Characterizing two solar-type Kepler subgiants with asteroseismology: KIC10920273 and KIC11395018
G. Dogan, T. S. Metcalfe, S. Deheuvels, M. P. Di Mauro, P. Eggenberger, O. L. Creevey, M. J. P. F. G. Monteiro, M. Pinsonneault, A. Frasca, C. Karoff, S. Mathur, S. G. Sousa, I. M. Brandao, T. L. Campante, R. Handberg, A.O. Thygesen, K. Biazzo, H. Bruntt, E. Niemczura, T. R. Bedding, W. J. Chaplin, J. Christensen-Dalsgaard, R. A. Garcia, J. Molenda-Zakowicz, D. Stello, J. L. Van Saders, H. Kjeldsen, M. Still, S. E. Thompson, J. Van Cleve
aa r X i v : . [ a s t r o - ph . S R ] N ov Draft version March 6, 2018
Preprint typeset using L A TEX style emulateapj v. 12/16/11
CHARACTERIZING TWO SOLAR-TYPE
KEPLER
SUBGIANTS WITH ASTEROSEISMOLOGY:KIC 10920273 AND KIC 11395018
G. DO ˘GAN , , , T. S. METCALFE , , , S. DEHEUVELS , , M. P. DI MAURO , P. EGGENBERGER , O. L.CREEVEY , , , M. J. P. F. G. MONTEIRO , M. PINSONNEAULT , , A. FRASCA , C. KAROFF , S.MATHUR , S. G. SOUSA , I. M. BRAND ˜AO , T. L. CAMPANTE , , R. HANDBERG , A.O. THYGESEN , ,K. BIAZZO , H. BRUNTT , E. NIEMCZURA , T. R. BEDDING , W. J. CHAPLIN , , J.CHRISTENSEN-DALSGAARD , , R. A. GARC´IA , , J. MOLENDA- ˙ZAKOWICZ , D. STELLO , J. L. VANSADERS , , H. KJELDSEN , M. STILL , S. E. THOMPSON , and J. VAN CLEVE High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,Denmark Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106, USA Space Science Institute, Boulder, CO 80301, USA Department of Astronomy, Yale University, PO Box 208101, New Haven, CT 06520-8101, USA INAF-IAPS, Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy Geneva Observatory, University of Geneva, Maillettes 51, 1290 Sauverny, Switzerland Universit´e de Nice, Laboratoire Cassiop´ee, CNRS UMR 6202, Observatoire de la Cˆote d’Azur, BP 4229, 06304 Nice cedex 4, France IAC Instituto de Astrof´ısica de Canarias, C/ V´ıa L´actea s/n, E-38200 Tenerife, Spain Universidad de La Laguna, Avda. Astrof´ısico Francisco S´anchez s/n, 38206 La Laguna, Tenerife, Spain Centro de Astrof´ısica and DFA-Faculdade de Ciˆencias, Universidade do Porto, Portugal Ohio State University, Department of Astronomy, 140 W. 18th Ave., Columbus, OH 43210, USA INAF, Osservatorio Astrofisico di Catania, via S. Sofia, 78, 95123, Catania, Italy Zentrum f¨ur Astronomie der Universit¨at Heidelberg, Landessternwarte, K¨onigstuhl 12, 69117 Heidelberg, Germany INAF - Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131, Napoli, Italy Instytut Astronomiczny, Uniwersytet Wroc lawski, ul. Kopernika 11, 51-622 Wroc law, Poland Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK Laboratoire AIM, CEA/DSM – CNRS - U. Paris Diderot – IRFU/SAp, Centre de Saclay, 91191 Gif-sur-Yvette Cedex, France Bay Area Environmental Research Institute / NASA Ames Research Center, Moffett Field, CA 94035, USA and SETI Institute / NASA Ames Research Center, Moffett Field, CA 94035, USA
Draft version March 6, 2018
ABSTRACTDetermining fundamental properties of stars through stellar modeling has improved substantiallydue to recent advances in asteroseismology. Thanks to the unprecedented data quality obtained byspace missions, particularly CoRoT and
Kepler , invaluable information is extracted from the high-precision stellar oscillation frequencies, which provide very strong constraints on possible stellar modelsfor a given set of classical observations. In this work, we have characterized two relatively faint stars,KIC 10920273 and KIC 11395018, using oscillation data from
Kepler photometry and atmosphericconstraints from ground-based spectroscopy. Both stars have very similar atmospheric properties;however, using the individual frequencies extracted from the
Kepler data, we have determined quitedistinct global properties, with increased precision compared to that of earlier results. We foundthat both stars have left the main sequence and characterized them as follows: KIC 10920273 is aone-solar-mass star ( M = 1 . ± . M ⊙ ), but much older than our Sun ( τ = 7 . ± .
47 Gyr), whileKIC 11395018 is significantly more massive than the Sun ( M = 1 . ± . M ⊙ ) with an age close tothat of the Sun ( τ = 4 . ± .
23 Gyr). We confirm that the high lithium abundance reported forthese stars should not be considered to represent young ages, as we precisely determined them to beevolved subgiants. We discuss the use of surface lithium abundance, rotation and activity relations aspotential age diagnostics.
