A spectroscopic test of the rotational modulation origin of periodic \emph{Kepler} photometric variability of A-type stars
aa r X i v : . [ a s t r o - ph . S R ] S e p MNRAS , 1– ?? (2020) Preprint 10 September 2020 Compiled using MNRAS L A TEX style file v3.0
A spectroscopic test of the rotational modulation origin ofperiodic
Kepler photometric variability of A-type stars
J. Sikora, , , ⋆ G. A. Wade, J. Rowe, Department of Physics, Engineering Physics & Astronomy, Queen’s University, Kingston, ON Canada, K7L 3N6 Department of Physics and Space Science, Royal Military College of Canada, PO Box 17000 Kingston, Ontario, Canada, K7K 7B4 Department of Physics and Astronomy, Bishop’s University, Sherbrooke, Qu´ebec, Canada, J1M 1Z7
Submitted 2020 Mth. XX
ABSTRACT
High-precision space-based photometry obtained by the
Kepler and
TESS missionshas revealed evidence of rotational modulation associated with main sequence (MS)A and late-B type stars. Generally, such variability in these objects is attributed toinhomogeneous surface structures (e.g. chemical spots), which are typically linkedto strong magnetic fields ( B &
100 G) visible at the surface. It has been reportedthat ≈
44 per cent of all A-type stars observed during the
Kepler mission exhibitrotationally modulated light curves. This is surprising considering that .
10 per centof all MS A-type stars are known to be strongly magnetic (i.e. they are Ap/Bp stars).We present a spectroscopic monitoring survey of 44 A and late-B type stars reportedto exhibit rotational modulation in their
Kepler light curves. The primary goal ofthis survey is to test the hypothesis that the variability is rotational modulation bycomparing each star’s rotational broadening ( v sin i ) with the equatorial velocities( v eq ) inferred from the photometric periods. We searched for chemical peculiaritiesand binary companions in order to provide insight into the origin of the apparentrotational modulation. We find that 14 stars in our sample have v sin i > v eq and/orhave low-mass companions that may contribute to or be responsible for the observedvariability. Our results suggest that more than 10 per cent of all MS A and late-Btype stars may exhibit inhomogeneous surface structures; however, the incidence rateis likely .
30 per cent.
Key words:
Stars: early-type, Stars: magnetic, Stars: rotation
Magnetic chemically peculiar stars (mCP or Ap/Bp stars)are known to exhibit horizontally inhomogeneous distribu-tions of various chemical elements within their atmospheres(e.g. Khokhlova & Riabchikova 1975; Kochukhov et al.2002; Silvester et al. 2014). The formation of these intense,long-lived chemical abundance non-uniformities (i.e. “chem-ical spots”) can largely be attributed to the presence ofstrong, organized magnetic fields that are visible at the star’ssurface (Michaud 1970; Michaud et al. 1981). One conse-quence of such chemical spots is that the star’s flux is locallyredistributed from the UV to longer wavelengths throughline blanketing and backwarming (Kochukhov et al. 2005a); ⋆ Based on observations obtained at the Canada-France-HawaiiTelescope (CFHT) which is operated by the National ResearchCouncil of Canada, the Institut National des Sciences de l’Universof the Centre National de la Recherche Scientifique of France, andthe University of Hawaii. when coupled with the star’s rotation, chemical spots oftenlead to periodic photometric variability (e.g. Krtiˇcka et al.2009, 2012, 2015).Currently, only a relatively small subsample of main se-quence (MS) A- and late B-type stars are known to exhibitlow-frequency photometric variability that can convincinglybe attributed to chemical spots. This subsample is domi-nated by the known mCP stars (e.g. Adelman & Woodrow2007; Bernhard et al. 2015), which account for ∼
10 per centof all MS A-type stars (Wolff 1968; Smith 1971; Sikora et al.2019a). Therefore, the detection of rotational modulation as-sociated with a MS A/late-B star generally implies a highprobability that the star is magnetic (Buysschaert et al.2018; David-Uraz et al. 2019, 2020). Based on an analysisof the high-precision light curves yielded by the
Kepler mis-sion, Balona (2013) reported the surprising discovery that ≈
44 per cent of A-type stars in the
Kepler sample exhibitphotometric variability that is consistent with a rotationalmodulation origin. These findings suggest that the accepted © J. Sikora et al. fraction of MS A-type stars that host detectable magneticfields may be underestimated by ≈
400 per cent. Such adramatic underestimation is made plausible by the fact thatthe reported photometric amplitudes, which have a meanvalue of 0 .
06 mmag, are significantly lower than the detec-tion limits associated with previous all-sky time series pho-tometry surveys (e.g. typical
Hipparcos photometric mea-surement uncertainties are & β UMa, and θ Leo – have been found to host ultra-weakmagnetic fields with strengths . ∼
100 times weakerthan the weakest fields found on mCP stars) (Ligni`eres et al.2009; Petit et al. 2011; Blaz`ere et al. 2016). Low-contrastspots presumably caused by inhomogeneous chemical ortemperature distributions have also recently been inferredfrom spectroscopic observations of Vega (B¨ohm et al. 2015;Petit et al. 2017). Two important questions related toVega’s spectroscopically detected spots currently remainunanswered: (1) are they directly linked to the star’s mag-netic structure (i.e. are the spot sizes and locations corre-lated with Vega’s complex magnetic field topology) and (2)do the spots modulate Vega’s brightness with its rotation?The discovery of ultra-weak fields on A-type stars in con-junction with the reported discovery of a large sample ofA stars that seem to exhibit rotationally modulated lightcurves has led to speculation that many, or perhaps all,MS A-type stars host detectable magnetic fields (those withstrengths & ∼ ≈ Kepler sample. Pedersen et al. (2017) carried out a study of 33 ofthese flaring A stars and concluded that, in at least 19 cases,the detected flares are not intrinsic to the A stars, but ratherthey are more likely attributable to low-mass companions orcontamination from background stars. Whether the detectedrotational modulation is intrinsic to the A stars themselvesand, if so, whether it is produced by chemical and/or tem-perature spots are important unanswered questions, partic-ularly in the context of stellar magnetism. Balona (2017) and more recently, Balona (2019), have begun to addressthe first of these questions using published projected rota-tional velocities ( v sin i ). These two studies identified 223MS A- and late B-type stars with effective temperaturesof 8 000 −
15 000 K that (1) appear to exhibit rotationallymodulated
Kepler , K2 , or TESS light curves and (2) haveavailable v sin i values. It was concluded that the majority ofthis subset have v sin i . v eq where v eq is each star’s equato-rial velocity derived under the assumption that the identifiedphotometric period corresponds to the rotation period (e.g.Fig. 1 of Balona 2019).In the following, we present the results of a spectro-scopic survey of a sample of MS A-type stars identified byBalona (2013) as exhibiting rotationally modulated Kepler light curves. The goals of this survey are (i) to test the rota-tional modulation hypothesis by measuring each star’s v sin i value and comparing with v eq inferred under the assumptionthat the photometric periods are the rotation periods, (ii)to search for line profile variability that may be a diagnosticof surface structures similar to that associated with Ap/Bpstars, (iii) to search for strong chemical peculiarities thatmay provide insight into the origin of the observed variabil-ity, and (iv) to search for the presence of low-mass binarycompanions, where such detections may provide an alterna-tive explanation for the variability (i.e. the variability maybe intrinsic to such companions or it may be associated withorbital motions or tidal distortions).In Sections 2 and 3 we introduce the sample of stars in-cluded in our survey along with the observations used in thisstudy. The derivation of each star’s fundamental parameters,chemical abundances, and radial velocities derived from pho-tometric and spectroscopic measurements are presented inSect. 4. In Sect. 5, we present our search for radial velocityvariability that is indicative of binary companions. Sect. 6presents a new analysis of the Kepler light curves that ex-hibit variability that is hypothesized to be due to rotationalmodulation. In Sect. 7, we compare the derived rotationalbroadening parameters with implied equatorial velocities inorder to test whether the periods inferred from the
Kepler light curves can plausibly be attributed to each star’s ro-tation period. Finally, in Sections 8 and 9 we discuss theresults of our survey and the conclusions that can be drawn.
Our sample was selected from the list of 875 targets iden-tified by Balona (2013) as exhibiting rotational modulationin their
Kepler light curves. The full sample of stars thatBalona (2013) searched for such signatures consisted of all
Kepler targets with effective temperatures ( T eff ) listed in the Kepler
Input Catalogue (KIC) (Brown et al. 2011) as beingbetween 7 500 and 10 000 K (1 974 stars). This T eff range wasadopted in order to approximately select all stars with A9 toA0 spectral types. The T eff values reported in the KIC arederived from photometric observations obtained primarilyusing SDSS griz filters. Brown et al. (2011) note that theywere unable to obtain u observations for all of the starswithin the Kepler field due to the prohibitively long expo-sure times that were required and, as a result, they considerthe effective temperatures included in the KIC with values > MNRAS , 1– ????
