Testing abundance-age relations beyond solar analogues with Kepler LEGACY stars
Thierry Morel, Orlagh L. Creevey, Josefina Montalban, Andrea Miglio, Emma Willett
AAstronomy & Astrophysics manuscript no. ms © ESO 2020November 10, 2020
Testing abundance-age relations beyond solar analogues with
Kepler
LEGACY stars (cid:63)
Thierry Morel , Orlagh L. Creevey , Josefina Montalbán , Andrea Miglio , and Emma Willett Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Quartier Agora, Allée du 6 Août19c, Bât. B5C, B4000-Liège, Belgiume-mail: [email protected] Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304Nice Cedex 4, France School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UKReceived 19 August 2020 ; accepted 6 November 2020
ABSTRACT
The prospects of using abundance ratios as stellar age indicators appear promising for solar analogues, but the usefulness of thistechnique for stars spanning a much wider parameter space remains to be established. We present abundances of 21 elements in asample of 13 bright FG dwarfs drawn from the
Kepler
LEGACY sample to examine the applicability of the abundance-age relations tostars with properties strongly departing from solar. These stars have precise asteroseismic ages that can be compared to the abundance-based estimates. We analyse the well-known binary 16 Cyg AB for validation purposes and confirm the existence of a slight metalenhancement ( ∼ / ingestion. We draw attention to systematicerrors in some widely-used catalogues of non-seismic parameters that may significantly bias asteroseismic inferences. In particular,we find evidence that the ASPCAP T e ff scale used for the APOKASC catalogue is too cool for dwarfs and that the [Fe / H] values areunderestimated by ∼ ff ective temperature, metallicity, and / ormass. We find that the seismic and abundance-based ages di ff er on average by 1.5-2 Gyrs, while taking into account a dependencywith one or two stellar parameters in the calibrations leads to a global improvement of up to ∼ ∼ ff sets between our abundances and those used to construct the calibrations or to the choice of the set oftheoretical isochrones. The conclusions above are supported by the analysis of literature data for a larger number of Kepler targets.For this extended sample, we find that incorporating a T e ff dependency largely corrects for the fact that the abundance-based ages arelower / larger with respect to the seismic estimates for the cooler / hotter stars. Although investigating age dating methods relying onabundance data is worth pursuing, we conclude that further work is needed to improve both their precision and accuracy for stars thatare not solar analogues. Key words.
Asteroseismology – Stars: fundamental parameters – Stars: abundances
1. Introduction
Stellar ages play a central role in various fields of astrophysics.Yet, it is one of the most di ffi cult properties to determine withgood accuracy in field stars. Although traditionally used in thepast to tag young or old stellar populations, [Fe / H] is now recog-nised not to be a reliable age proxy for stars in the Galactic discs(e.g. Edvardsson et al. 1993). However, several studies of so-lar twins / analogues spanning a narrow range of parameters haverecently unveiled remarkable correlations between stellar ages(derived from isochrone fitting) and other abundance ratios (e.g.da Silva et al. 2012; Bedell et al. 2018; Nissen 2015, 2016; Nis-sen et al. 2017, 2020; Adibekyan et al. 2016; Spina et al. 2016,2018; Tucci Maia et al. 2016; Jofré et al. 2020; Lin et al. 2020). (cid:63) Based on observations made at Institut Pytheas / Observatoirede Haute Provence (CNRS), France. Table B.1 is onlyavailable in electronic form at the CDS via anonymousftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/???/???
Abundance ratios of elements produced through di ff erent nucle-osynthesis channels (e.g. [Y / Mg]) are particularly sensitive toage because the relative amounts of ejecta released in the inter-stellar medium (ISM) strongly varied along the evolution of theGalaxy (e.g. Spina et al. 2016). The proposed relations betweenthe abundance ratios and age generally extend over ∼
10 Gyrsand are relatively tight (scatter of ∼ ff erent stel-lar populations and help tackling some important issues, such asthe formation history of our Galaxy. This question is particularlyrelevant and timely in view of the flood of stellar abundancesthat are already delivered by several medium- to high-resolutionspectroscopic surveys (e.g. Buder et al. 2020; Gilmore et al.2012). A wealth of information into the processes that shapedour Galaxy is encoded in these data, but reconstructing the time-line of events is essential. The sensitivity of the abundance-age relations against the stellar properties remains largely unex- Article number, page 1 of 23 a r X i v : . [ a s t r o - ph . S R ] N ov & A proofs: manuscript no. ms plored. As a matter of fact, recent studies have warned that careshould be exercised when applying the abundance-age relationsderived from solar analogues to stars with characteristics (espe-cially metallicity) significantly di ff erent from those of the Sun,not belonging to the Galactic thin disc, or even located outsidethe solar circle (e.g. Feltzing et al. 2017; Titarenko et al. 2019;Casali et al. 2020; Delgado Mena et al. 2019). The last point is aconcern because old stars are believed to have radially migratedover large distances across the Galaxy (e.g. Sellwood & Bin-ney 2002). Although these dependencies cannot be ignored, theyare currently neither well understood nor well quantified. Someabundance ratios appear to be relatively insensitive to some ofthese issues (e.g. [Y / Mg] compared to [Sr / Mg]; Nissen et al.2020), but the dependency may be quantitatively di ff erent de-pending on the nucleosynthesis details of the pair of chemicalelements involved.Another line of progress is related to the fact that the cali-brations are based on isochrone ages that fully rely on the pre-dictions of evolutionary models and are of limited applicabil-ity for unevolved dwarfs. Even in favourable cases (e.g. sub-giants), the accuracy in the age determination of field stars isseverely limited by systematic uncertainties (see discussion in,e.g., Sahlholdt et al. 2019). In contrast, ages from asteroseismol-ogy can be derived with good confidence even for stars on orslightly o ff the zero-age main sequence (ZAMS), and are be-lieved to be precise at the ∼ ff ected by chromospheric activity for stars typically youngerthan 4-5 Gyrs (Spina et al. 2020). Calibrating the abundance ra-tios against seismic ages would thus constitute a step forward,but the current samples of seismic targets with very precise abun-dances are still too limited in size. First attempts in that direc-tion were made by Nissen et al. (2017, hereafter N17) who de-termined the abundances of 12 elements in 10 solar-metallicitystars with an age of at most 7 Gyrs observed by the Kepler satel-lite. Their results support the abundance-isochrone age relationsfor solar analogues / twins, although a larger scatter in the calibra-tions is noticeable. One of the reasons may be that their targetsare significantly hotter and more evolved than the Sun.We present in this paper the abundances of 21 elements ina sample of 13 bright Kepler targets spanning a wide parameterrange and with precise seismic ages extending up to ∼
12 Gyrs tofurther examine the performance of the abundance-age relationsfor stars with properties departing from solar. Our abundancesprobe the main nucleosynthesis production channels (iron-peak, α - and neutron-capture) and also include lithium. These prime Kepler targets are amongst the main-sequence, solar-like starswith the best set of fundamental parameters (e.g. mass or age)known from asteroseismology. Our spectroscopic constraintscomplement the seismic data and will also aid further theoret-ical modelling.
2. Selection of targets and basic properties
Our targets are drawn from the so-called
Kepler
LEGACY sam-ple made up of a total of 66 solar-like dwarfs and subgiants(Lund et al. 2017; Silva Aguirre et al. 2017). They have all beenthe subject of intensive and long-duration (up to 4 years) obser-vations by the
Kepler satellite, and are currently the stars withthe best seismic parameters available. In particular, as discussedin Sect. 4, two independent studies derived the ages to 5-10%precision through a thorough modelling of these data. Becausethe amplitude of solar-like oscillations is proportional to stellarluminosity, this sample is biased towards stars hotter and more evolved than the Sun. However, stars at the end of the core-hydrogen burning phase that exhibit modes of mixed characterwere excluded (Lund et al. 2017; Silva Aguirre et al. 2017).To adequately investigate the dependency of the abundance-age relations as a function of [Fe / H] (e.g. Feltzing et al. 2017;Skúladóttir et al. 2019; Casali et al. 2020), we first choose starsin Buchhave & Latham (2015) with a metallicity deviating bymore than 2 σ (i.e. 0.2 dex) from solar. Our stars span a widerange of [Fe / H] values: from –0.95 to + / N) considerations, and13 stars with V <
10 mag could eventually be observed. Threenear-solar metallicity stars (KIC 6106415 and 16 Cyg AB) withhigh-precision abundance studies in the literature (N17, and ref-erences therein) were included in this sample for validation pur-poses. Except 16 Cyg B (Cochran et al. 1997), none of our tar-gets is known to host planets or is a
Kepler
Object of Interest(KOI).Very precise radial velocities (RVs) are obtained from our in-strument cross-correlation (CCF) data (a G2 mask was used). Acomparison with the line-of-sight (LOS) values of Lund et al.(2017) reveals clear RV changes in KIC 7871531 and KIC12317678 for which we obtain values larger by ∼ − , respectively. The variations for the former are small,but significant at the ∼ σ level. We therefore identify these twostars as single-lined (SB1) binaries. The binarity of our targetshas recently been investigated on the basis of Apache Point Ob-servatory Galactic Evolution Experiment (APOGEE) data. First,Price-Whelan et al. (2018) applied on the DR14 dataset a methodoptimised for sparse multi-epoch RV observations, but classi-fied KIC 7871531 as single. On the other hand, El-Badry et al.(2018b) used another approach and fitted DR13 spectra to singleout RV variables and stars with composite spectra (SB2). Threeof our targets are flagged as single stars (KIC 3656476, KIC6603624, and KIC 8694723), whereas three others are classifiedas SB2’s (KIC 7871531, KIC 9965715, and KIC 12317678). Nosignificant RV variations are apparent in our CCF data of KIC9965715, which were acquired over two consecutive nights. Theanalysis of El-Badry et al. (2018b) confirms the binary nature ofKIC 7871531 and KIC 12317678, although we do not detect thesignature of the secondaries in our optical spectra. The APOGEEobservations cover the near-infrared (near-IR) H -band, and aretherefore sensitive to cool companions. We computed the abso-lute magnitudes of the primaries in the V band, M V , assuming Gaia
DR2 parallaxes (Gaia Collaboration et al. 2018). Althoughthe E ( B − V ) values are compatible with zero within the uncer-tainties, we also used reddening estimates from the 3D mapsof Lallement et al. (2018) . For the secondaries, we used thesolar-metallicity isochrones of Spada et al. (2013) with a mixing-length parameter, α = M V assuming the seis-mic masses and ages of the primaries discussed in Sect. 4, alongwith the mass ratios of El-Badry et al. (2018b). We concludethat the companions do not contribute to more than ∼
10% to theflux in the optical. No visual companions have been detected inten of our targets (including KIC 7871531 and KIC 12317678)from adaptive optics imaging in the visible (Schonhut-Stasiket al. 2017). If confirmed, the composite nature of the spectrais expected to bias our abundances by about 0.05 dex at most The LOS values of Lund et al. (2017) for these two stars are basedon spectroscopic data obtained in early June 2014: see
Kepler
Commu-nity Follow-up Observing Program (CFOP) website at https://cfop.ipac.caltech.edu/home/ . See online tool at https://stilism.obspm.fr/ .Article number, page 2 of 23. Morel et al.: Abundance-age relations in the
Kepler
LEGACY sample
Table 1.