Subject headings: asteroseismology — stars: evolution — stars: fundamental parameters — stars:individual (KIC 10920273, KIC 11395018) — stars: solar-type INTRODUCTION
Classical modeling of single stars mostly relies on fit-ting to the atmospheric properties obtained throughspectroscopic and/or photometric observations, such aseffective temperature, surface gravity, and elementalabundances. This yields a large number of possible mod-els covering a wide range of values for the fundamentalproperties, particularly in the absence of independent
Electronic address: [email protected] radius and luminosity measurements. Asteroseismologyhas been revolutionizing stellar modeling as a result ofthe precise and accurate inferences on stellar structurethat have been made possible by a new generation of as-teroseismic observations. Stellar fundamental properties,particularly mass and radius, can be determined to a fewpercent uncertainty, even when only the average seismicparameters are used as additional constraints, with theprecision on age determination being as good as ∼ Kepler stars). Moreover, the individual frequencies allow us toobtain information about the stellar interiors.Asteroseismology has proved very effective in con-straining the stellar age and the evolutionary stage withthe help of specific features seen in the oscillation spectra.Mixed modes are particularly important in this regard.As a star evolves, the frequencies of the p modes decreasedue to increasing stellar size, while the g-mode frequen-cies increase. By the time the star moves off the mainsequence, the g- and p-mode trapping cavities are closerto each other, which results in the interaction of the twotypes of modes as they go through “avoided crossings”.The modes affected by this interaction are referred toas mixed modes due to having g-mode characteristics inthe deep interior, and p-mode characteristics near thesurface, of the star (see Osaki 1975 and Aizenman et al.1977 for an introductory discussion). They are sensi-tive to the central conditions and hence encode infor-mation from the core, where the chemical compositionchanges due to the nuclear reactions driving the evolu-tion of the star. Since the timescales of avoided cross-ings are very small compared to the stellar evolutionarytimescale, mixed modes provide very strong constraintson the stellar age (see, e.g., Deheuvels & Michel 2011;Metcalfe et al. 2010; Benomar et al. 2012 for recent anal-yses).
Kepler is a space telescope with a diameter of 0.95 mthat has been providing high-quality photometric datasince the beginning of its operations in May 2009 (see,e.g., Borucki et al. 2010; Koch et al. 2010; Chaplin et al.2010, 2011). The mission’s primary objective is to searchfor Earth-sized planets through the transit method.Asteroseismology is being used to characterize a sub-sample of stars, some of which host planets.
Kepler monitors more than 150,000 stars, and ∼ ∼
10 months of the mission (Gilliland et al. 2010;Chaplin et al. 2011). Solar-like oscillations were de-tected in at least 500 of those survey stars (Chaplin et al.2011). A subsample ( ∼ Kepler bandpass) andKIC 11395018 (kepmag=10.76 mag) are among a hand-ful of asteroseismic targets that were observed continu-ously from the start of science operations. Consequently,extended timeseries were available from early in themission, making both stars attractive targets for aster-oseismic analysis (Campante et al. 2011; Mathur et al.2011). We also acquired ground-based spectra in orderto characterize these stars. They are G-type stars withvery similar spectroscopic properties, especially T eff andlog g (see, Sect. 2.1), so it is difficult to discriminate be- tween models for the two stars using classical approaches.These approaches include matching the position of thestar in the Hertzsprung-Russell (H-R) diagram in theform that shows luminosity versus effective temperatureas the star evolves, or alternatively in the log g - T eff di-agram, given that the luminosity cannot be calculatedusing the available observations.We present the observational data employed to char-acterize our stars in Section 2, our modeling approach inSection 3, and the results in Section 4, while Section 5provides a summary and conclusions. OBSERVATIONAL CONSTRAINTS
Atmospheric properties
Atmospheric properties of KIC 10920273 andKIC 11395018 were obtained from observations with theFIES spectrograph (Frandsen & Lindberg 1999) at theNordic Optical Telescope (NOT on La Palma, Spain),at medium resolution (R ∼ σ error boxes inFig. 1). We adopted a set of atmospheric constraints foreach star that were closest to the mean of results fromseveral methods described by Creevey et al. (2012). Thisapproach was preferred for the sake of reproducibility,rather than using the mean values. However, we didnot restrict our model-searching space to less than 3- σ uncertainty around these constraints; therefore, theselected values represent well the overall results of thespectroscopic analysis.The spectroscopic υ sin i values for KIC 10920273 andKIC 11395018 are 1 . ± . − and 1 . ± . − ,respectively (Creevey et al. 2012). These low values in-dicate either slow rotation or low inclination angle i , al-though the latter is statistically unlikely. Rotational pe-riods from modulation of the Kepler data due to spotson the surfaces of the two stars were measured to be ∼
27 days for KIC 10920273 (Campante et al. 2011), and ∼
36 days for KIC 11395018 (Mathur et al. 2011). Thesignal-to-noise ratios (SNR) of the peaks used for thesemeasurements, particularly for KIC 10920273, were low,so the results should be used with caution. Mathur et al.(2011) inferred i > ◦ for KIC 11395018 based on com-bining the rotational frequency splittings with the mea-sured rotation period, which implies this star to be aslow rotator. A wider range of inclinations was pos-sible for KIC 10920273 (Campante et al. 2011). How-ever, the modulation in the light curve has been detectedwith less uncertainty in the new analysis performed usinglonger time series including Kepler
Q9 and Q10 data ofKIC 10920273 (Garc´ıa et al., private communication). Ifconfirmed, this would be consistent with a relatively highinclination angle, and slow rotation. The effects of thecentrifugal force are negligible for slowly rotating stars. However, rotational mixing may lead to changes in the If, counter to our expectations, one of these stars were to beconfirmed as a fast rotator, rotational effects on the oscillationfrequencies would have to be taken into account (see Su´arez et al.2010 for a detailed analysis of the effects of centrifugal distortionon solar-like oscillations). Currently, there is no robust detection ofrotational frequency splittings that can be included as constraintsin our modeling.
HARACTERIZING KIC 10920273 AND KIC 11395018 3
TABLE 1Adopted atmospheric constraints for KIC 10920273and KIC 11395018 (from Creevey et al. 2012)
Star T eff (K) log g [Fe/H]KIC 10920273 5790 ±
74 4 . ± . − . ± . ±
100 4 . ± .