Input Catalogue (KIC) (Brown et al. 2011) as beingbetween 7 500 and 10 000 K (1 974 stars). This T eff range wasadopted in order to approximately select all stars with A9 toA0 spectral types. The T eff values reported in the KIC arederived from photometric observations obtained primarilyusing SDSS griz filters. Brown et al. (2011) note that theywere unable to obtain u observations for all of the starswithin the Kepler field due to the prohibitively long expo-sure times that were required and, as a result, they considerthe effective temperatures included in the KIC with values > MNRAS , 1– ???? (2020) pectroscopic survey of Kepler
A stars Based on the list of 875 rotationally variable A-typestars identified by Balona (2013), we selected those tar-gets with
V < / N ∼
100 per pixel at a wavelength of ≈ < − . Based on the spectral typesthat were found in the literature, 23 of the 44 stars in oursample are late-B (B7-B9) or early-A (A0-A2) type stars;the majority of the 27 stars in the sample with publishedluminosity classes are identified as MS stars (class IV or V)while 3 are identified as being evolved (class III). One of the44 sample stars is a mid-B type star, 9 are mid- to late-A(A3-A9) type stars, and no spectral types could be found forthe remaining 11 stars. The sample also contains two knownAm stars (KIC 8692626 and KIC 8703413), one known Bpstar (KIC 8324268), and two known Be stars (KIC 7131828and KIC 3848385). The extracted spectral types are listedin Table 1. This study primarily makes use of two observational datasets:
Kepler photometric light curves and high-resolutionspectroscopic measurements obtained with ESPaDOnS atCanada France Hawaii Telescope (CFHT).
Kepler photometric measurements
The photometric variability associated with the 44 starsin our sample was first reported by Balona (2013) basedon light curves obtained with the
Kepler spacecraft(Borucki et al. 2010). The passband of the filter used forthese photometric measurements has an effective wavelengthof 5 800 ˚A and an effective width of ≈ Kepler ’s CCD pixels is 4” (Koch et al. 2010).All of the
Kepler observations used for the 44 starsin this study were obtained from the Mikulski Archive forSpace Telescopes (MAST) following data release 25. Weused all of the available long cadence light curves (i.e. ∆ t =30 min), which were taken over a time period of 4 yrs fromMay 2, 2009 to May 11, 2013. The majority of the availablelight curves (41 of the 44 stars) span time periods between 13and 18 observing quarters, where each quarter spans approx-imately 3 months; two of the 44 stars have light curves span-ning 6 and 9 quarters (KIC 3629496 and KIC 3848385, re-spectively) while measurements spanning only a single quar-ter (quarter 3) are available for KIC 11600717.We used the PDC SAP light curves, which have beenprocessed by the Kepler team to correct for various er-rors such as the removal of outliers and systematic trends(Smith et al. 2012). All PDC SAP flux measurements that https://archive.stsci.edu/kepler/ were stored as ‘NaN’ were removed. We carried out addi-tional post-processing, which involved removing (i) remain-ing outliers and (ii) long-term trends occuring over timescales on the order of each observing quarter; the de-trendingwas carried out by fitting first- or second-order polynomialsto the flux measurements spanning each quarter and divid-ing by the resulting fit. The Stokes I spectroscopic measurements presented in thisstudy were obtained using the ESPaDOnS ´echelle spectro-graph installed at the 3 . R ∼
65 000) and is optimizedfor a wavelength range of approximately 3 600 −
10 000 ˚A.All of the measurements were reduced using the Upenapipeline feeding the libre-esprit reduction code describedby Donati et al. (1997).One of the goals of this study is to detect radial veloc-ity variability induced by the presence of binary compan-ions. Given that the orbital periods of any such binary com-panions are unknown, we needed to obtain multiple mea-surements of each sample star’s radial velocity over bothshort ( . &
100 d) timescales. We adoptedthe observing strategy employed by Lampens et al. (2018)in which the time between consecutive observations is grad-ually increased from . ∼ ∼
30 d. Atotal of 319 Stokes I observations were obtained over a timeperiod spanning ≈
11 months from Jan. 31, 2018 to Dec.21, 2018; each of the 44 stars in our sample was observedbetween 6 and 9 times. Eighty-five per cent of the 319 ob-servations have S / N >
75 per pixel while the median S/N is100 per pixel. The observations are summarized in Table 1where we list the total number of times each target was ob-served, the maximum and median achieved S/Ns, the S/Nassociated with each star’s averaged spectrum, and the min-imum and maximum time (∆ t min and ∆ t max , respectively)between any two observations. We derived fundamental parameters for the stars in our sam-ple (effective temperatures, surface gravities, metallicities,radii, masses, etc.). This was carried out using both thehigh-resolution spectroscopic observations obtained for thisstudy and published multi-colour photometric observationsobtained using various filters.
Spectral modelling of the Stokes I ESPaDOnS observationswas carried out using the Grid Search in Stellar Parameters( gssp ) code (Tkachenko 2015). This code uses grids of pre-computed
LLmodels atmospheric models (Shulyak et al.2004) in order to generate grids of synthetic spectra thatspan a range of parameters. The parameters that can bevaried include effective temperature ( T eff ), surface gravity(log g ), metallicity ([M/H]), rotational broadening ( v sin i ),and microturbulence ( ξ ). Individual chemical abundancescan also be fit one element at a time. The χ values associ-ated with each model in the defined grid are computed and MNRAS , 1– ?? (2020) J. Sikora et al.
Table 1.
The sample of 44 A- and B-type stars included in this study. Columns 1 to 4 list each target’s KIC number, HD number,spectral type, and V magnitude. Column 5 lists the number of Stokes I observations obtained with ESPaDOnS for each star. Columns 6and 7 list the maximum and median achieved S/N per pixel at a wavelength of 5 500 ˚A associated with the observations while column 8lists the S/N per pixel associated with each star’s averaged spectrum. Columns 9 and 10 list the minimum and maximum time betweenany two observations.KIC HD Sp. V No. Max Median Combined ∆ t min ∆ t max ID num. Type (mag) obs. S/N pxl − S/N pxl − S/N pxl − (d) (d)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)1572201 182757 8.54 7 116 95 254 1.0 2852859567 184217 B9.5IV-V a a b c d c e c f e g g h e g h i j c i i h i e k e e d i i i i i a Tkachenko et al. (2013), b Sato & Kuji (1990), c Skiff (2014), d Davis et al. (1973) e Frasca et al. (2016), f Kharchenko (2001), g Niemczura et al. (2017), h Niemczura et al. (2015) i Pedersen et al. (2017), j Grillo et al. (1992), k Wilson (1953) the minimal χ model can be determined along with eachparameter’s uncertainty based on the associated χ inter-vals.The spectral modelling analysis consisted of the follow-ing three steps.(i) We derived T eff , log g , [M/H], and v sin i values by fittingH β and H γ line profiles, which are both relatively sensitive to changes in T eff and log g for A and late-B type stars.KIC 3848385, KIC 5371784, and KIC 7131828 are Be starsand exhibit significant emission within their H β and H γ lineprofiles (these cases are discussed in more detail in Sect. 4.5)and as a result, their Balmer lines could not be modelled us-ing gssp . In these three cases, we used narrow 21 ˚A-widthspectral windows centered at 4 475 ˚A to derive T eff , log g , MNRAS , 1– ????
The sample of 44 A- and B-type stars included in this study. Columns 1 to 4 list each target’s KIC number, HD number,spectral type, and V magnitude. Column 5 lists the number of Stokes I observations obtained with ESPaDOnS for each star. Columns 6and 7 list the maximum and median achieved S/N per pixel at a wavelength of 5 500 ˚A associated with the observations while column 8lists the S/N per pixel associated with each star’s averaged spectrum. Columns 9 and 10 list the minimum and maximum time betweenany two observations.KIC HD Sp. V No. Max Median Combined ∆ t min ∆ t max ID num. Type (mag) obs. S/N pxl − S/N pxl − S/N pxl − (d) (d)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)1572201 182757 8.54 7 116 95 254 1.0 2852859567 184217 B9.5IV-V a a b c d c e c f e g g h e g h i j c i i h i e k e e d i i i i i a Tkachenko et al. (2013), b Sato & Kuji (1990), c Skiff (2014), d Davis et al. (1973) e Frasca et al. (2016), f Kharchenko (2001), g Niemczura et al. (2017), h Niemczura et al. (2015) i Pedersen et al. (2017), j Grillo et al. (1992), k Wilson (1953) the minimal χ model can be determined along with eachparameter’s uncertainty based on the associated χ inter-vals.The spectral modelling analysis consisted of the follow-ing three steps.(i) We derived T eff , log g , [M/H], and v sin i values by fittingH β and H γ line profiles, which are both relatively sensitive to changes in T eff and log g for A and late-B type stars.KIC 3848385, KIC 5371784, and KIC 7131828 are Be starsand exhibit significant emission within their H β and H γ lineprofiles (these cases are discussed in more detail in Sect. 4.5)and as a result, their Balmer lines could not be modelled us-ing gssp . In these three cases, we used narrow 21 ˚A-widthspectral windows centered at 4 475 ˚A to derive T eff , log g , MNRAS , 1– ???? (2020) pectroscopic survey of Kepler
A stars Figure 1.
