 30 and 70 per cent of the break-upvelocity (Reese et al 2017) The lack of significant regulari-ties in the majority of the amplitude spectra of young δ Sctstars presented in this work is consistent with the conclu-sions presented by Reese et al (2017) On the other thoseδ Sct stars with regularities consistent with consecutive ra-dial order p modes may provide valuable constraints of rota-tion and inclination of δ Sct stars based on the discussion byReese et al (2017) which are typically difficult parametersto determine for δ Sct stars

Other added complications when studying the ampli-tude spectra of δ Sct stars are non-linearity in the form ofmode coupling between pulsation modes and non-linearityin the form of harmonics and combination frequencies

(Breger amp Montgomery 2014 Bowman 2017) which createpseudo-regularities and dense forest-like amplitude spectraIt is also important to note that the pulsation modes in δ Sctstars can potentially cover a large range of radial orders suchthat the expected frequency separation between consecutiveradial order p modes in a star is not a constant This can beunderstood from the transition from moderate to high val-ues of n coinciding with the transition for which the asymp-totic approximation applies This produces a non-constantvalue for the frequency separation across several radial or-ders in δ Sct stars further complicating the task or findingregularity in a starrsquos amplitude spectrum (see eg Bregeret al 2009) All the discussed effects act towards blurringthe mean of the normalised amplitude spectrum for an en-semble of δ Sct stars which explains the lack of regularityin the average amplitude spectrum shown in the rows abovethe white space in the bottom panel of Fig 8

Many δ Sct stars are observed to have low-frequencypeaks in their amplitude spectra (Balona 2014 Balona et al2015) which can be identified as combination frequencies orindependent g-mode pulsations (see eg Saio et al 2018)The full scientific potential of these hybrid stars is yet to beestablished with direct measurements of rotation and angu-lar momentum transport needed in A stars needed to fill thegap between B and F stars and constrain theoretical mod-els (Van Reeth et al 2016 Aerts et al 2017b) Asteroseis-mic modelling of the most-promising δ Sct stars for whichmode identification is possible using the stellar evolutioncode MESA (Paxton et al 2011 2013 2015) and the pulsa-tion code GYRE (Townsend amp Teitler 2013) will constraintheoretical models of interior physics in these stars (see egVan Reeth et al 2018) The ensemble of 963 δ Sct stars ob-served continuously by the Kepler Space Telescope with anunprecedented photometric precision is the best data setsfor studying the interior physics of A stars and will remainof great use for many years to come

ACKNOWLEDGEMENTS

DMB would like to thank the Kepler science team for pro-viding such excellent data and Prof Conny Aerts Dr PeterPapics and Dr Timothy Van Reeth for useful discussionsFunding for the Kepler mission is provided by NASArsquos Sci-ence Mission Directorate The Kepler data presented in thispaper were obtained from the Mikulski Archive for SpaceTelescopes (MAST) Support for MAST for non-HST datais provided by the NASA Office of Space Science via grantNNX09AF08G and by other grants and contracts The re-search leading to these results has received funding fromthe European Research Council (ERC) under the EuropeanUnionrsquos Horizon 2020 research and innovation programme(grant agreement No670519 MAMSIE) This research hasmade use of the SIMBAD database operated at CDS Stras-bourg France the SAONASA Astrophysics Data Systemand the VizieR catalogue access tool CDS StrasbourgFrance We thank the referee for their comments that im-proved the manuscript

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Bouabid M-P Dupret M-A Salmon S Montalban J Miglio

A Noels A 2013 MNRAS 429 2500

Bowman D M 2016 PhD thesis Jeremiah Horrocks InstituteUniversity of Central Lancashire Preston UK

Bowman D M 2017 Amplitude Modulation of Pulsation Modesin Delta Scuti Stars Springer doi101007978-3-319-66649-5

Bowman D M Kurtz D W 2014 MNRAS 444 1909

Bowman D M Holdsworth D L Kurtz D W 2015 MNRAS449 1004

Bowman D M Kurtz D W Breger M Murphy S JHoldsworth D L 2016 MNRAS 460 1970

Breger M 2000a in Szabados L Kurtz D eds Astronomical

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Breger M 2000b in Breger M Montgomery M eds Astronom-ical Society of the Pacific Conference Series Vol 210 Delta

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Breger M Bregman J N 1975 ApJ 200 343