Basic properties of the targets. The determination of the stellar parameters is described in Sects. 4 and 5.1.
KIC ID Other IDs Spectral type V Binary status Population T e ff [K] log g [cgs] ξ [km s − ] [Fe / H]KIC 3656476 G5 IV 9.55 ... thin disc (?) 5680 ±
15 4.233 ± ± ± ±
24 4.266 ± ± ± ±
19 4.296 ± ± ± ±
20 4.325 ± ± ± / SB2 a thin disc 5510 ±
15 4.477 ± ± ± ±
15 4.545 ± ± ± ±
23 4.497 ± ± ± ±
56 4.112 ± ± ± b ... thick disc 5985 ±
35 4.329 ± ± ± a thin disc 6335 ±
40 4.280 ± ± ± ±
13 4.293 ± ± ± ±
15 4.358 ± ± ± / SB2 a thin disc 6550 ±
112 4.061 ± ± ± Notes. ( a ) The SB2 classification is based on the analysis of near-IR APOGEE spectra by El-Badry et al. (2018b), but note that we do not detectthe signature of the secondaries in our optical spectra. ( b ) In R band. (El-Badry et al. 2018a). Leaving the visual binary 16 Cyg aside,there are to our knowledge no signs of binarity in the other tar-gets. For KIC 8006161, in particular, no RV changes are detectedin our data secured two days apart. This lack of variability is sup-ported by the long-term CORAVEL monitoring of Halbwachset al. (2018).We used Gaia
DR2 astrometric and proper motion data (GaiaCollaboration et al. 2018), along with our RVs, to compute thespace components of the targets relative to the local standardof rest (see Morel et al. 2003). Following Bensby et al. (2014),we then estimated the probability that the stars belong to a givencomponent of the Galaxy. We find compelling evidence that KIC8760414 displays kinematic properties consistent with a thick-disc membership (it is about 7 times less likely to belong to thehalo). The situation is much more ambiguous based on kinemat-ical information alone for three other stars with an age above8 Gyrs. We find that they are about twice more likely to be-long to the thin disc. However, given that kinematical proper-ties do not allow a clean separation between the populations(e.g. Bensby et al. 2014), additional criteria — including theabundance pattern and the age — must be considered. As KIC8760414, KIC 7970740 exhibits enhanced abundances of the α elements ([ α / Fe] ∼ + α sequence defined by thick-disc stars inthe [Fe / H]-[ α / Fe] plane (e.g. Reddy et al. 2006). Its chemicalproperties therefore also support a thick-disc membership. Wewill consider below that these two stars, which turn out to bethe oldest and most metal-poor objects in our sample, are mem-bers of the thick disc. It can be noted that their age (10.7 and12.0 Gyrs; Sect. 4) is consistent with this Galactic componenthaving formed about 11 Gyrs ago (Silva Aguirre et al. 2018;Miglio et al. 2020; Montalbán et al. 2020). The two other thick-disc candidates (KIC 3656476 and KIC 6603624) have largemetallicities, but solar abundances of the α elements. This in-dicates that they do not belong to the population of metal-rich, α -enhanced stars (the so-called h α mr stars) claimed to be dis-tinct by Adibekyan et al. (2011). It is likely that these metal-rich,solar [ α / Fe] stars with an age of 8-9 Gyrs migrated from the in-ner Galaxy (e.g. Miglio et al. 2020). The rest of the sample areclearly thin-disc members.Our spectral analysis might be a ff ected by chromospheric ac-tivity, although the impact is, as expected, much larger for (very)young objects (e.g. Flores et al. 2016; Yana Galarza et al. 2019).As recently shown by Spina et al. (2020), the e ff ect on spectrallines can be significant for stars with log R (cid:48) HK (cid:38) –5.0, where R (cid:48) HK is an activity proxy determined through measurements of the Defined as the unweighted mean of the Mg, Si, and Ti abundances. Ca ii H + K emission-line fluxes. A homogeneous determinationof the activity level is provided for 11 of our targets by Breweret al. (2016). The log R (cid:48) HK values are found to tightly cluster ina range roughly solar (from –5.21 to –4.89), even for the onlytwo targets that are much younger than the Sun (the early-F starsKIC 9965715 and KIC 12317678; see Sect. 4). A much higheractivity level at the time of our observations cannot be ruled out,as illustrated by the strong activity cycle of KIC 8006161 (Kieferet al. 2017; Karo ff et al. 2018), but activity is not expected to bea major concern in our sample largely dominated by stars withan age similar to that of the Sun or older.The basic properties of our targets are summarised in Table1.
3. Observations
The observations were secured with the échelle, fibre-fed SO-PHIE spectrograph installed at the 1.93-m telescope of the Ob-servatoire de Haute Provence (OHP, France) during the period10-12 July 2018. The observations were carried out in the ob-ject + sky configuration. The spectra in High-E ffi ciency (HE)mode have a resolving power, R , of about 40,000 and a completewavelength coverage from about 3875 to 6945 Å. Two to threeexposures were obtained for the faintest targets and co-added ac-cording to their S / N. The final S / N per pixel ranges from 213 to350 at ∼ / N above 100 (the meanS / N is 168) and a satisfactory blaze correction. Observations ofa point-like source have to be preferred (e.g. Gray et al. 2000),while co-adding spectra from various reflecting bodies is not anissue (e.g. Bedell et al. 2014).Initial processing steps were carried out with the SOPHIE re-duction pipeline (DRS). All spectra were corrected for RV shiftsbased on the DRS CCF data. The spectrograph is simultaneouslyfed through two circular optical fibres, one illuminated by thetarget and the other one by the nearby sky. The spectrum ob-tained through the latter aperture was used for the sky subtrac-tion of the stellar exposure. Finally, the mean spectra were splitup into segments with a typical length of about 70 Å and a wave-length coverage carefully chosen in such a way that the contin-uum level could be adequately fit with low-order cubic spline orLegendre polynomials. To ensure the highest level of homogene- Note that our spectroscopic observations of KIC 8006161 were acci-dentally obtained during an activity minimum (Karo ff et al. 2018).Article number, page 3 of 23 & A proofs: manuscript no. ms ity, the same wavelength bounds and fitting functions were usedfor all stars. Standard tasks implemented in the IRAF softwarewere used for the final reduction steps.
4. Asteroseismic data
We make use below of the age, mass, M , and surface gravity,log g , determined from asteroseismology. Two teams indepen-dently performed a seismic modelling of the oscillation frequen-cies and / or their separation ratios estimated from the full-length,short-cadence Kepler dataset (Lund et al. 2017). While the anal-ysis of Creevey et al. (2017, hereafter C17) is based on a singlepipeline (
AMP : Asteroseismic Modeling Portal; Metcalfe et al.2009), Silva Aguirre et al. (2017, hereafter SA17) followed an-other approach and used six di ff erent modelling procedures. Asdiscussed by SA17, the results provided by the various pipelinesare consistent despite the variety of codes employed or the dif-ferent sensitivity of the seismic diagnostics to surface e ff ects, forinstance. We thus assume in the following the unweighted meanof all their individual values. It can be noted that C17 only usedfrequency ratios that are largely insensitive to the uncertain mod-elling of the near-surface layers. Stellar parameters of Kepler dwarfs and subgiants determined through a detailed modellingof the full frequency dataset have to be preferred over those in-ferred from the global seismic quantities because they are moreprecise by a factor of a few. Notwithstanding, there is no evi-dence for the LEGACY stars for systematic di ff erences signifi-cantly exceeding the uncertainty level (Serenelli et al. 2017).There is an overall good agreement between the parame-ters of interest determined by C17 and SA17 (see Fig. 1), buta systematic di ff erence for M and log g is apparent for the mostevolved Kepler
LEGACY stars with log g (cid:46) / optimisation procedures or choice of input physics.We simply note that the main di ff erences between the modelsin SA17 and C17 is the exploration of initial helium abundanceand mixing length. Some of the pipelines of SA17 fix the mix-ing length or restrict it, whereas others impose a chemical en-richment law. Their C2kSMO method (Lebreton & Goupil 2014)is the closest to that employed by C17. An enrichment law isnot imposed in C17, and high initial helium mass fractions, Y i ,and low masses are occasionally obtained (e.g. KIC 8694723 andKIC 9965715 for which Y i = Y i near to 0.257 and 0.267, and the newmasses are revised upwards to 1.08 and 1.09 M (cid:12) . In any case,this M - Y i correlation does not impact the age. Furthermore, therelatively small di ff erences observed for our sample between theresults obtained by C17 and SA17 have no impact on our con-clusions. Given that there is also no obvious reason to prefer oneset of values over the other, we simply adopt in the following theaverage values quoted in Table 2.A di ff erent procedure was adopted for KIC 9965715 becausethe significantly di ff erent and more reliable spectroscopic con-straints we obtain (as argued in Sect. 7.1) led us to redeterminethe seismic parameters of this star. Two independent modellingapproaches referred to in the following as ASTEC+ADIPLS and
CLES+LOSC were performed. Full details are provided in Ap-pendix A. The best-fit parameters obtained are shown in the bot- IRAF is distributed by the National Optical Astronomy Observato-ries, operated by the Association of Universities for Research in As-tronomy, Inc., under cooperative agreement with the National ScienceFoundation.
Fig. 1.
