Asteroseismology of red giants from the first four months of Kepler data: Fundamental parameters
T. Kallinger, B. Mosser, S. Hekker, D. Huber, D. Stello, S. Mathur, S. Basu, T.R. Bedding, W.J. Chaplin, J. De Ridder, Y.P. Elsworth, S. Frandsen, R.A. Garcia, M. Gruberbauer, J.M. Matthews, W.J. Borucki, H. Bruntt, J. Christensen-Dalsgaard, R.L. Gilliland, H. Kjeldsen, D.G. Koch
aa r X i v : . [ a s t r o - ph . S R ] O c t Astronomy&Astrophysicsmanuscript no. 15263 c (cid:13)
ESO 2018October 2, 2018
Asteroseismology of red giants from the first four months ofKepler data: Fundamental stellar parameters
T. Kallinger , , B. Mosser , S. Hekker , D. Huber , D. Stello , S. Mathur , S. Basu , T. R. Bedding , W. J. Chaplin , J.De Ridder , Y. P. Elsworth , S. Frandsen , R.A. Garc´ıa , M. Gruberbauer , J. M. Matthews , W.J. Borucki , H.Bruntt , J. Christensen-Dalsgaard , R.L. Gilliland , H. Kjeldsen , and D. G. Koch Institute for Astronomy (IfA), University of Vienna, T¨urkenschanzstrasse 17, 1180 Vienna, Austria Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada LESIA, CNRS, Universit´e Pierre et Marie Curie, Universit´e Denis, Diderot, Observatoire de Paris, 92195 Meudon cedex, France School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia High Altitude Observatory, NCAR, P.O. Box 3000, Boulder, CO 80307, USA Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101, USA Instituut voor Sterrenkunde, K.U. Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium Danish AsteroSeismology Centre (DASC), Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C,Denmark Laboratoire AIM, CEA / DSM-CNRS, Universit´e Paris 7 Diderot, IRFU / SAp, Centre de Saclay, 91191, GIf-sur-Yvette, France Institute for Computational Astrophysics, Department of Astronomy and Physics, Saint Marys University, Halifax, NS B3H 3C3,Canada NASA Ames Research Center, MS 244-30, Mo ff ett Field, CA 94035, USA Department of Physics and Astronomy, Building 1520, Aarhus University, 8000 Aarhus C, Denmark Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USAReceived ; accepted
ABSTRACT
Context.
Clear power excess in a frequency range typical for solar-type oscillations in red giants has been detected in more than 1 000stars, which have been observed during the first 138 days of the science operation of the NASA
Kepler satellite. This sample includesstars in a wide mass and radius range with spectral types G and K, extending in luminosity from the bottom of the giant branch upto high-luminous red giants, including the red bump and clump. The high-precision asteroseismic observations with
Kepler provide aperfect source for testing stellar structure and evolutionary models, as well as investigating the stellar population in our Galaxy.
Aims.
We aim to extract accurate seismic parameters from the
Kepler time series and use them to infer asteroseismic fundamentalparameters from scaling relations and a comparison with red-giant models.
Methods.
We fit a global model to the observed power density spectra, which allows us to accurately estimate the granulation back-ground signal and the global oscillation parameters, such as the frequency of maximum oscillation power. We find regular patterns ofradial and non-radial oscillation modes and use a new technique to automatically identify the mode degree and the characteristic fre-quency separations between consecutive modes of the same spherical degree. In most cases, we can also measure the small separationbetween l =
0, 1, and 2 modes. Subsequently, the seismic parameters are used to estimate stellar masses and radii and to place thestars in an H-R diagram by using an extensive grid of stellar models that covers a wide parameter range. Using Bayesian techniquesthroughout our entire analysis allows us to determine reliable uncertainties for all parameters.
Results.
We provide accurate seismic parameters and their uncertainties for a large sample of red giants and determine their astero-seismic fundamental parameters. We investigate the influence of the stars’ metallicities on their positions in the H-R diagram. Finally,we study the red-giant populations in the red clump and bump and compare them to a synthetic population. We find a mass andmetallicity gradient in the red clump and clear evidence of a secondary-clump population.
Key words. stars: late-type - stars: oscillations - stars: fundamental parameters
1. Introduction
Studying solar-type oscillations has proved to be a powerful wayto test the physical processes in stars (e.g. Christensen-Dalsgaard2004) that are similar to our Sun and also to the more evolved redgiants, which represent the future of our Sun. The turbulent mo-tions in the convective envelopes of these stars produce an acous-tic noise that can stochastically drive (and damp) resonant p-mode oscillations, typically with small amplitudes. On the otherhand, the global properties of solar-type oscillations, such as the
Send o ff print requests to : [email protected] frequency range where they are excited to observable amplitudesand their characteristic spacings, are predominantly defined bythe stellar mass and radius. By using accurate asteroseismic data,it should therefore be possible to constrain fundamental param-eters to levels of precision that would otherwise be impossible.This has important applications in, for example, exoplanet stud-ies, which depend on firm knowledge of the fundamental param-eters of the host star. Asteroseismic data can put tight constraintson the absolute radii of transiting planets, or determine the age ofan exoplanetary system (e.g. Christensen-Dalsgaard et al. 2010). Kepler red giants
An obvious requirement for such asteroseismic studies is theavailability of accurate observational data. The first indicationsof solar-type oscillations in G and K-type giants were based onground-based observations in radial velocity (Arcturus: Merline1999; ξ Hya: Frandsen et al. 2002; ǫ Oph: De Ridder et al. 2006)and photometry (M67: Stello et al. 2007), which largely su ff eredfrom low signal-to-noise data sets and aliasing. The periods ofsolar-type oscillations in red giants range from hours to days andhence call for long and preferably uninterrupted observationsto resolve the oscillations, which can be done best from space.Space-based detection were made with the star tracker of the Wide Field Infrared Explorer satellite (WIRE; e.g. Buzasi et al.2000; Retter et al. 2003), the
Hubble Space Telescope (HST;e.g. Edmonds & Gilliland 1996; Kallinger et al. 2005;Stello & Gilliland 2009),
Microvariability and Oscillationof Stars (MOST; Barban et al. 2007; Kallinger et al. 2008a,b),and the
Solar Mass Ejection Imager (SMEI; Tarrant et al.2007). Significant improvement in quality and quantity ofthe observations came from the 150-day long observationswith the
Convection, Rotation and planetary Transits satellite(CoRoT), which provided clear detections of radial and non-radial oscillation modes in numerous stars (De Ridder et al.2009; Hekker et al. 2009; Carrier et al. 2010; Kallinger et al.2010; Mosser et al. 2010). Most recently, the NASA
KeplerMission has demonstrated its great asteroseismic potential toobserve solar-type oscillations in red giants (Bedding et al.2010; Hekker et al. 2010b; Stello et al. 2010). We refer to ourcompanion papers presenting a more detailed study of theasteroseismic observables (Huber et al. 2010) and a comparisonof global oscillation parameters derived using di ff erent methods(Hekker et al. 2010c).The oscillation spectrum of a solar-type oscillating starpresents a pattern of modes with nearly regular frequency spac-ings, where the signature of these spacings carries informationabout the internal structure of the star. The large frequency spac-ing ( ∆ ν ), for example, is the frequency di ff erences between con-secutive overtones having the same spherical degree ( l ), and isrelated to the acoustic radius and therefore to the mean den-sity of the star (Brown et al. 1991; Kjeldsen & Bedding 1995).Another directly accessible seismic parameter, the frequency ofmaximum oscillation power ( ν max ), is related to the acoustic cut-o ff frequency and therefore, in the adiabatic case and under theassumption of an ideal gas, defined by the surface gravity and ef-fective temperature of the star (e.g. Kjeldsen & Bedding 1995).Estimating fundamental parameters from these seismic pa-rameters has become an important application of asteroseismicobservations. Recent investigations in this context were madeby Stello et al. (2008), who analysed 11 bright red giants ob-served with the WIRE satellite. They compared traditional meth-ods to determine stellar masses with a new method, that uses thee ff ective temperature, the Hipparcos parallaxes and their mea-surement for ν max , to estimate an asteroseismic mass. The A2Zpipeline (Mathur et al. 2010) uses two di ff erent methods to esti-mate the mass and radius of a star: one based on the scaling lawsand the other one that starts with the measurement of ∆ ν and usesa pre-calculated grid of evolutionary models to obtain an initialguess of the fundamental parameters of the star. For the lattermethod, a minimisation algorithm is performed to estimate theradius and the mass with a higher accuracy (Creevey et al. 2007).Basu et al. (2010) presented the Yale-Birmingham (YB) method,which aims to deduce precise stellar radii from a combinationof seismic and conventional variables. Within the context of theasteroFLAG hare-and-hounds exercises for the Kepler Mission ,Stello et al. (2009b) summarised other methods, which provide
Fig. 1.
