Asteroseismology of red giants from the first four months of Kepler data: Global oscillation parameters for 800 stars
D. Huber, T. R. Bedding, D. Stello, B. Mosser, S. Mathur, T. Kallinger, S. Hekker, Y. P. Elsworth, D. L. Buzasi, J. De Ridder, R. L. Gilliland, H. Kjeldsen, W. J. Chaplin, R. A. Garcia, S. J. Hale, H. L. Preston, T. R. White, W. J. Borucki, J. Christensen-Dalsgaard, B. D. Clarke, J. M. Jenkins, D. Koch
aa r X i v : . [ a s t r o - ph . S R ] O c t Accepted for publication in the Astrophysical Journal
Preprint typeset using L A TEX style emulateapj v. 11/10/09
ASTEROSEISMOLOGY OF RED GIANTS FROM THE FIRST FOUR MONTHS OF
KEPLER
DATA: GLOBALOSCILLATION PARAMETERS FOR 800 STARS
D. Huber , T. R. Bedding , D. Stello , B. Mosser , S. Mathur , T. Kallinger , S. Hekker , Y. P. Elsworth ,D. L. Buzasi , J. De Ridder , R. L. Gilliland , H. Kjeldsen , W. J. Chaplin , R. A. Garc´ıa , S. J. Hale ,H. L. Preston , T. R. White , W. J. Borucki , J. Christensen-Dalsgaard , B. D. Clarke , J. M. Jenkins ,and D. Koch Accepted for publication in the Astrophysical Journal
ABSTRACTWe have studied solar-like oscillations in ∼
800 red-giant stars using
Kepler long-cadence photome-try. The sample includes stars ranging in evolution from the lower part of the red-giant branch to theHelium main sequence. We investigate the relation between the large frequency separation (∆ ν ) andthe frequency of maximum power ( ν max ) and show that it is different for red giants than for main-sequence stars, which is consistent with evolutionary models and scaling relations. The distributionsof ν max and ∆ ν are in qualitative agreement with a simple stellar population model of the Keplerfield, including the first evidence for a secondary clump population characterized by M & M ⊙ and ν max ≃ − µ Hz. We measured the small frequency separations δν and δν in over 400 stars and δν in over 40. We present C-D diagrams for l = 1, 2 and 3 and show that the frequency separationratios δν /∆ ν and δν /∆ ν have opposite trends as a function of ∆ ν . The data show a narrowing ofthe l = 1 ridge towards lower ν max , in agreement with models predicting more efficient mode trappingin stars with higher luminosity. We investigate the offset ǫ in the asymptotic relation and find a clearcorrelation with ∆ ν , demonstrating that it is related to fundamental stellar parameters. Finally, wepresent the first amplitude- ν max relation for Kepler red giants. We observe a lack of low-amplitudestars for ν max & µ Hz and find that, for a given ν max between 40 − µ Hz, stars with lower ∆ ν (and consequently higher mass) tend to show lower amplitudes than stars with higher ∆ ν . Subject headings: stars: oscillations — stars: late-type INTRODUCTION
Stars with convective envelopes show solar-like oscil-lations that are sensitive to the physical processes gov-erning their interiors (see, e.g., Brown & Gilliland 1994;Christensen-Dalsgaard 2004). Following the success ofhelioseismology, the detection of such oscillations in a Sydney Institute for Astronomy (SIfA), School ofPhysics, University of Sydney, NSW 2006, Australia;[email protected] LESIA, CNRS, Universit´e Pierre et Marie Curie, Universit´eDenis, Diderot, Observatoire de Paris, 92195 Meudon cedex,France High Altitude Observatory, NCAR, P.O. BOX 3000, Boul-der, CO 80307, USA Department of Physics and Astronomy, University of BritishColumbia, Vancouver, Canada Institute for Astronomy, University of Vienna, 1180 Vienna,Austria School of Physics and Astronomy, University of Birming-ham, Edgbaston, Birmingham B15 2TT, UK Eureka Scientific, 2452 Delmer Street Suite 100, Oakland,CA 94602-3017, USA Instituut voor Sterrenkunde, K.U.Leuven, Belgium Space Telescope Science Institute, 3700 San Martin Drive,Baltimore, Maryland 21218, USA Danish AsteroSeismology Centre (DASC), Department ofPhysics and Astronomy, Aarhus University, DK-8000 Aarhus C,Denmark Laboratoire AIM, CEA/DSM-CNRS, Universit´e Paris 7Diderot, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-Yvette,France Department of Mathematical