A large sample of calibration stars for Gaia: log g from Kepler and CoRoT
O. L. Creevey, F. Thévenin, S. Basu, W. J. Chaplin, L. Bigot, Y. Elsworth, D. Huber, M. J. P. F. G. Monteiro, A. Serenelli
aa r X i v : . [ a s t r o - ph . GA ] A p r Mon. Not. R. Astron. Soc. , 1– ?? (2002) Printed 4 June 2018 (MN L A TEX style file v2.2)
A large sample of calibration stars for Gaia: log g from Kepler and CoRoT fields
O. L. Creevey ⋆ , F. Th´evenin , S. Basu , W. J. Chaplin , L. Bigot , Y. Elsworth ,D. Huber , M. J. P. F. G. Monteiro and A. Serenelli Laboratoire Lagrange, CNRS, Universit´e de Nice Sophia-Antipolis, Nice, 06300, France. Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101, USA. School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK. NASA Ames Research Center, Moffett Field, CA 94035, USA Centro de Astrof´ısica and Faculdade de Ciˆencias, Universidade do Porto, 4150-762 Porto, Portugal Institute of Space Sciences (CSIC-IEEC), Campus UAB, Facultad de Ciencias, Torre C5, parell 2, Bellaterra, Spain.
ABSTRACT
Asteroseismic data can be used to determine stellar surface gravities with preci-sions of < .
05 dex by using the global seismic quantities h ∆ ν i and ν max along withstandard atmospheric data such as T eff and metallicity. Surface gravity is also one ofthe four stellar properties to be derived by automatic analyses for 1 billion stars fromGaia data (workpackage GSP_Phot ). In this paper we explore seismic data from mainsequence F, G, K stars ( solar-like stars ) observed by the
Kepler spacecraft as a poten-tial calibration source for the methods that Gaia will use for object characterisation(log g ). We calculate log g for some bright nearby stars for which radii and masses areknown (e.g. from interferometry or binaries), and using their global seismic quantitiesin a grid-based method, we determine an asteroseismic log g to within 0 .
01 dex of thedirect calculation, thus validating the accuracy of our method. We also find that errorsin adopted atmospheric parameters (mainly [Fe/H]) can, however, cause systematicerrors on the order of 0.02 dex. We then apply our method to a list of 40 stars to deliverprecise values of surface gravity, i.e. uncertainties on the order of 0.02 dex, and we findagreement with recent literature values. Finally, we explore the typical precision thatwe expect in a sample of 400+
Kepler stars which have their global seismic quantitiesmeasured. We find a mean uncertainty (precision) on the order of better than 0.02 dexin log g over the full explored range 3 . < log g < .
6, with the mean value varyingonly with stellar magnitude (0 . − .
02 dex). We study sources of systematic errorsin log g and find possible biases on the order of 0.04 dex, independent of log g andmagnitude, which accounts for errors in the T eff and [Fe/H] measurements, as wellas from using a different grid-based method. We conclude that Kepler stars provide awealth of reliable information that can help to calibrate methods that Gaia will use, inparticular, for source characterisation with
GSP_Phot where excellent precision (smalluncertainties) and accuracy in log g is obtained from seismic data. Key words: asteroseismology – stars: fundamental parameters – stars: late-type –surveys: Gaia – surveys:
Kepler – Galaxy: fundamental parameters –
Large-scale surveys provide a necessary homogenous set ofdata for addressing key scientific questions. Their science-driven objectives naturally determine the type of observa-tions that will be collected. However, to fully exploit the ⋆ E-mail:[email protected] survey, complementary data, either of the same type butmeasured with a different instrument or of a different ob-servable, needs to be obtained. Combining data from severallarge-scale surveys can only result in the best exploitationof both types of data. c (cid:13) O. L. Creevey et al.
The ESA Gaia mission is due to launch in Autumn2013. Its primary objective is to perform a 6-D mapping ofthe Galaxy (3 positional and 3 velocity data) by observingover 1 billion stars down to a magnitude of V = 20. Themission will yield distances to these stars, and for about20/100 million stars distances with precisions of less than1%/10% will be obtained.Gaia will obtain its astrometry by using broad band G photometry . The spacecraft is also equipped with aspectrophotometer comprising both a blue and a red prismBP/RP, delivering colour information. A spectrometer willbe used to determine the radial velocities of objects as faras G = 17 (typical precisions range from 1–20 kms − ),and for the brighter stars ( G <
11) high resolution spec-tra (R ∼ GSP_Phot whose objectives are to obtain stel-lar properties for 1 billion single stars by using the G bandphotometry, the parallax π , and the spectrophotometric in-formation BP/RP (Bailer-Jones 2010). The stellar proper-ties that will be derived are effective temperature T eff , sur-face gravity log g , and metallicity [Fe/H], and also extinc-tion A G in the astrometric G band to each of the stars.Liu et al. (2012) discuss several different methods that weredeveloped to determine these parameters using Gaia dataand we refer to this paper and references within for de-tails. In brief, they discuss the reliability of determiningthe four parameters by using simulations, and in particular,they conclude that they expect typical precisions in log g on the order of 0.1 - 0.2 dex for main sequence late-typestars, with mean absolute residuals (true value minus in-ferred value from simulations) no less than 0.1 dex for starsof all magnitudes, see Figure 14 and 15 of Liu et al. (2012).We note that the stellar properties derived by GSP_Phot willbe used as initial input parameters for the workpackage de-voted to detailed spectroscopic analysis of the brighter tar-gets
GSP_Spec (Recio-Blanco et al. 2006; Bijaoui et al. 2010;Kordopatis et al. 2011; Worley et al. 2012) using the RadialVelocity Spectrometer data (Katz 2005). Spectroscopic de-terminations of log g , T eff and [Fe/H] in general can havelarge correlation errors, and if log g is well constrained, T eff ,[Fe/H] and chemical abundances can be derived much moreprecisely.The different algorithms for GSP_Phot discussed byLiu et al. (2012) used to determine the stellar properties inan automatic way have naturally been tested on syntheticdata. However, to ensure the validity of the stellar proper-ties, a set of 40 bright benchmark stars have been compiledand work is still currently underway to derive stellar proper-ties for all of these in the most precise, homogenous manner(e.g. Heiter, et al. in prep.). Unfortunately, these benchmarkstars will be too bright for Gaia, and so a list of about 500primary reference stars has also been compiled. The idea isto use precise ground-based data and the most up-to-datemodels (known from working with the benchmark stars) todetermine their stellar properties as accurately (correct) and For stars fainter than V ∼
17, the radial velocities will not beavailable. The G photometric scale is similar to V . Figure 1.