20 0 . ± . properties of the models even for slowly rotating stars,because the efficiency of this mixing is more directly re-lated to differential rotation in stellar interiors ratherthan to surface rotational velocities (Pinsonneault et al.1990; Eggenberger et al. 2010). Studying the impact ofrotation on post-main-sequence stars would require a de-tailed discussion of the effects of rotational mixing onthe chemical gradients in the central parts of the star.These influence the asteroseismic properties of the mod-els, particularly the mixed modes. In the specific case ofthe evolved post-main-sequence stars modeled here, how-ever, we expect that these effects on the chemical gradi-ents would already be erased, as found by Miglio et al.(2007) for models of 12 Bootis A in the thick-shell-H-burning phase (see also the discussion of the effectsof microscopic diffusion in the subgiant HD 49385 byDeheuvels & Michel 2011). We therefore have not in-cluded the rotational effects for most of the analyses (seeSection 3).When the atmospheric properties alone are considered,these two stars are very similar. Due to the degen-eracy inherent in the H-R diagram analysis (see, e.g.,Fernandes & Monteiro 2003), it is not possible to deter-mine the global stellar properties with sufficiently highprecision to study their detailed characteristics withoutthe help of seismic data, which we now discuss. Fig. 1.—
Log g -T eff diagram for KIC 10920273 andKIC 11395018. Surface gravity, g , is in cgs units. Spectroscopicconstraints given in Table 1 are shown by 1- and 2- σ error boxes(dotted; blue for KIC 10920273 and red for KIC 11395018). Evolu-tionary tracks of two models indicated by star symbols (SA1 andBA1 from Tables 4 and 5) are plotted using the same color code.The points with error bars represent the weighted means and thestandard deviations of the asteroseismic determinations (see Ta-bles 4 and 5). Asteroseismic data
We used
Kepler data from observations made in theperiod from 2009 May to 2010 March, i.e., from the com-missioning run (Q0) through Quarter 4 (Q4). The formalfrequency resolution is ∼ µ Hz. From the power spec-tra, Campante et al. (2011) and Mathur et al. (2011)reported individual frequencies for KIC 10920273 andKIC 11395018, based on analyses performed by severalteams. The final sets of results included a minimal and amaximal list of frequencies, where the former were thoseagreed upon by more than half of the fitters and the latterwere those agreed upon by at least two fitters. There-fore, the frequencies that are in the maximal list butnot the minimal list are less certain. For details of thefrequency-extraction techniques and the selection meth-ods, we refer the reader to Campante et al. (2011) andMathur et al. (2011). The analysis of each star resultedin the extraction of up to a total of 25 individual os-cillation frequencies for radial ( l = 0), dipole ( l = 1),and quadrupole ( l = 2) modes, including several mixedmodes. These mixed modes carry information from thecore and hence provide stronger constraints on the evolu-tionary stage of the stars, as discussed in Section 1. Westarted by searching for models using the minimal-listof frequencies and then extended our analysis to includeadditional frequencies from the maximal lists. MODELING APPROACH
Asteroseismic modeling is performed by optimizing thestellar model parameters to match the observed seismicquantities and also the classically observed (or derived)stellar properties, such as effective temperature, surfacegravity, surface metallicity (along with radius, mass andluminosity, when available). The seismic quantities in-clude, but are not limited to, the average large and smallfrequency separations , the frequency of maximum powerin the oscillation spectrum ( ν max ), and the individual os-cillation frequencies. Naturally, the individual frequen-cies provide the most detailed information and the high-est precision in the derived stellar properties (see, e.g.,Metcalfe et al. 2010 and Mathur et al. 2012).We used the individual oscillation frequencies and theatmospheric properties ( T eff , log g , and [Fe/H]) as con-straints to carry out the stellar model optimization. Asan initial guess for the parameter space to be searched,we used preliminary results of the mass determinationfrom the analysis of Creevey et al. (2012), which were inagreement with their final results within the uncertain-ties. They derived stellar properties using the averageseismic quantities together with the atmospheric con-straints. The final values given by Creevey et al. (2012)were 1 . ± . M ⊙ for KIC 10920273 and 1 . ± . M ⊙ for KIC 11395018.Five teams participated in the modeling of these starsusing a variety of evolutionary codes and fitting methods.Most of the methods were either based on searching forthe best-fitting model in a grid specifically computed forthis analysis or on using a pre-existing grid to determinethe general area of the stellar properties in the parame- The large frequency separation is ∆ ν n,l = ν n,l − ν n − ,l , andthe small frequency separation is δν n,l = ν n,l − ν n − ,l +2 , where ν n,l is the frequency of the mode with spherical degree l and radialorder n . Dogan et al.ter space before going into further refinement process forindividual stars. One team used the Asteroseismic Mod-eling Portal (AMP), which is a pipeline analysis tool thatoptimizes the seismic and non-seismic properties globallyusing a genetic algorithm. AMP starts the model-searchwith four random independent sets of initial parametersand performs the search over a large parameter space(Metcalfe et al. 2009; Woitaszek et al. 2009). The vari-ety of codes and methods employed give us an estimate ofthe external uncertainties inherent in the analysis. Thelist of codes and the configurations regarding the inputphysics are presented in Table 2.The individual fitting methods also differed slightly.ASTEC1 calculated grids of models within the 3- σ un-certainty of the non-seismic constraints and performedthe optimization by a 2-step process, refining the gridsseveral times in the second step guided by the seismic χ values – described by