Comparison for the
Kepler
LEGACY sample between the stellarmass ( top panel ), age ( middle panel ) and surface gravity ( bottom panel )determined by C17 and SA17. The di ff erences for a given parameterare the values of C17 minus those of SA17. The stars in our sample areshown as red, filled circles. KIC 9965715 is shown for completeness asa cross. tom part of Table 2, and the straight mean values adopted in thefollowing. Compared to C17 and SA17, we find a mass revisedupwards by ∼
7% and that the star is ∼
30% younger.The comparison between the results of various pipelines sug-gests that the seismic ages used in our study are precise. This issupported by the fact that, for instance, inversion techniques pro-vide an age for 16 Cyg AB (in the range 7.0-7.4 Gyrs; Buldgenet al. 2016) fully compatible with the values adopted. However,we caution that the accuracy of seismic ages cannot be firmlyevaluated, although they appear to be relatively robust againstthe choice of the input physics when detailed seismic informa-tion is available (e.g. Lebreton & Goupil 2014). It is reassur-ing that both C17 and SA17 closely reproduced the solar age tomuch better than ∼
5. Abundance analysis
Except log g (see above), the atmospheric parameters and abun-dances of 21 chemical elements were determined from a line-by-line di ff erential analysis relative to the Sun, plane-parallel,1D MARCS model atmospheres (Gustafsson et al. 2008) andthe 2017 version of the line-analysis software MOOG originallydeveloped by Sneden (1973).
Article number, page 4 of 23. Morel et al.: Abundance-age relations in the
Kepler
LEGACY sample
Table 2.
Summary of seismic parameters for the sample. The parameters obtained by C17 and SA17 for KIC 9965715 are given for completeness:our updated values can be found in the bottom part of the table.
Literature values
Name M [M (cid:12) ] Age [Gyr] log g [cgs]C17 SA17 Adopted C17 SA17 Adopted C17 SA17 AdoptedKIC 3656476 1.101 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± This study
Name M [M (cid:12) ] Age [Gyr] log g [cgs] ASTEC+ADIPLS CLES+LOSC
Adopted
ASTEC+ADIPLS CLES+LOSC
Adopted
ASTEC+ADIPLS CLES+LOSC
AdoptedKIC 9965715 1.07 ± + . − . ± ± + . − . ± ± + . − . ± The line list is taken from Reddy et al. (2003), as it wasshown that the abundance analysis described below leads to ane ff ective temperature, T e ff , in good agreement for the G2 and K1main-sequence stars α Cen AB (Morel 2018) and the G0 sub-giant β Hyi (unpublished) with interferometric-based estimates(Kervella et al. 2017; Heiter et al. 2015). In addition, [Fe / H] iswithin the range of the commonly accepted values for these threestars (e.g. Jofré et al. 2014).The equivalent widths (EWs) were measured manually as-suming Gaussian profiles (multiple fits were used for well-resolved blends). Features with an unsatisfactory fit or signifi-cantly a ff ected by telluric features based on the atlas of Hinkleet al. (2000) were discarded. Strong spectral features with RW = log (EW / λ ) > –4.80 were also excluded provided that a su ffi -cient number of lines are left after this operation. These lines arebest avoided for di ff erent reasons, e.g., they lie on the non-linearpart of curve of growth or are formed in the upper atmosphereand particularly sensitive to chromospheric activity (Spina et al.2020). A “constrained” analysis whereby the surface gravity is fixed tothe very precise seismic value (Table 2) was enforced. The modelparameters ( T e ff , microturbulence, [Fe / H], and mean abundanceratio of the α elements with respect to iron, [ α / Fe]) were itera-tively modified until the following conditions are simultaneouslyfulfilled: (1) the Fe i abundances exhibit no trend with RW; (2)the mean abundances derived from the Fe i and Fe ii lines areidentical; and (3) [Fe / H] and [ α / Fe] are consistent with the modelvalues. The α -element abundance of the model varied depend-ing on [Fe / H] following Gustafsson et al. (2008). For instance,[ α / Fe] = + / H] = –0.4. Although excitation balanceof iron is not formally fulfilled in our constrained analysis, itcan be noted that the slope between the Fe i abundances and thelower excitation potential (LEP) deviates from zero by less than ∼ σ in our sample. The only clear exception is KIC 6603624 forwhich it appears that the adopted seismic log g is lower by ∼ T e ff and log g where held fixed to 5777 K and4.44, respectively, whereas the microturbulence, ξ , was left as afree parameter.Following N17, we corrected the Fe i abundances from de-partures from local thermodynamic equilibrium (LTE). We made use of Spectrum Tools to interpolate the non-LTE correc-tions as a function of the stellar parameters. The calculationsare described in Bergemann et al. (2012) and are available fora representative set of lines (about two thirds of the total). Thiswas deemed unnecessary for the Fe ii -based abundances becausenon-LTE e ff ects are negligible. Even for Fe i , the di ff erential cor-rections are small (less than 0.02 dex), and taking them into ac-count has very little impact on the parameters derived throughiron ionisation balance.Freezing log g to the seismic value has a much more pro-found e ff ect on our results. We find that relaxing this constraintwould result in log g , T e ff and [Fe / H] values larger on averageby 0.11 dex, 50 K and 0.03 dex, respectively. Our choice of aconstrained analysis is motivated by the fact that severe discrep-ancies between the spectroscopic log g and the reference valueare occasionally observed on a star-to-star basis (e.g., up to 0.5dex in the extreme case of KIC 12317678). There is still no con-sensus among the community as to whether such an analysis hasto be preferred for seismic targets (e.g. Doyle et al. 2017). Aconstrained analysis is undoubtedly more precise, but whether itis also more accurate is unclear. Our study does not shed newlight on this issue. We simply note, as argued in Sect. 7.2, thata comparison with the handful of long-baseline interferometricmeasurements available lends some support to the cooler T e ff scale resulting from the constrained analysis. This is consistentwith the recent claim based on a larger sample of interferometricbenchmark targets that a constrained analysis is more accurate,especially for F-type stars (Gent et al. 2020). Our line list for elements other than iron is taken from Reddyet al. (2003). However, for some key elements (Mg, Al, and Zn)we substituted this line list with that of Meléndez et al. (2014)because not enough lines are included. This was also done forCo because not all lines in Reddy et al. (2003) have hyperfinestructure (HFS) data available. HFS and isotopic splitting weretaken into account for Sc, V, Mn, Co, and Cu using atomic datafrom the Kurucz database and assuming the Cu isotopic ratio ofRosman & Taylor (1998). The barium abundance is solely basedon Ba ii λ http://nlte.mpia.de . Available at http://kurucz.harvard.edu/linelists.html .Article number, page 5 of 23 & A proofs: manuscript no. ms own tests on the Sun using the atomic data of Prochaska et al.(2000).We do not discuss further the abundances derived for Si ii andCr ii because they are based on a single feature, whereas those ofthe corresponding neutral species rely on up to 6 lines. However,despite the uncertainty plaguing the Si ii - and Cr ii -based abun-dances, ionisation balance is fulfilled for these two elements inthe vast majority of cases (Fig. 2). Fig. 2.
Comparison for Si ( top panel ) and Cr ( bottom panel ) betweenthe mean abundances yielded by the neutral and singly-ionised ion. Themean di ff erence is given in each panel (the number in brackets is thenumber of stars the calculation is based on). A representative error baris also shown. The determination of the lithium abundance relied on a spec-tral synthesis of Li i λ v sin i values of Bruntt et al. (2012) and computed the macro-turbulence from the calibrations as a function of T e ff of Brunttet al. (2010). Tests using the broadening parameters from theworks above, but also from Doyle et al. (2014), show that theexact choice of these quantities is irrelevant. For the Sun, we ob-tain an absolute lithium abundance, log (cid:15) (cid:12) (Li) =+ ff set (see also Sect. 8).Our non detection for five stars is also consistent with the find-ings of Bruntt et al. (2012) and Beck et al. (2017).For a number of reasons, we do not attempt to correct ourabundances for the combined e ff ect of departures from LTE andtime-dependent convection that would require in principle todrop the rudimentary assumption of 1D. First, performing full3D, non-LTE radiative transfer calculations is extremely tedious.As a result, theoretical predictions are currently only availablefor a few elements and very often do not span the whole pa-rameter space of our sample (Amarsi et al. 2019b; Amarsi &Asplund 2017; Nordlander & Lind 2017; Gallagher et al. 2020;Bergemann et al. 2019; Mott et al. 2020). To our knowledge,spectral line formation under non-LTE has not even been inves-tigated for some key elements in the context of our study (e.g. Y). Correcting for either non-LTE or
3D e ff ects is not necessar-ily recommended because the corrections might be of oppositesign and counterbalance each other (e.g. Amarsi et al. 2019a). Asa result, there is no guarantee that the corrected abundances aremore accurate. Second, and more importantly, the abundance-age relationships that are the main focus of this paper are builton the assumptions of 1D and LTE (e.g. Delgado Mena et al.2017). Should our abundance data be used in a wider context,we caution that care should be exercised for some elements thatare known to be particularly sensitive to non-LTE and / or 3D ef-fects, especially in the metal-poor regime (e.g. Mn; Bergemannet al. 2019).
6. Results
The atmospheric parameters and chemical abundances are givenin Tables 1 and B.1, respectively. The random uncertainties arecomputed following common practice (see Morel 2018). For theabundances, the line-to-line scatter, σ int , and the uncertaintiesarising from errors in the stellar parameters are added in quadra-ture. For iron, σ int typically amounts to 0.04 dex. For the el-ements with a single diagnostic line, we assume σ int = i lines.We explored to what extent the choice of another familyof model atmospheres would a ff ect our results. Namely, we re-peated the analysis using ATLAS9 Kurucz models (Castelli &Kurucz 2003) for 5 representative stars spanning the parame-ter space of our sample in terms of T e ff , log g , and [Fe / H]: KIC7871531, KIC 8006161, KIC 8694723, KIC 8760414, and 16Cyg A. We find in all cases very small T e ff di ff erences not ex-ceeding 5 K. Although the use of Kurucz models leads to sys-tematically lower microturbulent velocities (on average by 0.035km s − ), this is paralleled in the Sun. As a net result, we find thatall the abundance ratios di ff er by a negligible amount (virtuallyin all cases below 0.01 dex).