High SNR power/frequency spectrum of a
Kepler solar-like star KIC 6603624. Some individual frequencies ν aredenoted with their degree l and radial order n in parenthesis(Appourchaux et al. 2012). The reference radial order n usuallycorresponds to orders similar to 20 for these stars. We also showthe individual large and small frequency separations, ∆ ν ( l, n ) and δν ( l, n ) respectively, for three cases, and the approximate fre-quency corresponding to the maximum power in the spectrum, ν max . For much lower SNR spectra, the mean value h ∆ ν i of thelarge frequency separations and ν max can usually be determinedeven if individual frequencies ν can not be resolved. precisely (small uncertainties) as possible. These primaryreference stars will be observed by Gaia and thus will serveas a set of calibration stars. A third list of secondary ref-erence stars has also been compiled. These consist of about5000 fainter targets.In the last decade or so, much progress in the fieldof asteroseismology has been made, especially for stars ex-hibiting Sun-like oscillations. These stars have deep outerconvective envelopes where stochastic turbulence gives riseto a broad spectrum of resonant oscillation modes (e.g.Ulrich 1970, Leibacher & Stein 1971, Brown & Gilliland1994, Salabert et al. 2002, Bouchy & Carrier 2002). Thepower spectrum of such stars can be characterised by someglobal seismic quantities; h ∆ ν i , ν max , and h δν i . The quantity h ∆ ν i is the mean value of the large frequency separations ∆ ν l,n = ν l,n − ν l,n − where ν l,n is a resonant oscillationfrequency with degree l and radial order n , ν max is the fre-quency corresponding to the maximum of the bell-shapedfrequency spectrum, and h δν i is the mean value of the smallfrequency separations δν l,n = ν l,n − ν l +2 ,n − . Figure 1 showsthe power (frequency) spectrum of a solar-like star, observedby the Kepler spacecraft, depicting these quantities.Even when individual frequencies can not be determinedfrom the frequency spectra both h ∆ ν i and ν max can stillbe extracted quite robustely. Many methods have been de-veloped to do this using Kepler -like data (Bonanno et al.2008; Huber et al. 2009; Mosser & Appourchaux 2009;Hekker et al. 2010; Campante et al. 2010; Mathur et al.2010b; Karoff et al. 2010) and these have been compared inVerner et al. (2011) (see references therein). The global seis-mic quantities have been shown to scale with stellar param-eters such as mass, radius, and T eff e.g. Brown & Gilliland(1994); Bedding & Kjeldsen (2003); Stello et al. (2008);Huber et al. (2011); Bedding (2011); Miglio et al. (2012);Silva Aguirre et al. (2012). By comparing the theoreticalseismic quantities with the observed ones over a large grid of c (cid:13) , 1– ?? large seismic sample of calibration stars for Gaia stellar models, very precise determinations of log g ( < . < Kepler field ofview, ∼
100 square-degrees, centered on galactic coordinates76.32 ◦ , +13.5 ◦ . Kepler is a NASA mission dedicated to char-acterising planet-habitability (Borucki et al. 2010). It ob-tains photometric data of approximately 150,000 stars witha typical cadence of 30 minutes. However, a subset of stars(less than 1000 every month) acquire high-cadence data witha point every 1 minute. This is sufficient to detect andcharacterise Sun-like oscillations in many stars. Verner et al.(2011) and Chaplin et al. (2011) recently showed the detec-tions of the global seismic quantities for a sample of 600 F,G, K main sequence and subgiant (V/IV) stars with typicalmagnitudes 7 < V <
12, while both CoRoT and
Kepler haveboth shown their capabilities of detecting these same seismicquantities in 1000s of red giants, e.g. Hekker et al. (2009);Kallinger et al. (2010); Bedding et al. (2010); Baudin et al.(2011); Mosser et al. (2012); Stello et al. (2013).With the detection of the global seismic quantities inhundreds of main sequence stars (and 1000s of giants), the
Kepler field is very promising for helping to calibrate Gaia
GSP_Phot methods. In particular, the global seismic quan-tities deliver one of the four properties to be extracted byautomatic analysis, namely log g . Gai et al. (2011) studiedthe distribution of errors for a sample of simulated stars us-ing seismic data and a grid-based method based on stellarevolution models. They concluded that log g derived fromseismic properties ( “seismic log g ” ) is almost fully indepen-dent of the input physics in the stellar evolution modelsthat are used. More recently, Morel & Miglio (2012) com-pared classical determinations of log g to those derived alonefrom a scaling relation (see Eq.[2]), and concluded that themean differences between the various methods used is ∼ g . While some studies have focussed on comparingseismic radii or masses with alternative determinations, forexample, Bruntt et al. (2010), no study has been done fo-cussing on both the accuracy and precision of a seismic log g using one or more grid-based methods for stars with inde-pendently measured radii and masses. The accuracy andprecision in log g for these bright stars has also not beentested while considering precisions in data such as those ob-tained by Kepler . Such a study could validate the use ofseismic data as a calibration source for Gaia. For the rest ofthis work the term precision refers exclusively to the deriveduncertainty in log g , while accuracy refers to how true thevalue is.With these issues in mind, the objectives of this paperare to (i) test the accuracy and precision of a seismic log g from a grid-based method using bright nearby targets forwhich radii and masses have been measured (Sect. 3), (ii)determine log g for an extended list of stars whose globalseismic properties and atmospheric parameters are availablein the literature using the validated method (Sect. 4), and(iii) study the distribution of log g and their uncertainties of http://kepler.nasa.gov Table 1.
Summary of the frequently used notation in this study.Notation Definition g gravitylog g logarithm of gT eff effective temperature h ∆ ν i mean large frequency separation ν max frequency of maximum power[Fe/H] metallicity R stellar radius M stellar mass σ uncertainty in log gs systematic error in log gA G extinction in the Gaia G bandr stellar magnitude in the SDDS system over 400 F, G, K V/IV Kepler stars derived by a grid-basedmethod while concluding on realistic uncertainties (preci-sions) and possibles sources of systematic errors for this po-tential sample of Gaia calibration stars (Sect. 5). In thiswork, our analysis is restricted to stars with log g > g . log g In this section we summarise the various methods that areused to determine the surface gravity (or the logarithm ofthis log g ) of a star. Comparing each of these methods di-rectly would be the ideal approach for unveiling shortcom-ings in our models (systematic errors) and reducing uncer-tainties by decoupling stellar parameters. log g from independentdeterminations of mass and radius The most direct method of determining log g involves mea-suring the mass M and radius R of a star in an independentmanner. Surface gravity g is calculated using Newton’s Lawof Gravitation: g = GM/R where G is the gravitationalconstant. For detached eclipsing spectroscopic binaries, both M and R can be directly measured by combining photometric andradial velocity time series (Ribas et al. 2005; Creevey et al.2005, 2011; He lminiak et al. 2012). The orbital solution issensitive to the mass ratio and the individual M sin i ofboth components, where i is the inclination. The photomet-ric time series displays eclipses (when the orbital plane hasa high enough inclination) that are sensitive to i and therelative R . Once i is derived, then the individual M aresolved. Kepler’s Law relates the orbital period of the systemΠ, the system’s M , and the separation of the components.Π is known from either eclipse timings or observing a full c (cid:13) , 1– ?? O. L. Creevey et al. radial velocity orbit. Once the individual M are known thenΠ scales the system (providing the separation) and thus theindividual R , and g can be then calculated. R is measured by combining the angular diameter θ from in-terferometry with the distance to the star. The distance (orits inverse the parallax) has been made available using datafrom the Hipparcos satellite for stars with V < ∼ T eff with mea-surements of bolometric flux (Silva Aguirre et al. 2012), orfrom calibrated relations using photometry (Kervella et al.2004). Using R and h ∆ ν i a model-independent mass deter-mination can be obtained by using the asteroseismic relationwhich links mean density h ρ i and h ∆ ν ih ∆ ν ih ∆ ν i ⊙ ≈ r ρρ ⊙ = p ( M/M ⊙ ) / ( R/R ⊙ ) (1)where h ∆ ν i ⊙ is the solar value (e.g. Kjeldsen & Bedding1995; Huber et al. 2011). When high signal-to-noise ratio seismic data are available,individual oscillation frequencies (see Fig. 1) can be usedto do detailed modelling, and hence determine M (e.g.Do˘gan et al. 2010; Brand˜ao et al. 2011; Bigot et al. 2011;Metcalfe et al. 2012a; Miglio et al. 2012). When combinedwith an independently measured R , the uncertainties inmass σ ( M ) can reduce to <
3% (Creevey et al. 2007;Bazot et al. 2011; Huber et al. 2012). If an independentlydetermined R is not available, then stellar modelling alsoyields R to between 2 – 5% and M with less precision. How-ever, this method depends on the physics in the interior stel-lar models unlike the methods mentioned above, and usingdifferent input physics may result in different values of mass.Typical uncertainties/accuracies in M does not usually ex-ceed about 5% for bright targets when more constraints areavailable, and this translates to less than a 0.02 dex error inlog g for stars that we consider in this work. As g is sensi-tive to seismic data, then R and M are correlated and log g can de determined with a precision of ∼ .