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Breger M Lenz P Pamyatnykh A A 2009 MNRAS 396 291

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Brown T M Latham D W Everett M E Esquerdo G A

2011 AJ 142 112

Cantiello M Mankovich C Bildsten L Christensen-Dalsgaard

J Paxton B 2014 ApJ 788 93

Chaplin W J Miglio A 2013 ARAampA 51 353

Christensen-Dalsgaard J 2000 in Breger M Montgomery M

eds Astronomical Society of the Pacific Conference Series Vol210 Delta Scuti and Related Stars p 187

Cox J P 1963 ApJ 138 487

Degroote P et al 2010 Nature 464 259

Dupret M A Grigahcene A Garrido R Gabriel M ScuflaireR 2004 AampA 414 L17

Dupret M A Grigahcene A Garrido R Gabriel M ScuflaireR 2005 AampA 435 927

Eggenberger P et al 2017 AampA 599 A18

Garcıa Hernandez A et al 2009 AampA 506 79

Garcıa Hernandez A et al 2013 AampA 559 A63

Garcıa Hernandez A Martın-Ruiz S Monteiro M J P F GSuarez J C Reese D R Pascual-Granado J Garrido R

2015 ApJ 811 L29

Gilliland R L et al 2010 PASP 122 131

Grigahcene A Dupret M-A Gabriel M Garrido R ScuflaireR 2005 AampA 434 1055

Grigahcene A et al 2010 ApJ 713 L192

Handler G 2009 MNRAS 398 1339

Hekker S Christensen-Dalsgaard J 2017 AampARv 25 1

Holdsworth D L et al 2014 MNRAS 439 2078

Holdsworth D L Saio H Bowman D M Kurtz D W SefakoR R Joyce M Lambert T Smalley B 2018 MNRAS

Houdek G 2000 in Breger M Montgomery M eds Astronom-

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Houdek G Dupret M-A 2015 Living Reviews in Solar Physics

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Huber D et al 2014 ApJS 211 2

Koch D G et al 2010 ApJ 713 L79

Kurtz D W et al 2005 MNRAS 358 651

Kurtz D W Saio H Takata M Shibahashi H Murphy S J

Sekii T 2014 MNRAS 444 102

Kurtz D W Shibahashi H Murphy S J Bedding T R Bow-man D M 2015 MNRAS 450 3015

Lee Y-H Kim S S Shin J Lee J Jin H 2008 PASJ 60 551

Maeder A 2009 Physics Formation and Evolution of Rotating

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McNamara D H 2000 in Breger M Montgomery M eds As-

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Meynet G Ekstrom S Maeder A Eggenberger P Saio H

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K Neiner C Lignieres F Green J J eds Lecture Notesin Physics Berlin Springer Verlag Vol 865 Lecture Notes

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Michel E et al 2017 in European Physical JournalWeb of Conferences p 03001 (arXiv170503721)

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Montgomery M H OrsquoDonoghue D 1999 Delta Scuti Star

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Murphy S J Bedding T R Niemczura E Kurtz D W Smalley

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Grassitelli L Wang E S 2016 MNRAS 459 1201

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eds Lecture Notes in Physics Berlin Springer Verlag Vol 523

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Bloemen S Southworth J 2014 AampA 570 A8

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This paper has been typeset from a TEXLATEX file prepared by

the author


1 Introduction

11 Regularities in the amplitude spectra of Sct stars

2 Compiling ensembles of Kepler Sct stars

21 Extracting Sct stellar parameters

22 Extracting Sct pulsation parameters

3 Revisiting the edges of the Classical Instability Strip with Kepler data

4 Characterising the pulsation properties of Sct stars observed by Kepler



43 Low-frequency variability in Sct stars



5 Regularities in the amplitude spectra of Sct star stars

6 Discussion and Conclusions


stellar structure and evolution The dominant pulsations inδ Sct stars are self excited by the opacity (κ) mechanism op-erating in the He ii ionisation zone (Cox 1963 Breger 2000bAerts et al 2010) which produces pressure (p) modes withperiods as long as eight hours and short as 15 min (Uytter-hoeven et al 2011 Holdsworth et al 2014) These p modestypically have low radial orders and are most sensitive tothe surface layers of a star Another form of coherent modeexcitation in δ Sct stars is from turbulent pressure which isable to excite p modes with radial orders between 7 le n le 10with higher pulsation mode frequencies for stars within theclassical instability strip (Houdek 2000 Antoci et al 2014Xiong et al 2016)