Examples of the fits to the observed average H γ and H β profiles. The black curves correspond to the observations while thered curve corresponds to the synthetic spectra. [M/H], and v sin i values. This region was selected becauseof the fact that it encompasses various spectral lines thatare, to various degrees, sensitive to both T eff and log g : forA-type stars with 7 000 . T eff .
10 000 K this spectral re-gion includes Fe i , Fe ii , and Mg ii lines while for early-A andlate-B type stars with 10 000 . T eff .
15 000 K this regionincludes He i and Mg ii lines. We note that the 21 ˚A-widthspectral window was found to be less sensitive to changesin T eff and log g compared to H β and H γ , which is demon-strated by the typically larger uncertainties yielded by the gssp code: the median T eff and log g uncertainties for thosethree stars exhibiting Balmer line emission are 1 250 K and1 . . ξ = 2 km s − during this first step of the analysis.(ii) Multiple 100 ˚A-width spectral windows ranging from4 900 ˚A to 5 900 ˚A were then fit in order to place additionalconstraints on [M/H] and v sin i and to derive ξ ; T eff andlog g were fixed at the values derived during the first step ofthe analysis.(iii) Finally, the 100 ˚A-width spectral windows used in the pre-vious step were fit while varying individual chemical abun-dances with all other parameters fixed.Prior to fitting, the observed spectra were normalizedusing two methods depending on whether or not Balmerlines were being fit. In both cases, we used the un-normalizedspectra that were reduced with libre-esprit (Donati et al.1997). The normalization of the H β and H γ profiles was car-ried out by selecting narrow wavelength regions spanning ∼ λ ). These regions have wavelengths centered at approxi-mately λ ±
40 ˚A and are therefore outside of the Lorentzian-broadened region of the Balmer line profiles. A first orderpolynomial function was then fit across the line within theselected regions and the entire Balmer line profile was nor-malized to the resulting fit. The spectral windows that do not include Balmer lines were normalized using a polynomialfit – typically of 1 st or 2 nd degree – to the continuum.Initial parameter grids used during the execution of gssp were centered on T eff = T KICeff (where T KICeff is theeffective temperature listed in the Kepler Input Catalog;Brown et al. 2011), log g = 4 . / H] = 0 .
0, and ξ = 2 . − while initial v sin i values were estimatedby eye. These initial grids spanned 1 000 K in T eff , 0 . g , 0 . . − in ξ , and 50 km s − in v sin i . We adopted grid resolutions of ∆ T eff = 100 K,∆ log g = 0 . / H]) = 0 .
1, ∆ ξ = 0 . − ,and ∆ v sin i = 1 km s − . After each execution of the gssp code, the parameter grids were re-centered on the best-fitting values and, if necessary, the range of the grids wasexpanded until either (i) upper and lower 1 σ confidencelimits were reached or (ii) either physically plausible limitswere reached (e.g. we expect MS A- and late B-type starsto have 5 000 < T eff <
20 000, log g > . LLmodels atmospheric models werereached (i.e. T eff ∈ [5 000 ,
20 000] K, log g ∈ [3 . , .
0] (cgs),[M/H] ∈ [ − . , . ξ ∈ [0 ,
20] km s − , v sin i > − ).For those executions of the gssp codes requiring large pa-rameter ranges, the computation time was slightly reducedfor practical purposes by decreasing the grid resolution. InFig. 1 we show several examples of the obtained fits to theH β and H γ lines.The fitting of individual chemical abundances describedin step 3 of the spectral modelling analysis was carriedout by first identifying those elements that exhibit suffi-ciently deep lines within the 100 ˚A-width spectral windowbeing fit. This was done by referring to spectral line listsprovided by the Vienna Atomic Line Database (VALD)for specified T eff and log g values (Piskunov et al. 1995;Ryabchikova et al. 2015). Any elements found to have lineswith unbroadened normalized depths > .
05 were selectedto be fit. Initial abundances were assigned based on the de-rived [M/H] values. For each element, the gssp routine was
MNRAS , 1– ?? (2020) J. Sikora et al.
Figure 2.
Fits to the 21 ˚A-width spectral windows centered at4 475 ˚A for the three stars in our sample that were found to ex-hibit strong Balmer line emission. As in Fig. 1, the black curvescorrespond to the observed average spectra and the red curvescorrespond to the synthetic spectra. Note that the abundancesof individual chemical elements were not varied during this stepof the analysis, which likely contributes to the poor fit obtainednear 4 468 ˚A for KIC 3848385 and KIC 7131828. carried out twice: once with a wide and low-resolution grid([X/H] ∈ [ − , −
2] and ∆ [X / H] = 1 .
0) and a second time us-ing a narrower, higher-resolution [X/H] grid (∆[X / H] = 0 . χ value obtained from the firstiteration. This was done in order to reduce the total com-putation time involved in the analysis.The final parameters ( T eff , log g , [M/H], v sin i , and ξ )and chemical abundances derived from the spectral mod-elling analysis were obtained by fitting each star’s averagedobserved spectrum. The averaged spectra typically have S/Nvalues two to three times higher than the individual obser-vations (see Table 1). They were obtained by first subtract-ing any radial velocity shift associated with each individualobserved spectrum (the derivation of the radial velocitiesis discussed below in Sect. 4.1.1). The normalized spectrawere then interpolated onto a common abscissa and the av-erage flux, weighted by the measurement uncertainties, wascomputed; care was taken to ensure consistent normaliza-tion between the individual spectra. In Fig. 2, we show thefits to the 21 ˚A-width spectral windows centered at 4 475 ˚Athat were used for the derivation of T eff and log g of thethree stars exhibiting Balmer line emission. We comparedthe best-fitting T eff values derived from the Balmer line pro-files ( T Balmereff ) and from the 4 475 ˚A region ( T ) for thestars that do not exhibit Balmer line emission. Althoughthe values are nearly all in agreement within the estimateduncertainties, we do see that the T values are systemat-ically higher than the T Balmereff values by .
500 K.The final adopted abundancies were taken to be theweighted average of the values derived by fitting each of theaveraged spectral windows where the weights were obtained
Figure 3.
Examples of fits to the 100 ˚A-width spectral regionsderived during step 3 of the spectral modelling analysis. The blackcurves correspond to the average observed spectra, the solid-redand dashed-blue curves correspond to the best-fitting syntheticspectra derived with and without fitting individual abundances,respectively. from the 1 σ χ intervals yielded by the gssp code. In theelectronic version of this paper, we list the derived T eff , log g ,[M/H], v sin i , and ξ values. In three cases, ξ values andindividual chemical abundances could not be derived due tolow S/Ns and/or high v sin i values, which did not allow foruseful constraints to be obtained by fitting the 100 ˚A-widthspectral windows. The 41 of 44 stars in our sample for whichat least one chemical abundance was derived are discussedbelow in Sect. 4.2. In Fig. 3, we show several examples ofthe fits to various 100 ˚A-width spectral windows. We derived the radial velocities using two methods. The firstmethod involved a comparison between the observed spec-tra and the synthetic models, which was carried out in twosteps. First, we calculated synthetic models using the initialparameters ( T eff , [M/H], v sin i , and ξ ) as described above inSect. 4.1. These models were then used to roughly estimatethe radial velocities by eye. Higher quality spectral fits weresubsequently derived using the three-step modelling anal-ysis described in Sect. 4.1 while incorporating the roughlyestimated radial velocities.The spectral models that were derived for each ofthe observed spectra were subsequently used to refine theroughly estimated radial velocities. This involved re-fittingthe spectral models to the observed spectra while vary-ing the radial velocity of each model along with two re-normalization parameters. The re-normalization parameters MNRAS , 1– ????
