Fundamental Properties of Stars using Asteroseismology from Kepler & CoRoT and Interferometry from the CHARA Array
D. Huber, M. J. Ireland, T. R. Bedding, I. M. Brandão, L. Piau, V. Maestro, T. R. White, H. Bruntt, L. Casagrande, J. Molenda-Żakowicz, V. Silva Aguirre, S. G. Sousa, T. Barclay, C. J. Burke, W. J. Chaplin, J. Christensen-Dalsgaard, M. S. Cunha, J. De Ridder, C. D. Farrington, A. Frasca, R. A. García, R. L. Gilliland, P. J. Goldfinger, S. Hekker, S. D. Kawaler, H. Kjeldsen, H. A. McAlister, T. S. Metcalfe, A. Miglio, M. J. P. F. G. Monteiro, M. H. Pinsonneault, G. H. Schaefer, D. Stello, M. C. Stumpe, J. Sturmann, L. Sturmann, T. A. ten Brummelaar, M. J. Thompson, N. Turner, K. Uytterhoeven
aa r X i v : . [ a s t r o - ph . S R ] S e p accepted for publication in ApJ Preprint typeset using L A TEX style emulateapj v. 08/22/09
FUNDAMENTAL PROPERTIES OF STARS USING ASTEROSEISMOLOGY FROM
KEPLER & COROT ANDINTERFEROMETRY FROM THE CHARA ARRAY
D. Huber , M. J. Ireland , T. R. Bedding , I. M. Brand˜ao , L. Piau , V. Maestro , T. R. White ,H. Bruntt , L. Casagrande , J. Molenda- ˙Zakowicz , V. Silva Aguirre , S. G. Sousa , T. Barclay ,C. J. Burke , W. J. Chaplin , J. Christensen-Dalsgaard , M. S. Cunha , J. De Ridder , C. D. Farrington ,A. Frasca , R. A. Garc´ıa , R. L. Gilliland , P. J. Goldfinger , S. Hekker , S. D. Kawaler , H. Kjeldsen ,H. A. McAlister , T. S. Metcalfe , A. Miglio , M. J. P. F. G. Monteiro , M. H. Pinsonneault ,G. H. Schaefer , D. Stello , M. C. Stumpe , J. Sturmann , L. Sturmann , T. A. ten Brummelaar ,M. J. Thompson , N. Turner , and K. Uytterhoeven accepted for publication in ApJ ABSTRACTWe present results of a long-baseline interferometry campaign using the PAVO beam combiner at theCHARA Array to measure the angular sizes of five main-sequence stars, one subgiant and four red giantstars for which solar-like oscillations have been detected by either
Kepler or CoRoT. By combininginterferometric angular diameters, Hipparcos parallaxes, asteroseismic densities, bolometric fluxes andhigh-resolution spectroscopy we derive a full set of near model-independent fundamental propertiesfor the sample. We first use these properties to test asteroseismic scaling relations for the frequencyof maximum power ( ν max ) and the large frequency separation (∆ ν ). We find excellent agreementwithin the observational uncertainties, and empirically show that simple estimates of asteroseismicradii for main-sequence stars are accurate to . T eff = 4600 − − ±
32 K (with a scatter of 97 K) and − ±
31 K (witha scatter of 93 K), respectively. Finally we present a first comparison with evolutionary models,and find differences between observed and theoretical properties for the metal-rich main-sequencestar HD 173701. We conclude that the constraints presented in this study will have strong potentialfor testing stellar model physics, in particular when combined with detailed modelling of individualoscillation frequencies.
Subject headings: stars: oscillations — stars: late-type — techniques: photometric — techniques:interferometric Sydney Institute for Astronomy (SIfA), School of Physics, Uni-versity of Sydney, NSW 2006, Australia NASA Ames Research Center, MS 244-30, Moffett Field, CA94035, USA NASA Postdoctoral Program Fellow; [email protected] Department of Physics and Astronomy, Macquarie University,NSW 2109, Australia Australian Astronomical Observatory, PO Box 296, Epping,NSW 1710, Australia Centro de Astrof´ısica and Faculdade de Ciˆencias, Universidadedo Porto, Rua das Estrelas, 4150-762 Porto, Portugal Department of Physics and Astronomy, Michigan State Uni-versity, East Lansing, MI 48823-2320, USA Stellar Astrophysics Centre, Department of Physics and As-tronomy, Aarhus University, Ny Munkegade 120, DK-8000 AarhusC, Denmark Research School of Astronomy & Astrophysics, Mount StromloObservatory, The Australian National University, ACT 2611, Aus-tralia Astronomical Institute of the University of Wroc law, ul.Kopernika 11, 51-622 Wroc law, Poland Bay Area Environmental Research Institute/NASA Ames Re-search Center, Moffett Field, CA 94035 SETI Institute/NASA Ames Research Center, MS 244-30,Moffett Field, CA 94035, USA School of Physics and Astronomy, University of Birmingham,Birmingham B15 2TT, UK Instituut voor Sterrenkunde, K.U.Leuven, Belgium Center for High Angular Resolution Astronomy, Georgia StateUniversity, PO Box 3969, Atlanta, GA 30302, USA INAF Osservatorio Astrofisico di Catania, Italy Laboratoire AIM, CEA/DSM-CNRS, Universit´e Paris 7Diderot, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-Yvette,France INTRODUCTION
The knowledge of fundamental properties such as tem-perature, radius and mass of stars in different evolu-tionary phases plays a key role in many applicationsof modern astrophysics. Examples include the im-provement of model physics of stellar structure andevolution such as convection (see, e.g. Demarque et al.1986; Monteiro et al. 1996; Deheuvels & Michel 2011;Trampedach & Stein 2011; Piau et al. 2011), the cal-ibration of empirical relations such as the color-temperature scale for cool stars (see, e.g., Flower 1996;Ram´ırez & Mel´endez 2005; Casagrande et al. 2010), andthe characterization of physical properties and habit-able zones of exoplanets (see, e.g., Baines et al. 2008; Space Telescope Science Institute, 3700 San Martin Drive,Baltimore, Maryland 21218, USA Astronomical Institute ’Anton Pannekoek’, University of Am-sterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Department of Physics and Astronomy, Iowa State University,Ames, IA 50011 USA Space Science Institute, Boulder, CO 80301, USA Department of Astronomy, Ohio State University, OH 43210,USA High Altitude Observatory, NCAR, P.O. Box 3000, Boulder,CO 80307, USA Instituto de Astrofisica de Canarias, 38205 La Laguna, Tener-ife, Spain Dept. Astrof´ısica, Universidad de La Laguna (ULL), Tenerife,Spain
D. Huber et al.van Belle & von Braun 2009; von Braun et al. 2011a,b).Many methods to determine properties of single fieldstars are indirect, and therefore of limited use for im-proving stellar models. Asteroseismology, the studyof stellar oscillations, is a powerful method to deter-mine properties of solar-type stars such as the meanstellar density with little model dependence (see, e.g.,Brown & Gilliland 1994; Christensen-Dalsgaard 2004;Aerts et al. 2010). Additionally, long-baseline interfer-ometry can be used to measure the angular sizes ofstars which, in combination with a parallax, yields alinear radius and, when combined with an estimate forthe bolometric flux, provides a direct measurement of astar’s effective temperature (see, e.g., Code et al. 1976;Boyajian et al. 2009; Baines et al. 2009; Boyajian et al.2012a,b; Creevey et al. 2012). Therefore, the combina-tion of both methods in principle allows a determina-tion of radii, masses and temperatures of stars with littlemodel dependence.While the potential of combining asteroseismologyand interferometry has been long recognized (see, e.g,Cunha et al. 2007), observational constraints have sofar restricted an application for cool stars to rela-tively few bright objects (North et al. 2007; Bruntt et al.2010; Bazot et al. 2011). Recent technological advances,however, have changed this picture. The launches ofthe space telescopes CoRoT (