02 dex with avery slight dependence on the physics in the models, e.g.Metcalfe et al. (2010). log g The surface gravity of a star is usually derivedfrom an atmospheric analysis with spectroscopic data(e.g. Th´evenin & Jasniewicz 1992; Bruntt et al. 2010;Lebzelter et al. 2012; Sousa et al. 2012). There are two usualapproaches for deriving atmospheric parameters (log g , T eff ,and [Fe/H]). The first approach is based on directly com-paring a library of synthetic spectra with the observed one,usually in the form of a best-fitting approach. A shortcom-ing of this method is that combinations of parameters can produce similar synthetic spectra so that many correlationsbetween the derived parameters exist. The more classicalmethod for determining atmospheric parameters relies onmeasuring the equivalent widths of iron lines (or other chem-ical species). This method assumes local thermodynamicequilibrium (LTE) and requires model atmospheres. Once T eff is determined (by requiring that the final line abun-dance is independent of the excitation potential or for starswith T eff > g is the only parameter controlling the ionisation bal-ance of a chemical element in the photospheric layers, whichacts on the recombination frequency of electrons and ions. g is then determined by requiring a balance between differ-ent ionized lines e.g. Fe-I and Fe-II. Spectroscopically de-termined log g can have large systematic errors especiallyfor more metal-poor stars where NLTE effects must betaken into account. This can change log g by ∼ log g derived from Hipparcos data An alternative method for determining log g relies on know-ing the distance d (in pc) to the star from astrometry, itsbolometric flux (combining these gives the luminosity of astar) and T eff . Then substituting M and log g for R in Ste-fan’s Law, one obtains the following log g = − . M +0 . V +BC V ) − d +4 log T eff , where V is the de-reddened V magnitude and BC V the bolometric correctionin V , and M (in M ⊙ ) a fixed value that is estimated fromevolution models, e.g. Th´evenin et al. 2001; Barbuy et al.2003. This Hipparcos log g is often used as a fixed parameterfor abundance analyses of stars. Typical uncertainties are noless than 0.08 dex where especially the error in M is large,and errors from d and T eff are not insignificant. We notethat Gaia will deliver unmistakeably accurate distances formuch fainter stars and these will provide a much improvedlog g using this method for individual stars. log g from evolutionary tracks When H-R diagram constraints are available ( T eff , L , metal-licity) then stellar evolution tracks can be used to pro-vide estimates of some of the other parameters of the star,e.g. mass, radius, and age. While correlations exist betweenmany parameters, e.g. mass and age, these correlations alsoallow us to derive certain information with better precision,e.g. mass and radius gives g . Exploring a range of modelsthat pass through the error box thus allows us to limit thepossible range in log g (e.g. Creevey et al. 2012b). log g Apart from performing detailed modelling using asteroseis-mic data (Sect. 2.1.3), one can rely on grids of stellar modelsto estimate stellar properties such as mass and radius withprecisions on the order of 2–12%. However, because astero-seismic data are extremely sensitive to the ratio of these twoparameters, then very precise determinations of log g and h ρ i can be obtained in an almost model-independent man-ner (e.g. Gai et al. 2011, in Sect. 5.3 we address this issue). c (cid:13) , 1– ?? large seismic sample of calibration stars for Gaia Such a grid-based asteroseismic log g can be obtained in thefollowing manner: a large grid of stellar models that spansa wide range of mass, age, and metallicity is constructed.Each model in the grid has a corresponding set of theoret-ical observables, such as log g , T eff , individual frequencies ν n,l , h ∆ ν i and ν max . A scaling relation is used to obtain ν max : ν max ν max , ⊙ ≈ ( M/M ⊙ )( R/R ⊙ ) p T eff /T eff , ⊙ (2)where ν max , ⊙ is the solar value (e.g. Vandakurov 1968;Tassoul 1980; Kjeldsen & Bedding 1995). The quantity h ∆ ν i can be obtained either from Eq. 1 or calculated directlyfrom the oscillation frequencies derived from the structuremodel. Differences, however, on the order of 2% in h ∆ ν i may be found by adopting one or the other method (e.g.,White et al. 2011, see also Mosser et al. 2013). A set of in-put observed data, e.g. {h ∆ ν i , T eff , and [Fe/H] } , is com-pared to the theoretical one, and the models that give thebest match to the data are selected and the value of log g and its uncertainty is derived using these best selected mod-els. Such a precise value for this seismic log g comes, in fact,from the very close relation between ν max and the cut-offfrequency, and recent work has made progress on under-standing this relation (Belkacem et al. 2011). Several seis-mic grid-based approaches for determining stellar proper-ties have been discussed and applied in the recent litera-ture (Stello et al. 2009; Quirion et al. 2010; Basu et al. 2010;Gai et al. 2011; Creevey et al. 2012a; Metcalfe et al. 2012a;Silva Aguirre et al. 2012), and for the rest of this paper weuse the term seismic log g to refer specifically to the grid-based method for determining log g . log g .3.1 Observations and direct determination of log g In order to test the reliability of an asteroseismically deter-mined log g , the most correct method is to compare it tolog g derived from mass and radius measurements of stars ineclipsing binaries (Sect. 2.1.1), apart from the Sun. Unfortu-nately the number of stars whose masses and radii are knownfrom binaries, where seismic data are also available, is quitelimited. For this sample of stars we have α Cen A and B,and Procyon. Following this, we rely on the combination ofasteroseismology and interferometry to determine log g , anduse the scaling relation which links density to the observedproperties of h ∆ ν i and R to provide an independent M mea-surement (Sect. 2.1.2). However, since this scaling