Examples of fits to the 100 ˚A-width spectral regionsderived during step 3 of the spectral modelling analysis. The blackcurves correspond to the average observed spectra, the solid-redand dashed-blue curves correspond to the best-fitting syntheticspectra derived with and without fitting individual abundances,respectively. from the 1 σ χ intervals yielded by the gssp code. In theelectronic version of this paper, we list the derived T eff , log g ,[M/H], v sin i , and ξ values. In three cases, ξ values andindividual chemical abundances could not be derived due tolow S/Ns and/or high v sin i values, which did not allow foruseful constraints to be obtained by fitting the 100 ˚A-widthspectral windows. The 41 of 44 stars in our sample for whichat least one chemical abundance was derived are discussedbelow in Sect. 4.2. In Fig. 3, we show several examples ofthe fits to various 100 ˚A-width spectral windows. We derived the radial velocities using two methods. The firstmethod involved a comparison between the observed spec-tra and the synthetic models, which was carried out in twosteps. First, we calculated synthetic models using the initialparameters ( T eff , [M/H], v sin i , and ξ ) as described above inSect. 4.1. These models were then used to roughly estimatethe radial velocities by eye. Higher quality spectral fits weresubsequently derived using the three-step modelling anal-ysis described in Sect. 4.1 while incorporating the roughlyestimated radial velocities.The spectral models that were derived for each ofthe observed spectra were subsequently used to refine theroughly estimated radial velocities. This involved re-fittingthe spectral models to the observed spectra while vary-ing the radial velocity of each model along with two re-normalization parameters. The re-normalization parameters MNRAS , 1– ???? (2020) pectroscopic survey of Kepler
A stars Figure 4.
Examples of those LSD profiles found to exhibit variability that is likely intrinsic to the A stars (left column) or is relatedto radial velocity variations of one or more stellar components (middle and right columns). The profiles shown for KIC 7974841 werecalculated using the 4 000 K line masks, which allow the cooler secondary and tertiary components to be more clearly identified; all ofthe other LSD profiles that are shown were calculatecd using the A star’s derived fundamental parameters. are simply coefficients of a first-order (i.e. linear) polyno-mial function; the observed spectra are divided by this first-order polynomial in order to minimize any discrepancy thatmay have been introduced by the initial continuum nor-malization procedure. We found that including these twore-normalization parameters yielded higher-quality fits andthus, more accurate radial velocities. The fitting analysiswas done using the mpfitfun
Levenberg-Marquardt algo-rithm implemented in idl .Uncertainties in the radial velocities were esti-mated by employing a block bootstrapping method (e.g.Lahiri & Lahiri 2003) involving the best-fitting syntheticmodel and observed spectra residuals. The models were di-vided into blocks spanning approximately 5 per cent of thewavelength range. Equally-sized blocks were then randomlysampled from the residuals and added to the model spec-trum blocks; radial velocities were re-derived from these newdata sets. This process was repeated 500 times and the un-certainty in each radial velocity measurement was taken tobe the standard deviation of the resulting distribution. Thismethod was used rather than simple random sampling withreplacement in order to account for correlated errors asso-ciated with the models. The final radial velocities and theirassociated uncertainties were calculated from the weightedaverage of the individual measurements.The second method involved generating Least-Squares Deconvolution (LSD) profiles (Donati et al. 1997;Kochukhov et al. 2010). LSD is a cross-correlation techniquein which a large number of spectral lines are combined, re-sulting in a single line pseudo-profile of higher S/N com-pared to any individual line. We generated the LSD profilesusing line masks consisting of spectral line data providedby VALD (Piskunov et al. 1995; Ryabchikova et al. 2015).Specifically, the line data were obtained using VALD’s ‘Ex-tract Stellar’ request in which the star’s T eff , log g , and ξ values are specified along with a wavelength range and adepth threshold. We used the T eff and log g values derivedin Sect. 4.1, ξ = 2 km s − , a wavelength range of 4 000 ˚A to7 000 ˚A, and a depth threshold equal to 5 per cent of the con-tinuum. Each line mask was then modified in such a way asto remove all Balmer lines, metallic lines appearing within the broad wings of the Balmer lines, and stellar spectral linesstrongly contaminated by telluric lines.The higher S/Ns obtained using the LSD technique canalso be used to detect line profile variability or evidenceof additional stellar components that may not be apparentin the spectra (see Fig. 4 for examples). For KIC 5461344,KIC 5880360, and KIC 8324268, changes in the line pro-files are apparent while no clear radial velocity shifts weredetected (based on both a visual inspection and on the vari-ability metric discussed in Sect. 5); therefore, it is plausiblethat the variability is intrinsic to the A stars (e.g. it is ev-idence of spots or pulsations) rather than being attributedto the spectral signatures of stellar companions. Variabil-ity is also apparent in the LSD profiles of KIC 5436432,KIC 6106152, KIC 7383872, and KIC 8692626 that is likelydue to stellar companions based on the fact that (1) betweentwo and three distinct components are visible and that (2)the components all exhibit large-scale radial velocity shifts.For each of the 44 stars in our sample, we computed addi-tional LSD profiles using a line mask generated for a coolstar with T eff = 4 000 K, log g = 5 . ξ = 2 km s − .These profiles were used to identify additional (cooler) stel-lar components in KIC 7974841 and KIC 10815604. In Fig.4 we show several examples of the LSD profiles that werefound to exhibit variability.Radial velocities for all of the stellar components thatwere identified within the LSD profiles were derived by fit-ting a Gaussian function to each component thereby allow-ing the center of each line to be identified. For those casesin which (i) more than one component is visible in the as-sociated LSD profiles and (ii) each component’s core canbe easily identified, multiple Gaussian functions were fit.Uncertainties in each radial velocity measurement were esti-mated by bootstrapping the residuals. This involved gener-ating 100 bootstrapped data sets through random samplingof the residuals with replacement and adding these to theGaussian fit; the fitting procedure was then carried out oneach of the bootstrapped data sets and the uncertaintieswere taken to be 3 times the standard deviation of the re-sulting distribution. We note that for such a bootstrappinganalysis, it is preferable to carry out a larger number of boot- MNRAS , 1– ?? (2020) J. Sikora et al. strapped data sets (e.g. 500); however, we found that thedistributions obtained using fewer bootstrapped data setsexhibit comparable standard deviations to those obtainedusing a larger number of bootstrapped data sets. We there-fore used 100 data sets in order to reduce the computationtime.Comparing the v r values derived using the two meth-ods (i.e. using spectral models and using LSD profiles), wefind that the values and uncertainties ( σ v r ) are generally inagreement. The median σ v r values derived using the spectralmodelling and LSD methods is 2 . − , and 2 . − ,respectively. All of the radial velocities derived from boththe spectra and the LSD profiles are listed in the electronicversion of this paper. The search for radial velocity variabil-ity and its interpretation is presented below in Sect. 5. As discussed in Sect. 4.1, [M/H] values (metallicities) werederived during step one of the spectral modelling analysis.The metallicity roughly corresponds to the overall abun-dance of those elements heavier than He relative to the so-lar abundance table. A more detailed analysis is required tomore accurately derive each individual element’s chemicalabundance, particularly in the case of chemically peculiarstars, which often exhibit significant abundance enhance-ments of only a small number of elements.We derived chemical abundances for 41 of the 44 stars inour sample for between 1 and 12 elements per star depend-ing largely on the S/N values associated with the averagedobservations and the star’s v sin i values (lower S/N valuesand higher v sin i values generally resulted in the abundancesof fewer elements being constrained). For nearly all of the41 stars, the abundances of Si and/or Fe-peak elements suchas Fe and Cr were able to be derived primarily because ofthe prevalance of these lines within the T eff range spannedby our sample. In a number of cases, the atmospheric abun-dances of rare earth elements such as Y, La, and Nd wereable to be derived; this is particularly useful considering thatAp/Bp stars commonly exhibit significant enhancements inthese particular abundances that can be several orders ofmagnitude higher than the abundances found in the Sun(e.g. Preston 1974; Smith 1996; Ghazaryan et al. 2018).For the majority of the elements that were selected forabundance determination using the procedure described inSect. 4.1, only upper limits could be established. In thesecases, the upper limits that were obtained were typicallyfound to be very high (i.e. above the chemical abundance ex-cesses typically associated with CP stars; Ghazaryan et al.2018) and are therefore not reported here. For those ele-ments for which both lower and upper limits could be de-rived, we report the weighted average. Generally, the abun-dance uncertainties yielded by the gssp code are asymmet-ric; therefore, the weighted averages and their associateduncertainties were computed using the method described byBarlow (2004).We find that the majority of the 41 stars for whichchemical abundances were derived have abundances that areapproximately consistent with solar values or are slightly un-derabundant relative to solar values (Asplund et al. 2009).However, 9 of the 41 stars have obvious overabundances oftwo or more elements; the derived abundances for these 9 Figure 5.