Equation (3). ASTEC2 ex-plored the models, which included turbulent diffusion,and calculated individual models guided by the frequen-cies. CESAM looked for models reproducing the firstavoided crossing as an initial requirement and then per-formed an optimization using χ -minimization to deter-mine stellar mass and age (see Deheuvels & Michel 2011for details of this method). The Geneva stellar evolutioncode was used to compute grids of rotating models withan initial velocity of 50 km s − on the zero-age main se-quence (ZAMS). This value results in surface velocitiesthat are typically lower than 10 km s − at the end of themain sequence (MS) for a solar-type star that is assumedto undergo magnetic breaking on the MS due to thepresence of a convective envelope (Krishnamurthi et al.1997). The initial parameters used by each team aregiven in Table 3.Oscillation frequencies of low-degree modes were calcu-lated by LOSC (Scuflaire et al. 2008) for stellar modelscomputed by CESAM, while the Aarhus Adiabatic Pul-sation Package (ADIPLS, Christensen-Dalsgaard 2008b)was used to calculate the frequencies for all of the othermodels.In relation to the oscillation frequencies, there is awell-known offset between the observed and the modelfrequencies for the Sun and solar-type stars, due to in-accurate representation of the near-surface layers in themodels. To address this issue, we used the empirical cor-rection suggested by Kjeldsen et al. (2008), who showedthat the difference between the observed and calculatedsolar frequencies, which gets larger with increasing fre-quency, can be fitted by a power law: ν obs ( n, − ν best ( n,
0) = a (cid:18) ν obs ( n, ν (cid:19) b , (1)where ν obs ( n,
0) and ν best ( n,
0) are the observed and bestmodel frequencies with degree l = 0 and radial order n , ν is a constant frequency usually chosen to be thefrequency at maximum oscillation power, a is the size ofthe correction at ν and can be calculated for each model,and b is the exponent to be determined.The right-hand-side of Equation (1) is the correctionterm to be added to the acoustic (p-mode) frequenciesof the best models. The mixed modes, however, are lesssensitive to the properties of the near-surface layers sincemuch of their energy is confined to the stellar center. In other words, we need to apply a smaller near-surface cor-rection to the frequency of a mixed mode than to a pmode with a similar frequency. Following Brand˜ao et al.(2011), we scaled the magnitude of the correction in-versely with Q nl , the inertia of a given mode normalizedby the inertia of a radial ( l = 0) mode at the same fre-quency (see, e.g., Aerts et al. 2010). Note that the inertiaof a mixed mode is much higher than that of a p mode.The correction to be applied to all calculated frequenciesis then of the form: ν corr ( n, l ) = ν best ( n, l ) + a (cid:18) Q nl (cid:19) (cid:18) ν best ( n, l ) ν (cid:19) b , (2)where ν corr represents the corrected model frequencies.Notice that ν obs ( n,
0) on the right-hand-side of Equa-tion (1) is replaced by the best model frequencies in orderto allow us to correct the frequencies outside the rangeof observed radial modes (see, Brand˜ao et al. 2011, fordetails).The solar value of the exponent b was calculatedby Kjeldsen et al. (2008) to be 4.90 using the GOLFdata (Lazrek et al. 1997) and the solar “Model S” ofChristensen-Dalsgaard et al. (1996). This value wasfound to range from 4.40 to 5.25 for the same model,depending on the number of radial orders included inthe calibration, but a was found to vary by less than0.1 µ Hz in all cases. Kjeldsen et al. (2008) suggestedthat the solar b value may be used for solar-like starsand this approach was successfully applied to β Hyi(Brand˜ao et al. 2011), HD 49385 (Deheuvels & Michel2011), KIC 11026764 (Metcalfe et al. 2010) and a sam-ple of
Kepler stars (Mathur et al. 2012). In this workASTEC1, ASTEC2, and Geneva codes adopted the solarvalue b = 4 .
90 from Kjeldsen et al. (2008) for calculat-ing the correction term, while AMP adopted b = 4 . b = 4 .
25, the calibrated value using the GOLF data(Gelly et al. 2002). Given how little a varies for a rel-atively large range of b for a given model, as discussedabove, using slightly different b values for model-fittinghas a negligible impact on the results. RESULTS AND DISCUSSION
Global properties
To select the best models, we defined the two normal-ized χ measures shown in Equations (3) and (4), whichallowed us to evaluate the qualities of the fits for theatmospheric parameters and the seismic parameters sep-arately. The seismic measure was χ = 1 N X n,l (cid:18) ν obs ( n, l ) − ν corr ( n, l ) σ ( ν obs ( n, l )) (cid:19) , (3)where N is the number of observed frequencies, ν corr ( n, l )represents the near-surface-corrected model frequencieswith spherical degree l and radial order n , ν obs ( n, l ) arethe observed frequencies, and σ ( ν obs ( n, l )) are the uncer-tainties on the observed frequencies. The measure forthe atmospheric properties was χ = 13 X (cid:18) P obs − P mod σ (P obs ) (cid:19) , (4)HARACTERIZING KIC 10920273 AND KIC 11395018 5where P= { T eff , log g , [Fe/H] } and the subscripts “obs”and “mod” represent the observed and model properties,respectively, with σ (P obs ) denoting the observational un-certainties. The values of [Fe/H] for the models werecalculated using the formula [Fe / H] = log(
Z/X ) mod − log( Z/X ) ⊙ , where the solar value was adopted fromGrevesse & Noels (1993) as ( Z/X ) ⊙ = 0 . χ values . We also present models fittedusing more or fewer frequencies than those in the mini-mal lists in order to see whether the model-fitting resultschange considerably. In each case, we calculated the χ in the tables using only the frequencies that were com-mon constraints for all of the models, in order to achievea consistent evaluation of the models.There is a good agreement between the observed fre-quencies and the model frequencies. The quality of thefits can be seen in the ´echelle diagrams for a sample ofmodels (Fig. 2). Overall frequency patterns, includingthe dipolar mixed modes, are matched quite well. Thefact that χ > . χ ( & . X c = 0 .