Convection, Rotation andplanetary Transits , Baglin et al. 2006a,b) and
Kepler (Borucki et al. 2010; Koch et al. 2010) has increasedthe number of stars with detected solar-like oscillationsto several thousands, providing a large sample span-ning from the main-sequence to He-core burning redgiant stars (De Ridder et al. 2009; Hekker et al. 2009;Gilliland et al. 2010b; Chaplin et al. 2011). At the sametime, the development of highly sensitive instrumentssuch as the PAVO beam combiner (Ireland et al. 2008)at the CHARA Array (ten Brummelaar et al. 2005) havepushed the sensitivity limits of long-baseline interferom-etry, bringing into reach the brightest objects for whichhigh-quality space-based asteroseismic data are available.Using these recent advances, we present a systematiccombined asteroseismic and interferometric study of low-mass stars spanning from the main-sequence to the redclump. TARGET SAMPLE
Our target sample was selected to optimize the combi-nation of asteroseismology and interferometry given theobservational constraints, while also covering a large pa-rameter space in stellar evolution. The majority of ourstars were taken from the sample analyzed by the KeplerAsteroseismic Science Consortium (KASC). We selectedfour unevolved stars, which are among the brightest oscil-lating solar-type stars observed by
Kepler . Note that ourinterferometric results for θ Cyg (Guzik et al. 2011) and16 Cyg A&B (Metcalfe et al. 2012) will be presented else-where. For the
Kepler giant sample, four of the brightestred giants with the best Hipparcos parallaxes were se-lected. Finally, the main-sequence stars HD 175726 andHD 181420 in our sample are located in the CoRoT fieldtowards the galactic center, and were among the firstCoRoT main-sequence stars with detected oscillations(Barban et al. 2009; Mosser et al. 2009). Note that ourPAVO campaign is also targeting solar-type oscillators in
Fig. 1.—
H-R diagram with the position of the target stars cal-culated using spectroscopy, photometry and Hipparcos parallaxes.Solar metallicity BaSTI evolutionary tracks from 0.6-2.6 M ⊙ insteps of 0.1 M ⊙ are shown as grey lines. The dashed line marksthe approximate location of the cool edge of the instability strip. the CoRoT field in the galactic anti-center, such as the F-star HD 49933 (Appourchaux et al. 2008; Benomar et al.2009; Kallinger et al. 2010a), which has already beensubject to interferometric follow-up (Bigot et al. 2011).However, due to poor weather conditions during the win-ter seasons on Mt. Wilson, not enough data has yet beencollected for these targets.In the remainder of this section we summarize the ba-sic parameters of our target stars derived using classicalmethods and measurements available in the literature.Table 1 lists the complete target sample of our study,with spectral types taken from the HD catalog. Nineof the ten stars in our sample have atmospheric parame-ters derived from modeling several hundred lines in high-resolution spectra using the VWA package (Bruntt et al.2010), as presented by Bruntt (2009), Bruntt et al.(2012) and Thygesen et al. (2012). These are also listedin Table 1. For HD 189349, we have analyzed a spec-trum obtained with the NARVAL spectrograph at thePic du Midi Observatory using three different methods:VWA (Bruntt et al. 2010), ROTFIT (Frasca et al. 2003,2006) and the method described by Santos et al. (2004),Sousa et al. (2006) and Sousa et al. (2008). In two ofthe three methods the surface gravity was fixed to thevalue calculated from asteroseismic scaling relations (seenext section). The resulting spectroscopic parametersfor each method are listed in Table 2, and we haveadopted a weighted mean of all three methods, givenin Table 1, for the remainder of this paper. Note thatfor HD 173701 spectroscopic parameters have also beenpublished by Valenti & Fischer (2005), Mishenina et al.(2004) and Kovtyukh et al. (2003), which are also listedin Table 2 for comparison. The published values are ingood agreement with the values adopted here.All stars in our sample have measured Hipparcos par-allaxes (van Leeuwen 2007), with uncertainties rangingfrom ∼ TABLE 1Fundamental properties of target stars using available literature information. Stars are separated into main-sequenceand subgiant stars (top) and red giants (bottom).
HD KIC Sp.T.
V B − V Spectroscopy Hipparcos T eff log g [Fe/H] π (mas) L/L ⊙ .
514 0 .
878 5390(60) 4.49(3) +0 . – G5V 6 .
711 0 .
571 6070(45) 4.53(4) − . .
205 0 .
558 5990(60) 4.31(3) − . – F2V 6 .
561 0 .
434 6580(105) 4.26(8) +0 . .
022 0 .
800 5264(60) 3.70(3) − . .
520 0 .
500 6230(60) 4.32(3) − . .
014 1 .
171 4706(80) 2.60(1) +0 . .
040 0 .
994 4898(80) 2.62(1) − . .
188 1 .
012 4940(80) 2.81(1) +0 . .
305 0 .
878 5118(90) 2.4(1) − . B and V magnitudes are Tycho photometry (Perryman & ESA 1997) converted into the Johnson system using the calibrationby Bessell (2000). Spectroscopic parameters were adopted from Bruntt et al. (2012), Bruntt (2009) and Thygesen et al.(2012). Spectroscopic parameters for HD 189349 are the weighted average of three results presented in this work (see text andTable 2). Brackets indicate the uncertainties on a parameter (note that this notation has been adopted throughout the paper). ple are at distances <
60 pc and hence reddening isexpected to be negligible (see Molenda- ˙Zakowicz et al.2009; Bruntt et al. 2012). Hence, we assumed zero red-dening for all unevolved stars with an uncertainty of0.005 mag. For the giants, we have estimated redden-ing by comparing observed colors to synthetic photome-try of models matching the spectroscopic parameters inTable 1, as described in more detail in Section 3.3. Toestimate an uncertainty, we have compared these valuesto E ( B − V ) values listed in the Kepler Input Catalog(KIC, Brown et al. 2011) for nearby stars and to esti-mates from the 3-D extinction model by Drimmel et al.(2003). The mean scatter between these methods for allstars is 0.02 mag, which we adopt as our uncertainty in E ( B − V ) for the giants in our sample. Finally, we usedthe spectroscopically determined effective temperaturesand metallicities to estimate a bolometric correction foreach star using the calibrations by Flower (1996) andAlonso et al. (1999) with appropriate zero-points as dis-cussed in Torres (2010), yielding the stellar luminositygiven in the last column of Table 1. Figure 1 shows anH-R diagram of our target stars, according to the prop-erties listed in Table 1, together with solar-metallicityBaSTI evolutionary tracks (Pietrinferni et al. 2004). OBSERVATIONS
Asteroseismology
The asteroseismic results presented in this paper arebased on observations obtained by the
Kepler andCoRoT space telescopes. Both satellites deliver near-uninterrupted, high S/N time series which are ideallysuited for asteroseismic studies. In this paper, we fo-cus on two global parameters: the frequency of max-imum power ( ν max ) and the large frequency separa-tion (∆ ν ). These are frequently used to determinefundamental properties of main-sequence and red gi-ant stars (see, e.g., Stello et al. 2009b; Kallinger et al.2010b,c; Chaplin et al. 2011; Hekker et al. 2011a,b;Silva Aguirre et al. 2011; Huber et al. 2011). For a gen-eral introduction to solar-like oscillations, we refer the TABLE 2Atmospheric parameters for stars from different methods.