The derived chemical abundances, measured relativeto the Sun’s chemical abundances (Asplund et al. 2009), of thestars with detected overabundancies of at least one element; thoseelements for which only upper bounds were derived are not shown. stars are shown in Fig. 5. The strongest overabundances areassociated with KIC 8324268, which is identified in the liter-ature as a Bp star; we observed strong overabundances of Si,Cr, Fe, and Nd with a high significance ( & σ ). Furthermore,this star exhibits line profile variability that was clearly de-tected in the LSD profiles, which is commonly associatedwith Bp stars.Three of the 9 stars found to be overabundant invarious elements are identified in the literature as Amstars (KIC 8692626, KIC 8703413, and KIC 9349245) werefound to have enhanced abundances of various elementsincluding Cr, Mn, Fe, Ni, Y, and Ba. Six of the 9 starsare not identified in the literature as being CP stars.KIC 5461344, KIC 7050270, and KIC 7974841 all have over-abundances of Mn and exhibit relatively narrow spectrallines ( .
30 km s − ) suggesting that they may belong tothe HgMn class of CP stars. We note that KIC 5461344also exhibits weak line profile variability, which is evidentin the LSD profiles (see Fig. 4). Such a phenomenon has MNRAS , 1– ????
30 km s − ) suggesting that they may belong tothe HgMn class of CP stars. We note that KIC 5461344also exhibits weak line profile variability, which is evidentin the LSD profiles (see Fig. 4). Such a phenomenon has MNRAS , 1– ???? (2020) pectroscopic survey of Kepler
A stars been previously found to be associated with a number ofHgMn stars (e.g. Ryabchikova et al. 1999; Adelman et al.2002; Kochukhov et al. 2005b). We carried out the synthetic energy distribution (SED)fitting analysis of available photometric observations andpublished distance estimates in order to derive eachstar’s radius and luminosity. Photometric observations ob-tained using various filters are available for all of the44 stars in our sample. This includes Johnson B and V measurements, Tycho B T and V T measurements (ESA1997), 2MASS J , H , and K S measurements (Cohen et al.2003), and in several cases, Str¨omgren uvby measurements(Hauck & Mermilliod 1998) and/or Geneva UB BB V V G measurements (Rufener & Nicolet 1988). Distances to 43of the 44 stars have been published by Bailer-Jones et al.(2018) based on Gaia
DR2 parallax measurements; no dis-tance estimate could be found for one of the 44 stars,KIC 5371784. The uncertainties associated with the dis-tances are relatively low: the maximum uncertainty is ≈ ≈ T eff , log g , and [M/H] at the valuesderived from the spectroscopic observations (see Sect. 4.1)while allowing the radius ( R ) to vary. We note that the gridof model SEDs have a range in [M/H] from − . .
5. ForKIC 8324268, we derived a value of [M / H] = 0 .
8, which fallsoutside of this range; in this case we used the [M / H] = 0 . V band ( A V ) are re-ported by Mathur et al. (2017) for 42 of the 44 stars in oursample and by (Brown et al. 2011) for the remaining 2 stars.The corresponding E( B − V ) values reported by these stud-ies, which range from E( B − V ) = 0 .
006 mag to 0 .
113 mag,depend in part on the estimated T eff , log g , and [M/H] val-ues; therefore, inaccuracies in the fundamental stellar pa-rameters will propagate to the E( B − V ) values. Given thatwe have derived estimates of T eff , log g , and [M/H] spectro-scopically, we opted to include E( B − V ) as a free param-eter (where we impose the constraint that E( B − V ) > R . The dereddening was carried out usingthe method described by Cardelli et al. (1989). The lumi-nosities were then calculated using the Stefan-Boltzmannrelation with the T eff values and the radii derived from thisanalysis.The inferred R and L values may be impacted, to someextent, by binarity: if the primary star has a stellar compan-ion, its true R and L values will be lower than the derivedvalues due to the contribution from the companion. Assum-ing that (i) any companion has a lower radius and effectivetemperature than that of the primary component and that(ii) the T eff values derived from the spectra are accurate(Sect. 4.1), the primary star’s true R and L values may beoverestimated by factors of √
2, respectively. Inthe 6 cases in which the spectral signatures of at least onedimmer companion was detected, their contributions to thetotal spectrum are significantly lower than those of the pri-mary component; therefore, it is unlikely that the adopted R and L values are significantly affected by binarity. For 42 of the 44 stars in our sample, revised T eff , log L ,and R values have been published by Mathur et al. (2017)and Berger et al. (2018) based in part on the Gaia
DR2 cat-alog (Gaia Collaboration et al. 2018). Comparing the T eff values reported in these catalogs ( T liteff ) with the values de-rived in this study ( T speceff , Sect. 4.1), we find reasonableagreement where T speceff . ≈ T speceff − T liteff ) increases roughly mono-tonically to ∼ T eff values reported in theKIC are likely unreliable above 9 000 K (Brown et al. 2011)and (2) the T eff values reported by Mathur et al. (2017) forthe 42 sample stars have simply been increased by 223 K rel-ative to the KIC values. The radii derived in this study arelargely discrepant with those values reported in the KIC.However, we find reasonable agreement between the radiireported by Berger et al. (2018) and our values likely due tothe fact that both studies rely on distance estimates derivedfrom Gaia
DR2 parallax measurements (Bailer-Jones et al.2018).
The masses and ages of the stars in our sample were de-rived by comparing each star’s Hertzsprung-Russell diagram(HRD) position as implied by its T eff and L values with pub-lished model evolutionary tracks. We used the grids pub-lished by Mowlavi et al. (2012) and Ekstr¨om et al. (2012).The grid computed by Mowlavi et al. (2012) has a highermass and metallicity resolution but only extends from zeroage main sequence masses of 0 . . M ⊙ . The majorityof our stars (approximately 75 per cent) have masses lessthan 3 . M ⊙ ; in these cases, we used the high resolutiongrid to compute the masses and ages. For the stars with T eff and L values falling outside of the ranges associated withthe evolutionary tracks computed by Mowlavi et al. (2012)(and thus, with masses & . M ⊙ ), we used the lower resolu-tion grid computed by Ekstr¨om et al. (2012). In both cases,only the non-rotating models were used.The method by which the masses, ages, and their asso-ciated uncertainties were derived for the stars in our sampleis described by Sikora et al. (2019a). In Fig. 6, we show theHRD along with the solar-metallicity, non-rotating modelscomputed by Ekstr¨om et al. (2012) for reference. The de-rived masses and ages are listed in the electronic version ofthis paper. As noted in Sect. 4.1, three of the stars in our sample,KIC 3848385, KIC 5371784, and KIC 7131828, were foundto exhibit strong, apparently non-variable Balmer line emis-sion. In Fig. 7, we show the averaged H α line profiles forthese three stars compared with model photospheric profilesgenerated using each star’s derived fundamental parameters.It is evident that, in each of the three cases, the peak (orpeaks, in the case of KIC 5371784 and KIC 7131828, whichare characterized by two symmetric peaks on either side ofthe Balmer line’s core) of the emission lies within the extentof the Doppler-broadened line core.Balmer line profiles similar to those shown in Fig. 7 are MNRAS , 1– ?? (2020) J. Sikora et al.
Figure 6.
HRD associated with the stars composing the spectro-scopic survey (open blue) using T eff and log L values derived inthis study. Filled grey symbols correspond to the stars identifiedby Balona (2013) as exhibiting variability that is consistent withrotational modulation; the T eff and log L values of these stars aretaken either from the catalog compiled by Mathur et al. (2017)and Berger et al. (2018) (816 stars) or from the KIC (Brown et al.2011) (59 stars). known to be associated with classical Be stars – rapidly ro-tating MS B-type stars that host hot, gaseous Kepleriandecretion disks (Slettebak 1988; Porter & Rivinius 2003;Rivinius et al. 2013). The diversity of Be star Balmer lineemission profiles is understood to be related principally tothe inclination angle ( i ) of the star’s rotation axis as pro-posed by Struve (1945) (e.g. Fig. 3 of Slettebak 1979) anddemonstrated in Fig. 6 of Sigut et al. (2015); based on theirBalmer line emission profiles, we would expect KIC 3848385to have the lowest i , KIC 5371784 to have the highest i , and for KIC 7131828 to have an intermediate i value.This can be tested using the derived v sin i values, R val-ues, along with the P rot values inferred from the Kepler light curves (Sect. 6). No distance estimate is available forKIC 5371784 and thus, its stellar radius could not be de-rived. For KIC 3848385, we find i = 11 ± i = 20 ± i values differ is consistent with thegeometrical model proposed by Struve (1945). We also notethat Be star Balmer line emission is known to be variableover timescales as long as ∼ decades (Rivinius et al. 2013);considering that the three stars in our sample were observedover timescales ∼ − As noted in Sect. 4.1.1, we detected at least two stellar com-ponents associated with six stars in our sample. Additional
Figure 7.