Sciences, University of SouthAfrica, Box 392 UNISA 0003, South Africa NASA Ames Research Center, MS 244-30, Moffett Field,CA 94035, USA SETI Institute, NASA Ames Research Center, MS 244-30,Moffett Field, CA 94035, USA variety of stars holds great promise for improving ourunderstanding of stellar structure and evolution.The traditional goal of asteroseismology is the accu-rate measurement of individual mode frequencies, whichcan be used to test stellar physics by comparing them tofrequencies predicted by models. In a more general ap-proach, global oscillation parameters can be used. Theseinclude the average frequency separations, which are di-rectly related to properties of the sound speed in the stel-lar interior and therefore to fundamental stellar parame-ters. Other parameters are the amplitude and central fre-quency of the oscillation envelope, which are importantfor understanding the physics of driving and dampingof these modes. The measurement of these parameterspresents a valuable addition to classical methods such asspectroscopy, and a powerful tool to systematically studystellar evolution when oscillations are detected in a largeensemble of stars.Compared to main-sequence stars, red giants pulsatewith larger amplitudes and longer periods, therefore re-quiring less sensitivity but longer and preferably con-tinuous time series for unambiguous detections. Thefirst attempts to detect oscillations in G and K giantswere focused on nearby targets such as Arcturus (Smith1983; Cochran 1988; Belmonte et al. 1990). This wasfollowed by campaigns targeting single stars and clus-ters using precise ground-based Doppler spectroscopy(Frandsen et al. 2002; De Ridder et al. 2006) and pho-tometry (Stello et al. 2007), as well as using space-basedphotometers such as the
HST (Edmonds & Gilliland1996; Gilliland 2008; Stello & Gilliland 2009),
WIRE
D. Huber et al.(Buzasi et al. 2000; Retter et al. 2003; Stello et al. 2008),
MOST (Barban et al. 2007; Kallinger et al. 2008a,b) and
SMEI (Tarrant et al. 2007).A breakthrough was achieved by
CoRoT with unam-biguous detections of radial and non-radial modes in ∼
800 red giants (De Ridder et al. 2009; Hekker et al.2009; Carrier et al. 2010). With a maximum time serieslength of 150 d but limitations due to the low-Earth orbitof the satellite, the
CoRoT detections were focused onlow-mass He-core burning stars (the red clump). Usingthese data, Miglio et al. (2009) performed the first popu-lation study using global oscillation parameters and con-cluded that the distributions were qualitatively in agree-ment with the current picture of the star formation ratein our galaxy. Mosser et al. (2010) subsequently useda larger number of detections in the
CoRoT sample toinvestigate correlations of various oscillation parameters.A new era of “ensemble asteroseismology” was re-cently entered with the launch of the
Kepler space tele-scope (Gilliland et al. 2010). First results for red gi-ants were based on 34 d of data (Bedding et al. 2010b;see also Stello et al. 2010; Hekker et al. 2010b) whichdemonstrated the enormous potential of
Kepler data.While Bedding et al. (2010b) focused on a sample of low-luminosity giants, the goal of this paper is to systemati-cally investigate global oscillation parameters in the com-plete
Kepler red-giant sample using data spanning up to138 d. We refer to our companion papers for the compar-ison of global oscillation parameters derived using differ-ent methods (Hekker et al. 2010c) and the asteroseismicdetermination of stellar masses and radii (Kallinger et al.2010b). OBSERVATIONS AND DATA ANALYSIS
The
Kepler space telescope was launched in March2009 with the principal science goal of detecting Earth-like planets around solar-like stars through the observa-tion of photometric transits.
Kepler employs two ob-servation modes, sampling data either in 1 min (short-cadence) or 29.4 min (long-cadence) intervals. For ourstudy of pulsations in red giants we used
Kepler long-cadence data (Jenkins et al. 2010), which have a Nyquistfrequency of 283 µ Hz.