HD KIC T eff log g [Fe/H] Ref173701 8006161 5390(60) 4.49(3) 0.34(6) 1 ∗ ∗ ∗ ∗ log g fixed to asteroseismic value. (1) Bruntt et al. (2012),(2) Valenti & Fischer (2005), (3) Mishenina et al. (2004),(4) Kovtyukh et al. (2003), (5) this paper: a - VWA(Bruntt et al. 2010), b - Santos et al. (2004), Sousa et al.(2006) and Sousa et al. (2008), c - ROTFIT (Frasca et al.2003, 2006). reader to the review by Bedding (2011).Figure 2 presents the power spectrum for each star,sorted by the frequency of maximum power ( ν max ). Inmost cases, a clear power excess due to solar-like oscil-lations is visible. A summary of the datasets used inour analysis, as well as the derived asteroseismic param-eters, is given in Table 3. The analysis of Kepler starsis based on either short-cadence (Gilliland et al. 2010a)or long-cadence (Jenkins et al. 2010) data up to Q10,which were corrected for instrumental trends as describedin Garc´ıa et al. (2011). Global asteroseismic parameterswere extracted using the automated analysis pipeline byHuber et al. (2009), which has been shown to agree wellwith other methods (Hekker et al. 2011c; Verner et al.2011). Due to the length and very high S/N of the
Kepler data, the modes are resolved and uncertainties on ν max and (particularly) ∆ ν are dominated by the adoptedmethod (e.g., the range over which ∆ ν is determined)rather than measurement errors. To account for this, weadded in quadrature to the formal uncertainties an uncer-tainty based on the scatter of different methods used bySilva Aguirre et al. (2012) for short-cadence data and by D. Huber et al. Fig. 2.—
Power density spectra for all stars in our sample, sorted by the frequency of maximum power ( ν max ). Note the change in x-axisscale for main-sequence (left column), subgiant (top right column) and red-giant stars (four bottom right panels). Note that the high peakat ∼ µ Hz for HD 187637 is a known artefact of
Kepler short-cadence data (Gilliland et al. 2010a).
Huber et al. (2011) for long-cadence data. The analysisby Huber et al. (2011) was based on data spanning fromQ0-6, which in most cases was sufficient to resolve themodes and reliably estimate ν max and ∆ ν (Hekker et al.2012). In general, the uncertainties on the asteroseismicparameters for most Kepler stars are negligible comparedto the uncertainties on other observables. A notable ex-ception is HD 189349, with a relatively large uncertaintyof ∼
4% in the large frequency separation. Inspection ofthe power spectrum shows that the modes for this starare very broad, making a determination of ∆ ν difficult.We speculate that the unusually broad modes may berelated to the low metallicity of this object, but a more in-depth analysis is beyond the scope of this paper.For the two CoRoT stars in our sample, we have re-analyzed publicly available data using the method de-scribed in Huber et al. (2009). Our results for HD181420are in good agreement with the values published byBarban et al. (2009). For HD175726, our analysis didnot yield significant evidence for regularly spaced peaks,and yielded only marginal evidence for a power excessat 1900 ± µ Hz. Mosser et al. (2009) have argued thatthis power excess is compatible with solar-like oscillationsand showed evidence for a large variation of ∆ ν with fre-quency, which could be responsible for the null-detectionin our analysis. We have adopted the published valueundamental Properties of Stars using Asteroseismology and Interferometry 5 TABLE 3Asteroseismic observations and measured parameters.
HD KIC Data T (d) Duty cycle (%) ν max ( µ Hz) ∆ ν ( µ Hz)173701 8006161 Kepler SC Q5-10 557 92 3619(98) 149.3(4)175726 – CoRoT SRc01 28 90 1915(200) 97.2(5)177153 6106415 Kepler SC Q6-8,10 461 70 2233(60) 104.3(3)181420 – CoRoT LRc01 156 90 1574(31) 75.1(3)182736 8751420 Kepler SC Q5,7-10 557 75 568(15) 34.6(1)187637 6225718 Kepler SC Q6-10 461 91 2352(66) 105.8(3)175955 10323222 Kepler LC Q0-10 880 91 46.7(1.1) 4.86(3)177151 10716853 Kepler LC Q1-7,9-10 869 77 48.8(1.1) 4.98(7)181827 8813946 Kepler LC Q1-10 869 91 73.1(1.2) 6.45(7)189349 5737655 Kepler LC Q1-10 869 91 29.9(1.1) 4.22(16) Detection adopted from Mosser et al. (2009). for ∆ ν by Mosser et al. (2009) and a value for ν max cor-responding to the maximum of the power excess in thespectrum, with a conservative uncertainty of 10%. Interferometry
Interferometric observations were made with the Pre-cision Astronomical Visible Observations (PAVO) beamcombiner (Ireland et al. 2008) at the Center for High An-gular Resolution Astronomy (CHARA) on Mt. Wilsonobservatory, California (ten Brummelaar et al. 2005).Operating at a central wavelength of λ = 0 . µ m withbaselines up to 330 m, PAVO at CHARA is one of thehighest angular-resolution instruments world-wide.A complete description of the instrument was givenby Ireland et al. (2008), and we summarize the basic as-pects here. The light from up to three telescopes passesthrough vacuum tubes and into a series of optics to com-pensate the path difference. The beams are then colli-mated and passed through a non-redundant mask whichacts as a bandpass filter, and spatially modulated in-terference fringes are formed behind the mask. The in-terference pattern is then passed through a lenslet ar-ray and a prism, producing fringes in 16 segments onthe CCD detector, each being spectrally dispersed inseveral independent wavelength channels. Major advan-tages of the PAVO design are high sensitivity (with alimiting magnitude of R ∼ ∼ −
330 m.Interferometric observations require careful calibrationof the observed visibilities. Ideally, this is achieved byobserving bright, unresolved point sources as closely aspossible to the target object in time and distance. ForPAVO observations of targets as small as in our case, thismeans calibrating with late B to early A stars since at thePAVO magnitude limit these stars are distant enough tohave significantly smaller diameters (0.1–0.15 mas) thanour target stars. Table 4 lists all calibrators that wereused in our analysis. Expected sizes are calculated usingthe V − K relation by Kervella et al. (2004) for dwarf andsubgiant stars. V band magnitudes have been taken from TABLE 4Calibrators used for interferometric observations.
HD Sp.T. V − K E ( B − V ) θ V − K ID171654 A0V -0.067 0.036 0.141 c174177 A0V 0.249 0.020 0.191 gh176131 A2V 0.345 0.012 0.155 ac176626 A2V 0.084 0.026 0.146 ac177959 A3V 0.451 0.029 0.152 b178190 A2V 0.381 0.027 0.157 bd179095 A0V -0.069 0.022 0.129 gh179124 B9V 0.280 0.095 0.146 d179483 A2V 0.316 0.028 0.144 e179733 A0V 0.211 0.038 0.117 ac180138 A0V 0.075 0.045 0.128 c180501 A0V 0.147 0.027 0.117 gh180681 A0V 0.112 0.031 0.111 acei183142 B8V -0.462 0.060 0.093 ei184147 A0V 0.007 0.019 0.121 egi184787 A0V 0.034 0.017 0.154 cf188252 B2III -0.461 0.047 0.155 ce188461 B3V -0.461 0.109 0.095 efj189845 A0V 0.136 0.053 0.127 fj190025 B5V -0.230 0.157 0.084 j190112 A0V 0.067 0.027 0.113 fList of dropped calibrators: HD179395, HD181939,HD182487, HD184875, HD189253; “ID” refers to the ID ofthe target star for which the calibrator has been used (seecolumn 3 of Table 5). the Tycho catalog and were converted into the Johnsonsystem using the calibration by Bessell (2000). K magni-tudes were adopted from 2MASS (Skrutskie et al. 2006).Interstellar reddening for each calibrator was estimatedusing the extinction model by Drimmel et al. (2003).Although we have checked each calibrator in the lit-erature for possible multiplicity, rotation and variabil-ity prior to observations, our data show that roughly1/4 of all observed calibrators are more resolved thanexpected, and therefore potentially unsuitable for cali-bration. These calibrators are listed at the bottom ofTable 4. Possible reasons for this include previously un-detected binary systems and rapid rotation causing de-viations from spherical symmetry. D. Huber et al. TABLE 5Interferometric observations and measured parameters.