Relative flux for the Q0, Q1, and Q2 data ofKIC 6838420. The top panel shows the original time series (greypoints) with the 2nd order polynomial fits overlaid (black lines).The bottom panel shows the residual time series.stellar radii based on the observed large frequency separationand conventional observables. The basic principle of the YB andasteroFLAG methods is the same. They compare observed seis-mic parameters ( ν max and / or ∆ ν ) and other observables ( T e ff , V , π , log g , metallicity, etc.) to those of stellar models, where theseismic parameters of the models are determined from scalingrelations or adiabatic model frequencies. If the input parame-ters are well defined, these methods enable very precise esti-mates for the stellar radius. Most of the red giants observed withCoRoT and Kepler , however, are rather faint and, although theseismic parameters can be determined with high precision, ad-ditional constraints are generally very uncertain, if available atall. To account for this, Kallinger et al. (2010) (hereafter PaperI) presented a modified approach for 31 red giants observed forabout 150 days with the CoRoT space telescope. They exclu-sively used the measured seismic parameters ν max and ∆ ν to de-rive estimates for stellar fundamental parameters from the afore-mentioned scaling relations and a grid of solar-calibrated red-giant models, without making use of any other input parameters.They also indicated that their mass and radius determination isrelatively insensitive to the metallicity and / or evolutionary stageof the investigated red giants.In this paper, we largely follow the approach of Paper I butfor red giants that have been observed during the first ∼ Kepler satellite. We fit a globalmodel to the power density spectra of the high-precision pho-tometric time series to measure ν max . We use an improved ap-proach to determine the large and small frequency spacings,which also provides an automated identification of the modedegree. We apply the same methods to determine ν max and ∆ ν to SOHO / VIRGO (
Variability of solar IRradiance and GravityOscillations ; Frohlich et al. 1997) data to measure the solar ref-erence values needed as an input for the scaling relations. Wecompare the measured seismic parameters with those deter-mined from a multi-metallicity red-giant model grid (by usingthe scaling relations) to derive a reliable stellar mass and radiusand a reasonable e ff ective temperature and luminosity for the Kepler red giants
Fig. 2.
Left panel : Power density spectra for a sample of red giants observed with
Kepler . Black lines indicate the global model fitand dotted lines show the global model plotted without the Gaussian component, which serve as a model for the background signal.Dashed lines indicate the background components. KIC numbers are given in the upper right corners.
Right panel : Residual powerdensity spectra shifted to the central frequency (given in absolute numbers in the plots) of our model to determine the frequencyseparations and normalised to the large frequency separation. Black lines correspond to the best fitting model. The dashed line marksthe midpoint between adjacent l =
2. Observations
The NASA
Kepler Mission (Borucki et al. 2008, 2010) waslaunched in March 2009 with the primary goal of searching fortransits of Earth-sized planets in and near the habitable zonesof Sun-like stars. The satellite houses a 95-cm aperture modi-fied Schmidt telescope that points at a single field in the con-stellation Cygnus for the entire mission lifetime ( > ff erential photometer with a wide fieldof view that continuously monitors the brightnesses of about150,000 stars. This makes it an ideal instrument for astero-seismology and the Kepler Asteroseismic Science Consortium(KASC) has been set up to study many of the observed stars(see Gilliland et al. 2010 for an overview and first results).In this paper we concentrate on the long-cadence (29.4 min-utes sampling; Jenkins et al. 2010) data that have been collectedwithin the astrometric and asteroseismic programmes during thecommissioning phase (Q0; ∼ ∼ ∼ http: // kepler.asteroseismology.com giants in the Kepler Input Catalogue (KIC; Latham et al. 2005).The combined time series consist of about 5900 or 5430 mea-surements and span a total duration of about 138 or 127 days,depending on the availability of Q0 data.In Fig. 1 we show the relative flux for a typical red giant.Whereas the Q0 and Q1 time series show only long term trends,the Q2 data reveal a more complex behaviour. The time seriesappears to consist of five parts with sudden jumps at the transi-tions, which are instrumental. Additionally there is a step gra-dient at least at the beginning of the second and fourth “subset”(BJD - 2 450 000 ≃ Kepler had to be cooleddown again after safe-mode operations. To account for these in-strumental artefacts, we split the time series into 7 subsets (Q0,Q1, and 5 subsets for Q2). We tried several approaches to modelthe gradients, but this turned out to be quite di ffi cult as the ac-tual shape di ff ered from star to star, even increasing gradientshave been found in some cases. Therefore, we simply removedthe leading data points, including the steepest part of the gradi-ent for the first, second, and fourth subsets of the Q2 data. Intotal we rejected about 5.5 days of measurements, degrading theoverall duty cycle from about 91 to 87%, with only minor conse-quences on the spectral window function. Finally, we subtracteda second-order polynomial fit from each subset. The resultingtime series is shown in the bottom panel of Fig. 1. This approach Kepler red giants does, of course, suppress any intrinsic long period signal. Theshortest subset is about 11 days long, which means that we filterout signal below about 1 µ Hz.The long-cadence data from
Kepler that are accessible toKASC consist of two major samples. Firstly, the so-called as-trometric reference stars (Batalha et al. 2010; Monet et al. 2010)comprising about 1 000 stars that have been selected to be distant(and therefore having a small parallax), but bright (and there-fore being mostly giants) and unsaturated stars in a
Kepler mag-nitude range of 11.0–12.5 mag, which are uncrowded and uni-formly distributed over the focal plane. Secondly, about 1 300stars that have been selected for asteroseismology by the var-ious working groups of KASC according to di ff erent criteriasuch as their presumed membership to a cluster or due to theircolour index. We computed the power density spectra between1 µ Hz and the Nyquist frequency ( ∼ µ Hz) for all stars andsearched them visually (i.e., by eye) for red-giant characteris-tics. We found a total of 1041 stars (670 astrometric and 371asteroseismic) that show both a clear power excess hump withregularly spaced peaks and a background that decreases towardshigher frequencies. We identified them as red giants for the sub-sequent analysis.