0) but have quite different characteristics, as seenin Tables 4 and 5. The fact that KIC 10920273 is an oldsolar analog (with one solar mass and near-solar metal-licity) makes it an interesting target for further studies.A typical solar model would turn off from the MS ataround 9-10 Gyr. However, the metallicity and partic-ularly the helium abundance alter this age estimate. Inthis case it is the high helium abundance that affectsthe MS turn-off age more than the low metallicity. Themodels with higher helium abundance behave similar tothose with higher mass (higher luminosity), following anevolutionary track similar to that of a higher-mass star,and hence have shorter MS lifetimes.The results in Tables 4 and 5 were weighted by thegoodness of the seismic fit, i.e. the inverse of χ . Thisway, any misleading contribution due to coarse grids iseliminated. Although this does not provide a direct mea-surement of the systematic uncertainties, it still allows usto estimate the order of magnitude of the external errorsexpected from using different inputs, codes, and fittingmethods. A similar determination of the systematic er- The labels of the models presented in Tables 4 and 5 start with“S” and “B” for KIC 10920273 and KIC 11395018, which stand for“Scully” and “Boogie” – the nicknames of the stars within KeplerAsteroseismic Science Consortium, Working Group 1. rors for the case of bright
Kepler stars 16 Cyg A and Bwas carried out by Metcalfe et al. (2012) using differentevolutionary codes and fitting methods. The uncertain-ties we determined are mostly greater due to the lowerSNR in the data of our faint stars. Systematic errorsin determination of stellar properties using grid-basedpipelines caused by different observational constraintsand different input physics were discussed more gener-ally for a few
Kepler stars by Creevey et al. (2012). Wediscuss the uncertainties further in the next section.
Fig. 2.— ´Echelle diagrams for the selected models ofKIC 10920273 (SA3, upper panel) and KIC 11395018 (BD, lowerpanel). The radial orders are plotted horizontally (see Bedding2012) and the vertical axes indicate the frequencies at the middle ofeach order. The background is a gray-scale map of the power spec-trum from the observations, with the observed frequencies shownby filled symbols with 1- σ error bars. Minimal-list frequencies areplotted in red, while the additional maximal-list frequencies areplotted in blue. Open symbols are used for the model frequencies,with smaller size implying larger normalized mode inertia. Circles,triangles, and squares represent l = 0, l = 1, and l = 2 modesrespectively. Discussion of uncertainties
To evaluate the typical uncertainties caused by dif-ferent input physics further, we calculated some addi-tional models and small grids using KIC 10920273 as a Dogan et al.test case. We selected SB1 as our base model. Keepingthe input parameters (mass,
Z/X , Y , α ) fixed, we firstexplored the effects of changing one single ingredient ofinput physics at a time. We also changed the value of α while keeping everything else fixed. On every new evo-lutionary sequence, we calculated the frequencies of themodels that had all atmospheric properties (log g , T eff ,and [Fe/H]) within 3 σ of the observed values. We thenselected the model that best matched the observed fre-quencies. In the cases of including core overshoot, chang-ing the convection treatment (to CGM formulation), andusing several different nuclear rates (Bahcall et al. 1995;Parker 1986; Adelberger et al. 1998), the new models re-produced the observed frequencies as well as the basemodel with an age difference of only 0.7% at most (whichcorresponds to changes in radius of < .
2% and in log g of < . χ comparable to that of the base model, wecarried on with the analysis. These cases resulted fromincluding diffusion and gravitational settling of helium,using two different versions of the low-temperature opac-ities given in Table 2, and varying the value of mixinglength parameter (in the range of 1.6–2.0). For each ofthese cases, we computed additional small grids aroundthe base model by varying all the input parameters, inorder to see how much the output properties were differ-ent for two models with different input physics but sim-ilar frequencies. We then calculated the weighted meanand standard deviation in the same way as in the orig-inal analysis. The standard deviation in this case rep-resents the typical uncertainties caused by using a fixedset of input physics, hence decreasing the level of model-dependance in the results substantially. The mean val-ues for age, luminosity, radius, T eff , and log g calculatedfrom the additional analysis agreed with the original re-sults within 1 σ , while their standard deviations were ofthe same order as those qiven in Table 4. This confirmsthat the uncertainties presented here are realistic. More-over, the resulting values of radius and log g from theadditional analysis are essentially the same as the orig-inal results. This is reassuring given the importance ofasteroseismology in determining the radius in a robustway.The internal uncertainties were different for eachmethod. However, the dominant source of uncertainty isthe non-uniqueness of the solution rather than the sta-tistical errors. Parameter correlations allow a trade-offbetween parameters, leading to different families of so-lutions that are almost equally good seismic fits. Theeffective range of these correlations is narrowed substan-tially, but not eliminated, by the use of seismic data. Weused a single method (AMP) to evaluate the uniquenessof the best models for both stars, since the uniquenessdepends on the specific constraints adopted in each case.AMP finds the lowest value of χ in the entire searchspace. Along the way it also identifies the secondaryminima – which can be far away from, but not muchworse than, the formally best solution. For both stars,these secondary minima (SA2 and BA2 in Tables 4 and 5)are marginally worse than the best solutions found using the same set of inputs (SA1 and BA1), but the valuesof the mass and helium abundance are quite different.This reflects the well-known mass-He degeneracy (see,e.g., Lebreton et al. 1993; Fernandes & Monteiro 2003;Metcalfe et al. 2009), which is not entirely lifted, evenwith the help of asteroseismology. The mass is con-strained more strongly than it would be without aster-oseismic data but in the absence of external constraintson the helium abundance, we cannot choose one partic-ular model. Consequently, we used both the primary-and secondary-minimum solutions from AMP in the cal-culation of the weighted means. The range of the AMPresults leads to relatively large “uniqueness uncertain-ties” in the fitted stellar properties. These are deter-mined as follows for KIC 10920273, and KIC 11395018,respectively: 3%, and 7% in stellar mass; 8%, and 21%in the initial metallicity ( Z/X ) i ; 10%, and 25% in Y i ;7%, and 8% in age; and 1%, and 2% in radius. Note thatthe larger uncertainties in the mass and chemical com-position for KIC 11395018 reflect the fact that it has two observed avoided crossings, which provide more strin-gent constraints such that neighboring models are signif-icantly worse. Only a relatively large jump along the pa-rameter correlations yielded a secondary minimum witha comparable seismic fit (cf. Metcalfe et al. 2010). Whenwe evaluate the uncorrelated statistical uncertainties foreach of these minima with a local analysis using singu-lar value decomposition (SVD), we found very small er-rors (e.g., as low as 5 × − % for mass), reflecting thesteep χ surfaces corresponding to these results. Al-though these ’local’ uncertainties are real, i.e., changingthe values of the parameters by the correlated ’tiny’ un-certainties causes a large increase in the χ ( > ν = 1135 . ± . µ Hz), while the other one was identi-fied as a quadrupole mode (with ν = 873 . ± . µ Hz)that was tagged as a possible mixed mode introduced aposteriori (Campante et al. 2011). Both SA4 and SB2were selected using this alternative frequency set forKIC 10920273, and the agreement between the model andobservations was not affected substantially. Therefore,we cannot ascertain whether these two peaks are stellarin origin. Nevertheless, we do not completely rule outthe possibility of these peaks being stellar as some of themodels that result from using alternative frequency setsdo contribute to the weighted mean values significantly.Furthermore, we note that the peak at ν = 873 . µ Hz,which is tagged as a possible quadrupole mixed mode, isin the middle of the frequencies of a dipole mixed modeand a quadrupole mode in most of our models. So theHARACTERIZING KIC 10920273 AND KIC 11395018 7observed peak may correspond to a dipole mixed mode,which, according to the models, has a relatively low nor-malized mode inertia indicating an observable amplitude.