HD KIC ID Scans/Nights Baselines µ R θ UD θ LD θ ( V − K ) θ IRFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3 presents the calibrated squared-visibility mea-surements as a function of spatial frequency for all targetsin our sample, with a summary of observations given inTable 5. We have collected at least three independentscans for each target over at least two different nights,and the visibilities of each target were calibrated with atleast two different calibrators (see also Table 4). Notethat each scan typically produces a measurement of vis-ibility in 20 independent wavelength channels, resultingin a total of ∼ V = (cid:18) − µ λ µ λ (cid:19) − × (cid:20) (1 − µ λ ) J ( x ) x + µ λ ( π/ / J / ( x ) x / (cid:21) , (1)with x = πBθ LD λ − . (2)Here, V is the visibility, µ λ is the linear limb-darkeningcoefficient, J n ( x ) is the n -th order Bessel function, B is the projected baseline, θ LD is the angular diame-ter after correction for limb-darkening, and λ is thewavelength at which the observation was made. Linearlimb-darkening coefficients in the R band for our tar-gets were estimated by interpolating the model grid ofClaret & Bloemen (2011) to the spectroscopic estimatesof T eff , log g and [Fe/H] (Table 1) for a microturbulentvelocity of 2 km s − . Uncertainties on the limb-darkeningcoefficients were estimated from the difference in themethods presented by Claret & Bloemen (2011). Thechoice of the limb-darkening model has little effect onthe final fitted angular diameters. Detailed 3-D hydro-dynamical models by Bigot et al. (2006), Chiavassa et al.(2010) and Chiavassa et al. (2012) for dwarfs and giantshave shown that the differences to simple linear limbdarkening models are 1% or less in angular diameter forstars with near solar-metallicity. For a moderately re-solved star with V ∼ .
5, a 1% change in angular di-ameter would arise from a change of less than 1% in V , which is less than our typical measurement uncertainties.The procedure used to fit the model and estimate theuncertainty in the derived angular diameters was de-scribed by Derekas et al. (2011). In summary, Monte-Carlo simulations were performed which took into ac-count uncertainties in the adopted wavelength calibra-tion (0.5%), calibrator sizes (5%), limb-darkening coef-ficients (see Table 5), as well as potential correlationsacross wavelength channels. The resulting fitted angulardiameters of each target, corrected for limb-darkening,are given in Table 5. We also give the uniform-disc di-ameters in Table 5, which were derived by setting µ λ = 0in Equation (1).A few comments on our derived diameters are neces-sary. Firstly, one calibrator in our sample (HD 179124),which is the main calibrator for HD 181420, was recentlyfound to be a rapidly rotating B star with v sin i =290 km/s (Lefever et al. 2010). This introduces an extrauncertainty on the estimated calibrator diameter. Wehave accounted for this by assuming a 20% uncertaintyin the calibrator diameter, which roughly corresponds tothe maximum change in the average diameter expectedfor rapid rotators (Domiciano de Souza et al. 2002). Sec-ondly, a few of our target stars (e.g. HD 187637) are onlyabout 50% bigger in angular size than their calibrators.This means that the uncertainties on the derived diam-eters will be strongly influenced by the assumed uncer-tainties of the calibrator diameters, which in our caseare 5%. While such an uncertainty is reasonable com-pared to the scatter in the photometric calibrations (see,e.g., Kervella et al. 2004), the diameter measurement it-self will only be scientifically useful if the uncertaintyin the measured diameter is smaller than the precisionof indirect techniques. Further data at longer baselineswith smaller calibrators will be needed to reduce the un-certainties for these targets.Indirect techniques to estimate angular diameters in-clude surface brightness relations (see, e.g., van Belle1999; Kervella et al. 2004) and the infrared fluxmethod (IRFM, see, e.g., Ram´ırez & Mel´endez 2005;Casagrande et al. 2010). Figure 4 compares our mea-sured angular diameters with predictions using the( V − K ) surface-brightness relation for dwarfs and sub-giants by Kervella et al. (2004) and the IRFM methodcoupled with asteroseismic constraints, as described inSilva Aguirre et al. (2012). For the ( V − K ) relation weundamental Properties of Stars using Asteroseismology and Interferometry 7 Fig. 3.—
Squared visibility versus spatial frequency for all stars in our sample. Red solid lines show the fitted limb-darkened disc model.The order of panels is the same as in Figure 2. Note that the error bars for each star have been scaled so that the reduced χ equals unity. have adopted a 1% diameter uncertainty for all stars(Kervella et al. 2004). We find good agreement for allstars for both methods, with a residual mean of − ± ±
2% for ( V − K ) and IRFM, respectively, bothwith a scatter of 5%. Our results therefore seem to con-firm that the relation by Kervella et al. (2004) is alsovalid for red giants, as suggested by Piau et al. (2011),and that combining the IRFM method with asteroseismicconstraints, as done by Silva Aguirre et al. (2012), yieldsaccurate diameters for both evolved and unevolved stars.These tests of indirect methods are encouraging. Weemphasize that interferometry remains an important toolto validate these methods for a wider range of evolution-ary states, chemical compositions, and distances. The ( V − K ) relation, for example, is based on an empiricalrelation calibrated using nearby stars that does not takeinto account potential spread due to different chemicalcompositions, and is only valid for de-reddened magni-tudes. An illustration of the importance of using interfer-ometry is HD 181827, which shows a significantly smallermeasured diameter than predicted from ( V − K ). Thissmaller diameter is also in agreement with asteroseismicresults, which suggest a smaller radius (see Section 4.1). Bolometric Fluxes
To estimate bolometric fluxes for our target sample,we first extracted synthetic fluxes from the MARCSdatabase of stellar model atmospheres (Gustafsson et al. D. Huber et al.