7. Validation of stellar parameters
A comparison with the results of Buchhave & Latham (2015)obtained with the Stellar Parameters Classification tool (
SPC )is particularly relevant because, with the exception of KIC9965715 and 16 Cyg AB, their T e ff and [Fe / H] values wereadopted by C17 and SA17 to perform their seismic modelling. Aconstrained analysis was also enforced by adopting as priors theseismic gravities of Chaplin et al. (2014). The results of Buch-have & Latham (2015) are based on data obtained with the TRESéchelle spectrograph installed on the 1.5-m telescope of the FredLawrence Whipple Observatory (Mt. Hopkins, Arizona). Thespectra have similar characteristics as ours ( R ∼ / N is modest. Despite this dif-ference in terms of data quality, there is a very close agreementbetween the stellar parameters without any evidence for system-atic discrepancies (Fig. 3).In sharp contrast, our T e ff and [Fe / H] values for KIC 9965715are discrepant with those of di ff erent origin used by C17 and In the range 30-110 per resolution element at 5110 Å based on infor-mation about the observing runs in Furlan et al. (2018).Article number, page 6 of 23. Morel et al.: Abundance-age relations in the
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LEGACY sample
SA17 to model the
Kepler data. We find that the star is lessmetal poor and much hotter, which is much more in line witha F2 V classification and in agreement with the conclusions ofMolenda- ˙Zakowicz et al. (2013) and Brewer et al. (2016). Onthe other hand, Compton et al. (2018) reported problems whentrying to model this star with the T e ff adopted by C17 and SA17.Overall, one can note a poorer level of agreement betweenour values and those quoted in the APOCASK catalogue(Serenelli et al. 2017), either derived from the analysis of near-IRAPOGEE DR13 spectra with the APOGEE Stellar Parametersand Chemical Abundances Pipeline (ASPCAP) or, to a lesserextent, from Sloan Digital Sky Survey (SDSS) griz photome-try. The ASPCAP T e ff scale is indeed known to be too cool fordwarfs and subgiants (e.g. Serenelli et al. 2017; Martinez et al.2019), which likely explains the [Fe / H] discrepancy also seen inFig. 3. Generally speaking, it is commonplace to find noticeabledi ff erences between [Fe / H] determinations based on optical ornear-IR spectra, or even between the results of di ff erent pipelinesapplied to APOGEE data (Sarmento et al. 2020). Fig. 3.
Upper panels: di ff erences (this study minus literature) betweenour T e ff and [Fe / H] values and those adopted by C17 and SA17 for theirseismic modelling.
Bottom panels: same as upper panels, but for theSDSS and ASPCAP non-seismic constraints adopted by Serenelli et al.(2017). The adopted values are the preliminary
Kepler
AsteroseismologyScience Consortium (KASC) estimates available at the time of the anal-ysis (Silva Aguirre, private communication) and are not taken from Pin-sonneault et al. (2012, 2014), as erroneously quoted in C17 and SA17. APOKASC is a joint collaboration between APOGEE and theKASC.
We present in Table 3 a global comparison with other liter-ature studies. Molenda- ˙Zakowicz et al. (2013) and Furlan et al.(2018) used various methods. For the former work, there are dis-crepancies amounting to up to almost 0.5 dex between the spec-troscopic and seismic gravities. One can note that a cooler / hotter T e ff scale is often associated to lower / larger surface gravities onaverage. This is a well-known consequence of the degeneracyin the determination of these two quantities from spectroscopy(e.g. Torres et al. 2012) that implies that the T e ff scale is sensi-tive to the presence of systematic biases in the determination oflog g . Compared to our results, the T e ff and log g di ff erences in-deed appear to be positively correlated for the data of Molenda-˙Zakowicz et al. (2013) and Brewer et al. (2016). This is also truefor the SPC and combined results of Furlan et al. (2018). We finda good agreement with our T e ff scale when allowance is madefor the fact that the spectroscopic gravities for the unconstrainedstudies above deviate from the very precise seismic estimates.Nonetheless, there is evidence that our metallicities are on aver-age larger by ∼ / or more metal-poor stars. This is not sur-prising because a robust determination is much more di ffi cult toachieve. T e ff scale Four stars in our sample have a nearly model-independent T e ff determined from a combination of absolute flux andlimb-darkened CHARA angular diameter measurements: KIC6106415 and KIC 8006161 (Huber et al. 2012), as well as 16Cyg AB (White et al. 2013). We do not discuss the CHARA ob-servations of 16 Cyg AB by Boyajian et al. (2013) because theywere carried out without the PAVO beam combiner whose usehas been claimed to yield more reliable measurements for starsthat are not well resolved (e.g. Casagrande et al. 2014; Karovi-cova et al. 2018). The T e ff scale of Boyajian et al. (2013) andWhite et al. (2013) appear indeed discrepant, with the formerbeing ∼
100 K cooler.To add a comparison point at higher e ff ective temperatures,we analysed the Gaia benchmark β Vir (F9 V) following exactlythe same steps as the main targets. The reference T e ff (6083 ± g (4.10 ± / N ∼ R ∼ / N of 330. Weobtain for β Vir: T e ff = ±
35 K and [Fe / H] = + ± / H] = + ± g lower by 0.01 dex only. The T e ff and [Fe / H] values are hence very similar.Figure 4 shows a comparison between our e ff ective tempera-tures and those derived from absolute flux and long-baseline in-terferometric measurements. The values obtained by N17 fromiron ionisation balance are also added. Except for 16 Cyg AB,there is indication that the spectroscopic values (either ours or Article number, page 7 of 23 & A proofs: manuscript no. ms
Table 3.
Comparison with stellar parameters in the literature. The mean di ff erences are this study minus literature. N is the number of stars incommon. The T e ff and [Fe / H] values of Buchhave & Latham (2015) were used to estimate the seismic ages of most of our targets. Note that theyprovide the mean metallicity, not [Fe / H]. For Molenda- ˙Zakowicz et al. (2013), we averaged for a given star and method (either
ROTFIT or MOOG )the results obtained with di ff erent spectrographs. Study and tool (cid:104) ∆ T e ff (cid:105) [K] (cid:104) ∆ log g (cid:105) (cid:104) ∆ [Fe / H] (cid:105) N Bruntt et al. (2012) a VWA + ±
79 –0.01 ± + ± ROTFIT + ± + ± + ± MOOG –43 ±
101 –0.12 ± + ± a SPC + ± + ± + ± b SME + ± + ± + ± SPC + ± + ± + ± Kea + ± + ± + ± SpecMatch + ± + ± + ± Newspec + ± + ± ± + ± + ± + ± Notes. ( a ) Constrained analysis. ( b ) Two entries with di ff erent IDs are given in the catalogue for KIC 5184732, KIC 7970740, and KIC 8006161:we chose the values based on the spectrum with the highest S / N. from N17 for KIC 6106415) are systematically larger. Therehave been previous reports of systematically hotter T e ff scalesdetermined from spectroscopy (e.g. Boyajian et al. 2013; Breweret al. 2016). Definitive conclusions can hardly be drawn consid-ering the paucity of data points and the recognition that, as dis-cussed above, there is some concern about the robustness of theinterferometric data. In particular, we find that the interferomet-ric radii of KIC 6106415 and KIC 8006161 tend to be larger thanthe seismic estimates of both C17 and SA17. The di ff erence isparticularly outstanding for KIC 6106415, and was ascribed bySA17 to the fact that it is poorly spatially resolved and possi-bly a ff ected by calibration problems. The limb-darkened angu-lar diameters, θ LD , of 16 Cyg AB are larger than those of KIC6106415 and KIC 8006161 ( θ LD ∼ β Viris well resolved ( θ LD ∼ T e ff discrepancy observed in the two stars that arebarely resolved may solely be explained by an overestimation oftheir angular diameters. Reddening should not be an issue giventhat these stars are all located within 50 pc and comfortably lieinside the Local Bubble. In any case, the cooler T e ff scale ob-tained when fixing log g (see Fig. 4) seems to support the choicemade in Sect. 5.1 of a constrained analysis (see also Gent et al.2020).
8. Validation of chemical abundances
The bright, G1.5 V + G3 V binary system 16 Cyg AB is a targetof great interest in the context of asteroseismology (e.g. Buld-gen et al. 2016; Verma et al. 2014; Bazot et al. 2019; Farniret al. 2020), and has been the subject of numerous high-precisionabundance studies. Interestingly, there is evidence for a smallmetallicity di ff erence between the two components that might beattributable to the formation and / or ingestion of planets. The ex-istence of a slight enhancement in metals ( ∆ [Fe / H] ∼ ff erence repre-sents a stringent test of the precision of our results. We compare Fig. 4.
Comparison between our ionisation temperatures (constrainedand unconstrained shown as filled and open black circles, respectively)and those in the literature derived from interferometric and absolute fluxmeasurements (filled red squares). The blue crosses show the results ofN17 determined from iron ionisation balance. The stars are ordered asa function of increasing T e ff . in Appendix C our results to those of N17 and Tucci Maiaet al. (2019) that are based on data of exceptional quality ( R = / N ∼ ff ering by only ∼ ff erence of the abundances with respect to hydrogen of ∼ It should be noted that non-LTE corrections were applied to mostelements by N17. Unfortunately, their LTE abundances cannot be com-puted because the mean di ff erential corrections are not provided on astar-to-star basis. However, as discussed by N17, they are exceedinglysmall for solar analogues and probably irrelevant for the purpose of ourdiscussion. Their uncertainties for elements other than iron also refer to[X / Fe], but we regard them in the following as being representative ofthose for [X / H].Article number, page 8 of 23. Morel et al.: Abundance-age relations in the
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LEGACY sample
The G0 V star KIC 6106415 was studied by N17. We findan ionisation T e ff lower by ∼
35 K. A comparison between theabundance data is shown in Fig. 5, as a function of the 50%condensation temperature for a solar-system composition gas( T c ; Lodders 2003). Although there is a slight, systematic o ff set( ∼ / H] values, the scatter is very small( ∼ ff set a ff ecting the absolute abun-dances may have several causes (e.g. continuum placement). Thelargest discrepancy ( ∼ (cid:104) ∆ [X / Fe] (cid:105) (this study minus N17) = –0.007 ± Fig. 5.
Upper panel : abundance pattern of KIC 6106415 with respect tohydrogen, [X / H], as a function of T c . Our [X / H] results and those of N17are shown as filled and open circles, respectively. A dotted, horizontalline is drawn at our [Fe / H] value. The solid line shows our weighted,linear fit of [X / H] as a function of T c . The fit obtained by N17 is over-plotted as a dashed line. Lower panel : di ff erences (our values minustheirs) with respect to N17, δ [X / H]. The average δ [X / H] value is given(the number of elements in common is indicated in brackets). To guidethe eye, a dotted line is drawn at δ [X / H] = Although the tests described above suggest that our abun-dance results are precise at the 0.02-0.03 dex level for early Gdwarfs, we warn the reader that this conclusion cannot be ex-tended to the stars in our sample with properties significantlydeparting from solar. It is because the e ff ects of various physicalphenomena (departures from LTE, convection inhomogeneities,atomic di ff usion) no longer cancel out to first order through ourdi ff erential analysis.For a more general validation of the elemental abundances,we consider the comprehensive studies of the Kepler targets un-dertaken by Bruntt et al. (2012) and Brewer et al. (2016). Thesurface gravity was held fixed to the seismic value in the for-mer work. Both analyses are based on spectral synthesis per-formed with (semi-)automatic tools. The results are comparedto ours in Fig. 6. Although some outliers are evident (e.g. KIC12317678 when compared to Brewer et al. 2016), there is a rea-sonable agreement with only a slight o ff set in general for theabundance ratios with respect to hydrogen (of the order of 0.04dex on average when considering the median di ff erences). Themost conspicuous di ff erence is the lower carbon abundances weobtain compared to Bruntt et al. (2012), but their uncertaintiesare quite large for some stars (up to 0.15 dex). However, whether or not we exclude this element, we tend to find a slightly betteragreement with Brewer et al. (2016).