Observed (black) and synthetic (red) H α profiles forthe three stars in our sample exhibiting strong, apparently non-variable emission. The vertical dashed lines indicate the extentof the Doppler-broadened core; in all three cases, the peak of theemission falls within these limits. multi-star systems were also inferred based on radial velocityvariations of the primary star.We searched the derived radial velocities for variabil-ity by first calculating generalized Lomb-Scargle (GLS) pe-riodograms as described by Zechmeister & K¨urster (2009).These periodograms were calculated using an oversamplingfactor of 10 and a maximum frequency of 1 / ∆ t min . The fre-quency exhibiting the maximum power was identified andused as the initial guess for the best-fitting orbital frequency( f orb = 1 /P orb ). We evaluated the significance of any v r variability by comparing the sinusoidal fit associated withthe best-fitting frequency yielded by the GLS periodogramto that of a constant (i.e. straight line) fit. The χ differ-ence (∆ χ ) between the fits was then computed and thosecases in which ∆ χ corresponds to values > σ were con-sidered to be significant and thus, members of multi-starsystems. Statistically significant v r variability was detectedfor 6 of the stars in our sample for which no companionscould be spectroscopically identified. Including those starswith spectroscopically identified companions, we concludethat at least 11 of the 44 stars (25 ±
11 per cent) in our sam-ple are members of spectroscopic multi-star systems. Thisis consistent with the ≈ −
45 per cent incidence rate ofspectroscopic binaries among intermediate-mass field stars(e.g. Duchˆene & Kraus 2013).The GLS periodogram normalized power associatedwith false alarm probabilities (FAPs) of 1 per cent for each
MNRAS , 1– ????
MNRAS , 1– ???? (2020) pectroscopic survey of Kepler
A stars Figure 8.
Examples of the v r measurements phased by the best-fitting orbital periods associated with the primary component(KIC 8390826), the primary and secondary components (KIC 6106152), and the secondary and tertiary components (KIC 7974841).Black circles, red squares, and blue circles correspond to the primary, secondary, and tertiary components, respectively. The significanceof v r variability ( σ ) of the primary components are listed in the top right corner of each panel. set of radial velocities were estimated using a Monte Carlosimulation (e.g. Cumming 2004). In general, a large number( &
10) of possible orbital periods are associated with eachof the 11 detected multi-star systems. This is likely relatedto the small number of observations obtained for each tar-get. Nonetheless, we refined the best-fitting orbital periodsinferred from the periodograms and derived orbital param-eters to fit the v r curves. This was done by defining a gridof orbital periods centered on the estimated P orb value witha grid resolution ranging from 10 − to 10 − d. Keplerian v r curves were then fit to each set of v r measurements using the mpfitfun Levenberg-Marquardt algorithm implemented in idl . The fitting carried out for each P orb grid value involvedfour free parameters: the semi-amplitude ( K ), the systemicvelocity ( v ), the longitude of periastron ( ω ), and the ec-centricity ( e ). The solution yielding the minimum χ valuewas then adopted. In the case of KIC 5436432, the best-fitting orbital solution yielded e ≈ .
7, which led to largeuncertainties in v and K . The small number of available v r measurements do not allow for significant constraints tobe placed on KIC 5436432’s eccentricity; therefore, we en-forced e < . f [ M ]) and minimum com-panion masses ( M min B ) for the six targets for which only theprimary component’s radial velocities could be measured. Inthe cases of KIC 5436432 and KIC 7383872, the three identi-fied components all exhibit statistically significant variabil-ity, however, no consistent orbital period associated withthe components could be derived. We therefore only fit theprimary component v r curves and report the derived f ( M )and M min B values. Additional observations are required in or-der to constrain the orbital configuration. For KIC 6106152,the GLS periodograms associated with the primary and sec-ondary v r measurements both exhibit a maximum power at ≈ . P orb was derived along with the massratio ( q = M B /M A ). Neither the primary or the secondarycomponents of KIC 10815604 were found to exhibit statisti-cally significant variability, suggesting that P orb ≫ | v r,B − v r,C | will vary with half the orbital period.We generated a GLS periodogram using the | v r,B − v r,C | val-ues, identified the best-fitting period, and labeled the B andC components based on the phased v r,B and v r,C curves.An orbital solution was then derived along with minimummasses of the two components. No variability was detectedfor the primary component, suggesting that the gravitation-ally bound secondary and tertiary components orbit the pri-mary component with a period ≫ v r measurements. The derived orbital periods, v r semi-amplitudes, systemic velocities, eccentricities, and secondaryand tertiary component mass constraints are listed in Ta-ble 2. The 44 A- and B-type stars in our sample have been identi-fied by Balona (2013) as exhibiting variability that is consis-tent with rotational modulation. We analyzed the available
Kepler light curves in order to (i) verify the presence of suchvariability and (ii) to derive high-precision estimates of whatare believed to be these stars’ rotation periods.We searched for statistically significant variability byfirst calculating Lomb-Scargle (LS) periodograms (Lomb1976; Scargle 1982; Press et al. 2007) from the post-processed
Kepler light curves (we used LS periodogramsrather than the weighted GLS periodograms that were calcu-lated for the v r measurements due to the prohibitively longcomputation times required to generate GLS periodogramsfrom the light curves, which typically consist of & mea-surements). The iterative pre-whitening procedure describedby Degroote et al. (2009) was then carried out. This involvedselecting the highest amplitude signal that is apparent in theLS periodogram, fitting a sinusoidal model given by∆ Kp ( t ) = c + n f X j =1 A j sin [2 π ( f j t − φ j )] , (1)to the original light curve where the j indices are associatedwith each of the n f extracted frequencies, and then calcu-lating a new LS periodogram from the resulting residuals. MNRAS , 1– ?? (2020) J. Sikora et al.
Table 2.
Derived orbital parameters for those systems found exhibiting statistically significant v r variability. Note that for each system,more than one possible orbital period was identified; additional measurements are necessary to better constrain the orbtial paramters.The first block lists the parameters derived only from the primary component’s v r measurements, the second block (KIC 6106152) liststhe parameters derived using both the primary and secondary components, and the third block (KIC 7974841) lists the parametersderived using the secondary and tertiary components where no v r variability associated with the primary component was detected.KIC ID P orb (d) v (km s − ) e K (km s − ) f ( M ) M min2 ( M ⊙ )( × − M ⊙ )4567097 6 . − . ± . < . . ± . < . . . − . ± . > . ± ±
17 0 . . − . ± . . ± . ± < .
03 0 . . − ± < . ± < . . − ± < . ± < . . − . ± . < .
03 25 . ± . < . . − . ± . < . . ± . < . . . − ± < . ± <
80 0 . P orb (d) v (km s − ) e K (km s − ) K (km s − ) q M ( M ⊙ )6106152 27 . − ± . ± .
07 19 ± . ± . . ± .
03 0 . ± . P orb (d) v (km s − ) e K (km s − ) K (km s − ) q M min2 ( M ⊙ ) M min3 ( M ⊙ )7974841 2 . − . ± . . ± .
004 85 . ± . . ± . . ± .
06 0 .
62 0 . Each parameter in Eqn. 1 ( c , A j , and φ j ) is re-fit for eachnew frequency that is extracted. This process may be car-ried out indefinitely leading to the extraction of both realastrophysical signals in addition to apparent signals thatare introduced by noise. In order to avoid this, we adoptedthe stopping criterion proposed by Van Reeth et al. (2015)in which the A j values are compared with their associatedamplitudes obtained from the LS periodogram. If these twovalues differ by more than 50 per cent, the pre-whiteningprocedure is terminated. We also imposed an additional con-straint such that a maximum of 100 frequencies would beextracted, which reduced the computational time requiredfor the analysis.Many of the periodograms generated from the lightcurves exhibit relatively large amplitude peaks at f . . − that increase in amplitude with decreasing f ; thesesignals are likely due to systematic instrumental effects (e.g.quarterly gaps for data downloads, focus drift, quarterly ro-tations of the spacecraft, Balona 2011; Murphy 2012) andwere removed from the lists of extracted frequencies. In sev-eral cases, the LS periodograms were found to contain re-gions of densely distributed, high-amplitude frequencies; inthese cases, we removed all of the frequencies except forthe one exhibiting the highest amplitude. The S/Ns of theextracted frequencies were estimated by assuming that thefinal LS periodogram calculated from the residuals of thefinal sinusoidal model (Eqn. 1) consists predominantly ofnoise (i.e. it is assumed that no real, high-amplitude sig-nals remain in the final residuals). We then average the pe-riodogram over frequency bins 0 . − in width and di-vide each extracted frequency’s amplitude by the local am-plitude of the binned LS periodogram. For most of thelight curves, the adopted pre-whitening frequency extrac-tion analysis yielded multiple independent, low-frequency(0 . . f .