Fig. 4.—
Fractional differences between angular diameters mea-sured with PAVO to diameters determined using the ( V − K )surface-brightness relation by Kervella et al. (2004) (upper panel)and using the infrared flux method with asteroseismic constraints,as described in Silva Aguirre et al. (2012) (lower panel). Black di-amonds show main-sequence and subgiants stars, and red trianglesshow giant stars. HD numbers of each target are labelled in theupper panel. − for plane-parallel models (unevolved stars)and 2 km s − for spherical models with a mass of1 M ⊙ for red giants. We then multiplied the syn-thetic stellar fluxes by the filter responses for theJohnson-Glass-Cousins U BV RIJHKL , Tycho B T V T and 2MASS JHK s systems and integrated the result-ing fluxes to calculate synthetic magnitudes for eachMARCS model. Filter responses and zeropoints weretaken from Bessell & Murphy (2012) ( U BV RI , B T V T ),Cohen et al. (2003) (2MASS), and Bessell et al. (1998)( JHKL ). We note that synthetic photometry calculatedusing MARCS models has previously been validated us-ing observed colors in stellar clusters (Brasseur et al.2010; VandenBerg et al. 2010). To check the influenceof the chosen mass for the spherical models, we have re-peated the above calculations for typical red giant modelswith T eff = 5000 K and log g = 2 −
3. The fractional dif-ferences in the integrated flux for each filter for massesranging from 0 . − M ⊙ was found to be less than 0.5%in all bands, and are therefore negligible for our analysis.The amount of photometry in the literature for oursample is unfortunately small. The targets are generallytoo faint to have reliable magnitudes in the Johnson-Glass-Cousins system, and they are too bright to have afull set of SDSS photometry in the KIC. To ensure consis-tency of our bolometric fluxes we only used photometrythat is available for all stars in our sample, namely Ty-cho2 B T V T and 2MASS JHK s magnitudes. The adoptedphotometry and uncertainties are listed in Table 6.To calculate bolometric fluxes, we largely followed theapproach described in Alonso et al. (1995). For each tar-get star, we first found the six models bracketing thespectroscopic determinations T eff , log g and [Fe/H], as Fig. 5.—
Spectral energy distibutions of Procyon, the Sun andArcturus to test our method to determine bolometric fluxes. Blacklines are MARCS models with parameters as given in each panel,smoothed with a constant spectral resolution λ/ ∆ λ ∼
200 (cor-responding to a width of ∼ BV JHK photometry is shown as red diamonds and bluesquares, respectively. All fluxes have been normalized to 1 in theband with highest flux for a given star. The inset shows the re-sult of Monte-Carlo simulations to estimate the uncertainty in thebolometric flux as described in the text. Dashed and dotted linesshow the literature values and 1- σ uncertainties. given in Table 1. We then transformed the synthetic B T V T JHK s magnitudes of each model into fluxes, andnumerically integrated these fluxes using the pivot wave-length for each filter response, calculated as describedby Bessell & Murphy (2012) (note that this choice of areference wavelength is independent of the spectral typeconsidered). The numerical integration yielded an esti-mate f int , which we then compared to the true bolometricflux, f bol = σT , where σ is the Stefan-Boltzmann con-stant. This yielded a correction factor c = f int /f bol foreach of the six models, which gave the percentage of fluxincluded when integrating the photometry over discretewavelengths. The final bolometric flux was then calcu-lated by integrating the observed fluxes the same wayas the model fluxes, and dividing the resulting estimateby the correction factor c found by interpolating the sixcorrections factors to the spectroscopic estimates of T eff ,log g and [Fe/H]. Note that this interpolation was neces-sary because the step size of the MARCS grid is typicallylarger than the uncertainties of the spectroscopic param-undamental Properties of Stars using Asteroseismology and Interferometry 9 Fig. 6.—
Spectral energy distributions for all stars in our sample. Black lines are the MARCS models with parameters as given in eachpanel, smoothed with a constant spectral resolution λ/ ∆ λ ∼
200 (corresponding to a width of ∼ B T V T JHK bands, respectively. All fluxes have been normalized to 1 in the bandwith highest flux for a given star. The order of panels is the same as in Figure 2. eters. Uncertainties in the derived bolometric fluxes werefound by perturbing the input photometry and the spec-troscopic parameters according to their estimated uncer-tainties (see Tables 1 and 6), repeating the procedure5000 times, and taking the standard deviation of the re-sulting distribution.To test this approach, we have used the same methodfor three bright stars that span a similar range of evolu-tionary stages as our sample and for which bolometricfluxes have been well determined: Procyon, the Sun,and Arcturus. Since Tycho and 2MASS photometryis not available for such bright stars, we have used
BV JHK photometry to mimic the available informa- tion for our target sample. Photometry has been takenfrom the General Catalog of Photometric Data (GCDP,Mermilliod et al. 1997) for Procyon and Arcturus, andfrom Colina et al. (1996) for the Sun. Figure 5 showsthe spectral energy distributions (SEDs) of all threestars, comparing the MARCS model that best matchesthe physical parameters of each star (black line) to theobserved and synthetic fluxes in the
BV JHK bands(red and blue squares, respectively). Note that theMARCS models have been smoothed to a spectral res-olution of λ/ ∆ λ ∼
200 for better visibility. The insetsshow the distributions of the Monte-Carlo simulationsdescribed above compared to the literature values of the0 D. Huber et al.
TABLE 6Broadband photometry, estimated reddening and bolometric fluxes for all target stars.
HD KIC B T V T J H K E ( B − V ) F bol (10 − erg s − cm − ) MARCS ATLAS+ θ ATLAS+ V T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bolometric flux (dashed line) and their 1- σ uncertain-ties (dotted lines). Literature bolometric fluxes havebeen taken from Ram´ırez & Allende Prieto (2011) forArcturus, Aufdenberg et al. (2005) and Fuhrmann et al.(1997) for Procyon, and we have adopted an effectivetemperature of 5777 ± σ , with a maximum deviation of ∼ σ for Arcturus.Figure 6 shows the SEDs of all target stars with the ap-propriate models for each star, and Table 6 lists our bolo-metric fluxes based on the procedure described above.We note that for the red giants in our sample, inter-stellar reddening cannot be neglected. To estimate red-dening using the SED, we adopted the reddening law byO’Donnell (1994) (see also Cardelli et al. 1989) and iter-ated over E ( B − V ) to find the observed colors that bestfit the colors of the six models bracketing the spectro-scopic parameters in Table 1. We then again interpolatedto the spectroscopic T eff , log g and [Fe/H] values, anal-ogously to the correction factor described above. Thederived reddening estimates for the giants are listed inTable 6.To further test these results, we have used an in-dependent method to determine bolometric fluxes forfour stars by combining publicly available flux-calibratedELODIE spectra (Prugniel et al. 2007), broadband pho-tometry and ATLAS9 models (Castelli & Kurucz 2003,2004). We started by calculating a grid of ATLAS9 mod-els in the 3- σ error box of the spectroscopically deter-mined T eff , log g and [Fe/H] (see Table 1). Each modelspectrum was then multiplied by the B T V T JHK s filterpassbands and integrated over all wavelengths to com-pute a synthetic flux in each band. Model fluxes werethen calibrated into fluxes received on Earth using ei-ther the measured angular diameter or the Tycho V T magnitude. To find the model that best fits the photo-metric data we then compared the grid of model fluxeswith the observed fluxes, calculated using the same ze-ropoints as in the procedure described above. Finally,the bolometric flux of each star was determined by inte-grating the ELODIE spectrum between 390 and 680 nmtogether with the synthetic ATLAS9 model (covering thewavelength ranges <
390 nm and >
680 nm) that best fitsthe observed photometry.To estimate uncertainties the above procedure was re-peated 100 times, drawing random values for the ob- served photometry given in Table 6, and adding the stan-dard deviation of the resulting distribution in quadra-ture to the uncertainty of the total flux of the ELODIEspectra. The final values for the two different cali-bration methods are given in Table 6. The derivedbolometric fluxes agree well with the estimates fromMARCS models, reassuring us that the model depen-dency and adopted method have little influence com-pared to the estimated uncertainties. We note that wehave also compared our bolometric fluxes with estimatesderived from the infrared flux method, as described inSilva Aguirre et al. (2012). Again, we have found goodagreement with our estimates within the quoted uncer-tainties. FUNDAMENTAL STELLAR PROPERTIES
Asteroseismic Scaling Relations
The large frequency separation of oscillation modeswith the same spherical degree and consecutive radial or-der is closely related to the mean density of star (Ulrich1986): ∆ ν ∝ M / R − / . (3)Additionally, Brown et al. (1991) argued that the fre-quency of maximum power ( ν max ) for solar-like starsshould scale with the acoustic cut-off frequency, whichwas used by Kjeldsen & Bedding (1995) to formulate asecond scaling relation: ν max ∝ M R − T − / . (4)Provided the effective temperature of a star is known,Equations (3) and (4) allow an estimate of the stellarmass and radius. This can be done by either combin-ing the two equations (the so-called direct method, seeKallinger et al. 2010c) or by comparing the observed val-ues of ∆ ν and ν max with values calculated from a gridof evolutionary models (the so-called grid-based method,see Stello et al. 2009b; Basu et al. 2010; Gai et al. 2011).Our interferometric observations, presented in the Sec-tion 3.2, allow us to test Equations (3) and (4). Using theHipparcos parallaxes in combination with the angular di-ameters, we have calculated linear radii for our sampleof stars, which are listed in Table 7. These are comparedto asteroseismic radii calculated using Equations (3) and(4) (using T eff values taken from Table 1) in Figure 7.undamental Properties of Stars using Asteroseismology and Interferometry 11 Fig. 7.—
Comparison of stellar radii measured using interfer-ometry and calculated using asteroseismic scaling relations. Blackdiamonds show our
Kepler and CoRoT sample, and red asterisksshow several bright stars as indicated in the plot for comparison.The dashed line marks the 1:1 relation.