relation isused explicitly in the grid-based method, we have opted toomit stars where this method provides the mass, except forthe solar twin 18 Sco, which was included because of its sim-ilarity to the Sun. To complete the list of well-characterisedstars we have then chosen some targets for which an interfer-ometric R has been measured and detailed seismic modellinghas been conducted to determine the star’s M by several au-thors (Sect. 2.1.3). The stars which fall into this group areHD 49933 and β Hydri. The seven stars are listed in orderof log g in Table 2 along with h ∆ ν i , ν max , T eff , [Fe/H], M ,and R . When several literature values are available these are also listed. The final column in the table gives the direct value of log g as derived from M and R . For HD 49933 and β Hydri we adopt the weighted mean values of log g whichare 4.214 and 3.958 dex respectively (see Table 2) and theseare summarized in column 2 of Table 3. For the rest of thepaper we refer to these determinations of log g as the direct determinations. log g We use the grid-based method, RadEx10, to deter-mine an asteroseismic value of log g (Creevey et al.2012a). The grid was constructed using the ASTECstellar evolution code (Christensen-Dalsgaard 2008) withthe following configuration: the EFF equation of state(Eggleton et al. 1973) without Coulomb corrections, theOPAL opacities (Iglesias & Rogers 1996) supplemented byKurucz opacities at low temperatures, solar mixture fromGrevesse & Noels (1993), and nuclear reaction rates fromBahcall & Pinsonneault (1992). Convection in the outer con-vective envelope is described by the mixing-length theory ofB¨ohm-Vitense (1958) and this is characterised by a variableparameter α MLT (where l = α MLT H p , l is the mixing-lengthand H p the pressure scale height). When a convective coreexists, there is an overshoot layer which is also characterisedby a convecctive core overshoot parameter α ov and this isset to 0.25 (an average of values used recently in the lit-erature). We also ignore diffusion effects, although we notethat for accurate masses and ages this needs to be taken intoaccount.The grid considers models with masses M from 0.75 –2.00 M ⊙ and ages t from ZAMS to subgiant. The initialmetallicity Z i spans 0.003 – 0.030 in steps of ∼ . X i is set to 0.71: this correspondsto an initial He abundance Y i = 0 . − . α MLT = 2 . α MLT to varye.g. Creevey et al. (2012b); Bonaca et al. (2012).To obtain the grid-based model stellar properties, e.g.log g , M , and t , we perturb the set of input observationsby scaling its input error with a number drawn randomlyfrom a normal distribution and add this to the input (ob-served) value. We compare the perturbed observations to themodel ones and select the model that matches best. This isrepeated 1,000 times to yield model distributions of best-matching stellar parameters. In this method, h ∆ ν i is calcu-lated using Eq. 1 and the input observations consist primar-ily of h ∆ ν i , ν max , T eff , and [Fe/H], although other inputs arepossible, for example, L or R . The model distributions arethen fitted to a Gaussian distribution and the fitted stellarproperty and its uncertainty ( σ ) are defined as the centralvalue and the standard deviation σ of the Gaussian fit. Inthis work we consider just the derived value of log g . We determine a seismic log g for the reference stars usingthe method explained above. We consider different setsof input data in order to test the effect of the differentobservations on the accuracy (and precision) of a seismic c (cid:13) , 1– ?? O. L. Creevey et al.
Table 2.
Properties of the reference starsStar h ∆ ν i ν max T eff [Fe/H] R M log g ( µ Hz) (mHz) (K) (dex) (R ⊙ ) (M ⊙ ) (dex) α CenB 161.5 ± a a ± b +0.25 ± b ± c ± d ± ± e
18 Sco 134.4 ± a a ± a ± a ± a ± a ± ± a b ± c ± ± d α CenA 105.6 a a ± b +0.24 ± b ± c ± d ± b HD 49933 85.66 ± a a ± b –0.35 ± b b b ± b ± c ± d –0.38 d ± d ± d ± ± e ± e ± ± e ± e ± ± a b ± c –0.05 ± d ± e ± f ± β Hydri 57.24 ± a a ± b –0.10 ± c ± b ± b ± d ± e ± +0 . , f − . ± ± g –0.03 ± +0.07 g ± g ± g ± g References: a Kjeldsen et al. (2005), b Porto de Mello et al. (2008), c Kervella et al. (2003), d Pourbaix et al. (2002), e Bigot et al.(2006), a Bazot et al. (2011), a Taking the average of Table 3 from Toutain & Froehlich (1992), b Kjeldsen & Bedding (1995), c Grevesse & Sauval (1998), d ⊙ =1.98919e30 kg, 1R ⊙ =6.9599e8 km, G = 6 . e
11 m kg − s − (NIST database), a Bouchy & Carrier (2002), b Quirion et al. (2010), a Using the l = 0 modes with Height/Noise > b Kallinger et al. (2010) log g is the mean and standard deviation of the values cited from their Table 1, c Gruberbauer et al.(2009), d Bigot et al. (2011), e Creevey & Bazot (2011), a Eggenberger et al. (2004), b Marti´c et al. (2004), c Fuhrmann et al. (1997), d Allende Prieto et al. (2002), e Kervella et al. (2004), f Girard et al. (2000), a Bedding et al. (2007), b North et al. (2007), c Bruntt et al. (2010), d Brand˜ao et al. (2011), e Do˘gan et al. (2010), f Fernandes & Monteiro (2003), g da Silva et al. (2006) log g :(S1) {h ∆ ν i , ν max , T eff , [Fe/H] } ,(S2) {h ∆ ν i , ν max , T eff } , and(S3) {h ∆ ν i , T eff } ,For the potential sample of Gaia calibration stars,[Fe/H] will not always be available, and in some cases, ν max is difficult to determine for very low S/N detections.The observational errors in our sample are very smalldue to the brightness (proximity) of the star, so we alsoderive an asteroseismic log g while considering observationalerrors that we expect for Kepler stars (see Verner et al. 2011and Figure 5). We consider three types of observational er-rors while repeating the exercise:(E1) the true measurement errors from the literature,(E2) typically “good” errors expected for these stars, i.e. σ ( h ∆ ν i ) = 1.1 µ Hz, σ ( ν max ) = 5%, σ ( T eff ) = 70 K, and σ ([Fe / H]) = 0 .
08 dex (see Sect. 5.1), and(E3) “not-so-good” measurement errors, primarily consider-ing the fainter targets (V ∼ σ ( h ∆ ν i ) = 2.0 µ Hz, σ ( ν max ) = 8%, σ ( T eff ) = 110 K, and σ ([Fe / H]) = 0 .