3. Power spectra modelling
The power spectra of solar-type oscillations have characteristicfeatures. Besides an instrumental white noise component, theyshow a frequency-dependent background signal. This signal canbe represented by several super-Lorentzian functions with in-creasing characteristic frequencies and decreasing characteristicamplitudes. Each of these components is believed to representa separate class of physical process such as stellar activity andthe di ff erent scales of granulation, and most of them are stronglyconnected to the turbulent motions in the convective envelope.On top of the background signal, one finds additional power dueto pulsation in a broad hump. This power excess arises froma sequence of stochastically excited and damped oscillations,which correspond to high-overtone radial and non-radial acous-tic modes. The mode amplitudes are defined by the excitationand damping and are modulated by a broad envelope. The centreof the envelope is usually called the frequency of maximum os-cillation power ( ν max ) and its shape is approximately Gaussian. ν max The background signal in the power spectra of solar-type oscilla-tions can be modelled by the sum of super-Lorentzian functions, P ( ν ) = P A i / (1 + (2 πντ i ) c i ), with ν being the frequency, A i , τ i ,and c i being the characteristic amplitudes, timescales, and theslopes of the background model. This model was first introducedby Harvey (1985) to characterise the solar background signal.In Paper I, it was shown that the solar background model alsoworks for the power spectra of red giants. Due to the larger radiiof red giants compared to the Sun, the amplitudes and timescalesof the background components are quite di ff erent but the model,particularly the slopes of the components are the same. Here, wefollow the approach of Paper I and model the observed powerdensity spectra with a superposition of white noise, the sum of Note that this function is frequently referred to as power law orHarvey-like model. It is, however, clearly not a power law and Harvey(1985) originally used a Lorentzian. We therefore suggest the name“super-Lorentzian” with the power 4, which is sometimes used in op-tics. super-Lorentzian functions, and a power excess hump approxi-mated by a Gaussian: P ( ν ) = P n + X i π a i / b i + ( ν/ b i ) + P g exp − ( ν max − ν ) σ g (1)where P n corresponds to the white noise contributions and a i isthe rms amplitude of the i th background components. The pa-rameter b i corresponds to the frequency at which the power ofthe component is equal to half its value at zero frequency andis called the characteristic frequency. P g , ν max , and σ g are theheight, the central frequency, and the width of the power ex-cess hump, respectively. Note that σ g is about 1.18 times theHWHM. For our sample of red giants, the frequency coverageof the Kepler observations allowed us to model up to three back-ground components.The only di ff erence compared to the model in Paper I is thenumerator in the background models. Originally, a single param-eter, A , was used which corresponds to the power at frequencyequal to zero of the given component. However, tests have shownthat fitting a and b instead of fitting A and b , with A = π a / b ,yields a more robust fit and allows a more accurate measurementof the characteristic frequencies. This notation also makes moresense physically because a corresponds to the variance that thesignal produces in the time domain and can easily be related tothe observed total energy of, e.g., granulation.We used a Bayesian Markov-Chain Monte-Carlo (MCMC)algorithm to fit the global model to the power density spectra.See Paper I and Gruberbauer et al. (2009) for a detailed descrip-tion. Briefly, the algorithm automatically samples a wide param-eter space and delivers probability density distributions for allfitted parameters and their marginal distributions, from whichwe computed the most probable values and their 1 σ uncertain-ties. For the parameter limits, we followed a slightly modifiedapproach to that in Paper I. During the fitting process we kept ν max within ±
25% of the value inferred from the visual inspec-tion of the spectrum. The width of the power excess was allowedto vary between 5% and 50% of the initial guess of ν max , where Fig. 3.
The uncertainty in ν max in actual value (right axis) andin units of the frequency resolution as a function of the ratiobetween the height-to-background ratio (HBR) and the width ofthe power excess, σ g . The line indicates a power-law model forthe lower envelope. Kepler red giants δν δν ν max ∆ν relative uncertainty (%) p e r c e n t a g e o f s t a r s Fig. 4.
Histograms of the relative uncertainties for ν max , the largefrequency separation ∆ ν , and the small separations δν and δν .the lower limit prevented the algorithm from fitting the Gaussianto a single frequency bin in the spectrum. The frequency param-eters b i were allowed to vary from 0 to 1.5 times ν max , with thecondition b > b > b , where the indices indicate consecutivebackground components. The amplitude parameters a i were keptbetween 0 and 10 times the square root of the highest peak powerin the spectra. P g was allowed to vary from zero to 10 times theaverage power in the spectrum around the initial guess for ν max ,and P n was kept between 0.5 and 2 times the average power atthe high frequency end of the spectrum. The left panel of Fig. 2shows examples of power density spectra with the correspondingfits.An important aspect of our analysis is to understand the un-certainties of the determined parameters. One might expect thatthe white noise, and therefore the brightness of a star, is respon-sible for a significant part of the uncertainty in ν max but we donot find any correlation with the magnitude. On the other hand,there is a clear correlation between σ ν max and the ratio of theheight-to-background ratio (HBR) to the width of the power ex-cess ( σ g ), where we define the HBR as the ratio between theheight of the power excess ( P g ) and the background signal at ν max . In other words, we can more accurately determine the cen-tre of a narrow power excess hump with a large HBR than thecentre of a broad hump with a small HBR. This is illustrated inFig. 3 where we plot the absolute value of σ ν max (right axis) as afunction of HBR / σ g . Tests with subsets of the Kepler time series(and data sets from CoRoT) have shown that σ ν max is also di-rectly proportional to the frequency resolution of the time series,which is defined as the inverse data set length. To account forthis we plot σ ν max in units of the frequency resolution ( ν res ) onthe left axis in Fig. 3. From that, we can define a simple relationfor the lower limit of the uncertainty in ν max as σ ν max = ν res + HBR /σ g ) / ! . (2) For the solar case we would expect an uncertainty in ν max ofabout 5.3 µ Hz, which is in good agreement with the value of5.23 µ Hz found in Sect. 3.3 for SOHO / VIRGO data. Given thisrelatively simple error law we are confident that our uncertain-ties for ν max are reliable and do mainly reflect the constraints ofthe observations and not of our method. It is also interesting tosee that the uncertainty of ν max is very much defined by the staritself, since HBR / σ g is largely intrinsic to the star, for a giveninstrument and observing time. Nevertheless, we mention thatthe error law is purely phenomenological and might not be validoutside the range we use it.A histogram of the relative uncertainties in ν max is given inthe top panel of Fig. 4, showing a clear peak at about 2%. Thisis not surprising as a large fraction of the analysed red giantsare red-clump stars having a very similar ν max , and therefore asimilar width and height-to-background ratio of the power ex-cess. We see that, for almost all stars, we could determine ν max to within 4% and, for about half of our sample, to within 2%.A potential problem for the subsequent analysis is that weassumed the power excess hump to be symmetric, so that an in-trinsic asymmetry might result in a systematic error. In Paper Iit was claimed that the asymmetry of the power excess humpis within the observational uncertainties of ν max , and thereforenegligible. However, that conclusion was based on the analysisof only 31 stars. Our sample is more than 30 times larger andshould give a statistically more significant conclusion. We com-puted the weighted mean frequency, ν wm , in the frequency rangeof pulsation ( ν max ± σ g ), where we used the residual power af-ter correcting for the background signal as weight. We found ν wm consistently shifted towards higher frequencies comparedto ν max by 3.1 ± ν max of about 2.3%. This is, however, not a problemin our subsequent analysis since we find the same shift of about3% in the solar data (see Sect. 3.3), and as long as we compare ν max values that are determined in the same way, we do not haveto take into account asymmetries in the power excess humps. In the next step we used the white noise and background com-ponents of the global model (dotted lines in Fig. 2) to correct thepower density spectra for the background signal, leaving onlythe oscillation signal and white noise. The second parameter thatcan directly be determined from the observed power spectrumis the large frequency separation, ∆ ν . To determine ∆ ν , we usea similar approach as in Paper I and fit the following generalmodel to the residual power density spectrum over a frequencyrange spanning three radial orders around the frequency of max-imum oscillation power: P ( ν ) = P n + X i = − A i τ + ν − ( ν + i ∆ ν )] ( πτ ) + X j = − A j τ + ν − ( ν + j ∆ ν − δν )] ( πτ ) + X k = − , A k τ + ν − ( ν + k ∆ ν + δν )] ( πτ ) . (3)The model represents a sequence of eight Lorentzian profileswhose frequencies are parameterised by a central frequency, ν ,and three spacings, ∆ ν , δν , and δν , where the first, second,and third sum corresponds to three radial, three l =