Note the influence of T eff on Equation (4) is small: forsolar T eff a variation of 100 K causes only a 0.9% changein ν max , which is significantly smaller than our typicaluncertainties (see Table 3).The comparison in Figure 7 is very encouraging, show-ing an agreement between the two methods within 3- σ in all cases. The overall scatter about the residuals is ∼ σ and+0.8 σ compared to differences of -0.6 σ and -1.0 σ fromthe direct method, respectively.For comparison, Figure 7 also shows examples ofbright stars for which well constrained asteroseismicand interferometric parameters are available. We haveadopted values for ∆ ν and ν max from Stello et al.(2009a) and references therein, with uncertaintiesfixed to typical values of 1% in ∆ ν and 3% in ν max .Asteroseismic observations have been obtained fromthe MOST space telescope for ǫ Oph (Barban et al.2007; Kallinger et al. 2008), the CoRoT space-telescopefor HD 49933 (Appourchaux et al. 2008), and fromground-based Doppler observations for the remain-ing sample (Carrier & Bourban 2003; Kjeldsen et al.
Fig. 8.—
Comparison of ν max measured from asteroseismologyand calculated using independent measurements of R, M and T eff .Black diamonds show the Kepler and CoRoT sample, and red as-terisks show several bright stars as indicated in the plot for com-parison. The dashed line marks the 1:1 relation. ǫ Oph, Bazot et al. (2011) for 18 Sco, Bigot et al.(2011) for HD 49933 and from Bruntt et al. (2010) andreferences therein for the remaining sample. Parallaxeswere adopted from van Leeuwen (2007), except for α Cen A and B for which we have adopted the value byS¨oderhjelm (1999). Figure 7 again shows agreementwithin 3- σ in all cases. Excluding HD 175726 from oursample due to large uncertainties in the asteroseismicobservations, the residual scatter between asteroseismicand interferometric radii is 4% for dwarfs and 16% forgiants, with mean deviations of − ± ± . ν is on more solid ground than the scaling relationfor ν max , which only recently has been studied inmore detail observationally (see, e.g., Stello et al. 2009a;Mosser et al. 2010; White et al. 2011) and theoretically(Belkacem et al. 2011). To test Equation (4), we cancombine Equation (3) with the interferometrically mea-sured radii to calculate stellar masses, and combine thesewith T eff to calculate ν max . We compare these with the2 D. Huber et al.measured values in Figure 8. We again observe goodagreement within the error bars, with no systematic de-viation as a function of evolutionary status. Figure 8 alsodisplays a comparison with measured values for a sam-ple of bright stars, again showing good agreement withour results for the Kepler and CoRoT sample. We notethat Bedding (2011) has shown a similar comparison forbright stars, and noted a potential breakdown of the ν max relation for low-mass stars with ν max & µ Hz. Sincenone of the stars in our sample have ν max > µ Hz,we are unable to test this claim in our study.The large error bars for some stars in Figures 7 and8 may cast some doubt about the usefulness of inter-ferometry to test scaling relations. Indeed, for the redgiants in our sample the uncertainty in the interferomet-ric radius is completely dominated by the uncertaintyin the parallax. For these stars the PAVO data will bemost valuable to measure the effective temperature bycombining the angular diameter with an estimate of thebolometric flux, which can then be compared to indi-rect T eff estimates from broadband photometry and spec-troscopy (see next section). For most unevolved starsin the Kepler /CoRoT sample, our current uncertaintiesin the angular diameters are comparable to the paral-lax uncertainties. The bright star comparison sample,on the other hand, is dominated by the uncertainties inthe asteroseismic observables, which are much more diffi-cult to constrain from the ground or using smaller spacetelescopes. The fact that the asteroseismic uncertaintiesare almost negligible for the
Kepler /CoRoT sample ex-plains the somewhat counter-intuitive observation thatthe error bars in Figures 7 and 8 are similar for somestars of the
Kepler sample and for stars which are up to8 magnitudes brighter. This comparison underlines theimportance of obtaining precise asteroseismic constraintson bright stars for which constraints are available fromindependent observational techniques.
Spectroscopic and Photometric Temperatures
The measurement of the angular diameter θ LD of astar combined with an estimate of its bolometric flux f bol allows a direct measurement of the effective temperature: T eff = (cid:18) f Bol σθ (cid:19) / , (5)where σ is the Stefan-Boltzmann constant. We have usedour measured angular diameters presented in Section 3.2together with the bolometric flux estimates presented inSection 3.3 to calculate effective temperatures for oursample, which are listed in Table 7.The model dependency of effective temperatures cal-culated using Equation (5) is small, and hence suchestimates are important for calibrating indirect pho-tometric estimates such as the infrared flux method(see, e.g., Casagrande et al. 2010), as well as spectro-scopic determinations for which usually strong degen-eracies between T eff , log g and [Fe/H] exist (see, e.g.,Torres et al. 2012). Figure 9 compares the measuredeffective temperatures in our sample to estimates fromhigh-resolution spectroscopy (mostly using the VWApackage by Bruntt et al. (2010), see Table 1) as well asphotometric calibrations taken from Casagrande et al.(2010), Ram´ırez & Mel´endez (2005) and Bruntt et al. Fig. 9.—
Comparison of effective temperatures derived in thisstudy with spectroscopic estimates (panel a) and several photo-metric calibrations based on V − K s (panels b–d). Black diamondsare main-sequence stars, while red triangles show red giants. Notethat the relations by Casagrande et al. (2010) and Bruntt et al.(2012) are calibrated for main-sequence stars. (2012). We have chosen V − K s to calculate photomet-ric temperatures since this index usually gives the lowestresiduals as a temperature indicator for cool stars (see,e.g., Casagrande et al. 2010). The comparison in Fig-ure 9a shows good agreement of our temperatures withspectroscopy, with a residual mean of − ±
39 K witha scatter of 124 K for all stars, and − ±
33 K with ascatter of 97 K when excluding the F-star HD 181420 forwhich the angular diameter is not well determined. Wenote that this agreement is only slightly worse (with anincreased scatter by about 10 K) if we use the T eff valuesby Bruntt et al. (2012) and Thygesen et al. (2012) forwhich no asteroseismic constraints on log g were used.Bruntt et al. (2010) noted that a slight bias for spectro-scopic temperatures to be hotter than interferometric es-timates by ∼
40 K for a sample of nearby stars, which issomewhat confirmed by our results, although the scatteris significantly larger. Our result confirms that a combi-nation of spectroscopy and asteroseismology can be ap-plied for the accurate characterization of temperatures,radii and masses of much fainter stars, e.g. exoplanethost stars observed by the
Kepler mission.undamental Properties of Stars using Asteroseismology and Interferometry 13The photometric estimates shown in Figure 9b, 9cand 9d show slight systematic deviations. The calibra-tion by Casagrande et al. (2010) shows the best agree-ment, with only the coolest red giants being slightlyhotter than implied by our results. The calibrationby Ram´ırez & Mel´endez (2005) is the only one that di-rectly provides color-temperature relations calibrated forgiants. As already noted by Casagrande et al. (2010),the temperatures by Ram´ırez & Mel´endez (2005) seemto be systematically cooler than expected, and this isconfirmed by our results. Finally, the calibration givenby Bruntt et al. (2012) overestimates temperatures atthe cool end, which is again not surprising since theircalibration was based on main-sequence stars only, anddid not include corrections for lower surface gravitiesand different metallicities. Overall, we conclude thatphotometric estimates reproduce the measured temper-atures from interferometry well within the uncertain-ties, except for the giants where reddening is significant.We note that HD 173701 is the only star with sufficientSloan photometry to be included in the calibration byPinsonneault et al. (2012). The SDSS temperature, cor-rected for metallicity as described in Pinsonneault et al.(2012), is 5364 ±
100 K, in good agreement to the de-termined values here. Finally, we note that the effectivetemperatures presented in this section do not influencethe comparisons of the asteroseismic masses and radiicalculated in the previous section (which were calculatedusing spectroscopic T eff ), since the dependence of Equa-tion (4) on T eff is only small. Stellar Models
Detailed modelling will be deferred to a future pa-per, but we present some first basic comparisons for themost interesting cases here. We use the publicly avail-able BaSTI stellar evolutionary tracks (Pietrinferni et al.2004) with solar-scaled distribution of heavy elements(Grevesse & Noels 1993) and a standard mass loss pa-rameter of η = 0 . α MLT = 1 .