10 c d − ) signals, which may be attributed topulsations, rotationally variable companion stars, the orbitalperiod of those stars with detected companions, or a combi- nation of these phenomena. Several examples of these lightcurves and their associated periodograms are shown in Fig.9. In general, the rotation periods derived in this study( P rot ) are approximately consistent with those reported byBalona (2013) ( P rot , B13 ); however, in 4 cases (KIC 6106152,KIC 5461344, KIC 7974841, and KIC 8390826), the adopted P rot differs significantly from P rot , B13 (see Fig. 10). In thesecases, the differences are related to the identification of therotation frequencies in the periodograms: for those caseswith multiple independent peaks, we selected the peak withthe highest amplitude while Balona (2013) selected a loweramplitude peak.The frequencies and amplitudes of the peaks that arepresumed to be associated with each star’s rotation were re-fined using the analysis described by Sikora et al. (2019b) inwhich the light curves are fit using a sinusoidal function con-sisting of the rotation frequency and its first four harmonics.The rotation periods and the photometric amplitudes asso-ciated with these periods are listed in Table 3 along with theperiods reported by Balona (2013). All of the phased lightcurves and their associated LS periodograms are included inthe online version of this paper.
One of the primary goals of this study is to derive the rota-tional broadening parameters ( v sin i ) for each of the stars inour sample, which can in principle be used to falsify the hy-pothesis that photometric variability is due to the star’s ro-tation. This is carried out by comparing v sin i with the equa-torial velocity ( v eq ≡ πR/P rot ), which is derived by assum-ing that an identified photometric frequency ( f ) correspondsto the star’s rotation frequency (i.e. f = f rot = 1 /P rot ); if f does correspond to f rot , then rigid rotation requires that f > v sin i/ πR . MNRAS , 1– ????
One of the primary goals of this study is to derive the rota-tional broadening parameters ( v sin i ) for each of the stars inour sample, which can in principle be used to falsify the hy-pothesis that photometric variability is due to the star’s ro-tation. This is carried out by comparing v sin i with the equa-torial velocity ( v eq ≡ πR/P rot ), which is derived by assum-ing that an identified photometric frequency ( f ) correspondsto the star’s rotation frequency (i.e. f = f rot = 1 /P rot ); if f does correspond to f rot , then rigid rotation requires that f > v sin i/ πR . MNRAS , 1– ???? (2020) pectroscopic survey of Kepler
A stars Figure 9.
Examples of
Kepler light curves phased by the adoptedrotation periods (left panels) that were found to exhibit multi-ple independent, low-frequency signals in their LS periodograms(right panels). The red curve corresponds to the best-fitting sinu-soidal model (Eqn. 1). The adopted P rot is indicated in the peri-odograms by a red arrow. The vertical dashed blue line indicatesthe minimum frequency that can be attributed to the star’s rota-tion based on the derived v sin i values (i.e. rotation frequenciesmust be > v sin i/ πR ). The pink shaded regions that appear insome of the periodograms shown in here and in the online versioncorrespond to orbital frequencies with FAPs < v r measurements that exhibit statistically significant vari-ability. Note that the light curves have been downsampled by afactor of 20 to reduce the figure file size. In Fig. 11, we compare the v sin i and v eq values forthe 43 stars in our sample with derived v eq values. Asdiscussed previously, no distance estimate is available forKIC 5371784, which is required to derive R and by exten-sion, v eq . We find that 33 of the 43 stars (77 ±
25 per cent)exhibit v sin i < v eq within the estimated uncertainties andtherefore, the identified photometric periods can plausiblybe attributed to each star’s rotation period. In Figures 9 and10 as well as in those figures presented in the online versionof this paper, the periodograms are shown with the mini-mum f that can be attributed to the star’s rotation based onthe derived v sin i values (i.e. f min = v sin i/ πR ). It is clearthat for 9 of the 10 stars that have v sin i > v eq based on theadopted P rot and derived R (KIC 5436432, KIC 6106152,KIC 7383872, KIC 9392839, KIC 10724634, KIC 10815604,KIC 10974032, KIC 11189959, and KIC 12061741), thereare additional periodogram peaks with f > f min . Therefore,although the P rot values derived here along with those re-ported by Balona (2013) for these 9 stars are not physicallycompatible with each star’s rotation period, it is plausible Figure 10.
Kepler light curves phased by the adopted rotationperiods ( P rot ) (left panels) that differ significantly from those re-ported by Balona (2013) ( P rot , B13 ) along with their associatedLS periodograms (right panels). The red curve corresponds tothe best-fitting sinusoidal model (Eqn. 1). The value of 1 /P rot isindicated in the periodograms by a red arrow; a blue arrow indi-cates 1 /P rot , B13 . The vertical dashed blue lines and pink shadedregions are defined in Fig. 9. Note that the light curves have beendownsampled by a factor of 20 to reduce the figure file size. that those higher-frequency peaks do correspond to rotationperiods.We carried out an additional test of the rotational mod-ulation hypothesis by deriving inclination angles (sin i val-ues) for those 33 stars for which v sin i < v eq and comparingthe resulting distribution with that expected from a sam-ple of stars having randomly oriented rotation axes (e.g.Jackson & Jeffries 2010). In Fig. 12, we compare the cumu-lative distribution functions (CDFs) of these two samples.We find no statistically significant difference between thetwo distributions based on the derived Kolmogorov-Smirnov(KS) test statistic of 0.20 and accompanying p -value of 0.12. MNRAS , 1– ?? (2020) J. Sikora et al.
Table 3.
Rotation periods and
Kepler photometric amplitudesassociated with the 44 stars in our sample. Columns 1 to 4 list theKIC identifiers, rotation periods derived in this study, rotation pe-riods reported by Balona (2013), and the maximum photometricamplitudes inherent to the (presumably) rotationally modulatedvariability.KIC P rot P rot , B13 ∆ Kp max ID (d) (d) (mmag)(1) (2) (3) (4)1572201 2.36979(2) 2.370 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The discovery by Balona (2013) that approximately 44 percent of the MS A stars observed with
Kepler exhibit variabil-ity that is consistent with rotational modulation presents achallenge to our understanding of the physical constitutionof these objects. It suggests that a significantly higher frac-tion of MS A stars may host inhomogeneous surfaces – i.e.chemical or brightness spots – than previously believed. Cur-
Figure 11.
Comparison between the v sin i values derived fromthe spectroscopic observations and the v eq values derived usingeach star’s R value and P rot value inferred from the Kepler lightcurves. Points appearing in the top-left are inconsistent with ro-tational modulation while points in the lower-right are consis-tent with rotational modulation. Filled blue points correspond tothose stars for which chemical overabundances (i.e. higher thansolar) for at least one element were detected. The dashed linescorrespond to v eq derived using P rot and P rot /
2. Note that eachpoint has an assigned uncertainty and that in certain cases, theuncertainties are smaller than the size of the symbols. rently, the detection of chemical spots via both photometricrotational modulation and spectral line profile variabilityhas only been reported for Ap/Bp stars, which account for ∼
10 per cent of all MS A- and B-type stars, and for a smallnumber of relatively rapidly rotating HgMn stars. Therefore,verifying whether similar phenomena are associated with 44per cent of all MS A-type stars is necessary in order to ad-dress theoretical questions that have transpired over the pastdecade.In this study, we have obtained a large number of high-resolution ESPaDOnS Stokes I observations for 44 MS Aand late-B type stars that were previously identified byBalona (2013) as being rotationally variable. There are twoprimary goals of this survey. First, to test the hypothesisthat the photometric variability detected using Kepler is infact rotational modulation that is intrinsic to these objects.As described in Sect. 7, this was carried out by determiningwhether v sin i < v eq where v eq = 2 πR/P phot and P phot is theperiod of photometric variability that is hypothesized to cor-respond to P rot (e.g. Balona 2017, 2019). Furthermore, oursurvey has been designed to search for radial velocity vari-ability that may be indicative of low-mass ( . M ⊙ ) com-panions with convective envelopes, which commonly exhibitrotationally modulated light curves produced by star spots(e.g. McQuillan et al. 2012). The detection of such compan-ions provides a plausible, and arguably more probable, al-ternative explanation for the photometric variability. Thesecond primary goal is to search for strong chemical pecu- MNRAS , 1– ????
10 per cent of all MS A- and B-type stars, and for a smallnumber of relatively rapidly rotating HgMn stars. Therefore,verifying whether similar phenomena are associated with 44per cent of all MS A-type stars is necessary in order to ad-dress theoretical questions that have transpired over the pastdecade.In this study, we have obtained a large number of high-resolution ESPaDOnS Stokes I observations for 44 MS Aand late-B type stars that were previously identified byBalona (2013) as being rotationally variable. There are twoprimary goals of this survey. First, to test the hypothesisthat the photometric variability detected using Kepler is infact rotational modulation that is intrinsic to these objects.As described in Sect. 7, this was carried out by determiningwhether v sin i < v eq where v eq = 2 πR/P phot and P phot is theperiod of photometric variability that is hypothesized to cor-respond to P rot (e.g. Balona 2017, 2019). Furthermore, oursurvey has been designed to search for radial velocity vari-ability that may be indicative of low-mass ( . M ⊙ ) com-panions with convective envelopes, which commonly exhibitrotationally modulated light curves produced by star spots(e.g. McQuillan et al. 2012). The detection of such compan-ions provides a plausible, and arguably more probable, al-ternative explanation for the photometric variability. Thesecond primary goal is to search for strong chemical pecu- MNRAS , 1– ???? (2020) pectroscopic survey of Kepler
A stars Figure 12.