2, and two
Kepler red giants l = A i , A j , and A k are the individual rmsamplitudes. As in Paper I, we assume the same mode lifetime τ for all modes, which might not be true in reality, but this as-sumption has no impact on the determination of the spacingsand significantly stabilises the fit. We again used the BayesianMCMC algorithm to fit the model to the residual power den-sity spectrum. All mode amplitudes were allowed to vary inde-pendently between zero and 10 times the highest peak in theamplitude spectrum. This allowed the algorithm to account formissing modes or modes hidden in the noise. The mode life-time was sampled between 1 and 100 days. Most important herewere the parameter ranges for the spacings. With the conditionthat δν ≪ ∆ ν and δν < ∆ ν /
2, the model basically representsthe asymptotic relation (Tassoul 1980) for low-degree and high-radial order p modes. Consequently, δν and δν correspondto the small separations between adjacent l = l = ν o and ∆ ν to vary over a relativelywide range ( ν max ± σ g for ν o and 0.5 µ Hz to 2 σ g for ∆ ν ) the al-gorithm was able to automatically find the central mode that cor-responds to a radial mode if l = l = ∆ ν for all 1041 stars and found at leasttwo l = δν values for these stars but only accepted them for about one thirdof the total sample. This is because of the multiple dipole modesover a relatively broad frequency range per order, which occurdue to their mixed gravity / acoustic mode character (Dupret et al.2009), and makes it di ffi cult for our algorithm to obtain robustresults. We do not further investigate the small spacings here butrefer to Huber et al. (2010), where those results are presented.The oscillation spectra and the corresponding best fits forthe stars in Fig. 2 are illustrated in the right panel of that figure.As for our global fit parameters, we determined the most proba-ble parameters from the marginal distributions of the probabilitydensity delivered by the MCMC algorithm. Unlike for ν max , wewere not able to find a clear correlation between the uncertaintyin ∆ ν and any other parameter combination. We expect, how-ever, that σ ∆ ν depends on the frequency resolution, the signal-to-noise ratio of the individual modes and their lifetime. We couldnot reliably determine the mode lifetimes for many stars becausetheir mode profiles are undersampled, which means that the peakwidth due to the spectral window function of the observations isbroader than the actual profile width. In such a case our ampli-tudes and lifetimes are meaningless. The mode frequencies andtherefore the spacings are, however, not a ff ected by this phe-nomenon. Histograms for the uncertainties of the spacing pa-rameters are given in Fig. 4, showing that we can determine ∆ ν to within 1% for about 30% of our sample. Whereas the accu-racy of δν is mostly better than 10%, the relative errors for δν are relatively large. But one has to keep in mind that δν is wellbelow 1 µ Hz for a red giant and the absolute uncertainties of δν are still quite low.Originally, we would have had to exclude a number of starsfrom our sample because they pulsate with low frequenciesmaking it di ffi cult to reliably determine spacings from a fre-quency spectrum. There is, however, the so-called autocorrela-tion method described by Mosser & Appourchaux (2009), which Fig. 5.
Power density spectrum of a 1-year VIRGO time series(light-grey) and the corresponding global model fit (black line).The dark-grey line shows a smoothed (5 µ Hz boxcar filter) ver-sion from the average of nine consecutive 1-year VIRGO timeseries. The inset gives the probability density function for ν max with the vertical solid and dashed lines indicating the medianvalue and the 1 σ limits, respectively.measures ∆ ν from the first peak in the autocorrelation of the timeseries. This method is less a ff ected by the limited duration of theobservations than Fourier-based methods and is able to detectregular spacings down to a few times the frequency resolution(Mosser et al. 2010). To account for this, we used the ∆ ν valuesfrom the autocorrelation method for all stars with ν max < µ Hz( ∼
7% of the total sample) for the subsequent analysis.Finally, we cross-checked our results for ν max and ∆ ν withthose of other methods (Hekker et al. 2010a, Huber et al. 2009,Mathur et al. 2010, and Mosser & Appourchaux 2009) whichhave been used to determine the same seismic parameters forour sample (or subsample) of red giants. A direct comparison ofthe di ff erent methods shows that there are a number of outliersbut for most stars in our sample, at least one other method gavea value that is compatible with our results (i.e., within the uncer-tainties). Additionally, we have carefully checked by hand thereliability of the seismic parameters for all stars for which wefound a significant disagreement in the direct comparison (seeHekker et al. 2010c for the detailed comparison) and identifiedonly a few stars (less than 1%) which we had to eliminate fromour sample because their seismic parameters are ambiguous. The large frequency separation is related to the inverse soundtravel time through the star and therefore to the mean densityof the star. It scales as ∆ ν ⊙ ( M / R ) / from the solar case, with R and M being the total mass and the radius of the star, respec-tively, in solar units. An important point when using scaling rela-tions to estimate fundamental parameters is the definitions of thescaled seismic parameters. An often used solar reference value, Kepler red giants
Fig. 6.
Same as Fig. 5 but as delivered from our algorithm to determine the frequency spacings.
Top panels : Probability densityfunctions for the three spacing parameters and the central frequency of our model fit. Median values and 1 σ limits are indicated byvertical solid and dashed line, respectively. Bottom panel : Residual power density spectrum (light-grey) of a 1-year VIRGO timeseries and the corresponding model fit (black line). ∆ ν = µ Hz is based on the frequency di ff erence betweenthe radial modes with order n =
20 and 21 where the maximumoscillation power is seen (Toutain & Froehlich 1992). The fre-quency di ff erence of two single modes is di ffi cult to determinefor other stars and an average value of all (or some) observablemodes is often used (e.g, ∼ µ Hz for the Sun; Kjeldsen et al.2008). However, the frequency separation is a function of thefrequency itself (see e.g. Broomhall et al. 2009 for the Sun andMosser et al. 2010 for CoRoT red giants), and an average ∆ ν will depend on the actual number and frequency range of theobserved modes and is di ffi cult to compare for di ff erent obser-vations.The frequency of maximum oscillation power, on the otherhand, is related to the acoustic cut-o ff frequency in the stellar at-mosphere (e.g., Brown et al. 1991; Kjeldsen & Bedding 1995),which in turn scales as ν max , ⊙ ( M R − T e ff − / ) from the so-lar case, where often used values are ν max , ⊙ = µ Hz (e.g.,Kjeldsen & Bedding 1995) or 3100 µ Hz (e.g., Basu et al. 2010).Both seismic parameters scale with the stellar mass and radiusand can therefore be used to estimate R and M of a star from itsseismic parameters. But to use these seismic scaling relations ina consistent way we need values for the solar parameters, mea-sured in the same way as for our sample of red giants. Note thatthe seismic scaling relations are not laws of physics and possi-bly include some additional uncertainties. There are, however,strong indications that at least the ∆ ν -scaling is quite accuratefor cool stars like our sample of red giants (Basu et al. 2010;Stello et al. 2009a).We used a 1-year time series from the green channel ofthe SOHO / VIRGO data (Frohlich et al. 1997) obtained duringa solar activity minimum and fitted our global model (Eq. 1) tothe corresponding power density spectrum. The result is givenin Fig. 5, showing the original power density spectrum and asmoothed version of the average power density spectrum of nineconsecutive 1-year time series along with the best fitting model. Although the model is fitted to “only” a 1-year time series, italmost perfectly reproduces the average spectrum of the full 9-year time series. The reason why we do not use the 9-year timeseries is because of limited computer resources. Also shown isthe probability density function of the solar ν max parameter, fromwhich we determined ν max , ⊙ = ± µ Hz.In the next step we use the background part of the globalmodel fit to correct the power density spectrum and fit theasymptotic relation model to the residual spectrum. The solarp-mode profiles are naturally much better resolved than the p-modes in our sample of red giants and initial tests have shownthat the rotationally split components of the non-radial modessignificantly disturb the fit. To account for this we have includedrotationally split components for the l = σ limits. The bottom panels illustrate the residual powerdensity spectrum and the best model fit. We have determined ∆ ν ⊙ = ± µ Hz, δν , ⊙ = ± µ Hz, and δν , ⊙ = ± µ Hz. The uncertainties might appear unrealisticallysmall but they reflect a well defined case specifically chosen forour approach and should not be mistaken as “global” frequencyspacings of the Sun.