913 and an initial chemical composition of(
Y, Z ) = (0 . , . HD 182736
The star with the best constrained fundamental prop-erties in our sample is the subgiant HD 182736, with rel-ative uncertainties in temperature, radius and mass of0.7%, 2.6% and 7.7%, respectively. The fact that thebest observational result is achieved for the only sub-giant in our sample is not surprising: while the moredistant red giants generally have well constrained diam-eters due to their larger size, they suffer from a largeuncertainty in the parallaxes and effective temperaturesdue to their larger distance and significant reddening.On the other hand, main-sequence stars are generallytoo small to achieve a good precision on their measured
Fig. 10.—
Radius versus effective temperature with the positionof the subgiant HD 182736 shown as a red diamond. The solidline shows the BaSTI evolutionary model matching the metallic-ity from high-resolution spectroscopy and the mass determined inthis study. Dashed-dotted and dashed-triple-dotted lines show theeffect of varying the metallicity by 1 σ , while dotted and dashedlines show the same effect for varying the mass by 1 σ . The deter-mined mass and metallicity for HD 182736 are M = 1 . ± . M ⊙ and [Fe / H] = − . ± . diameters. Subgiants land in the ”sweet spot” betweenthese regimes, with angular sizes big enough for a precisemeasurement with PAVO and distances close enough tohave a well constrained Hipparcos parallax and negligiblereddening.Figure 10 shows a diagram of stellar radius versus ef-fective temperature with the position of HD 182736 ac-cording to the properties listed in Table 7 marked as ared diamond. The black solid line shows the evolution-ary track matching the determined mass and metallic-ity, calculated by quadratically interpolating the originalBaSTI tracks. Dashed-dotted and dashed-triple-dottedlines show the effect of varying the metallicity by 1 σ ,while dotted and dashed lines show the same effect forvarying the mass by 1 σ . The agreement between themodels and our observations is excellent, with a matchwithin 1 σ for both radius and temperature. We em-phasize that no fitting is involved in this comparison -the mass, radius, temperature and metallicity are deter-mined independently from the evolutionary tracks. Amore in-depth asteroseismic study using individual fre-quencies, in particular with respect to probing the corerotation rate using mixed modes (Deheuvels et al. 2012),combined with the results presented in this paper shouldyield powerful constraints for studying the structure andevolution of this evolved subgiant. HD 173701
Figure 11 shows the radius- T eff diagram for HD 173701,a metal-rich main-sequence star with relatively well con-strained properties. In this case, the agreement betweenBaSTI models and observations is poor. The differencecan be reconciled with a 3 − σ difference in mass andmetallicity, i.e. the star is more metal-rich and less mas-sive than implied from our observations. Indeed, theasteroseismic (but not model-independent) analyses byMathur et al. (2012) and Silva Aguirre et al. (2012) im-4 D. Huber et al. TABLE 7Fundamental properties of all Kepler and CoRoT stars in this study.
HD KIC T eff (K) [Fe/H] R/R ⊙ M/M ⊙ R/R ⊙ M / M ⊙ T eff (K)Spectroscopy ν max +∆ ν + T eff , sp π + Θ LD π + Θ LD + ∆ ν Θ LD + f bol . − . − . . − . − . . − . . − . Fig. 11.—
Radius versus effective temperature with the positionof the metal-rich main-sequence star HD 173701 shown as a reddiamond. Lines compare BaSTI and CESAM evolutionary trackswith different masses and initial Helium fractions (see text). Notethat each track starts at the zero-age main sequence (ZAMS). Thedetermined mass and metallicity for HD 173701 are M = 1 . ± . M ⊙ and [Fe / H] = +0 . ± . ply a mass of 1 . ± . M ⊙ and 0 . ± . M ⊙ forHD 173701, respectively, which would significantly im-prove the agreement. We also note that the ∼
100 K dif-ference to the spectroscopic T eff implies that the adoptedmetallicity may not be consistent with the interfero-metric T eff . However, as shown in Figure 11, even atthe spectroscopic temperature of 5390 K the position ofHD 173701 would still be slightly too cool for the massdetermined from the interferometric radius and astero-seismic density. Additionally, adopting a lower T eff inthe spectroscopic analysis would result in a lower metal-licity, and therefore enhance the disagreement betweenmodels and observations.A more interesting possibility is that the physical as-sumptions in the evolutionary models need to be adjustedto reproduce the properties of this star. To test this, wehave computed additional tracks using the 1D stellar evo-lution code CESAM (Morel & Lebreton 2008). We useopacities from Ferguson et al. (2005) for the metal repar- tition by Asplund et al. (2009), and NACRE nuclear re-action rates are adapted from Angulo et al. (1999). Themodels include diffusion and gravitational settling, andconvection is described using the mixing length theoryby B¨ohm-Vitense (1958) with a solar calibrated value of1.88. We have computed two models with the spectro-scopically determined metallicity of [Fe/H] = 0.34 and amass of 1.06 M ⊙ , once with solar-calibrated initial heliummass fraction Y = 0 . Y = 0 . R ′ HK ) = − .
87. Both the slower rotationperiod and solar-like activity do not seem to be com-patible with a decreased convection efficiency (smallermixing length parameter), which would be needed tobring the models in better agreement to our observa-tions. Additionally, a sub-solar helium mass fractionfor HD173701 does not seem to be compatible with theroughly linear helium-to-metal enrichment for metal-richstars (Casagrande et al. 2007), although the scatter inthis relation is large and studies of the Hyades have con-firmed that stars can be depleted in helium and at thesame time have a super-solar metallicity (Lebreton et al.2001; Pinsonneault et al. 2003).An alternative explanation could be related to inad-equate modeling of stellar atmospheres for metal-richstars. Systematics in these models would affect thebolometric flux and hence the determined effective tem-perature. Furthermore, systematic errors in the limb-darkening models for metal-rich stars would change thederived angular diameter, which influences the deter-mined radius, mass and effective temperature. Detailed3-D models by Bigot et al. (2006) for the metal-richK-dwarf α Cen B showed less significant limb-darkeningand hence slightly smaller diameters compared to sim-ple 1-D models, while Chiavassa et al. (2010) found dif-ferences up to 3% for models of metal-poor giants.Such differences are expected to be enhanced in visibileundamental Properties of Stars using Asteroseismology and Interferometry 15
Fig. 12.—
Same as Figure 10 but for the red giant starHD 175955. The determined mass and metallicity for HD 175955are M = 1 . ± . M ⊙ and [Fe / H] = +0 . ± . wavelengths (such as the observations presented here)compared to infrared observations (Allende Prieto et al.2002; Aufdenberg et al. 2005). Furthermore, compar-isons of 1-D to 3-D models have also yielded higher fluxesfor 3-D models, particularly at short wavelengths, whichcould lead to small inceases in the derived effective tem-perature (see, e.g., Aufdenberg et al. 2005; Casagrande2009). A higher effective temperature would bring bet-ter agreement to the evolutionary tracks and spectro-scopic estimates. More detailed modeling will be neededto confirm if refined estimates of limb-darkening, takinginto account the non-solar metallicity for HD 173701, canexplain the observed differences. HD 175955
Figure 12 presents a model comparison for HD 175955,a red giant with a well constrained angular diameter andthe most precise Hipparcos parallax. Gravity mode pe-riod spacings measured using asteroseismology have beenused to classify this star as a H-shell burning, ascendingred giant branch star (Bedding et al. 2011). Figure 12shows that the measured temperature of HD 175955 isslightly hotter than the position of the ascending RGBtracks, but overall in good agreement with its determinedmass and metallicity. Similar to HD 182736, a combina-tion of the constraints presented here with detailed as-teroseismic studies (such as the measurement of mixedmode rotational splittings to constrain the core rotationrate, see Beck et al. 2012) should allow a detailed the-oretical study of the internal structure and evolution ofthis star.