9. Discussion
Several linear relations directly linking isochrone ages and abun-dance ratios have been proposed for solar twins / analogues (e.g.Bedell et al. 2018; Nissen 2015; Tucci Maia et al. 2016). Suchlinear relationships may be an oversimplification (Spina et al.2016, 2018). It is also becoming increasingly clear that othervariables must be taken into account in order to apply them tostars with parameters strongly departing from the solar values(e.g. Feltzing et al. 2017). Recently, Delgado Mena et al. (2019,hereafter DM19) used a large sample of main-sequence FGKstars observed as part of the HARPS-Guaranteed Time Observa-tions (HARPS-GTO) program to empirically explore the depen-dency of the abundance-isochrone age relationships as a functionof T e ff , metallicity, and mass. They found that several abundanceratios, when combined with one or two stellar parameters, canbe used to infer stellar ages to within 1.5-2 Gyrs. They proposedthree types of relationships: 1D (age vs abundance ratio), 2D(age vs abundance ratio and either T e ff , [Fe / H], or M ) and finally3D (age vs abundance ratio and two of either T e ff , [Fe / H], and M ). The main rationale behind including a T e ff and M depen-dency is to capture the e ff ect of atomic di ff usion, although it canbe noted that using metal abundance ratios largely reduces itsimportance (Dotter et al. 2017). They concluded that 2D rela-tions perform better compared to those in 1D (especially when[Fe / H] or M are folded in) and that 3D relations do not lead toa major additional improvement. Furthermore, it was found that3D relations yield similar results regardless of the choice of thetwo stellar parameters. It should be noted that, contrary to somestudies (e.g. Nissen 2016), thin- and thick-disc stars were notseparated when constructing these calibrations. Thick-disc starsconstitute about 20% of the subsample of 354 stars with the mostprecise isochrone ages (uncertainty below 1.5 Gyr) built for thatpurpose. Therefore, we apply the calibrations to our whole sam-ple in the following. We discuss below the ages we obtain fromthese relationships and compare them to the seismic estimates.DM19 proposed several abundance ratios that can be used as“chemical clocks”, among which fourteen can be computed fromour data. Although we refrain from discussing the time evolutionof other abundance ratios in our small dataset, let us briefly com-ment on the behaviour of the predominantly slow ( s -) neutron-capture elements Sr and Y. As previously found (e.g. Spina et al.2018), a smooth decline of [Sr / Fe] and [Y / Fe] with increasingage is seen. However, there is some evidence for an upturn forthe two thick-disc stars with an age above ∼
10 Gyrs (Fig. 7). Thisis consistent with a picture in which there was a more vigorousproduction during the formation of the thick disc of s -elementsfrom low-mass asymptotic giant branch (AGB) stars relative toiron from Type Ia supernovae (e.g. Battistini & Bensby 2016).This is apparently not seen in Fig. 7 for Ba, while the only Ceabundance at old ages is very uncertain. See DM19 for a discus-sion of the behaviours of the light and heavy s -elements.We show in Fig. 8 how some abundance ratios selected byDM19 vary as a function of seismic age. The data are colourcoded as a function of T e ff , [Fe / H], and M . Previous 1D relationsproposed in the literature are also overplotted (DM19; Nissen2016; Nissen et al. 2020; Tucci Maia et al. 2016; Spina et al.2018; Bedell et al. 2018; Titarenko et al. 2019). We recall that, Article number, page 9 of 23 & A proofs: manuscript no. ms
Fig. 6.
Comparison between our abundance ratios and those of Bruntt et al. (2012) and Brewer et al. (2016). The abundance di ff erences are thisstudy minus literature. The vertical dashed line connects the extreme values. The box covers the first to third quartile of the data, while the thickhorizontal line inside the box shows the median. As recommended by Bruntt et al. (2012), we chose their Fe ii -based abundances for [Fe / H] (butused the Mg i , Si i , Ti i , and Cr i abundances). For KIC 5184732, KIC 7970740, and KIC 8006161, we chose the values of Brewer et al. (2016)corresponding to the spectrum with the highest S / N. The number of stars the calculation is based on is indicated for each abundance ratio. The redsymbols show the abundance ratios used as age indicators (Sect. 9).
Fig. 7.
Abundances of the neutron-capture elements relative to iron asa function of seismic age. The thin- and thick-disc stars are shown ascircles and squares, respectively. The solar age is shown as a verticaldotted line. with the exception of DM19 and Titarenko et al. (2019), theserelationships only apply to solar analogues and are not expectedto provide a good fit to our data. Also, some of them may not bevalid for old or thick-disc stars. The deviations with respect to1D calibrations may be ascribed to stellar parameters di ff eringfrom the solar values. A good example is [Y / Si] where the sys-tematic deviations clearly noticeable for the old stars are largelyremoved when using 2D or 3D relations. A more general discus-sion is provided below. We compare in Table 4 for the whole sample the seismicages and those obtained for each type of relation (e.g. 1D) andabundance indicator (e.g. [Y / Mg]). Both the uncertainties in theabundance ratios and in the calibrations were propagated to theages. The best abundance indicators proposed by DM19 di ff erdepending on the parametrisation. In this respect, it is interestingto note that [Y / Al], which was often discussed in the literature inthe context of solar analogues (e.g. Nissen 2016), is not amongthe ratios displaying the best 1D relation when considering themuch more heterogeneous sample of DM19.The young, F-star KIC 12317678 falls close to the upper T e ff boundary of the relationships of DM19 and has abundance ratiosthat are particularly uncertain. However, we find that includingit in our sample does not bias our results. For instance, removingit leads to mean deviations between the seismic- and abundance-based ages for the whole sample that di ff er very little (in therange ± ff erences is typically ∼ / Fe]) appear to be less precise than others. There is tentativeevidence that the age scatter decreases as the intrinsic quality of
Article number, page 10 of 23. Morel et al.: Abundance-age relations in the
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LEGACY sample - . . [ M g / F e ][ S i/ F e ] - . . [ T i/ F e ][ Z n / F e ] - . . [ S r / M g ] - . . Abundance ratio [ S r / A l ] - . . [ S r / S i ] - . . [ S r / T i ] S e i s m i c a g e [ G y r s ] - . . [ S r / Z n ][ Y / M g ] [ Y / A l ] [ Y / S i ] [ Y / T i ] S e i s m i c a g e [ G y r s ] [ Y / Z n ] T e ff [ K ] - . . [ M g / F e ][ S i/ F e ] - . . [ T i/ F e ][ Z n / F e ] - . . [ S r / M g ] - . . Abundance ratio [ S r / A l ] - . . [ S r / S i ] - . . [ S r / T i ] S e i s m i c a g e [ G y r s ] - . . [ S r / Z n ][ Y / M g ] [ Y / A l ] [ Y / S i ] [ Y / T i ] S e i s m i c a g e [ G y r s ] [ Y / Z n ] [ F e / H ] - . - . - . - . . . - . . [ M g / F e ][ S i/ F e ] - . . [ T i/ F e ][ Z n / F e ] - . . [ S r / M g ] - . . Abundance ratio [ S r / A l ] - . . [ S r / S i ] - . . [ S r / T i ] S e i s m i c a g e [ G y r s ] - . . [ S r / Z n ][ Y / M g ] [ Y / A l ] [ Y / S i ] [ Y / T i ] S e i s m i c a g e [ G y r s ] [ Y / Z n ] . . . . . F i g . . L e ft pan e l s : v a r i a ti ono f t h ea bund a n ce i nd i ca t o r s , a s a f un c ti ono f s e i s m i ca g e . T h e d a t aa r ec o l ou r c od e d a s a f un c ti ono f T e ff . T h e t h i n - a nd t h i c k - d i s c s t a r s a r e s ho w n a s c i r c l e s a nd s qu a r e s , r e s p ec ti v e l y . T h e s o l a r a g e i ss ho w n a s a v e r ti ca l do tt e d li n e . M i dd l e and r i gh t pan e l s : s a m ea s l e f t p a n e l s , bu t c o l ou r c od e d a s a f un c ti ono f[ F e / H ] a nd M , r e s p ec ti v e l y . L i n ea r D r e l a ti on s i n t h e lit e r a t u r e li nk i ng a bund a n ce r a ti o s a nd a g e s a r e ov e r p l o tt e d a s d a s h e d li n e s : B e d e ll e t a l . ( , r e d ) , D M ( b l u e ) , N i ss e n ( ) a nd N i ss e n e t a l . ( , cy an ) , S p i n ae t a l . ( , m ag e n t a ) , T it a r e nko e t a l . ( , ye ll o w ) , a nd T u cc i M a i ae t a l . ( , g r ee n ) . Article number, page 11 of 23 & A proofs: manuscript no. ms the 1D and 2D T e ff relations increases (as parametrised by thegoodness-of-fit measure, adj- R ; see DM19 for definition). It isnot seen for the other relations, but the adj- R range is muchsmaller. The scatter between the abundance- and seismic-basedages is reduced by typically ∼
20% when averaging the results ofall abundance ratios. As may be expected, combining the resultsof many chemical clocks leads to more precise ages, but the gainis relatively modest.Ages estimated from pair of elements with significantly dif-ferent condensation temperatures, T c , may be more sensitive tostar-to-star di ff erences in their abundance- T c trend (N17; Nis-sen & Gustafsson 2018). Namely, stars with the same age mighthave di ff erent volatile-to-refractory abundance ratios (e.g. Bi-azzo et al. 2015). In our case, one might thus expect the twoabundance indicators involving the volatile element Zn ([Sr / Zn]and [Y / Zn]) to yield less precise ages. However, it is not borneout by our results (Table 4).There is clear evidence that the calibrations provide system-atically younger ages compared to asteroseismology by ∼ /
3D e ff ects may be e ffi cientlyerased in that case through a di ff erential analysis whatever theline list. On the contrary, the combined e ff ect might be quanti-tatively di ff erent depending on the choice of the diagnostic lineswhen the reference star occupies another region of the parame-ter space. We obtain ages on average ∼ ff er-ences (e.g. in the T e ff scale) are also likely to play a role. In anycase, it shows that age deviations of the right order of magnitudemight be ascribed to small ( (cid:46) ff sets. It is particularly true for Sr and Y because these two el-ements enter most calibrations. However, systematic di ff erencesbetween various sets of theoretical isochrones are well known.DM19 made use of PARSEC isochrones (Bressan et al. 2012) toinfer the ages through Bayesian inference. Adopting another setof evolutionary models may be enlightening in this respect.Figure 10 illustrates the age deviations, ∆ A , as a function of T e ff , [Fe / H], and M . The Sun is overplotted to ensure that its ageis correctly reproduced and that no systematic age o ff sets arepresent. The size of the error bar in this particular case also pro-vides an indication of the age uncertainty arising from the cali-brations alone. There is no evidence in Fig. 10 that the thick-discstars are outliers. We define a quantity, δ A , which is a measureof the improvement brought about by using 2D or 3D relationsinstead of 1D ones. A positive value implies that the age is closerby δ A Gyrs to the reference seismic value when using a 2D or3D parametrisation (and vice versa). The relative performanceof the various relations is better appraised when examining the δ A values for the four abundance indicators in common to all re-lations: [Sr / Mg], [Sr / Ti], [Y / Mg], and [Y / Zn] (Fig. 11). The useof more sophisticated relations generally leads to a better agree-ment with the seismic ages ( δ A positive and reaching up to + ff ect is small (see also fig.13 of DM19). Fig. 9.