Kepler is located in an Earth-trailing orbit with space-craft rolls performed at quarterly intervals to redirect so-lar panels towards the Sun. Data are consequently sub-divided into quarters, starting with the initial commis-sioning run (10 d, Q0), followed by a short first quarter(34 d, Q1) and a full second quarter (90 d, Q2). The basisof our study consists of 1531 light curves for which datafrom all these quarters are available (total time span of137.9 d). We did not include stars thought to be mem-bers of the open clusters in the
Kepler field (NGC6791,NGC6811, NGC6819 and NGC6866).Before we extracted asteroseismic information from thetime series, two main instrumental artifacts had to beaddressed. Firstly, in some cases data from differentquarters show intensity discontinuities that are mainlycaused by pixel shifts after reorientation of the spacecraft(Jenkins et al. 2010). Additionally, two safe-mode eventsin Q2 caused intensity drifts due to thermal effects, af-fecting in total ∼ Fig. 1.—
Top panel: Kepler light curve of the oscillating redgiant KIC5515314. The different quarters of data used in thiswork are indicated.
Middle panel:
Amplitude spectrum of the lightcurve shown in the top panel. Note that the ordinate shows thesquare root of power density, where the latter is power multipliedby the effective length of the dataset (calculated as the inverseof the area under the spectral window).
Bottom panel:
Spectralwindow, shown at the same abscissa scale as the middle panel.Note that the height of the peak at zero frequency is 1. order polynomials to Q0, Q1 and five separate segmentsof Q2 which were unaffected by intensity jumps (see alsoKallinger et al. 2010b). Finally, we performed a simple4- σ outlier clipping to remove remaining outlying datapoints (in most cases < µ Hz. The spectral window shown inthe bottom panel of Figure 1 demonstrates the nearlycontinuous sampling of
Kepler data. The weak peaksat multiples of 3.96 µ Hz are caused by the angular mo-mentum dumping cycle of the spacecraft causing one re-jected datapoint every 2.9 d. At a level of less than 1% inamplitude, these aliasing artifacts are negligible for theanalysis considered in this paper.To extract oscillation parameters from the correctedtime series, several methods have been employedlobal oscillation parameters of red giants observed by
Kepler ∼ Kepler data, includingmixed modes, background noise and varying modelifetimes between 15–75 d.Before reporting our results, we provide a brief sum-mary of the global oscillation parameters used in thiswork, their physical interpretation and the principalmethods with which they were determined. GLOBAL OSCILLATION PARAMETERS
Frequency of maximum power ( ν max ) As first argued by Brown et al. (1991), ν max for sun-like stars is expected to scale with the acoustic cut-off fre-quency and can therefore be related to fundamental stel-lar parameters, as follows (Kjeldsen & Bedding 1995): ν max = M/M ⊙ ( T eff /T eff , ⊙ ) . L/L ⊙ ν max , ⊙ . (1)As shown by Stello et al. (2009) for stellar models andas observed in stars with well-determined fundamentalparameters (see, e.g., Stello et al. 2008), this scaling re-lation also holds for red giants. In these stars, ν max isa good indicator of the evolutionary stage, and rangesfrom ∼ µ Hz for high-luminosity giants to ∼ µ Hzfor H-shell-burning stars in the lower part of the red-giant branch. It is measured by determining the max-imum of the power excess after heavily smoothing overseveral orders or by fitting a Gaussian function to theexcess power.To illustrate our sample, Figure 2 shows all starsin a plot of ν max versus T eff . Effective temperatureshave been taken from the Kepler Input Catalog (KIC)(Latham et al. 2005) and we have overlaid solar-scaled ASTEC evolutionary tracks (Christensen-Dalsgaard2008). In addition to the red giants, we also show a sam-ple of main-sequence and sub-giant stars for which oscil-lations had been detected prior to the
Kepler mission (seeStello et al. 2009, and references therein). Figure 2 canbe viewed as an asteroseismic H-R diagram in which, inthe absence of parallaxes, we have used 1 /ν max instead ofluminosity. As can be seen from the evolutionary tracksin Figure 2, our sample spans a total range of masses ofapproximately 1–3 M ⊙ , with temperatures ranging from4200 to 5200 K. For a detailed study of the fundamen-tal parameters of the stars in the sample we refer to ourcompanion paper (Kallinger et al. 2010b). Frequency separations