Comparison with previous results
Comparing our results with those from the pipelineanalyses of Creevey et al. (also given here in Tables 4and 5), we see that the mass determinations from thepipeline analyses were higher, which led to lower age es-timates. We emphasize that the previous pipeline analy-ses used only the average seismic quantities, hence lack-ing additional information from the individual frequen-cies and being affected by the uncertainties of the scal-ing relations. Therefore, some deviation from their val-ues was expected. Nonetheless, it is reassuring that themass, radius, and age determinations of Creevey et al.(2012) are within 2- σ uncertainty limits of our resultsfor KIC 10920273, and within 1 σ for KIC 11395018. Wealso confirm the robust determination of log g using scal-ing relations and grid-based analyses (see Table 7 inCreevey et al. 2012), with which our results are in agree-ment within 1 σ . Additionally, we note that our resultsconfirm that the mass and radius determined using onlythe scaling relations (e.g., Mathur et al. 2012) providegood initial estimates for these properties (1 . ± . M ⊙ and 1 . ± . R ⊙ for KIC 10920273; 1 . ± . M ⊙ and2 . ± . R ⊙ for KIC 11395018).There is excellent agreement, for both stars, betweenour results and the mass estimates of Benomar et al.(2012), who used the coupling strength of the observedmixed modes to determine the masses of several sub-giants, including KIC 10920273 (1 . ± . ± . M ⊙ )and KIC 11395018 (1 . ± . ± . M ⊙ ).The most substantial improvement in this work comesfrom the use of individual frequencies which yield in-creased precision, with age being affected the most. Thepresence of the mixed modes in the data allowed us todetermine the age with 5-7% precision, although withsome model-dependency. This result is a major improve-ment on the 35-40% precision in age achieved using at-mospheric and mean seismic parameters. Both stars aredetermined to be post-main-sequence subgiants with nohydrogen left in their cores. Evolutionary tracks of theselected models are shown in Fig. 1.Although we did not restrict the parameter searchto be within 1- σ uncertainty around the spectroscopicconstraints, the weighted mean values of T eff from themodels are within 1- σ limit for both stars, while log g results are in agreement with the spectroscopic valueswithin 2 σ (see Fig. 1), and [Fe/H] within 1.5 σ . Ourlog g results are in excellent agreement with asteroseis-mic log g values obtained from scaling relations (givenby Creevey et al. (2012) and also in Tables 4 and 5). Wealso note that our temperature results are in good agree-ment with the revised photometric values for the KeplerInput Catalog (KIC) from Pinsonneault et al. (2012),who derived T eff = 5872 ±