12 dex. log g In Figure 2 we compare the grid-based seismic log g withthe direct log g for the seven stars. Each star is representedby a point on the abscissa, and the y-axis shows (seismic -direct) value of log g . There are three panels which repre-sent the results using the three sets of input data. We alsoshow for each star in each panel three results; in the bot- tom left corner these are marked by ’E1’, ’E2’, and ’E3’,and they represent the results using the different errors inthe observations. The black dotted lines represent seismicminus direct log g =
0, and the grey dotted lines indicate ± g is generally estimated to within 0.01 dex in accuracyand with a precision of 0.015 dex. This result clearly showsthe validity of the global seismic quantities and atmosphericparameters for providing accurate values of log g . It can alsobe noted that the precision in log g typically decreases as (1)the observational errors increase (from E2 – E3), and (2) theinformation content decreases (S1 – S2 – S3, for example).One noticeable result from Figure 2 is the systematicoffset in the derivation of log g for HD 49933 when we use[Fe/H] as input (S1). This could be due to an incorrect[Fe/H] or T eff , an error in the adopted direct log g or a short-coming of the grid of models. This star is known to be activeand Mosser et al. (2005) found clear evidence of spot signa-tures in line bisectors. More recently Garc´ıa et al. (2010) andSalabert et al. (2011) found frequency and amplitude vari-ations similar to those in the Sun, evidence of the presenceof a stellar cycle. Garc´ıa et al. (2010) derived an S-index ofabout 0.3 corresponding to a very active star, even thoughno detection of a magnetic field has yet been confirmed (P.Petit, private communication 2012). Fabbian et al. (2012)showed that the magnetic effects in stellar models affect thedetermination of the metallicity and this, in turn, will affectthe determination of other stellar parameters such as mass. c (cid:13) , 1– ?? large seismic sample of calibration stars for Gaia Figure 2.
Accuracy of method.
Seismic-minus-direct log g for the seven sample stars while considering different subsets of inputobservations (different panels, S1, S2, S3) and different observational errors (E1, E2, E3). See Sect. 3.3 for details. In their work, they considered large magnetic field strengthsof several hundreds of Gauss.Since no evidence of such a large magnetic field has yetbeen found for HD 49933 we propose another explanation.We consider the effect of the confirmed presence of spotson the effective temperature. By considering a spot area ofabout 20 percent of the stellar surface we found that the realeffective temperature (that of the non spotted surface of thestar) should increase by about 300 K with respect to thenon-spotted star. Adding 300 K to T eff results in a seismiclog g that increases by 0 .
011 dex for S1 (considering errorsE1 and E2), which makes the new value consistent with thedirect value within error bars. The increase of 0.011 dexcorresponds to a relative increase of 3.7 σ and 1.2 σ for botherror types E1 and E2. If we add this same 300 K for cases S2and S3, we also find an increase in log g , but smaller (0.009and 0.006 dex), corresponding to a relative increase of 0.6 σ and 0.4 σ , respectively. For all of the calculations we have assumed that the inputobservations are correct (accurate). While this is certainly
Table 3. log g derived by RadEx10 for the reference stars usingthe true measurement errors. ∆ g = log g − log g direct .Star log g log g direct ∆ g ∆ g (dex) (dex) (dex) ( σ ) α Cen B 4.527 ± ± ± α Cen A 4.312 ± ± ± β Hydri 3.957 ± more true for brighter nearby targets where high SNR datacan be obtained, the same cannot be said for fainter stars.In particular with spectroscopic data, the determination of T eff and [Fe/H] are correlated and depend on the analysismethods used and the different model atmospheres (see e.g.Creevey et al. 2012a; Lebzelter et al. 2012). Additionally formany stars a photometric temperature may be the onlyavailable one and while these estimates are very good, sys-tematic errors are still unavoidable (Casagrande et al. 2010; c (cid:13) , 1– ?? O. L. Creevey et al.
Figure 3.
Seismic-minus-direct values of log g for β Hydri whenthe observations with systematic errors included for cases 2 and3 are analysed by the seismic method. See Sect. 3.5.
Boyajian et al. 2012), in particular due to unknown redden-ning. Larger photometric errors also lead to larger errorson the temperatures. This is not only a problem for fainterstars. For example, for β Hydri we found two determinationsof T eff — an interferometric one and a spectroscopic one. Forthe spectroscopic T eff , the corresponding fitted metallicityfrom the atmospheric analysis will correlate with it.To study the effect of systematic errors in the obser-vations, we repeated our analysis for β Hydri while usingthree sets of input data that change only in T eff and [Fe/H].The first set (1) uses the interferometric value of T eff fromNorth et al. (2007) and their [Fe/H] (5872, –0.10), which weconsider as the most correct, the second set (2) uses (5964,–0.10), and the third set (3) uses (5964, –0.03) as given byda Silva et al. (2006). The results for S1 and S2 are shownin Figure 3. The lower panel shows that for a systematicerror in both T eff and [Fe/H] the accuracy decreases. Thetop panel shows that when we only consider the T eff infor-mation we get a smaller increase in the offset than when weconsider both T eff and [Fe/H]. One way of interpreting thisresult is by considering that in S2 there is much more weightassigned to the seismic data than the atmospheric data, andso an incorrect atmospheric parameter should not influencethe final result as much as in case S1 where the atmosphericparameters have more weight. In the latter case, an incorrect T eff with the correct [Fe/H] will necessarily shift the masseither up or down (in terms of the H-R diagram), and resultin a more displaced log g . However, in both cases, we seethat the offset does not exceed 0.02 dex. For determining aseismic radius and mass, however, a systematic error in theinput observations has a much more profound effect on theoffset. In this case biases on the order of up to 6% and 20%in radius and mass can be obtained (Creevey & Th´evenin2012). It must also be noted that a systematic error in theatmospheric parameters is going to have a much larger neg-ative effect when we use only the global seismic quantitiesinstead of performing a detailed seismic analysis with indi-vidual frequencies, where the latter have much more weightin the fitting process. log g from the globalseismic quantities for the reference stars We summarize log g for the sample stars in Table 3 derivedby RadEx10 using h ∆ ν i , ν max , T eff , and [Fe/H], and the true observational errors. We highlight the excellent agree-ment between our seismically determined parameters, andthose obtained by direct mass and radius estimates. log g ismatched to within ∼ g to within 1 σ . log g FOR ANEXTENDED LIST OF STARS
We apply our grid-based method to an extended list ofstars with measured global seismic quantities and atmo-spheric parameters. Table 4 lists the star name along withthe other measured parameters that are used as the input forour method. The first part of the table comprises primarilybright stars whose oscillation properties have been measuredeither from ground-based or spaced-based instrumentation(see references given in the table). For most of these starsno errors are cited for h ∆ ν i and ν max . The second part ofthe table lists a set of 22 solar-type stars observed by the Kepler spacecraft and studied in Mathur et al. (2012). Wehave taken the seismic and atmospheric data directly fromthis paper. To conduct a homogenous analysis of these starswe adopted a 1.1 µ Hz error on h ∆ ν i for all of the stars,and a 5% error on ν max , typical of what has been found forthe large sample of Kepler stars (see Huber et al. 2011 andSect. 5.1).Table 5 lists the derived value of log g and 2 σ uncertain-ties (to allow for round-up error) for each of the stars usingRadEx10. We show two values of log g ; the first is obtainedby using the four input constraints {h ∆ ν i , ν max , T eff ,[Fe/H] } and the second log g no[Fe / H] is obtained by omitting [Fe/H]from the analysis.Figure 4 top panel compares the derived values of log g for the Kepler stars with those determined using the individ-ual oscillation frequencies, as given by Mathur et al. (2012),with our 1 σ error bars overplotted. We see that the grid-based method provides log g consistent with those derivedfrom a detailed asteroseismic analysis, although a very smalltrend can be seen. For some of the stars they obtain a fittedinitial He abundance significantly below the accepted pri-mordial value, suggesting that the corresponding fitted massand radius may be slightly biased (grey squares). If we omitthese stars, then we fit a slope of -0.03 ± g values. White et al. (2011)and Mosser et al. (2013) point out that a discrepancy mayexist between the two different theoretical values of h ∆ ν i (using scaling relations or from individual frequencies), andthe two different approaches could also be responsible forthis trend. We see, however, that our values agree generallyto within 0.01 dex. Silva Aguirre et al. (2012) have analysed6 of these stars and the lower panel of Fig. 4 shows a com-parison between their log g values with ours. Fitting the dif-ferences between our results, we obtain a slope of 0.001 ± c (cid:13) , 1– ?? large seismic sample of calibration stars for Gaia Table 4.