Kepler red giants
4. Asteroseismic fundamental parameters
Our approach to determining an asteroseismic mass and radiusdepends on the aforementioned scaling relations for ν max and ∆ ν : ν max = ± µ Hz × MR − T − / (4) ∆ ν = (134 . ± . µ Hz) × M / R − / (5)with M and R in solar units and T = T e ff / K . Obviously the ν max scaling relation depends on the e ff ective temperature of thestar. Although there are temperatures available in the KIC formost of the analysed stars, we decided not to directly use themin our analysis (i.e., in the above scaling relations). The KICparameters ( T e ff , log g , and [Fe / H]) are mostly determined frommulti-colour photometry and they are calibrated to be correct ina statistical sense, i.e., the average values of a large sample ofstars are correct. However, a star-by-star comparison (e.g., forstars associated in a cluster) has shown that the individual valuescan have a large scatter (up to some 100 K in T e ff ). Consequently,adopting the KIC temperatures would add uncertainties of up to5% and 10% to our radius and mass determinations, respectively.Therefore, we also compared the seismic parameters of the ob-served stars and their KIC temperatures to those inferred fromstellar models, and then adopted the fundamental parametersfrom the model that best reproduced all input parameters. Thereare several ways to do this. In Paper I, an initial guess for the stel-lar mass and radius was determined using an average tempera-ture for stars on the red giant branch. The initial mass and radiusvalues were then compared to those of a grid of solar-calibratedred-giant models to get a better estimate for the temperature. Thenew temperature was adopted and the procedure repeated, con-verging to a certain location in the H-R diagram after a few it-erations and delivering a full set of fundamental parameters foreach star. However, these iteratively determined parameters alsodepend on the chemical composition and the evolutionary statusof the model (i.e., whether the model is a red-giant branch or anasymptotic-giant branch model). This ambiguity allows a givenset of seismic parameters to converge to di ff erent locations inthe H-R diagram if di ff erent model grids with, e.g., a di ff erentchemical composition are used. This uncertainty has, however,only small e ff ects on the mass and radius determination. For thee ff ective temperature and luminosity it adds about ±
150 K and ±
20% uncertainties, respectively.A more general approach was presented by Basu et al.(2010). They used a combination of seismic and conventionalstellar parameters and compared them to those of a multi-metallicity model grid, where the aforementioned scaling rela-tions are used to determine the seismic parameters for the mod-els. Basu et al. (2010) defined a model likelihood from the dif-ference between the model and input parameters and infer thestellar radius from the resulting likelihood function. From theirtests with a number of artificial stars they concluded that it isvery unlikely to get a reasonable estimate for a red giants’ ra-dius if no accurate temperature and parallax are available. Thisconclusion was, however, based on only a single red-giant teststar that is located high up the red giant branch and thereforequite far away from where most of the observed red giants areexpected (i.e. in the red clump). Additionally, they have assumeduncertainties for the seismic input parameters that are about tentimes larger than what we have determined for our sample of redgiants.Here, we follow the approach of Basu et al. (2010) but for-mulate our search for a best set of fundamental parameters in a Bayesian sense. Given a set of seismic input parameters ν max , obs and ∆ ν obs , and the Gaussian distribution of their uncertainties, σ ν max and σ ∆ ν , we define the likelihood that the seismic modelparameters, ν max , model and ∆ ν model , matches the observed ones as L ν max = √ πσ ν max exp − ( ν max , obs − ν max , model ) σ ν max ! (6) L ∆ ν = √ πσ ∆ ν exp − ( ∆ ν obs − ∆ ν model ) σ ∆ ν . (7)To simplify matters, we also assume a Gaussian error for the KICe ff ective temperatures and define the likelihood that the modelmatches the observed temperature as L T e ff = √ πσ T e ff exp − ( T e ff , KIC − T e ff , model ) σ T e ff , (8)with a typical value of 200 K for σ T e ff . In the Bayesian approachwe can assign an overall probability of the model M i with theposterior probability I matching the observed parameters D withrespect to the entire set of models according to Bayes theoremas p ( M i | D , I ) = p ( M i | I ) p ( D | M i , I ) p ( D | I ) (9)where p ( M i | I ) = N m (10)is the uniform prior probability for a specific model with N m being the total number of models, and p ( D | M i , I ) = L ( ν max , ∆ ν, T e ff ) = L ν max L ∆ ν L T e ff (11)is the likelihood function. The denominator of Eq. 9 is a normal-isation factor for the specific model probability in the form of p ( D | I ) = N m X j = p ( M j | I ) · p ( D | M j , I ) . (12)Since the uniform priors are the same for all models they cancelin Eq. 9, which simplifies to p ( M i | D , I ) = p ( D | M i , I ) P N m j = p ( D | M j , I ) . (13) Fig. 7.
The projected probability (left) and cumulative probabil-ity (right) distribution functions for the radius of the artificial teststar 6 (Basu et al. 2010). The full horizontal line correspond tothe median and the dashed lines give the ± σ confidence inter-val. Kepler red giants
Fig. 8.
The stellar radius as a function of mass for the sampleof red giants as directly determined from Eq. 4 and 5 using theKIC e ff ective temperatures (top) and from the comparison withthe stellar model grid (bottom).The resulting model probability distribution automatically trans-lates into most probable fundamental parameters and their un-certainties by constructing the marginal distribution for the cor-responding parameter. The normalised probability of the mostprobable parameters is therefore a measure of how likely they arewith respect to the other models of the specific grid. We stressthat it does not tell us how probable the parameters are in anabsolute sense, although formally it must be implicitly assumedthan one of the tested models is actually true. The probabilitymust be interpreted as being restricted to the space of the mod-els being considered, and their associated physics. This meansthat as soon as additional models are added to the model space,the probabilities will change.We also mention for completeness that since we use an uni-form prior that rates each model with the same prior probabil-ity our Bayesian approach is actually not very di ff erent from thelikelihood method of Basu et al. (2010). But as soon as we wouldadd additional informations such as, e.g., an initial mass func-tion, the advantages of the Bayesian technique would becomeobvious. But this is beyond the scope of the present paper andwe leave it to future investigations. The red-giant models used for our analysis were extracted fromthe canonical scaled-solar BaSTI isochrones (Pietrinferni et al.2004) in version 5.0.0 with a mass-loss parameter η = Available from http: // albione.oa-teramo.inaf.it / . to the He-core burning main sequence (red clump) and back upto the asymptotic giant branch (ABG) to an age of about 15 Gyr.We restricted the grid to models that have already left the mainsequence with initial chemical compositions of (Z, Y) = (0.008,0.256), (0.01, 0.259), (0.0198, 0.2734), (0.03, 0.288), and (0.04,0.303). The model mass ranges from 0.7 to 4 M ⊙ with steps oftypically 0.02 M ⊙ . We rejected models both with a mass above4 M ⊙ and with metallicities below Z = ff erentmetallicity grids. We therefore increased the resolution in chemi-cal composition to steps of 0.002 in Z via interpolating the e ff ec-tive temperature for models of approximately constant mass andradius but with di ff erent chemical composition. The correspond-ing luminosities were determined from the Stefan-Boltzmannlaw, L ∝ R T . The final grid used consists of about 1.4 mil-lion models based on about 340 000 original BaSTI models. To test our algorithm we use test star 6 from Basu et al. (2010)from their Table 1. The input ν max and ∆ ν are 21 ± ± µ Hz, respectively, where we adopt typical uncertaintiesfrom our sample of red giants. The resulting model probabilitiesand cumulative probability distributions for the model radius areillustrated in Fig. 7, from which we infer the most probable ra-dius and its uncertainty to be R = ± ⊙ , which is in goodagreement with the input radius of 21.44 R ⊙ (Basu et al. 2010).A more realistic test case is the red giant ǫ Oph, for whichan interferometric radius is available. With ν max = ± ∆ ν = ± µ Hz (Kallinger et al. 2008a), and a spectro-scopic T e ff = ±
100 K (De Ridder et al. 2006) we deter-mined a radius of 10.7 ± ⊙ , which is in good agreement withthe interferometric radius of 10.55 ± ⊙ (Mazumdar et al.2009). The other parameters are also in fairly good agree-ment with independent measurements. Our mass estimate of1.89 ± ⊙ compares well to stellar masses determined froma detailed modelling: 1.85 ± ⊙ (Mazumdar et al. 2009) and2.02 M ⊙ (Kallinger et al. 2008a). Even our luminosity estimateof 59 ± ⊙ is compatible with the luminosity of 58 ± ⊙ basedon the Hipparcos parallax (van Leeuwen 2007). In a first step, we have excluded 65 red giants which are associ-ated with clusters from our sample because they might bias thesubsequent analysis. Fig. 8 shows the radius as a function of themass for the remaining sample of red giants. Whereas in the toppanel, R and M are directly determined from the seismic scal-ing relations adopting the KIC temperatures, the bottom panelshows R and M as they follow from the Bayesian comparisonwith the stellar model grid. Both distributions include many starslocated in a narrow range around ten solar radii. Most of the starsin this range are expected to correspond to the red clump (e.g.,Miglio et al. 2009) and their large number is due to the di ff erentevolutionary rates of giant-branch stars. Whereas stellar evolu-tion takes place quite rapidly during the RGB phase and after the Kepler red giants
Fig. 10.