70 K for KIC 10920273, and T eff = 5650 ±
59 K for KIC 11395018.
Non-seismic age diagnostics
We discussed the asteroseismic constraints on the stel-lar age in Section 1. Here we discuss the implications ofrotation and stellar activity on the age, as well as thoseof the surface lithium abundance. Rotation and activity are potentially valuable diagnos-tics of stellar age. It was shown that the Ca + emissionluminosity, an indicator of stellar activity, decays roughlyas t − . for some cluster stars and the Sun (Skumanich1972), furthermore, rotational decay was shown to followthe same law. Large samples of stellar rotation periodshave been collected, and the Kepler mission promisesmany more. There is therefore substantial interest instellar rotation-mass-age, or gyrochronology, relations(see Barnes 2003, 2007; Mamajek & Hillenbrand 2008;Meibom et al. 2011; Epstein & Pinsonneault 2012). Wediscussed in Section 2.1 that relatively slow rotation isinferred for both stars. Slow rotation rates imply rel-atively old stars, which is consistent with our astero-seismic determinations. However, one would not expectmain-sequence spin-down relationships to apply directlyto the evolved stars. Therefore, we cannot use the ro-tation rates for these stars to infer their ages with theage-rotation relations established for MS stars; these re-lations need to be calibrated for more evolved stars usinglarger samples.We have analyzed the chromospheric activity in theCa ii HK lines and found the levels of activity in bothstars to be very low. Fig. 3 shows the Ca ii K and H linesof KIC 10920273 and KIC 11395018 compared to the Sun.The solar spectrum was obtained from the solar light re-flected by Ganymede, which was observed with HARPSin April 2007 , when the Sun was close to the minimumof its activity cycle. We accounted for the different re-solving power of HARPS (R ≃ , ≃ , Kepler stars have chro-mospheric activity levels comparable to the quiet Sun, orlower. These low activity levels are consistent with theold ages we infer from asteroseismology; however, therough nature of the empirical age-activity relations forpost-MS stars does not allow us to make a quantitativeanalysis to infer ages.Another independent determination of stellar age maybe obtained by measuring the Li content at the stel-lar surface. Lithium is easily destroyed in stellar inte-riors and is only produced under unusual circumstances;it has therefore been employed as an age indicator forlow-mass stars. Lithium can be directly depleted if thesurface convection zone is deep enough. It can also bemixed into the radiative interior, or it can be stored be-low the surface convection zone by microscopic diffusionprocesses. In standard stellar models, pre-main-sequencedepletion occurs for most low-mass stars when they havedeep convection zones, and it is most severe in lowermass stars (Bodenheimer 1965). In a qualitative sense,a detection of Li in very cool stars is a strong indica-tor of youth. However, standard models also predictthat stars of order 0.9 solar masses and higher wouldnot experience main-sequence Li depletion, and there isstrong evidence from open clusters for a steady decreasein Li as a function of time, even for stars more massivethan the Sun (see Zappala 1972; Pinsonneault 1997; andSestito & Randich 2005 for reviews.)It was also shown by Randich (2010) that, for a fraction Dogan et al.
Fig. 3.—
Chromospheric activity in the Ca ii K and Ca ii H lines (flux relative to the continuum) for KIC 10920273 (upper panels) andKIC 11395018 (lower panels). The solar spectrum (Ganymede taken in 2007 with HARPS) is overplotted with a dotted (red) line. Theresiduals between the stellar and solar spectra (at the bottom of each plot) show that the two stars have activity levels comparable to thequiet Sun or lower. of solar-like stars with effective temperatures between5750 K and 6050 K, Li is not further depleted after theage of ∼ N (Li)= 2 . ± . N (Li)= 2 . ± . N (Li) = log [ n (Li) /n (H)]+12, with n being thenumber density of atoms and log N (H) = 12 by defini-tion). Considering the empirical Li-age relation estab-lished by Sestito & Randich (2005), Creevey et al. thendetermined that the given Li abundances would indicatelow ages (1-3 Gyr for KIC 10920273 and 0.1-0.4 Gyr forKIC 11395018), which are incompatible with the astero-seismic ages they determined through the pipeline mod-eling (see Tables 4 and 5) performed using the averageasteroseismic quantities.However, in addition to the bi-modality mentionedabove, the age-Li relation has been shown to be validfor MS stars and does not necessarily extend to moreevolved stars. This makes the age determination using the Li abundance ambiguous. Thus, despite the highLi abundance, we are confident that these are indeedevolved stars that have left the main sequence, due to thepresence of the mixed modes in the observed oscillationspectra and as confirmed by our asteroseismic analysis. SUMMARY AND CONCLUSIONS
We performed asteroseismic modeling of two
Kepler stars, KIC 10920273 and KIC 11395018, for which wehave long seismic data sets ( > . ± . M ⊙ ) and an age of τ = 7 . ± .
47 Gyr,while KIC 11395018 has a mass of 1 . ± . M ⊙ and anHARACTERIZING KIC 10920273 AND KIC 11395018 9age very close to that of the Sun ( τ = 4 . ± .
23 Gyr).These results agree, at the 2- σ level for KIC 10920273and 1- σ level for KIC 11395018, with the properties de-termined using the average asteroseismic quantities. Theresults presented here are much more precise than thosefrom the average asteroseismic quantities, though, andthey are also more accurate as a result of using moreobservational information, i.e. individual frequencies,and stronger constraints extracted from the observations,such as the mixed modes. We confirm the robust deter-mination of log g from the average seismic quantities, asour results are within 1- σ uncertainty of the pipeline re-sults.We confirmed these stars to be subgiants (havingevolved off the main sequence) and this allowed us toresolve the disagreement between the seismic ages deter-mined from the pipeline analyses and the ages estimatedusing the lithium abundance and the empirical Li-agerelationship. Basically, the Li abundance cannot be em-ployed to estimate the ages of the subgiants. Similarly,existing age-rotation-activity relations can only be in-dicative for subgiants as these relations are calibratedmostly for the main-sequence stars. This must be takeninto account for gyrochronology studies.We will soon obtain longer data sets from Kepler formany more stars and our results are a good indication ofwhat we can achieve. We note that KIC 10920273 andKIC 11395018 are at the faint end of the
Kepler astero-seismic targets; hence, this work sets a lower limit to thequality of information we can expect from asteroseismol-ogy.