Measured seismic and atmospheric properties for an extended list of Sun-like oscillatorsStar Name h ∆ ν i ν max T eff [Fe/H]( µ Hz) ( µ Hz) (K) (dex)HD 10700 169.0 a a ± a -0.10 ± b b ±
80 0.15 ± c c ± d ± s s ± s +0.09 ± s HD 52265 98.3 ± l l ± l ± l HD 61421 55.0 1000 6494 ±
48 0.01 ± e e ±
80 -0.86 ± f f ± g ± h h ± d ± e e ±
80 -0.04 ± i i ±
80 0.32 ± j j ±
80 0.12 ± ± m m ± m -0.15 ± m HD 181420 75.0 n ± n ± o ± o HD 181906 87.5 ± p ± p ± p -0.11 ± p HD 186408 103.1 q q ± q ± q HD 186427 117.2 q q ± q ± q HD 185395 84.0 r r ± r -0.04 r HD 203608 120.4 k k ±
80 -0.74 ± e e ±
80 0.01 ± ± ±
20 6150 ±
70 -0.19 ± ± ±
25 5700 ±
70 0.32 ± ± ±
60 5840 ±
70 0.14 ± ± ±
20 5825 ±
70 0.39 ± ± ±
25 5710 ±
70 0.04 ± ± ±
20 5950 ±
70 -0.11 ± ± ±
20 5895 ±
70 -0.26 ± ± ±
50 5600 ±
70 0.26 ± ± ±
30 5830 ±
70 -0.01 ± ± ±
25 5815 ±
70 0.10 ± ± ±
25 6050 ±
70 -0.52 ± ± ±
140 5340 ±
70 0.38 ± ± ±
40 6000 ±
70 -0.15 ± ± ±
65 5960 ±
125 -0.30 ± ± ±
95 5765 ±
70 -1.19 ± ± ±
10 6300 ±
65 -0.47 ± ± ±
15 5900 ±
70 -0.10 ± ± ±
35 6015 ±
70 -0.21 ± ± ±
20 5705 ±
70 0.34 ± ± ±
15 5930 ±
52 ...KIC 12009504 88.10 ± ±
20 6060 ±
70 -0.09 ± ± ±
30 5950 ±
70 0.02 ± a Teixeira et al. (2009), b Vauclair et al.(2008), c Bouchy & Carrier (2003), d Th´evenin et al. (2005), e Bruntt et al. (2010), f Carrier et al. (2005), g North et al. (2009), h Carrier et al. (2005), i Bouchy et al. (2005), j Carrier & Eggenberger (2006), k Mosser et al. (2008), l Ballot et al. (2011), m Mathur et al. (2010a), n Barban et al. (2009), o Bruntt (2009), p Garc´ıa et al. (2009), q Metcalfe et al. (2012b), r Guzik et al. (2011), s Deheuvels et al. (2010). When h ∆ ν i is not explicitly given it is calculated from the highest amplitude l = 0 modes. Values for the KICstars are taken from Mathur et al. (2012). ± log g FOR A LARGE SAMPLEOF
Kepler
STARS OF CLASSES IV/V
Our primary objective was to test the accuracy of a seismiclog g by using bright nearby targets that have independentmass and radius measurements. We showed in Sect. 3.4 (seeFig. 2) that our accuracy should be on a level of 0.01 dex with a precision of ∼ {h ∆ ν i , ν max , T eff } for the small sample of stars covering the range 3 . < log g < .
6. In this section we investigate the precision in log g for asample of 403 V/IV stars (log g > Kepler spacecraft by employing the same analysis methods.In particular we pay attention to systematic errors by 1)using different sets of observational constraints, 2) compar-ing results using two different methods which incorporatedifferent stellar evolution codes and physics, and 3) we alsoshow the distribution of errors as a function of magnitude c (cid:13) , 1– ?? O. L. Creevey et al.
Table 5.
Derived seismic log g for an extended list ofSun-like oscillators. log g and log g no[Fe / H] refer to using {h ∆ ν i , ν max , T eff ,[Fe/H] } and {h ∆ ν i , ν max , T eff } as the constraintsin the analysis. For a homogenous analysis we adopted 1.1 µ Hzand 5% as the errors on h ∆ ν i and ν max . We list 2 σ uncertaintiesfor log g . Star Name log g log g no[Fe / H] (dex) (dex)HD 10700 4.55 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± and log g and summarize the uncertainties and systematicsas a function of magnitude. During the first 9 months of the
Kepler mission targetsto be monitored with a 1 minute cadence during 1 montheach were selected by the KASC . These stars were chosen Kepler
Asteroseismic Science Consortium, see http://astro.phys.au.dk/KASC/
Figure 4.
Comparison of log g derived by RadEx10 with (top)detailed seismic analyses (Mathur et al. 2012) for 22 Sun-likestars observed by Kepler , and (bottom) six common stars fromSilva Aguirre et al. (2012). based on information available in the
Kepler Input Cata-log , KIC, (Brown et al. 2011) and were expected to exhibitsolar-like oscillations. A total of 588 stars with values oflog g between 3.0 and 4.5 dex were analysed (Garc´ıa et al.2011) and had their global seismic quantities determined(Huber et al. 2011; Verner et al. 2011; Chaplin et al. 2011).In this paper we concentrate on a subset of 403 less-evolvedstars with log g values between 3.75 and 4.50 dex derivedfrom RadEx10, the range for which we have validated ourmethod.The global seismic quantities have been determined us-ing the SYD pipeline as described by Huber et al. (2009),which uses the reference values of h ∆ ν i ⊙ = 135.1 µ Hz and ν max ⊙ = 3,090 µ Hz. To avoid systematic errors (Chaplin.et al. in prep.), we adopted these same values in our grid.The uncertainties on the seismic quantities include a contri-bution from the scatter between different analysis pipelines(Verner et al. 2011, Chaplin et al. in prep.). Figure 5 showsthe cumulative distribution for the errors in the seismic ob-servations, h ∆ ν i (top) and ν max (bottom) in units of µ Hzand %, respectively, for the 403 stars. Our choice for abso-lute and relative errors is for consistency with units used inthe recent literature. We show the cumulative distributionof the errors in order to see the typical errors for 50% and80% of the sample, which justifies the errors that we usedin Sect. 3.3. These are less than 1.1/2.0 µ Hz for h ∆ ν i , andless than 5%/8% for ν max , respectively.The T eff derived by the KIC have been shown not to c (cid:13) , 1– ?? large seismic sample of calibration stars for Gaia Figure 5.