H-R diagrams showing our sample of red giants (black dots) with respect to di ff erent metallicity model grids (red dots - inthe online version only).He flash at the tip of the RGB, it significantly slows down dur-ing the quiescent He-core burning phase in the red clump. Henceone can expect to find more red-clump stars and therefore starswith a similar radius than RGB stars in a random sample of redgiants.Although the two methods to determine R and M give a simi-lar picture there are some important di ff erences. Firstly, the very-low-mass stars in the top panel are most likely artefacts because,apart from binary stars which have lost a significant fraction oftheir mass, the universe is not old enough for 0.5-0.7 solar massstars to have become 10-15 solar radii giants. Secondly, the dis-tribution of stars below the red clump seems to be more realis-tic in the bottom panel than in the top panel. The higher massstars evolve faster than the lower mass stars and therefore fewerhigher mass stars should be present in a random sample of gi-ants. Additionally, the error bars are in general smaller in the bot-tom panel and the red-clump feature is more pronounced. We aretherefore confident that the additional e ff orts in determining thefundamental parameters are justified because they enable a moredetailed picture of the stellar population on the giant branch. As our algorithm also delivers e ff ective temperatures and lumi-nosities we can put the analysed stars in a H-R diagram. Panel a of Fig. 9 shows the model-independent positions in the H-R di-agram, with the temperatures from the KIC and the luminositiesfollowing from the Stefan-Boltzmann law, with R directly deter-mined from the seismic scaling relations (by using the KIC tem-peratures). The other panels show the positions in the H-R dia-gram as they follow from the Bayesian comparison with the stel-lar model grid with the colour code indicating the mass, radius,age, and metallicity of the best fit models. The distribution ofstars in the H-R diagram reveals some interesting features. Themost distinctive one (denoted as “A” in panel b of Fig. 9) consistsof a large number of stars ( ∼
35% of the total sample) lining upat almost constant luminosity and corresponds to the red clump.A similar feature (B) is located at slightly lower luminosities andwith an almost constant temperature. It most likely correspondsto the so-called red bump, which is another characteristics of stellar evolution on the giant branch due to the outward-movinghydrogen-burning shell that burns through the mean molecularweight discontinuity left by the first dredge-up from the con-vective envelope causing a slight contraction of the star beforethe star starts to ascend the giant branch again. Significantly lesspopulated than the red clump and bump is the feature (C) thatcorresponds to the so-called secondary clump (Girardi 1999) in-cluding high-mass stars ( > ⊙ ) that settle as He-core burningstars at lower luminosities than the lower-mass red-clump stars.The secondary-clump population is of particular interest becauseits mass puts tight constraints on, e.g., the convective-core over-shooting or the recent history of star formation in the Galaxy(Girardi 1999). We refer to Huber et al. (2010) who report onthe signature of secondary-clump stars in the distributions of theseismic observables of red giants in the Kepler field of view (seealso Mosser et al. 2010 for CoRoT red giants).Combining the mass distribution in the H-R diagram (panel b in Fig 9) with the radius distribution (panel c in Fig 9) we inferthat the red clump consists of about 0.8 to 1.8 M ⊙ stars (withthe low-mass stars accumulated at the bottom of the red clump)with a radius between about 10 to 12 R ⊙ . The red bump, on theother hand, is dominated by 1 M ⊙ stars but with a lower averageradius than the red-clump stars. The secondary-clump covers asimilar radius range but includes stars with masses above about2 M ⊙ . Outside the red clumps and bump, the stars follow theusual mass distribution with increasing masses towards highertemperatures and luminosities. The detailed structure in the H-Rdiagram is almost not visible in the model-independent approach(panel a ). The uncertain KIC temperatures obviously blur thedistribution of the stars in the red clumps and bumps. However,we note that the model-independent approach tentatively showsthe mass gradient in the red clump as well.Although we cannot constrain the metallicity to better thanabout ±
50% for individual stars, the metallicity distributionshows some interesting trends (see panel e in Fig. 9). In the redclump, there is a metallicity gradient ranging from metal-poorstars at the bottom to metal-rich stars at higher luminosities. Andthe red bump tentatively consists of metal-enhanced stars.To analyse the red clump and bump in more detail we com-pare in Fig. 10 the observed features with di ff erent metallicity Kepler red giants grids. As indicated in panel d of Fig. 9, the red clump is domi-nated by He-core burning stars of the solar-metallicity grid (mid-dle panel of Fig. 10). The metallicity gradient in the red clumpis due to the position of the He-core burning main-sequence inthe di ff erent metallicity grids. Whereas the metal-poor modelsare shifted towards higher temperatures and therefore towardsthe bottom of the observed red clump, the metal-rich models ac-cumulate at the opposite side of the red clump. Similar can beseen for the red bump. But there, metal-enhanced models aremore consistent with the observed sample than solar-metallicitymodels.In Fig. 11 we show histograms for the relative uncertaintiesin mass and radius (bottom panel) showing that for about halfof our sample we can constrain the mass and radius to within7% and 3%, respectively. These uncertainties are dominated bythe observational uncertainties of the seismic input parametersand are only slightly a ff ected by the model related parameters.On the other hand, the e ff ective temperature and luminosity of amodel with a given mass and radius can significantly di ff er for,e.g., a di ff erent initial chemical composition or mixing lengthparameter. Compared to Paper I, where only a single metallic-ity grid was used, we cover a wide range in chemical composi-tion and therefore expect to get a more realistic uncertainty forthe H-R diagram position. Histograms for the relative uncertain-ties in e ff ective temperature, age, and luminosity are given in thetop panel of Fig. 11 showing that we can determine the e ff ectivetemperature, age and luminosity to within ±
110 K, ± ± The radius and mass distribution in our sample of red giants isgiven in Fig. 12. The radius distribution clearly shows two pop-ulations of stars that are located in the same region of the H-Rdiagram. The H-shell burning stars ascending the giant branchand the He-core burning stars in the red clump. The very dif-ferent rate at which they change their fundamental parameters(e.g., the radius) results in two superposed components of theirradius distribution. The main component is a broad Gaussian-like distribution with a maximum number of stars between 9.5and 10 R ⊙ . This component is dominated by RGB stars ascend-ing the giant branch and the maximum falls onto the averageradius of the red bump (see panel c of Fig. 9) but also includesthe secondary-clump stars. The RGB stars are superposed withthe sharp distribution of red-clump stars with their radii rang-ing from 10.5 to 11.5 R ⊙ . Also interesting is the excess of starswith a radius around 20 R ⊙ . If real, this clustering of stars wouldbe very interesting because it might indicate stars on the AGBwhose He-burning shells burn through the discontinuity left bythe second dredge up, which happens indeed at about 20 solarradii in solar metallicity models. But a significantly larger sam-ple would be needed to verify if the excess is real.The mass distributions in the bottom panel of Fig. 12 showsthat there are only very few low-mass stars in our sample. Theirnumber slowly increases between 0.8 to 1.5 M ⊙ with a small ex-cess between 1 to 1.2 M ⊙ , and rapidly drops for higher masses.To test for bias of our composite sample we computed the radiusand mass distributions in the subsamples (red and blue bars inFig. 12) and found no significant di ff erence for the radius distri- bution. The mass distributions, on the other hand, are di ff erent.Obviously, there are more 1.3 to 1.5 M ⊙ stars in the asteroseis-mic sample than in the astrometric sample showing its maximumbetween 1 to 1.2 M ⊙ . We expect the excess of “high-mass” starsin the asteroseismic sample to be due to the original selection ofthe stars.The detailed structure in the radius and mass distribution en-abled us to identify di ff erent stellar populations in our samplewhich we can compare with synthetic populations for the Kepler field of view. To do so, we used the synthetic red-giant popu-lation for the
Kepler field of view as presented by Huber et al.(2010) and computed with the stellar synthesis code TRILEGAL(Girardi et al. 2005) in the same way as by Miglio et al. (2009)for one of the CoRoT fields. The synthetic radius and mass dis-tributions are indicated as dashed lines in Fig. 12. Although thecomparison can not be done in an absolute sense as the observedsample is biased, the observed and synthetic distributions lookquite similar but show also some interesting di ff erences. Thered-clump feature in the synthetic radius distribution is slightlybroader and has its maximum at a lower radius compared to theobserved distribution. Additionally, the RGB component is lesspronounced, which is due to significantly less red-bump starsin the synthetic population. Since the stellar synthesis does notinclude stellar clusters, the main di ff erence in the mass distri-butions is due to fewer 1.3 to 1.5 M ⊙ stars in the synthetic dis-tribution. More interesting is, however, the di ff erence for high-mass stars ( > ⊙ ) indicating di ff erently populated secondaryclumps. Although these di ff erences potentially carry detailed in-formations about the star formation history in the Kepler field ofview and the associated physics, it would require detailed mod-elling to further investigate them, which is beyond the scope ofthis paper.