Cumulative distribution function (CDF) of sin i (where i are the inclination angles) for those stars in our sam-ple consistent with v sin i < v eq (dashed red curve). The solidblack curve shows the CDF expected from a distribution of starswith random inclination angles. The Kolmogorov-Smirnov (KS)test statistic comparing the two distributions is shown. liarities and/or strong spectral line profile variability that iscommonly associated with Ap/Bp stars. The results of thesetwo goals are broadly summarized in Table 4 and discussedin detail below.Our analysis indicates that 33 of the 43 stars in oursample for which v eq could be inferred have v sin i < v eq within the estimated uncertainties (i.e. their periods andline widths are consistent with the rotational modulation hy-pothesis). Our search for radial velocity variability yieldeddetections of stellar companions associated with 11 of the44 stars in our sample, which includes 4 of the 33 starswith v sin i < v eq . The companions were all found to haveminimum masses ranging from ≈ . M ⊙ (Table 2).Moreover, the spectroscopic signatures of the companionswere either not detected or were found to be relatively weakcompared to that of the respective primary component in-dicating that they are dimmer and less massive. Therefore,it would not be surprising for these companion stars to hoststar spots that could be responsible for producing the ob-served photometric variability.M dwarfs found in the Kepler field for which rotationalvariability has been detected exhibit typical flux amplitudes(∆ F M /F M ) ranging from ∼ . − F M /F M (i.e. the M dwarf’sintrinsic variability) that is required to produced the ob-served ∆ Kp amplitudes listed in Table 3. Using the de-rived T eff and R values for the stars in our sample andthe fact that mid-type M dwarfs have T eff ∼ R ∼ . R ⊙ (e.g. Hardegree-Ullman et al. 2019), we find amedian ∆ F M /F M = 4 . ≈
30 per cent), we find ∆ F M /F M < P phot for thosestars with identified companions may be associated with the Table 4.
Summary of the results of our survey. Column 2 indi-cates whether the photometric variability is consistent with rota-tional modulation (i.e. whether v sin i < v eq ≡ πR/P phot withinthe uncertainties). Columns 3 to 5 indicate whether line pro-file variability, chemical peculiarites, or companions with massespotentially . M ⊙ were detected. Column 6 indicates whether P phot coincides with possible orbital periods for those stars withdetected companions. Only 21 stars are listed; the remaining 23stars in the sample exhibit v sin i < v eq and no detected lineprofile variability, chemical peculiarities, or companions.KIC ID Rot Line CP? Low- M P phot var? var? comp? ≈ P orb ?(1) (2) (3) (4) (5) (6)5430514 Yes No Yes No5461344 Yes Yes Yes No5880360 Yes Yes No No7050270 Yes No Yes No8324268 Yes Yes Yes No9349245 Yes No Yes No4567097 Yes No No Yes No7974841 Yes No Yes Yes No8692626 Yes No Yes Yes No8703413 Yes No Yes Yes Yes5371784 ? No No No5436432 No No No Yes No6106152 No No No Yes No7383872 No No Yes Yes No8390826 No No No Yes Yes9392839 No No No No10724634 No No No Yes Yes10815604 No No No Yes No10974032 No No No No11189959 No No No Yes No12061741 No No No No orbital periods of these systems (e.g. the systems may be el-lipsoidal or eclipsing variables, Smalley et al. 2014). As isshown in the periodograms (Figures 9 and 10 along withthose included in the online version of this paper), 3 of the11 stars with detected companions have P phot values thatcoincide with possible orbital periods inferred from radialvelocity measurements. We conclude that, for those 4 starswith v sin i < v eq and with detected companions, it is plau-sible that the photometric variability is not in fact intrinsicto the A/late-B type stars, however, we cannot definitivelyrule out the rotational modulation hypothesis.Based on the observations obtained within this study,we conclude that the rotational modulation hypothesis re-mains a plausible explanation for the origin of the Kepler photometric variability associated with the 29 of 43 MSA/late-B type stars in our sample (67 ±
23 per cent) thathave v sin i < v eq and no detected low-mass companions.Balona (2013) reported that 875 out of the estimated 1 974MS A type stars observed with Kepler (44 ± ± MNRAS , 1– ?? (2020) J. Sikora et al. the fraction of known magnetic A and late-B type stars thatmight be expected to exhibit rotational modulation.Chemical abundances exceeding solar values in at leastone element were detected for 9 stars of our sample suggest-ing that they are CP stars; of these, five have v sin i < v eq and no companions, 3 have v sin i > v eq and detected com-panions, and 1 has v sin i > v eq and no detected companions.Out of the 5 CP stars with v sin i < v eq and no companions,only 1 (KIC 8324268) is identified in the literature as anAp/Bp star. Line profile variability was also detected for thisstar, along with the CP star KIC 5461344, which suggeststhat both are likely Ap/Bp stars and host strong magneticfields with strengths &
100 G. The remaining 3 CP stars with v sin i < v eq and no detected companions (KIC 5430514,KIC 7050270, and KIC 9349245) all exhibit enhanced Cr,which is most commonly associated with Ap/Bp stars (e.g.Ghazaryan et al. 2018). In summary, we consider these 5targets to be candidate magnetic Ap/Bp stars.As shown in Table 1, the sample contains 3 starsthat have Am spectral types reported in the literature(KIC 8351193, KIC 8692626, and KIC 8703413). No abun-dances could be derived for KIC 8351193 as a result of itshigh v sin i value of 150 km s − , which also suggests thatit has been misclassified considering that Am stars typ-ically have relatively low v sin i values (e.g. Hauck 1986;Zorec & Royer 2012). The two other Am stars both havedetected companions and v sin i values .
45 km s − . In bothof these cases the photometric variability can possibly beattributed to spots on the (presumably) low-mass compan-ion. For KIC 8703413, P phot coincides with a possible orbitalperiod suggesting that the system may be an ellipsoidal vari-able (e.g. Smalley et al. 2014).Removing the 5 Ap/Bp candidates along with thosestars with detected companions leaves a total of 24 starsin our sample that (1) exhibit photometric variability thatis consistent with rotational modulation (i.e. they have v sin i < v eq ) and (2) that are presumably not strongly mag-netic based on the non-detection of either strong chemicalpeculiarities or spectral line profile variability. Currently,the origin of the photometric variability in these objectsis unkown. It has been suggested that the variability maybe related to ultra-weak magnetic fields having strengths . .
24 per cent of allMS A/late-B type stars where we have applied the samestatistical argument that was used above.
We have presented a detailed high-resolution spectroscopicstudy of 44 MS A/late-B type stars that reportedly exhibitrotationally modulated
Kepler light cuves (Balona 2013).For 14 of these stars, we found that either (1) the derived v sin i values are inconsistent with the photometric periodsbeing attributed to the star’s rotation period or (2) werefound to have low-mass companions that may be produc- ing the observed variability. We conclude that 29 of the 43stars for which radii and v sin i values could be derived arelikely consistent with the rotational modulation hypothesissince they have v sin i < v eq and no detected low-mass com-panions. This allowed us to revise the previously reportedincidence rate of rotationally variable MS A type stars from44 ± ± V magnitude forthe stars in our sample is 8 . TESS mission, which is allowing for the detection ofphotometric variability that may be attributed to rotationalmodulation of significantly brighter A and B-type stars (e.g.Sikora et al. 2019b; Balona et al. 2019) than those locatedin the
Kepler field.The detection of either weak spots or weak magneticfields that may be visible at the surfaces of such starscan be achieved by obtaining a series of extremely highS/N ( & I or V spectra using high-resolutionspectropolarimeters (e.g. Ligni`eres et al. 2009; B¨ohm et al.2015) such as ESPaDOnS@CFHT or NARVAL@TBL. Suit-able targets for such a search will need to be bright, haverelatively short rotation periods (such that the rotation pe-riod can be densely sampled), and have intermediate v sin i values. Considering the high cost of detecting and character-izing these features, it may be most efficient to first obtainlow- to moderate-S/N spectroscopic snapshot observationsof bright TESS targets that are found to exhibit rotationalmodulation in order to identify suitable targets. These tar-gets may subsequently be followed up with the requisitehigh-S/N Stokes I or V observations. ACKNOWLEDGMENTS
GAW acknowledges support in the form of a DiscoveryGrant from the Natural Science and Engineering ResearchCouncil (NSERC) of Canada.
DATA AVAILABILITY
The data underlying this article are available in the articleand in its online supplementary material.
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