Cumulative distributions of the errors in h ∆ ν i (top)and ν max (bottom) for 403 Kepler stars with derived log g between3.75 and 4.5 dex. Data taken from Huber et al. (2011). be accurate on a star-to-star basis (Molenda- ˙Zakowicz et al.2011), and so the ground-based Kepler support photometry(Brown et al. 2011) was re-analysed by Pinsonneault et al.(2012) to determine more accurate T eff for the ensemble of Kepler stars. These are the temperatures that we adopt forour first analysis and we refer to them as T effPin . In theirwork they consider a mean [Fe/H] = – 0.20 ± Kepler stars to provide analternative determination of T eff . We also adopt their T eff determinations in order to study the effect of biases in tem-perature estimates on the derived value of log g . We referto these temperature estimates as T effirfm . Support spectro-scopic data have also been obtained for 93 of the stars andthe metallicities are presented in Bruntt et al. (2012). Wefurther include these data to study the effects of possible bi-ases arising from lack/inclusion of metallicity information. log g from different sets of observations Using our validated method described in Sect. 3.2, we cal-culated values of log g and their uncertainties for the sam-ple of 403 stars using the set of observations comprising { T effPin , h ∆ ν i , ν max } while adopting a mean [Fe/H] = – 0.20 ± g is shownin Figure 6. Here it can be seen that typical uncertainties inlog g for this set of 403 Kepler stars is below 0.02 dex (thereis one star with an error of 0.05 dex), with a mean value of0.015 dex.In Figure 7 we show the difference in the fitted log g while considering different input observational sets com-pared to the reference set ‘log g ref ’. The subsets are: Set Figure 6.
Distribution of uncertainties in log g σ (log g ) as a func-tion of log g for a sample of 403 Kepler stars, using RadEx10 withobservational constraints comprising { T effPin , h ∆ ν i , ν max } . h ∆ ν i and T eff Pin only and Set 2 whichconsiders h ∆ ν i , ν max , and T effirfm . We note that for all ofthe analyses [Fe/H] was constrained to –0.20 ± >
90% of the models of the grid.Inspecting the top panel of Fig. 7 (Set 1) one can seethat by omitting ν max as an observed quantity can resultin differences of over 0.05 dex for a very small percent-age of the stars, but the absolute difference between thefull set of results is +0.005 dex with an rms of 0.01 dex.Here, we note that several authors have investigated the re-lation between h ∆ ν i and ν max and find tight correlations(Bedding & Kjeldsen 2003; Stello et al. 2009). The uncer-tainties arising from a set of data with less constraints usu-ally increases and we indeed find an increase in the uncer-tainties σ of ∼ .
01 dex. The extra scatter of 0.01 dex istaken care of in the larger σ .Inspecting the lower panel of Fig. 7 (Set 2) one can seethat the T eff derived by using different photometric scalesresults in a mean difference in log g of -0.002 dex (i.e. nosignificant overall effect) with an rms scatter of 0.007 dex.This latter fact implies that we can expect to add just under0.01 dex to the error budget in log g by considering T eff derived from different methods. We found a similar result inSect. 3.5 for β Hydri.The mean value of the derived uncertainties ( σ ) in log g for Set 1 and 2 are 0.023 and 0.015 dex, respectively, whilethose for the reference set are 0.015 dex. The accuracy ofthese log g (if we consider the reference set to be correct) iswithin a precision of 1 σ .Figure 8 compares the derived log g using the refer-ence set of observations, to those with measured T eff and[Fe/H] from Bruntt et al. (2012). The absolute mean resid-ual is 0.002 dex and is highlighted by the dotted grey line.We find that log g can differ by up to 0.02 dex by includ-ing [Fe/H]. This 0.02 dex is also consistent with what wefound in Sect. 3.5 for β Hydri when we considered differentmetallicity constraints.
To investigate the possible source of systematics arisingfrom using a different evolution code and input physics,we determined log g using a second pipeline code, Yale- c (cid:13) , 1– ?? O. L. Creevey et al.
Figure 7.
Differences in log g using subsets of observational con-straints. The reference set comprise ( h ∆ ν i , ν max , T effPin ). Set 1and 2 comprise ( h ∆ ν i , T effPin ) and ( h ∆ ν i , ν max , T eff IR ), respec-tively. For all sets [Fe/H] = –0.20 dex. The ± σ error bars areaverage error bars measured over bins of 0.1 dex correspondingto Set X. See Sect. 5.2 for details. Figure 8.
Differences in the derived seismic log g from RadEx10between adopting [Fe/H] = –0.20 ± g )and adopting [Fe/H] from spectroscopic analyses by Bruntt et al.(2012), log g [Fe / H] . See Sect. 5.2 for details. Bham (Gai et al. 2011; Creevey et al. 2012a). Details of thecode can be found in the cited papers. Here, it suffices toknow that the method is very similar to that of RadEx10,but the evolution code is based on YREC (Demarque et al.2008) in its non-rotating configuration, with the follow-ing specifications: OPAL EOS (Rogers & Nayfonov 2002)and OPAL high-temperature opacities (Iglesias & Rogers1996) supplemented with low-temperature opacitites fromFerguson et al. (2005), and the NACRE nuclear reactionrates (Angulo et al. 1999). Diffusion of helium and heavy-elements were included, unlike RadEx10.Figure 9 shows the difference in the derivedvalue of log g between RadEx10 and Yale-Bham using {h ∆ ν i , ν max , T effPin ,[Fe/H]=–0.2 } . We show the normaliseddifference, i.e. divided by the Yale-Bham errors, which are Figure 9.
The difference between the values of log g obtainedby RadEx10 and Yale-Bham as a function of Radex10 log g andnormalised by the Yale-Bham errors. See Sect. 5.3 for details. Figure 10.
Distribution of derived uncertainties in the seismiclog g from RadEx10 for 403 Kepler
V/IV stars as a function ofSDDS r magnitude. very similar to the RadEx10 errors. As can be seen from thefigure the agreement in log g between the different methodsis within 1 σ or < .
01 dex for 98% of the stars and 2 σ forall stars except one. A small absolute difference of 0.005 isfound with an rms of 0.005 (units of σ ). This is most likelydue to the different physics adopted in the codes. A veryslight systematic trend is present with a slope of -0.005 ± log g as a function of magnitude In Figure 10 we show the uncertainties in log g as a functionof SDDS r magnitude (obtained from the KIC) while usingthe reference set of observations. As the figure shows, preci-sion improves with apparent brightness, where more reliablemeasurements have smaller error bars. We can expect typ-ical uncertainties in log g of less than 0.03 dex for the fullsample with a mean value of < .
02 dex.In Table 6 we summarize for different apparent mag-nitude bins 1) the mean uncertainties h σ i and 2) the mean c (cid:13) , 1– ?? large seismic sample of calibration stars for Gaia Table 6.