Mean ages of KIC 6106415 and 16 Cyg AB obtained using ourabundance data ( circles ), those of N17 ( squares ), and those of TucciMaia et al. (2019, triangles) . The shaded horizontal stripes show theseismic ages, along with their ± σ uncertainty. However, it is conceivable that a much more significant im-provement is achieved in some regions of the parameter space.For instance, the age derived for the most metal-poor LEGACYstar, KIC 8760414, is revised upwards by almost 4 Gyrs usingthe relations making use of [Fe / H], and is no longer severely un-derestimated compared to the seismic result. DM19 cautionedthat their relations are not robust for [Fe / H] (cid:46) –0.8. Whether theimprovement observed for KIC 8760414 is only a fluke or indi-cates that the relations are still of some applicability for metal-poor stars must be investigated further. The improvement arisingfrom the use of 2D or 3D relations might be buried in the noisefor the other stars in our sample with parameters closer to solar. To further examine the performance of the abundance-age rela-tions proposed by DM19, we consider an extended sample of
Kepler dwarfs and subgiants with a homogeneous determinationin the literature of both the abundances and the seismic proper-ties. Namely, we select stars in common between Brewer et al.(2016) and Serenelli et al. (2017). Abundances of the key ele-ments Sr and Y are not provided by Bruntt et al. (2012). Notethat Brewer et al. (2016) empirically corrected their abundancedataset for trends as a function of T e ff . We use in the follow-ing the seismic results of Serenelli et al. (2017) based on SDSSdata. We ignore stars with T e ff taken from Brewer et al. (2016)below 5300 K because they fall outside the validity domain ofthe calibrations of DM19. Four stars exhibit anomalously highabundance ratios involving yttrium: we exclude KIC 9025370because it is an SB2 (N17) and KIC 11026764 because signifi-cantly lower values were obtained by Metcalfe et al. (2010). Weretain the remaining two stars: KIC 3733735 and KIC 9812850.No additional information about a general enhancement of the s − process elements is available, but, in any case, the ages in- For the very young ( ∼ / Al] is veryhigh, which rather points to a problem with the Al abundance.Article number, page 12 of 23. Morel et al.: Abundance-age relations in the
Kepler
LEGACY sample
Table 4.
Unweighted mean age deviation with respect to seismic estimates (abundance-based minus seismic) for each abundance indicator andrelation (the number in brackets is the number of stars the calculation is based on). The bottom row provides the grand mean for a given relation.A blank indicates that the abundance ratio is not among the best age indicators for the relevant relation according to DM19.
Abundance indicator 1D 2D 3D T e ff [Fe / H] M T e ff and [Fe / H][Mg / Fe] –1.216 ± / Fe] –2.419 ± ± / Fe] –0.917 ± ± / Fe] 0.051 ± ± ± / Mg] –0.968 ± ± ± ± ± / Al] ... ... –0.789 ± / Si] –1.375 ± ± ± / Ti] –0.697 ± ± ± ± ± / Zn] ... ... –0.668 ± / Mg] –1.449 ± ± ± ± ± / Al] ... ... –1.012 ± ± / Si] –2.104 ± ± ± ± / Ti] –1.507 ± ± ± ± / Zn] –0.243 ± ± ± ± ± ± ± ± ± ± ∼ g values down to 3.5) than our LEGACYsample. The former feature likely explains the large proportion( ∼ T e ff and [Fe / H] values (AppendixA). In contrast, here there are systematic di ff erences between theclassical parameters adopted for the determinations of the chem-ical abundances and ages (see Fig. 12). For the stars selected, wefind that the T e ff scale of Brewer et al. (2016) is on average ∼ T e ff scale. It supports our previ-ous conclusion that these T e ff values are too low. In addition, themetallicities of Brewer et al. (2016) are ∼ ∼ T e ff and 3D relationships (the latter also involving T e ff ) quite e ffi ciently erase the clear trend between the age dif- ference and T e ff . The lack of improvement when incorporating[Fe / H] as an additional variable might be due to the near-solarmetallicity range. We find that negative δ A values for the 2D[Fe / H] calibration are generally associated to the most evolvedstars (log g (cid:46) x -axis at ∆ A ∼ –2 and + ∼
10. Summary and conclusions
Our non-seismic parameters (especially T e ff , [Fe / H], and [ α / Fe])complement the superb
Kepler oscillation data obtained for thesebright stars and will aid further seismic modelling. These are im-portant observational constraints that are necessary to exploit thefull potential of asteroseismology (see, e.g. Chaplin et al. 2014;Serenelli et al. 2017; Creevey et al. 2012; Valle et al. 2018).We classify two of our targets with an age exceeding 8 Gyrsas thick-disc members mostly based on chemodynamical argu-ments. Their chemical pattern significantly di ff erent from thesolar mixture (in particular an enhancement of the α elements)must be taken into account to achieve sensible seismic inferences(e.g. Ge et al. 2015; Li et al. 2020). Our results can also be usefulin other more specific contexts, such as constraining the Galac-tic helium enrichment law (e.g. Verma et al. 2019) or improvingthe modelling of the outermost stellar layers (e.g. Compton et al.2018). Two Li-detected stars, KIC 5184732 and KIC 8694723,also have a determination of their rotational period from Kepler light curves (García et al. 2014; Karo ff et al. 2013), which pavesthe way for a detailed theoretical modelling of their interior.We find evidence that a few of our targets are associated ina binary system with a large mass ratio. This suggests the exis-tence in the Kepler
LEGACY sample of a sizeable fraction of bi-naries. This is supported by the fact that N17 detected two SB2’sin their sample of ten stars from snapshot observations only. Itwould not be surprising considering the ubiquity of companionsin solar-like stars (e.g. Duquennoy & Mayor 1991). This aspect
Article number, page 13 of 23 & A proofs: manuscript no. ms
Fig. 10.
Mean age deviation, ∆ A , with respect to the seismic estimate (abundance-based minus seismic) when considering all the abundanceindicators, as a function of T e ff , [Fe / H], and M . The quantity δ A is a measure of the improvement brought about by using 2D or 3D instead of 1Drelations (see Sect. 9.1). The thin- and thick-disc stars are shown as filled and open circles, respectively. The Sun is shown as an open triangle. Therightmost panels show the breakdown of the ∆ A and δ A values. certainly deserves further investigation (i.e. a dedicated RV mon-itoring).As guidance for further ensemble seismic modelling of Ke-pler main-sequence stars, our comparison with the results of(semi-)automatic methods suggests that the study of Buchhave& Latham (2015) is a valuable source of e ff ective temperaturesand metallicities. For the elemental abundances, the works ofBruntt et al. (2012) and Brewer et al. (2016) appear to be of sim-ilar quality for most elements. However, our results tend to be incloser agreement with those of the latter study. We confirm thatthe ASPCAP e ff ective temperatures used for the APOKASC cat-alogue are systematically too cool for dwarfs. This is not com-pletely unexpected given that the pipeline is naturally optimised for red giants (García Pérez et al. 2016). The consequence is thata grid-based modelling that makes use of the SDSS photometric T e ff scale may be more reliable for relatively unevolved Kepler targets, as already claimed by Serenelli et al. (2017). However,one can note in Fig. 12 the existence of a systematic discrepancybetween the SDSS T e ff values and those of Brewer et al. (2016).We find evidence in our sample that the APOKASC metallici-ties for dwarfs (solely based on ASPCAP) are underestimated bytypically ∼ ∼ Article number, page 14 of 23. Morel et al.: Abundance-age relations in the
Kepler
LEGACY sample
Fig. 11.
Same as Fig. 10, but only for the four abundance indicators common to all the relations. analysis of KIC 9965715, even larger changes can be expectedif the quality of the classical parameters is mediocre.
We find that the quality of stellar ages empirically determinedfrom abundance data is currently limited by the precision ofthe calibrations, which is typically 1.5-2 Gyrs when venturingaway from stars closely resembling our Sun. Constructing morecomplex relationships that take into account a dependency withone or two stellar parameters does improve the situation, if onlyslightly. For instance, incorporating T e ff as an additional variableglobally improves the agreement between the abundance- andseismic-based ages for the APOKASC stars studied by Breweret al. (2016). The calibrations presently available appear signif- icantly less precise than what can be achieved for stars akin tothe Sun (precision often below 1 Gyr), but some abundance ra-tios may hopefully prove in the future more suitable than othersoutside the rather restricted domain defined by solar analogues.A concern is that applying relationships constructed from a spe-cific abundance dataset (such as that of Delgado Mena et al. 2017used by DM19) is prone to the presence of study-to-study zero-point abundance o ff sets that are virtually unavoidable (e.g. Jofréet al. 2017). More generally, the possible existence of systematicdi ff erences between isochrone and seismic ages must be inves-tigated further to improve the accuracy of the relationships (seeBerger et al. 2020 for an example of comparison).Constructing empirical abundance-age calibrations allowingone to accurately infer the age of stars spanning a wide rangein spectral type, evolutionary status and metallicity — let aloneborn in di ff erent parts of the Galactic discs characterised by dif- Article number, page 15 of 23 & A proofs: manuscript no. ms
Fig. 12. Di ff erences between the stellar parameters of Brewer et al.(2016) and Serenelli et al. (2017). The mean values for the SDSS andASPCAP T e ff scales are given in each panel. We assumed the ASPCAP[M / H] values that are the overall scaled-solar abundances. ferent star-formation histories — appears to be a daunting en-deavour. A large number of stars with solar-like oscillations willbe detected by the TESS satellite, but this sample will be largelydominated by evolved subgiants (Campante et al. 2016). Thefull accomplishment of this goal might thus await the releaseof seismic ages for thousands of bright main-sequence stars bythe PLATO space mission (Rauer & Heras 2018) complementedby a high-quality determination of their surface chemical com-position.