Additional Notes
We note that for a few stars the derived stellar proper-ties appear unphysical and are likely related to potentialobservational errors. For HD 175726, for example, themeasured linear radius combined with the asteroseismicdensity implies a mass of 0 . ± . M ⊙ , which seemsincompatible with its measured radius, temperature andsolar-metallicity. Using the spectroscopically determinedmetallicity and the radius and temperature from inter-ferometry, a comparison with BaSTI models indicates a mass of 1.07 M ⊙ , which would imply asteroseismic val-ues of ∆ ν ∼ µ Hz and ν max ∼ µ Hz. These val-ues are significantly different than the results found byMosser et al. (2009). The difference could be explainedby an undetected companion causing a significant errorin the parallax, or due to measurement errors in eitherthe interferometric or the asteroseismic analysis. Unfor-tunately no CoRoT follow-up observations are plannedfor HD 175726, and hence a resolution of this discrepancywill have to await independent future observations.The red giant HD 181827, on the other hand, has alarge mass which is difficult to reconcile with evolution-ary theory. Both the asteroseismic and interferometricconstraints are solid, hence pointing to a potential prob-lem with the Hipparcos parallax. Indeed, HD 181827 hasthe largest fractional parallax uncertainty in our sample(10%), leading to a large uncertainty on the radius andhence mass. We note that HD 181827 has been astero-seismically identified as a secondary clump star (Girardi1999; Bedding et al. 2011), corresponding to a massive( & M ⊙ ) He-core burning red giant. Our result of a sig-nificantly higher mass for HD 181827 compared to typicalred clump giants is hence qualitatively in agreement withits asteroseismically determined evolutionary state. CONCLUSIONS
We have presented interferometrically measured angu-lar diameters of 10 stars for which asteroseismic con-straints are available from either the
Kepler or CoRoTspace telescopes. Combining these constraints with par-allaxes, spectroscopy and bolometric fluxes, we presenta full set of near model-independent fundamental prop-erties for stars spanning in evolution from the main-sequence to the red clump. Our main conclusions fromthe derived properties are as follows:1. Our measured angular diameters show good agree-ment with the surface-brightness relation byKervella et al. (2004) and the IRFM coupled withasteroseismic constraints by Silva Aguirre et al.(2012), with an overall residual scatter of 5%.Our results seem to confirm that the relationby Kervella et al. (2004) and the method bySilva Aguirre et al. (2012) are also reasonably ac-curate for red giants.2. A comparison of interferometric to asteroseismicradii calculated from scaling relations shows excel-lent agreement within the uncertainties. While theuncertainties for giants are large due to the uncer-tainties in the parallaxes, our results empiricallyprove that asteroseismic radii for unevolved starsusing simple scaling relations are accurate to atleast 4%. A test of the ν max scaling relation alsoshows no systematic deviations as a function of evo-lutionary state within the observational uncertain-ties.3. A comparison of measured effective tempera-tures with estimates from modeling high-resolutionspectra (mostly using the VWA method, seeBruntt et al. 2010) and from the photometric in-frared flux method (see Casagrande et al. 2010)shows good agreement with mean deviations of6 D. Huber et al. − ±
32 K (with a scatter of 97 K) and − ±
31 K(with a scatter of 93 K), respectively, for stars be-tween T eff = 4600 − Stellar ObservationsNetwork Group , Grundahl et al. 2006), which will de-liver precise multi-site radial-velocity timeseries for as-teroseismology and exoplanet studies of nearby stars.On the other hand, the planned European space mis-sion
Gaia (Perryman 2003) will provide accurate par-allaxes for stars down to
V <
15, while potential up-grades of interferometers such as the CHARA Array withadaptive optics will push the sensitivity limits of inter-ferometric follow-up to fainter stars, therefore improv-ing the overlap with
Kepler and future space-based mis-sions such as TESS (
Transiting Exoplanet Survey Satel-lite , Ricker et al. 2009). The possibility of independentlyconstraining radii, effective temperatures, masses andmetallicities using asteroseismology, astrometry, interfer-ometry and spectroscopy for a large ensemble of stars tostudy stellar physics as well as to characterize potentiallyhabitable exoplanets is clearly the next step for con-tinuing the exciting revolution induced by CoRoT and
Kepler over the coming decades.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the
Kepler
ScienceOffice and everyone involved in the
Kepler mission for making this paper possible. Funding for the
Kepler
Mis-sion is provided by NASA’s Science Mission Directorate.The CHARA Array is funded by the National ScienceFoundation through NSF grant AST-0606958, by Geor-gia State University through the College of Arts and Sci-ences, and by the W.M. Keck Foundation. The CoRoTspace mission, launched on December 27th 2006, hasbeen developed and is operated by CNES, with the con-tribution of Austria, Belgium, Brazil, ESA (RSSD andScience Programme), Germany and Spain. DH is thank-ful to Karsten Brogaard, Pieter Degroote and BenoˆıtMosser for interesting discussions and comments on thepaper. DH, TRB and VM acknowledge support from theAccess to Major Research Facilities Program, adminis-tered by the Australian Nuclear Science and TechnologyOrganisation (ANSTO). DH is supported by an appoint-ment to the NASA Postdoctoral Program at Ames Re-search Center, administered by Oak Ridge AssociatedUniversities through a contract with NASA. I.M.B. issupported by the grant SFRH / BD / 41213 /2007 fromFCT /MCTES, Portugal. J M- ˙Z acknowledges the PolishMinistry grant number N N203 405139. S.G.S acknowl-edges the support from the Funda¸c˜ao para a Ciˆencia eTecnologia (grant ref. SFRH/BPD/47611/2008) and theEuropean Research Council (grant ref. ERC-2009-StG-239953). WJC acknowledges financial support from theUK Science and Technology Facilities Council (STFC).MC acknowledges funding from FCT (Portugal) andPOPH/FSE (EC), for funding through a Contrato Cincia2007 and the project PTDC/CTE-AST/098754/2008.SH acknowledges financial support from the NetherlandsOrganisation for Scientific Research (NWO). KU ac-knowledges financial support by the Spanish NationalPlan of R&D for 2010, project AYA2010-17803. Fundingfor the Stellar Astrophysics Centre is provided by TheDanish National Research Foundation. The research issupported by the ASTERISK project (ASTERoseismicInvestigations with SONG and Kepler) funded by the Eu-ropean Research Council (Grant agreement no.: 267864).This publication makes use of data products from theTwo Micron All Sky Survey, which is a joint project ofthe University of Massachusetts and the Infrared Pro-cessing and Analysis Center/California Institute of Tech-nology, funded by the National Aeronautics and SpaceAdministration and the National Science Foundation.
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