Summary of uncertainties σ and systematic errors s inthe seismic log g from the Kepler sample of stars as a function ofmagnitude. We give the mean values hi of each over the samplethat falls into the corresponding magnitude bin. The systematicerrors s have subscripts which refer to possible sources of errorsin the input T eff , [Fe/H], and the code that is used to generatethe stellar models (see Sects. 5.2 and 5.3 for details). In the finalrow we summarize the mean uncertainties and adopted systematicerrors for the full sample.r h σ i h s Teff i h s [Fe / H] i h s code i (mag) (dex) (dex) (dex) (dex)6 < r < < r <
10 0.014 0.004 0.009 0.00410 < r <
11 0.016 0.009 ... 0.00611 < r <
12 0.017 0.009 ... 0.0066 < r <
12 0.015 0.01 0.02 0.01 systematic errors h s i arising from adopting different tem-perature scales h s T eff i , metallicities h s [Fe / H] i and differentpipeline codes h s code i . Note that the sample size for the binwith r < s [Fe / H] for r <
9. In the last rowwe give the mean uncertainty and the adopted systematicerrors over the full sample.
Kepler stars
From the sample of
Kepler data with 3 . < log g < . σ ofbetween 0.01 and 0.02 dex (max 0.03 dex for 99% of stars)when h ∆ ν i , ν max , and T eff are used as the input constraintsfor the pipeline analysis. A typical systematic error s of 0.01dex can be added to account for a possible systematic errorin T eff , such as that obtained by applying different calibra-tion methods to photometry. Similarly, we can add a sys-tematic error of 0.02 dex due to an incorrect or lack of a[Fe/H] measurement. We also showed that using differentmodels and physics in the pipeline analyses yields resultsin log g consistent to within 0.01 dex i.e. almost no modeldependence.Eliminating one of the seismic indices from the set ofdata yields a zero offset in the results for the full sample ofstars and a typical scatter of 0.01 dex, although differencesof over 0.05 dex were found for <
1% of the stars.
The first objective of this study was to investigate if we candetermine log g reliably using global seismic quantities andatmospheric data. We showed that our method is reliable bycomparing our results with values of log g derived from di-rect mass and radius estimates of seven bright nearby stars.We then applied our method to a list of 40 Sun-like starsand derived log g to within 1 σ ( ∼ .
01 dex) of those fromMathur et al. (2012) and Silva Aguirre et al. (2012). We alsoshowed that for a sample of 400+
Kepler stars (6 < r mag <
12) with log g between 3.75 and 4.50 dex typical uncertain-ties σ of less than 0.02 dex can be expected and we estimatedsystematic errors s of no more than 0.04 dex arising fromerrors in T eff and [Fe/H] measurements and using differentcodes.All of the Kepler stars will be observed by the Gaiamission, and for this reason they provide a valid set of cali-bration stars, by constraining log g with precisions and accu-racies much better than spectroscopic or isochrone methodsprovide for the current list of calibration stars for Gaia. Theastrophysical parameters to be determined from Gaia datausing the astrometry, photometry and BP/RP spectropho-tometry are T eff , A G , [Fe/H], and log g . By ensuring that the GSP_Phot methods deliver log g consistent with the seismicvalue will reduce the uncertainties and inaccurcies in theother parameters. Moreover, an independent log g providesan extra constraint for the determination of the atmosphericparameters from GSP_Spec where degeneracies between T eff and log g severely inhibit the precision of atmosphericallyextracted parameters and chemical abundances.While in this paper we concentrated primarily on us-ing Kepler data to determine log g , we note that both theCoRoT and Kepler fields have great potential in other as-pects. For example, Miglio et al. (2012) and Miglio et al.(2013) determine distances, masses, and ages of red gi-ants from the CoRoT fields to constrain galactic evolutionmodels, while Silva Aguirre et al. (2012) develop a methodwhich couples asteroseismic analyses with the infrared fluxmethod to determine stellar parameters including effectivetemperatures and bolometric fluxes (giving angular diame-ters), and hence distances. The distances obtained by bothMiglio et al. (2013) and Silva Aguirre et al. (2012) can becompared directly with a Gaia distance for either calibra-tion of Gaia data or stellar models. By combining data fromCoRoT and
Kepler with those from Gaia we can also deter-mine extremely precise angular diameters by coupling a seis-mically determined radius with the Gaia parallax. Finally,these stars will also be excellent calibrators for the FLAMEworkpackage of Gaia, which aims to determine masses andages for one billion stars.
In this work we explored the use of F, G, K IV/V stars ob-served in the
Kepler field as a possible source of calibrationstars for fundamental stellar parameters from the Gaia mis-sion. Our first objective was to test the reliability/accuracyof a seismically determined log g , and using a sample of sevenbright nearby targets we proved that seismic methods are ac-curate (see Fig. 2) by obtaining results to within 0.01 dexof the currently accepted log g values. We showed, however,that the accuracy of the input atmospheric parameters doesplay a role in the accuracy of the derived parameters. For β Hydri we found that errors in the atmospheric parameters,and in particular [Fe/H], can change log g by 0.02 dex.We then applied our grid-based method RadEx10 toan extended sample of stars from the literature. We de-rived their seismic log g and these are given in Table 5.We showed that a grid-based log g is consistent with thevalues of log g obtained by detailed seismic analysis fromMathur et al. (2012) for 22 stars and in excellent agreement c (cid:13) , 1– ?? O. L. Creevey et al. with the results from Silva Aguirre et al. (2012) for the 6common stars. We find typical precisions in log g of 0.02dex. We finally studied the distribution of errors in log g fromtwo analysis methods for a sample of 403 Kepler stars with3 . < log g < .
5, and we obtained a typical uncertainty ofbetween 0.01 and 0.02 dex in log g for F, G, K IV/V stars(see Table 6). We can add a total of 0.04 dex as a system-atic error which arises from the adopted temperature scales(0.01 dex) and measured metallicities (0.02 dex) as well asthe grids of models used (0.01 dex), which differ in evolu-tion codes and input physics. The precisions in the data areunprecedented, and the systematic errors are much smallerthan those stemming from any other method, especially for7 < V <
12 stars (see e.g. Creevey et al. 2012a Fig 1 andTable 3 which compare spectroscopically derived log g forfive Kepler stars with V ∼ >
500 IV/V stars in the
Kepler field and some in the CoRoTfields that exhibit Sun-like oscillations, there are also 1000’sof red giants in both CoRoT and
Kepler fields with thesesame measured quantities, however, the accuracy of thesestars’ seismic log g is yet to be studied. ACKNOWLEDGEMENTS
OLC thanks Luca Casagrande and Victor Silva Aguirre formaking data available. SB acknowledges NSF grant AST-1105930. AMS is partially supported by the European UnionInternational Reintegration Grant PIRG-GA-2009-247732,the MICINN grant AYA2011-24704, by the ESF EURO-CORES Program EuroGENESIS (MICINN grant EUI2009-04170), by SGR grants of the Generalitat de Catalunya andby the EU-FEDER funds. WJC and YE thank the UKSTFC for grant funding to support asteroseismic research.MJPFGM acknowledges the support through research grant
PTDC/CTE-AST/098754/2008 , from FCT/MEC (Portugal)and FEDER (EC). OLC is a Henri Poincar´e Fellow at theObservatoire de la Cˆote d’Azur (OCA), funded by OCA andthe Conseil G´en´eral des Alpes-Maritimes.
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