Acknowledgements.
We would like to thank the anonymous referee forher / his careful reading of the manuscript and thoughtful suggestions. TMacknowledges financial support from Belspo for contract PRODEX PLATOmission development, and wishes to thank Werner Verschueren (Belspo)for allowing the funding of the mission at OHP. JM, AM, and EW ac-knowledge support from the ERC Consolidator Grant funding scheme(project ASTEROCHRONOMETRY, https: // Gaia ( ),processed by the Gaia
Data Processing and Analysis Consortium (DPAC, ). Fundingfor the DPAC has been provided by national institutions, in particular theinstitutions participating in the
Gaia
Multilateral Agreement. This article madeuse of AIMS, a software for fitting stellar pulsation data, developed in thecontext of the SPACEINN network, funded by the European Commission’sSeventh Framework Programme. This research has also made use of NASA’sAstrophysics Data System Bibliographic Services and the SIMBAD databaseoperated at CDS, Strasbourg (France).
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Kepler
LEGACY sample
Appendix A: Seismic analysis of KIC 9965715
Appendix A.1: First approach (
ASTEC+ADIPLS ) We approached the new interpretation of the data from KIC9965715 in two steps. The first step consisted in using a densegrid of BASTI stellar models (Pietrinferni et al. 2004; Hidalgoet al. 2018) to match the T e ff , (cid:104) ∆ ν (cid:105) , and ν max . The global seismicquantities are taken from Lund et al. (2017). These models usea chemical enrichment law, and we refer to the relevant papersfor details of the physics. We use a simple Bayesian approach in2D (mass, age) on grids of di ff erent metallicities with mild pri-ors that have little influence on our results: mass is restricted to[0.8,1.3] M (cid:12) , age is restricted to [0,10] Gyrs, and we use a prioron the evolution state defined as the relative amount of time spentin the specific phase of the star’s life (main sequence, subgiant,and giant). The uncertainties are obtained by marginalising overthe mass and age, and are defined as the 68% confidence inter-val centred on the median values. We also record the model ob-servables and so we can define an optimal surface gravity log g = ± ff erentmetallicity value.In a second step, we use the fitted parameters from the firststep as initial starting points for a detailed stellar modelling.The use of the global seismic quantities brings important con-straints on certain stellar parameters such as the surface gravityor density. However, for the best precision and improved accu-racy (scaling relations are still subject to some systematic errors)on other parameters such as mass and age, we must use the in-formation contained in the detailed seismic data. We adopted asimilar approach to that presented in C17 by using the frequencyratios ( r , r ) in the optimisation. These values are calculateddirectly from the individual frequencies from Lund et al. (2017),and we derive the covariance matrix C by performing simpleMonte-Carlo-like simulations.In the optimisation approach we calculate a Likelihood L = e − χ (A.1)where χ = ( x − x M ) T C − ( x − x M ) . (A.2)Here x are the observational constraints and x M are the corre-sponding observables from the model. We additionally includedthe constraint of the mean large and small frequency separations, (cid:104) ∆ ν (cid:105) and (cid:104) δν (cid:105) , by calculating this value within the range of theobserved frequencies. We also included the metallicity and T e ff constraint.For this second step of stellar modelling, we used the AarhusSTellar Evolution Code ( ASTEC ) and the Adiabatic Pulsationcode (
ADIPLS ) from Christensen-Dalsgaard (2008b,a). The set-up of the physics is as follows: we assumed a non-rotating,non-magnetic star. The opacities and equation-of-state tables aretaken from the OPAL collaboration (Iglesias & Rogers 1996;Rogers & Nayfonov 2002). Nuclear reaction rates were takenfrom Angulo et al. (1999) and we include the values obtainedby the LUNA collaboration for the N(p, γ ) O reaction (Formi-cola et al. 2004). We use the solar mixture of Grevesse & Sauval(1998). Apart from the fixed physics described above, the onlyother parameters that control the evolution of the star are themass, M , and the initial metallicity and helium mass fraction, Z i and Y i , where X i + Y i + Z i =
1, and X refers to the hydrogenmass fraction. We also need to set the mixing-length parameter α , defined as the ratio of the mean-free path of a fluid element, Table A.1.
Stellar parameters for KIC 9965715 using the first approach.The bottom rows give the adopted values. The solar global parametersare from IAU 2015 Resolution B3 (Mamajek et al. 2015). ( Y i , α ) (0.257, 2.14) (0.257, 2.04) (0.287, 1.94) M [M (cid:12) ] 1.12 ± ± + . − . R [R (cid:12) ] 1.29 ± ± ± + . − . + . − . ± L –10.1 –9.2 –8.3 M [M (cid:12) ] 1.07 ± ± g ± Fig. A.1.
Marginalised distributions of the mass and the age for the set( Y i , α ) = (0.287,1.94), scaled so the maximum is equal to unity. Theadopted stellar parameter (maximum likelihood) and the lower and up-per values (68% confidence interval) are shown by the dashed lines. and the pressure scale height (see Böhm-Vitense 1958). The fi-nal parameter that controls the current structure of the model isthe age.We approached our optimisation of the parameters by build-ing several small 3D grids ( M , Z i , and age) while fixing ( Y i , α )at discrete values. We show some of these results in Table A.1,where the first set of ( Y i , α ) are equivalent to the values used inthe BASTI library. The parameters are defined as the maximumlikelihood ones, and the uncertainties are calculated as the 68%confidence interval with the highest probability, as shown by thedashed lines in Fig. A.1. We find that the mass varies between1.05-1.13 M (cid:12) with a strong dependency on Y i . The optimal ageremains in the range of between 2.7-3.3 Gyrs. We adopt the val-ues from the third column and, to account for the other parame-ters, we add the di ff erences to the uncertainties in quadrature, toobtain M = ± (cid:12) and an age of 2.8 ± Appendix A.2: Second approach (
CLES+LOSC ) The stellar parameters of KIC 9965715 have also been estimatedby using the open-source code
AIMS (Asteroseismic Inference ona Massive Scale; Reese et al. 2016; Lund & Reese 2018; Ren-dle et al. 2019) that implements a Bayesian inference approach.
AIMS evaluates the posterior distributions of stellar parametersusing a Markov Chain Monte Carlo (MCMC) ensemble sampler(Foreman-Mackey et al. 2013), and selects stellar models that
Article number, page 19 of 23 & A proofs: manuscript no. ms best fit observational data by interpolating (evolutionary tracksand frequencies) in a pre-computed grid.The grid of stellar models at the base of the procedure wascomputed using the code
CLES (Scuflaire et al. 2008b; Gabrielet al. 2014), and the oscillation frequencies of radial ( (cid:96) = (cid:96) = ,
2) modes for each stellar model usingthe adiabatic oscillation code
LOSC (Scuflaire et al. 2008a). Thestellar models computation follows the evolution from the pre-main sequence up to the point where the hydrogen content at thestellar centre, X c , is below 10 − . The prescriptions for the inputphysics are the following: nuclear reaction rates of Adelbergeret al. (2011) or from NACRE (Angulo et al. 1999) if not avail-able. The equation of state is FreeEoS (Irwin 2012). We adoptthe solar metal mixture from Asplund et al. (2009), and opac-ity tables were built combining OPAL (Iglesias & Rogers 1996)opacity values at high temperature with those of Wichita StateUniversity for low temperature domain (Ferguson et al. 2005).Atmospheric boundary conditions are provided by the Krishna-Swamy’s T( τ ) law (Krishna Swamy 1966) and the correspondingatmosphere structure is added on the top of the interior model.Convection is treated with the “mixing length” formalism (Cox& Giuli 1968). The corresponding α parameter was kept fixedfor all the grids, and was derived from the solar calibration ( α = ff u-sion of chemical elements (Thoul et al. 1994).We assume extra-mixing at the boundaries of convective re-gions introducing core-overshooting ( ov = H p —pressure scale height), and undershooting below the convectiveenvelope ( ∼ H p ). In both cases, the chemical mixing is in-stantaneous and the temperature gradient in the mixed region isthe radiative one. Finally, microscopic di ff usion of chemical el-ements has been included in one of the grids.The grids are parameterised by mass (from 0.9 to 1.8 M (cid:12) with a step of 0.01 M (cid:12) ) and initial [Fe / H] (from –0.45 to 0.00,with a step of 0.05 dex), assuming an enrichment law ∆ Y / ∆ Z provided by the solar calibration and a primordial He abundance, Y p , of 0.2485 (Komatsu et al. 2011).The stellar parameters of KIC 9965715 are derived usingthese grids and the following observational constraints: T e ff and[Fe / H], as classical constraints, as well as the average ν max andthe individual frequencies of 40 modes ( (cid:96) = , , and 2) of Lundet al. (2017) as seismic data. We also perform a second set of op-timisations using as seismic information the values of frequencyseparation ratios ( r , r , r ) and the frequency of lowest orderradial mode. In the former case, we use the two-terms surfacee ff ect correction from Ball & Gizon (2014).The analysis of the results allows to rule out models with anextra-mixing as large as 0.2 H p . On the other hand, the over-shooting below the convective envelope has no e ff ect on thestellar parameters derived. From “Bayesian Evidence” values,models including microscopic di ff usion of helium and metals areclearly preferred over those without it.The results of the di ff erent optimisations we perform indi-cate an initial chemical composition close to the solar one. So,stellar parameters based on non-di ff usion grids tend to deviatefrom the observed surface chemical composition. A good matchof surface composition can be obtained at the cost of a mediocrefit of the seismic properties. The stellar parameters based on fre-quency separation ratios are in good agreement with those de-rived using the individual frequencies. The parameters for eachoptimisation are defined from the posterior distributions, as themaximum likelihood ones, and the uncertainties are calculatedas the 68% confidence interval with the highest probability. Wesummarise these results in Table A.2 in two sets of stellar param- Table A.2.
Stellar Parameters for KIC 9965715 using the second ap-proach. The bottom rows give the adopted values. The solar global pa-rameters are from IAU 2015 Resolution B3 (Mamajek et al. 2015).