We thank the entire
Kepler team, without whom these resultswould not be possible. Funding for this Discovery mission is pro- vided by NASA’s Science Mission Directorate. We also thank allfunding councils and agencies that have supported the activitiesof KASC Working Group 1. This article is based on observa-tions made with the Nordic Optical Telescope (NOT) operatedon the island of La Palma in the Spanish Observatorio del Roquede los Muchachos. GD gratefully acknowledges financial supportfrom the following institutions: Advanced Study Program (ASP)of the National Center for Atmospheric Research (NCAR), NASAunder Grant No. NNX11AE04G (together with MP and TSM),The Danish Council for Independent Research; and thanks Y.Elsworth, S. Hekker, M. St¸e´slicki, J. C. Su´arez, M. J. Thomp-son, and the anonymous referee for useful comments. NCAR ispartially supported by the National Science Foundation. Fund-ing for the Stellar Astrophysics Centre is provided by The Dan-ish National Research Foundation. The research is supportedin part by the ASTERISK project (ASTERoseismic Investiga-tions with SONG and Kepler) funded by the European ResearchCouncil (Grant agreement no.: 267864). AOT acknowledges sup-port from Sonderforschungsbereich SFB 881 ”The Milky Way Sys-tem” (subproject A5) of the German Research Foundation (DFG).IMB is supported by the grant SFRH / BD / 41213 /2007 fromFCT /MCTES, Portugal. IMB and MJPFG were supported inpart by grant PTDC/CTE-AST/098754/2008 from FCT-Portugaland FEDER. JM- ˙Z acknowledges the Polish Minstry grant num-ber N N203 405139. KB acknowledges the funding support fromthe INAF Postdoctoral fellowship. This research was carried outwhile OLC was a Henri Poincar´e Fellow at the Observatoire dela Cˆote d’Azur. The Henri Poincar´e Fellowship is funded by theConseil G´en´eral des Alpes-Maritimes and the Observatoire de laCˆote d’Azur. RAG has received funding from the European Com-munity’s Seventh Framework Programme (FP7/2007-2013) undergrant agreement No. 269194 (IRSES/ASK). SGS acknowledgesthe support from the Funda¸c˜ao para a Ciˆencia e Tecnologia (grantref. SFRH/BPD/47611/2008) and the European Research Coun-cil (grant ref. ERC-2009-StG-239953). TLC acknowledges finan-cial support from project PTDC/CTE-AST/098754/2008 fundedby FCT/MCTES, Portugal. WJC acknowledges financial supportfrom the UK Science and Technology Facilities Council. We ac-knowledge the KITP staff at UCSB for their warm hospitality dur-ing the research program “Asteroseismology in the Space Age”.This research was supported in part by the National Science Foun-dation under Grant No. PHY05-51164.REFERENCESAdelberger, E. G., Austin, S. M., Bahcall, J. N., et al. 1998,Reviews of Modern Physics, 70, 1265Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. 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Team Diffusion Convection Overshoot EOS Opacities Nuclear reaction& settling treatment (core) (high/low temperature) ratesAMP a (ASTEC) b He c MLT d no OPAL2005 e OPAL f /Alexander & Ferguson (1994) B & P (1992) g ASTEC1 none MLT no OPAL2005 OPAL/Ferguson et al. (2005) NACRE h ASTEC2 He & heavy elements MLT no OPAL2005 OPAL/Ferguson et al. (2005) NACRECESAM i none CGM j yes OPAL2005 OPAL/Alexander & Ferguson (1994) NACREGeneva k He & heavy elements l MLT yes OPAL2005 OPAL/Alexander & Ferguson (1994) NACREa) Metcalfe et al. (2009), b) Aarhus Stellar Evolution Code (Christensen-Dalsgaard 2008a), c) as described by Michaud & Proffitt (1993), d) Mixing length theory (B¨ohm-Vitense1958), e) Rogers & Nayfonov (2002), f) Iglesias & Rogers (1996), g) Bahcall & Pinsonneault (1992), h)Angulo et al. (1999), i) Morel (1997), j) Canuto-Goldman-Mazzitelli model forturbulent convection (Canuto et al. 1996), k) Eggenberger et al. (2008) l) Proffitt & Michaud (1991)
TABLE 3Parameter space searched by each team
Team M / M ⊙ Z/X Y α α ov AMP (ASTEC) 0.75–1.75 0.0026–0.079 0.22–0.32 α MLT =1.0–3.0 N/AASTEC1 1.00–1.60 0.01–0.07 0.24–0.32 α MLT =1.8 N/AASTEC2 1.2–1.4 0.025–0.046 0.26–0.30 α MLT =1.78–1.84 N/ACESAM N/A* 0.026–0.042 0.24–0.28 α CGM = 0.52–0.68 0.0–0.2Geneva 1.00–1.50 0.016–0.040 0.25–0.30 α MLT =1.8 0.1* For each given set of parameters (
Z/X, Y, α, α ov ), the method proposed by Deheuvels & Michel (2011) results in a precise estimate of the mass by using the observed large frequencyseparation and the frequency of the mixed modes. D oga n e t a l. TABLE 4
Fitted parameters for KIC 10920273.
Model
M/M ⊙ ( Z/X ) i Y i α t (Gyr) L/L ⊙ R/R ⊙ T eff (K) log g [Fe/H] X c χ χ SA1 (AMP) 1.00 0.0154 0.296 2.04 6.74 3.31 1.779 5844 3.937 − − a − b − − b − b c − − − ± ± ± ± ± ± d ± e a Maximal-list frequencies are used as input. b Two frequencies are excluded from the maximal-list frequencies (see text). c Canuto-Goldman-Mazzitelli (CGM) model for turbulent convection (Canuto et al. 1996) is used in this model. d This is the value when the average small frequency separation is also used as a seismic constraint. e Asteroseismic log g obtained from scaling relations (Table 7 of Creevey et al. 2012) TABLE 5
Fitted parameters for KIC 11395018.
Model
M/M ⊙ ( Z/X ) i Y i α t (Gyr) L/L ⊙ R/R ⊙ T eff (K) log g [Fe/H] X c χ χ BA1 (AMP) 1.23 0.034 0.301 1.94 4.46 4.40 2.158 5697 3.860 0.144 0.0 8.07 0.49BA2 (AMP) 1.32 0.028 0.241 1.94 4.84 4.53 2.210 5671 3.869 0.049 0.0 8.51 0.69BA3 (AMP) a a b c ± ± ± ± ± ± d ± e a Maximal-list frequencies are used as input. b No diffusion is included in this model unlike other ASTEC2 models. c Canuto-Goldman-Mazzitelli (CGM) model for turbulent convection (Canuto et al. 1996) is used in this model. d This is the value when the average small frequency separation is also used as a seismic constraint. e Asteroseismic log gg