Finally, we compared in Fig. 13 (upper panel) the seismicallydetermined e ff ective temperature with the KIC temperature.We find the seismic temperature systematically shifted towardslower temperatures by about 50 K (see linear fit and binned val-ues). The rms scatter (about 120 K) of the temperature di ff erenceis, however, within the assumed uncertainties for the KIC tem-peratures of 200 K (private communication T. Brown). The KICuncertainties seem therefore to be overestimated but one has tokeep in mind that our algorithm uses the KIC temperatures as aninput parameter and the two temperature estimates are thereforenot independent. A more meaningful indicator for the reliabilityof our fundamental parameter estimation is the surface gravity,which is also given in the KIC but not used in our analysis. Theseismic mass and radius directly translate to a surface gravityand a comparison between the seismic and KIC surface gravityis shown in the bottom panel of Fig. 13. Here we found a system-atic di ff erence indicated by a linear fit and binned values. For thered-clump stars (around log g = ff erence is negligiblebut drifts apart for stars above (towards lower surface gravity)and below (towards higher surface gravity) the red clump. Thedi ff erence is, however, less than 0.5 dex for the entire range, andtherefore within the uncertainties for the KIC parameters.We also compared our seismic log g values for a few starsin common with the spectroscopic study of Bruntt et al. (2010,A&A, in preparation). There is very good agreement with amean di ff erence (spectroscopy minus seismic log g ) of ∆ log g =+ . ± .
15. We quote the RMS scatter for the seven stars incommon.
Kepler red giants
5. Summary and conclusions
We have analysed high-precision photometric time series fromthe first four months of
Kepler observations and found morethan 1 000 stars that show a clear power excess in a frequencyrange typical for solar-type oscillations in red giants. We haveapplied robust and automated methods to accurately determinethe global seismic parameters ν max and ∆ ν , and provide an auto-mated identification of the mode degree as well as small spacingsfor about one half of our sample. We have analysed the uncer-tainties in our parameter determination and find a clear relationfor the uncertainty of ν max depending on the frequency resolu-tion but also on the height-to-background ratio and the width ofthe power excess, which are largely determined by the star it-self as long as the power excess is well above the white noise ofthe observations. We have applied our methods to solar data todetermine solar reference values for ν max and ∆ ν . The measuredseismic parameters were then compared to those of an exten-sive multi-metallicity grid of stellar models, where the seismicparameters of the models were determined from scaling rela-tions. A Bayesian approach for the search of a best fit betweenobserved and model parameters allowed us to derive realisticuncertainties for all fundamental parameters. In principle, wecould have estimated the fundamental parameters from the seis-mic scaling relations using the KIC temperatures. However, wefound strong indications that our analysis produced more accu-rate fundamental parameters and gives us a more detailed viewof the stellar populations in our sample of stars.We have placed the analysed stars in a H-R diagram andfound clear features in the distribution of the stars, which weidentified as the red clump, the secondary clump, and the redbump. We found a mass gradient in the red clump with the low-mass stars accumulated at the bottom of the red clump and thatthe red bump is dominated by 1 M ⊙ stars. Although we cannotdetermine the chemical composition reliably of individual starswe can conclude in a statistical sense that most of the red-clumpstars in our sample are similar to the Sun in terms of the theirinitial chemical composition with some indications for a metal-licity gradient that follows the mass gradient of the red clump.On the other hand we found that the bump stars are more consis-tent with metal-enhanced stars, which is surprising for a sampleof stars that is selected according to criteria that do not constrainthe chemical composition. A possible explanation could be thatthe mixing length parameter used to construct the models is toohigh. A slightly less e ffi cient convection would shift the solarmetallicity red-bump models towards lower temperatures in thedirection of the observed red bump.The large sample allowed us to investigate the detailed struc-ture of the radius and mass distribution of red giants in the Kepler field of view clearly showing the di ff erent populations. A com-parison with synthetic distributions indicated quantitative agree-ment but needs further investigations. With the present studyand what was presented by Huber et al. (2010) and Mosser et al.(2010) there are now detailed red-giant populations availablefor three di ff erent regions in the sky, which should be usedfor future detailed population synthesis studies as first done byMiglio et al. (2009).Finally, we mention that although the parameters uncertain-ties in our analysis are already quite realistic they still repre-sent only a lower limit. There are several e ff ects we do not yettake into account. For example, the frequency dependence ofthe large frequency spacing (cf Mosser et al. 2010) might play arole, as might the weak asymmetry of the power excess humps.To investigate this in more detail we have to wait until Kepler can provide significantly longer time series. On the other hand,we expect a larger e ff ect from the limitations of the used modelgrid. Although our grid covers a wide range in chemical com-position, which is one of the parameters that can significantlyinfluence the M - L - R - T e ff relation, there are other model param-eters such as overshooting or a better description of convection,that could change this relation as well and therefore could haveconsequences to our analysis. These e ff ects are di ffi cult to esti-mate and are still largely unexplored territory.For the individual seismic parameters we refer toHekker et al. (2010c) providing an online table for all red gi-ants observed with Kepler . However, our fundamental parameterestimates are not included in this table because they will con-tinuously be improved with the ongoing observations. But weencourage everybody who is interested in our results to requestthem personally from the lead author.
Acknowledgements.
Funding for the Kepler Mission is provided by NASA’sScience Mission Directorate. TK is supported by the Canadian Space Agency,the Austrian Research Promotion Agency, and the Austrian Science Fund.SH, YPE and WJC acknowledge support by the UK Science and TechnologyFacilities Council. DH acknowledges support by the Astronomical Society ofAustralia. The research leading to these results has received funding fromthe European Research Council under the European Community’s SeventhFramework Programme (FP7 / / ERC grant agreement n ◦ / /
04. The National Center for Atmospheric Research isa federally funded research and development center sponsored by the U.S.National Science Foundation. We acknowledge support from the AustralianResearch Council. The authors gratefully acknowledge the Kepler Science Teamand all those who have contributed to making the Kepler Mission possible.