Grids With di ff usion Without di ff usion M [M (cid:12) ] 1.20 ± ± ± ± R [R (cid:12) ] 1.315 ± ± g ± ± Z ± ± Y ± ± M [M (cid:12) ] 1.20 + . − . Age [Gyr] 2.41 + . − . log g + . − . eters depending on whether they are based on the model gridswith or without microscopic di ff usion. We adopt the values ofthe first column and inflate the uncertainties to take into accountthat the target could be up to 1 Gyr older if the constraint on thesurface chemical composition is relaxed. Appendix B: Abundance results
Article number, page 20 of 23. Morel et al.: Abundance-age relations in the
Kepler
LEGACY sample
Table B.1.
Abundance results for the stars in our sample.
KIC 3656476 KIC 5184732 KIC 6106415 KIC 6603624 KIC 7871531 KIC 7970740 KIC 8006161[Fe / H] a ± +
6) 0.41 ± +
6) –0.05 ± +
5) 0.27 ± +
6) –0.19 ± +
5) –0.47 ± +
6) 0.35 ± + (cid:15) (cid:12) (Li) < ± ± < < < < i / Fe] –0.02 ± ± ± ± ± ± ± i / Fe] 0.04 ± ± ± ± ± ± ± i / Fe] –0.01 ± ± ± ± ± ± ± i / Fe] 0.05 ± ± ± ± ± ± ± i / Fe] 0.03 ± ± ± ± ± ± ± i / Fe] –0.05 ± ± ± ± ± ± ± ii / Fe] b ± ± ± ± ± ± ± i / Fe] –0.01 ± ± ± ± ± ± ± i / Fe] b ± ± ± ± ± ± ± i / Fe] –0.02 ± ± ± ± ± ± ± i / Fe] b –0.05 ± ± ± ± ± ± ± i / Fe] b –0.04 ± ± ± ± ± ± ± i / Fe] 0.02 ± ± ± ± ± ± ± i / Fe] b ± ± ± ± ± ± ± i / Fe] 0.01 ± ± ± ± ± ± ± i / Fe] –0.14 ± ± ± ± ± ± ± ii / Fe] –0.09 ± ± ± ± ± ± ± ii / Fe] –0.03 ± ± ± ± ± ± ± ii / Fe] 0.00 ± ± ± ± ± ± ± i / Mg i ] –0.13 ± ± ± ± ± ± ± i / Al i ] –0.19 ± ± ± ± ± ± ± i / Si i ] –0.17 ± ± ± ± ± ± ± i / Ti i ] –0.12 ± ± ± ± ± ± ± i / Zn i ] –0.15 ± ± ± ± ± ± ± ii / Mg i ] –0.09 ± ± ± ± ± ± ± ii / Al i ] –0.15 ± ± ± ± ± ± ± ii / Si i ] –0.12 ± ± ± ± ± ± ± ii / Ti i ] –0.08 ± ± ± ± ± ± ± ii / Zn i ] –0.10 ± ± ± ± ± ± ± Table B.1.
Continued.
KIC 8694723 KIC 8760414 KIC 9965715 KIC 12069424 KIC 12069449 KIC 12317678(16 Cyg A) (16 Cyg B)[Fe / H] a –0.41 ± +
4) –0.95 ± +
4) –0.29 ± +
4) 0.11 ± +
5) 0.08 ± +
5) –0.19 ± + (cid:15) (cid:12) (Li) 2.09 ± ± ± ± < ± i / Fe] 0.01 ± ± ± ± ± ± i / Fe] –0.01 ± ± ± ± ± ± i / Fe] 0.03 ± ± ± ± ± ± i / Fe] –0.07 ± ± ± ± ± i / Fe] 0.00 ± ± ± ± ± ± i / Fe] 0.04 ± ± ± ± ± ± ii / Fe] b ± ± ± ± ± ± i / Fe] 0.04 ± ± ± ± ± ± i / Fe] b ± ± ± ± i / Fe] –0.10 ± ± ± ± ± ± i / Fe] b –0.22 ± ± ± ± ± ± i / Fe] b ... ... ... –0.04 ± ± ± i / Fe] –0.06 ± ± ± ± ± ± i / Fe] b –0.06 ± ± ± ± ± ± i / Fe] –0.06 ± ± ± ± ± ± i / Fe] –0.02 ± ± ± ± ± ± ii / Fe] –0.03 ± ± ± ± ± ± ii / Fe] 0.20 ± ± ± ± ± ii / Fe] 0.06 ± ± ± ± ± i / Mg i ] –0.05 ± ± ± ± ± ± i / Al i ] 0.05 ± ± ± ± ± i / Si i ] –0.01 ± ± ± ± ± ± i / Ti i ] –0.06 ± ± ± ± ± ± i / Zn i ] 0.04 ± ± ± ± ± ± ii / Mg i ] –0.06 ± ± ± ± ± ± ii / Al i ] 0.04 ± ± ± ± ± ii / Si i ] –0.03 ± ± ± ± ± ± ii / Ti i ] –0.07 ± ± ± ± ± ± ii / Zn i ] 0.03 ± ± ± ± ± ± Notes.
The number in brackets gives the number of lines the abundance ratio is based on. ( a ) Weighted average of the Fe i - and Fe ii -based abundances. ( b ) Corrected for HFS e ff ects. Appendix C: Comparison with literature results for16 Cyg AB
The detailed abundance pattern of the components in the well-studied binary system 16 Cyg AB has recently been investi-gated with high precision by a number of studies. These arethe two brightest solar analogues in the
Kepler field. Note- worthy is the fact that the primary is slightly more metal-richthan the secondary. The following results were obtained asa function of increasing significance level: ∆ [Fe / H] (A–B) =+ ± + ± + ± ff erence in metal content is unclear, but might be relatedto planetary formation and possibly subsequent engulfment. A Article number, page 21 of 23 & A proofs: manuscript no. ms
Jupiter-mass planet is actually known to orbit 16 Cyg B along avery eccentric ( e ∼ ff erential analysis of 16 Cyg A with respect to 16 Cyg B. Theparameters of 16 Cyg B in Table 1 were adopted. Our analysisprocedures strictly follow those employed relative to solar forthe other targets. Such an approach for binary components withclosely similar parameters is expected to increase the precisionof the results and, consequently, to better reveal subtle chemi-cal di ff erences (e.g. Ramírez et al. 2015). We indeed find that itleads to a better precision with respect to the solar analysis. Forinstance, the line-to-line scatter in the iron abundances is dividedby a factor of about two (here a mere ∼ Table C.1.
Abundance results for 16 Cyg A with respect to 16 Cyg B.
KIC 12069424 (16 Cyg A) ∆ T e ff [K] + ± ∆ log g [cgs] a –0.065 ± ∆ ξ [km s − ] + ± N ∆ [X / H] ∆ [X / Fe]Fe b + + ± i + ± ± i + ± + ± i + ± ± i + ± + ± i + ± + ± i + ± + ± ii c + ± ± i + ± ± i c + ± + ± i + ± ± i c + ± ± i c + ± ± i + ± + ± i c + ± + ± i + ± ± i + ± + ± ii + ± ± ii + ± + ± ii + ± ± ∆ [Y ii / Mg i ] + ± ∆ [Y ii / Al i ] –0.008 ± ∆ [Y ii / Si i ] –0.011 ± ∆ [Y ii / Ti i ] + ± ∆ [Y ii / Zn i ] –0.002 ± ∆ [Sr i / Mg i ] + ± ∆ [Sr i / Al i ] –0.005 ± ∆ [Sr i / Si i ] –0.008 ± ∆ [Sr i / Ti i ] + ± ∆ [Sr i / Zn i ] + ± Notes. N is the number of lines the abundance ratio is based on. ( a ) Given by the seismic values (see Table 2). ( b ) Weighted average of theFe i - and Fe ii -based abundances. ( c ) Corrected for HFS e ff ects. With respect to N17 and Tucci Maia et al. (2019), we obtain aslightly cooler T e ff scale by ∼
20 K on average. Figure C.1 showsa comparison between the abundance patterns of the two compo-nents. As can be seen, there is an overall remarkable agreementwith the literature data, with average di ff erences at the 0.01-dexlevel. On an element-to-element basis, the di ff erences are sys- tematically below 0.05 dex. Considering the weighted (by theinverse variance) average of all the abundances with respect tohydrogen, [X / H], we obtain (cid:104) ∆ [X / H] (cid:105) (A–B) = + ± σ signifi-cance level) and supports the slight enhancement in metals of theprimary. This conclusion is strengthened by the fact that ∆ [X / H](A–B) is systematically positive for the 20 elements (Fig. C.1).The stellar parameters of the two components are so close that,e.g., di ff erential di ff usion e ff ects are not expected to modify thisconclusion (Deal et al. 2015). The detection of such a smallmetallicity di ff erence between the two components indicates thata precision at the 0.02-0.03 dex level is realistically achieved forthe abundances of the solar analogues in our sample (once again,it does not pertain to stars outside of this category). It is an in-teresting conclusion in the context of large-scale spectroscopicsurveys, as it shows that such subtle abundance di ff erences canbe revealed even with data of limited quality — especially re-solving power — provided that adequate analysis strategies areimplemented. However, we are unable to find evidence for theincrease of ∆ [X / H] (A–B) as a function of T c reported by TucciMaia et al. (2019) and, to a lesser extent, by Laws & Gonzalez(2001) and N17. In contrast, the ∆ [X / H] (A–B)- T c behaviour wefind is completely flat (–0.21 ± × − dex K − ). This mightbe ascribed to our lower data quality (but see, e.g., Ramírez et al.2011). Article number, page 22 of 23. Morel et al.: Abundance-age relations in the
Kepler
LEGACY sample
Fig. C.1.
Abundance patterns with respect to hydrogen, [X / H], and di ff erences with respect to literature values (this study minus literature), δ [X / H], as a function of T c . The left and right panels show the comparison with N17 and Tucci Maia et al. (2019), respectively. We ignored themolecular-based C abundances of Tucci Maia et al. (2019). The results for 16 Cyg A, 16 Cyg B, and 16 Cyg A with respect to 16 Cyg B are shownin the top, middle and bottom panels, respectively. Our [X / H] results and those in the literature are shown as filled and open circles, respectively.A dotted, horizontal line is drawn at our [Fe / H] value. The solid lines show the weighted, linear fit of [X / H] as a function of T c . The fits obtainedin the literature for the full set of elements are overplotted as dashed lines (the trends for 16 Cyg AB with respect to the Sun are not available inTucci Maia et al. 2019). The average δ [X / H] values are given in the relevant panels (the number of elements in common is indicated in brackets).To guide the eye, a dotted line is drawn at δ [X / H] ==