The instability regions of the δ Sct and the lower-massγ Dor stars are predicted to overlap in the HR diagram(Dupret et al 2004 2005 Houdek amp Dupret 2015) withstars that simultaneously pulsate in p modes excited by theκ mechanism and gravity (g) modes excited by the convec-tive flux blocking (modulation) mechanism commonly re-ferred to as hybrid stars The exact nature the excitationmechanism of g modes and its physical relationship with theκ-mechanism in intermediate-mass stars is not fully under-stood (see eg Xiong et al 2016) with a significant fractionof δ Sct stars hotter than the γ Dor instability region ob-served to pulsate with g modes (Uytterhoeven et al 2011Bowman 2016 2017) Few hybrid stars were known priorto space photometry (see eg Handler 2009) The unprece-dented photometric precision duty cycle and total length ofthe Kepler mission (Borucki et al 2010 Koch et al 2010)revealed that many δ Sct stars are hybrid stars (Grigahceneet al 2010 Uytterhoeven et al 2011 Balona amp Dziembowski2011 Balona 2014) On the other hand it is also demonstra-bly true that pure δ Sct stars with no significant frequenciesbelow 5 dminus1 albeit rare exist (Bowman 2017)

An analysis of 750 A and F stars was carried out byUytterhoeven et al (2011) using 1 yr (Q0 ndash Q4) of Keplerdata and it was concluded that approximately 25 per centwere hybrid pulsators Later analyses of A and F stars us-ing longer time spans of Kepler data by Balona (2014) andBalona et al (2015) have found much higher fractions ofδ Sct stars with low frequencies in their amplitude spectrabut also demonstrate that less than fifty per cent of starswithin the classical instability strip pulsate This measuredincidence of pulsation amongst δ Sct stars should be consid-ered a lower limit with 50 per cent of A and F stars observedto pulsate using the detection threshold Kepler photometrywhich is of order a few micromag in the best of cases For manyA and F pulsators it is not established if observed low-frequency peaks are caused by pulsation modes rotationalmodulation combination frequencies or have some other as-trophysical cause (Bowman 2017) Recent studies by VanReeth et al (2016) and Saio et al (2018) have discoveredthat global normal modes of Rossby waves (r modes) whichcause temperature perturbations in a rotating star are vis-ible in A and F stars in the Kepler data set Specifically rmodes in rapidly rotating γ Dor stars exhibit a power excessat low frequency such that r modes with azimuthal order mproduce a group of frequencies slightly below m times a starrsquosrotation frequency (Saio et al 2018)

For chemically normal stars within the classical insta-bility strip it is not surprising that they are observed topulsate from the nature of the driving mechanism in δ Sct

stars (Murphy et al 2015) although it is also known thatapproximately half of stars within the classical instabilitystrip are not pulsating (Balona amp Dziembowski 2011) at thephotometric precision of order a few micromag in Kepler dataFurther work is needed to study the interplay between theflux blocking mechanism and the κ mechanism in A and Fstars with a focus on understanding the synergy between thepulsation excitation mechanisms and the observed minorityof δ Sct stars pulsating in purely p modes This providesmotivation for characterising the pulsation properties of alarge ensemble of δ Sct stars observed by Kepler presentedin this work

However one of the main difficulties encountered whenobserving δ Sct stars is the issue of identifying pulsationmodes in terms of their radial order n angular degree `and azimuthal order m The typical low-order p modes inδ Sct stars often do not show any regular spacing in theamplitude spectrum unlike the asymptotic high radial or-der p modes in low-mass solar-type stars mdash see Chaplin ampMiglio (2013) and Hekker amp Christensen-Dalsgaard (2017)for reviews of solar-like pulsators The rapid rotation modevisibility and our incomplete understanding of mode excita-tion in δ Sct stars make them one of the more challenginggroups of pulsating stars to study using asteroseismology(eg Reese et al 2017) Despite these complexities ongo-ing efforts to apply ensemble asteroseismology techniques toδ Sct stars are being made using similar methods to thoseused to study low-mass stars (Garcıa Hernandez et al 20092013 2015 Paparo et al 2016 Michel et al 2017)