References
Ballot, J., Garc´ıa, R. A., & Lambert, P. 2006, MNRAS, 369, 1281Barban, C., Matthews, J. M., De Ridder, J., et al. 2007, A&A, 468, 1033Basu, S., Chaplin, W. J., & Elsworth, Y. 2010, ApJ, 710, 1596Batalha, N. M., Borucki, W. J., Koch, D. G., et al. 2010, ApJL, 713, L109Bedding, T. R., Huber, D., Stello, D., et al. 2010, ApJL, 713, L176Borucki, W., Koch, D., Basri, G., et al. 2008, in IAU Symposium, Vol. 249, IAUSymposium, ed. Y.-S. Sun, S. Ferraz-Mello, & J.-L. Zhou, 17–24Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977Broomhall, A., Chaplin, W. J., Davies, G. R., et al. 2009, MNRAS, 396, L100Brown, T. M., Gilliland, R. L., Noyes, R. W., & Ramsey, L. W. 1991, ApJ, 368,599Buzasi, D., Catanzarite, J., Laher, R., et al. 2000, ApJL, 532, L133Carrier, F., De Ridder, J., Baudin, F., et al. 2010, A&A, 509, A73 + Christensen-Dalsgaard, J. 2004, Solar Physics, 220, 137Christensen-Dalsgaard, J., Kjeldsen, H., Brown, T. M., et al. 2010, ApJL, 713,L164Creevey, O. L., Monteiro, M. J. P. F. G., Metcalfe, T. S., et al. 2007, ApJ, 659,616De Ridder, J., Barban, C., Baudin, F., et al. 2009, Nature, 459, 398De Ridder, J., Barban, C., Carrier, F., et al. 2006, A&A, 448, 689Dupret, M., Belkacem, K., Samadi, R., et al. 2009, A&A, 506, 57Edmonds, P. D. & Gilliland, R. L. 1996, ApJL, 464, L157 + Frandsen, S., Carrier, F., Aerts, C., et al. 2002, A&A, 394, L5Frohlich, C., Andersen, B. N., Appourchaux, T., et al. 1997, Solar Physics, 170,1Gilliland, R. L., Brown, T. M., Christensen-Dalsgaard, J., et al. 2010, PASP, 122,131Girardi, L. 1999, MNRAS, 308, 818Girardi, L., Groenewegen, M. A. T., Hatziminaoglou, E., & da Costa, L. 2005,A&A, 436, 895Gizon, L. & Solanki, S. K. 2003, ApJ, 589, 1009Gruberbauer, M., Kallinger, T., Weiss, W. W., & Guenther, D. B. 2009, A&A,506, 1043Harvey, J. 1985, in ESA Special Publication, Vol. 235, Future Missions in Solar,Heliospheric & Space Plasma Physics, ed. E. Rolfe & B. Battrick, 199– + Hekker, S., Broomhall, A., Chaplin, W. J., et al. 2010a, MNRAS, 402, 2049Hekker, S., Debosscher, J., Huber, D., et al. 2010b, ApJL, 713, L187Hekker, S., Elsworth, Y., De Ridder, J., et al. 2010c, A&A (submitted)Hekker, S., Kallinger, T., Baudin, F., et al. 2009, A&A, 506, 465Huber, D., Bedding, T. R., Stello, D., et al. 2010, ApJ (submitted)
Kepler red giants
Huber, D., Stello, D., Bedding, T. R., et al. 2009, Communications inAsteroseismology, 160, 74Jenkins, J. M., Caldwell, D. A., Chandrasekaran, H., et al. 2010, ApJL, 713,L120Kallinger, T., Guenther, D. B., Matthews, J. M., et al. 2008a, A&A, 478, 497Kallinger, T., Guenther, D. B., Weiss, W. W., et al. 2008b, CoAst, 153, 84Kallinger, T., Weiss, W. W., Barban, C., et al. 2010, A&A, 509, A77 + Kallinger, T., Zwintz, K., Pamyatnykh, A. A., Guenther, D. B., & Weiss, W. W.2005, A&A, 433, 267Kjeldsen, H. & Bedding, T. R. 1995, A&A, 293, 87Kjeldsen, H., Bedding, T. R., & Christensen-Dalsgaard, J. 2008, ApJ, 683, L175Latham, D. W., Brown, T. M., Monet, D. G., et al. 2005, in Bulletin ofthe American Astronomical Society, Vol. 37, Bulletin of the AmericanAstronomical Society, 1340– + Mathur, S., Garc´ıa, R. A., R´egulo, C., et al. 2010, A&A, 511, A46 + Mazumdar, A., M´erand, A., Demarque, P., et al. 2009, A&A, 503, 521Merline, W. J. 1999, in Astronomical Society of the Pacific ConferenceSeries, Vol. 185, IAU Colloq. 170: Precise Stellar Radial Velocities, ed.J. B. Hearnshaw & C. D. Scarfe, 187– + Miglio, A., Montalb´an, J., Baudin, F., et al. 2009, A&A, 503, L21Monet, D. G., Jenkins, J. M., Dunham, E. W., et al. 2010, ArXiv e-printsMosser, B. & Appourchaux, T. 2009, A&A, 508, 877Mosser, B., Belkacem, K., Goupil, M. J., et al. 2010, ArXiv e-printsNigam, R., Kosovichev, A. G., Scherrer, P. H., & Schou, J. 1998, ApJL, 495,L115 + Pietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, ApJ, 612, 168Retter, A., Bedding, T. R., Buzasi, D. L., Kjeldsen, H., & Kiss, L. L. 2003, ApJL,591, L151Stello, D., Basu, S., Bruntt, H., et al. 2010, ApJL, 713, L182Stello, D., Bruntt, H., Kjeldsen, H., et al. 2007, MNRAS, 377, 584Stello, D., Bruntt, H., Preston, H., & Buzasi, D. 2008, ApJL, 674, L53Stello, D., Chaplin, W. J., Basu, S., Elsworth, Y., & Bedding, T. R. 2009a,MNRAS, 400, L80Stello, D., Chaplin, W. J., Bruntt, H., et al. 2009b, ApJ, 700, 1589Stello, D. & Gilliland, R. L. 2009, ApJ, 700, 949Tarrant, N. J., Chaplin, W. J., Elsworth, Y., Spreckley, S. A., & Stevens, I. R.2007, MNRAS, 382, L48Tassoul, M. 1980, ApJS, 43, 469Toutain, T. & Froehlich, C. 1992, A&A, 257, 287van Leeuwen, F. 2007, A&A, 474, 653
Fig. 9.
H-R diagram showing the location of our red-giant sam-ple as directly determined from Eq. 4 and 5 using the KIC ef-fective temperatures (panel a ) and from the comparison withthe stellar model grid (panel b to e ). The colour code (in theonline version only) indicates the stellar mass ( a and b ), ra-dius ( c ), age ( d ), and metallicity ( e ), where the radius scale hasbeen truncated below 6 and above 14 R ⊙ . Grey lines show solar-metallicity BaSTI evolutionary tracks. The boxes given in thelower right corners illustrate typical uncertainties. Kepler red giants
Fig. 11.
Histograms of the relative (and absolute for T e ff ) uncer-tainties for the asteroseismic fundamental parameter. p e r c e n t a g e o f s t a r s Fig. 12.
Histograms for the radius and mass distribution in oursample of red-giant stars compared to the distribution of a syn-thetic red-giant population.
Fig. 13.
Comparison between the seismically determined e ff ec-tive temperature and surface gravity and the corresponding KICparameters. Dashed lines indicate a linear fit and red square sym-bols (in the online version only) show binned values.ec-tive temperature and surface gravity and the corresponding KICparameters. Dashed lines indicate a linear fit and red square sym-bols (in the online version only) show binned values.