Significant progress in understanding the physics ofδ Sct stars has been made since the space photometry revo-lution facilitated by the CoRoT (Auvergne et al 2009) andKepler (Borucki et al 2010) space missions The rich pulsa-tion spectra of δ Sct stars offer the potential to probe physicsat different depths using asteroseismology thus these starscan provide useful constraints of stellar structure and evo-lution theory in an important transition region on the mainsequence (Breger 2000b Aerts et al 2010) The full scientificpotential of studying pulsation modes in δ Sct stars with as-teroseismology in the era of space photometry has yet to beexploited

11 Regularities in the amplitude spectra of δ Sctstars

It has been the goal of various studies to search for regu-larities in the amplitude spectra of δ Sct stars (eg Gar-cıa Hernandez et al 2009 2013 2015 Paparo et al 2016Michel et al 2017) with a particular focus of identifyingthe asymptotic frequency separation between modes of con-secutive high radial order known as the large frequencyseparation ∆ν However δ Sct stars often have high modedensities in their amplitude spectra caused by dozens andsometimes hundreds of pulsation mode frequencies makingit difficult to distinguish intrinsic pulsation modes This iscomplicated by non-linearity of pulsation modes in δ Sctstars in the form of harmonics nνi and combination frequen-cies nνi plusmn mνj where n and m take integer values (Papics2012 Kurtz et al 2015 Bowman 2017) These non-linearcombination frequencies can have high Fourier amplitudesbut do not represent intrinsic pulsation modes since theydescribe the non-sinusoidal shape of a starrsquos light curve in











A = A0 sinc( π

n

)= A0 sinc

(πν

νsamp

) (1)




























































































KIC


Revised KIC


Spectroscopy


































































et al (2017)













ACKNOWLEDGEMENTS


REFERENCES


































2011 AJ 142 112






Cox J P 1963 ApJ 138 487








2015 ApJ 811 L29












12
















doi101007978-3-642-33380-4 1


doi101051epjconf201716003001


Newsletter 13 28



B 2015 MNRAS 447 3948













224 41






















MNRAS 474 2774





2012 MNRAS 422 2960











the author


1 Introduction























A = A0 sinc( π

n

)= A0 sinc

(πν

νsamp

) (1)




























































































KIC


Revised KIC


Spectroscopy


































































et al (2017)













ACKNOWLEDGEMENTS


REFERENCES


































2011 AJ 142 112






Cox J P 1963 ApJ 138 487








2015 ApJ 811 L29












12
















doi101007978-3-642-33380-4 1


doi101051epjconf201716003001


Newsletter 13 28



B 2015 MNRAS 447 3948













224 41






















MNRAS 474 2774





2012 MNRAS 422 2960











the author


1 Introduction






































































































KIC


Revised KIC


Spectroscopy


































































et al (2017)













ACKNOWLEDGEMENTS


REFERENCES


































2011 AJ 142 112






Cox J P 1963 ApJ 138 487








2015 ApJ 811 L29












12
















doi101007978-3-642-33380-4 1


doi101051epjconf201716003001


Newsletter 13 28



B 2015 MNRAS 447 3948













224 41






















MNRAS 474 2774





2012 MNRAS 422 2960











the author


1 Introduction





































































et al (2017)













ACKNOWLEDGEMENTS


REFERENCES


































2011 AJ 142 112






Cox J P 1963 ApJ 138 487








2015 ApJ 811 L29












12
















doi101007978-3-642-33380-4 1


doi101051epjconf201716003001


Newsletter 13 28



B 2015 MNRAS 447 3948













224 41






















MNRAS 474 2774





2012 MNRAS 422 2960











the author


1 Introduction





















ACKNOWLEDGEMENTS


REFERENCES


































2011 AJ 142 112






Cox J P 1963 ApJ 138 487








2015 ApJ 811 L29












12
















doi101007978-3-642-33380-4 1


doi101051epjconf201716003001


Newsletter 13 28



B 2015 MNRAS 447 3948













224 41






















MNRAS 474 2774





2012 MNRAS 422 2960











the author


1 Introduction













































2011 AJ 142 112






Cox J P 1963 ApJ 138 487








2015 ApJ 811 L29












12
















doi101007978-3-642-33380-4 1


doi101051epjconf201716003001


Newsletter 13 28



B 2015 MNRAS 447 3948













224 41






















MNRAS 474 2774





2012 MNRAS 422 2960











the author


1 Introduction
















MNRAS 474 2774





2012 MNRAS 422 2960











the author


1 Introduction














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