According to the asymptotic relation for modes of lowangular degree l and high radial order n (Tassoul 1980;Gough 1986), frequencies of solar-like oscillations can bedescribed by a series of characteristic separations. Obser- Fig. 2.— ν max versus effective temperature for all red giantsin our sample (diamonds), as well as main-sequence and sub-giantstars studied by Stello et al. (2009) (squares). Error bars have beenomitted for clarity. The grey lines show solar-metallicity ASTEC evolutionary tracks (Christensen-Dalsgaard 2008) for a range ofmasses (note that the 0.8 M ⊙ track has been evolved beyond theage of the universe). The dashed line marks the approximate po-sition of the red edge of the instability strip. vationally, these separations can be expressed as follows(Bedding & Kjeldsen 2010): ν n,l = ∆ ν ( n + 12 l + ǫ ) − δν l . (2)Here, ∆ ν denotes the mean large frequency separationof modes with the same degree and consecutive order.∆ ν is directly related to the sound travel time acrossthe stellar diameter and probes the mean stellar density(Ulrich 1986). This means that ∆ ν is expected to scaleas follows:∆ ν = ( M/M ⊙ ) . ( T eff /T eff , ⊙ ) ( L/L ⊙ ) . ∆ ν ⊙ . (3)In Equation (2), δν l denotes the small frequency sepa-rations of non-radial modes relative to radial modes, asfollows: δν = ν n, − ν n − , , (4) δν = 12 ( ν n, + ν n +1 , ) − ν n, , (5) δν = 12 ( ν n, + ν n +1 , ) − ν n, . (6)Following Bedding et al. (2010b), we have used δν in-stead of the more commonly used δν due to the broad-ening of the l = 1 ridge in red giants caused by mixedmodes (Dupret et al. 2009; Deheuvels et al. 2010). Formain-sequence stars, small frequency separations are sen-sitive to variations of the sound speed gradient near thestellar core, which changes as the star evolves due to theincrease in its mean molecular weight.The phase constant ǫ in Equation (2) has contributionsfrom the inner and outer turning point of the modes. D. Huber et al.This is usually expressed as ǫ = + α , where α is thecontribution from the outer turning point, which is deter-mined by the properties of the near-surface region of thestar (Christensen-Dalsgaard & Perez Hernandez 1992). Maximum mode amplitude
The maximum mode amplitude is related to theturbulent convection mechanisms that excite anddamp the oscillations. Kjeldsen & Bedding (1995)found that model predictions of solar-like oscillationsby Christensen-Dalsgaard & Frandsen (1983) implied ascaling for velocity amplitudes of v osc ∝ (cid:18) LM (cid:19) s , (7)with s = 1 . s = 0 . A λ observed in photometry at wave-length λ is related to the velocity amplitude: A λ ∝ v osc λ T − r eff , (8)where r = 1 . r = 2 . s = 0 . r = 2 .
0) for cluster red giantsobserved with
Kepler , while Mosser et al. (2010) founda best fitting value of s = 0 . ± .
02 (assuming r = 1 . CoRoT red giants. As for ν max , mode amplitudesare usually determined by heavily smoothing the powerspectrum to eliminate variations caused by the stochasticnature of the signal (Kjeldsen et al. 2008) or by fittinga Gaussian function to the power excess envelope. Inboth cases, it is important to determine an accurate fitto the stellar background contribution. To account forthe effects of averaging from the 29.4 min integrations,we divided the observed amplitude with a sinc function: A real = A obs / sinc( π ν max ν Nyq ). RESULTS
The ∆ ν - ν max relation The growing number of detections of solar-like oscilla-tions across the HR diagram has revealed a tight power-law relation between ∆ ν and ν max :∆ ν = α ( ν max /µ Hz) β . (9)Using a fit to red giants and main-sequence stars,Stello et al. (2009) derived α = 0 . ± . µ Hz and β = 0 . ± . CoRoT red giants ( ν max < µ Hz) by Hekker et al.(2009) and subsequently by Mosser et al. (2010), withthe latter finding α = 0 . ± . µ Hz and β = 0 . ± . Kepler data allow us to investigate this relationfor the first time over a large part of the red giant branch.Figure 3 shows the relation for our sample. To derive the
Fig. 3.—
Upper panel: ∆ ν versus ν max for the Kepler red giants.The fitted relation is indicated by the red dashed line, with red dot-ted lines marking 3- σ uncertainties. The blue dashed-dotted lineshows the relation by Stello et al. (2009). Note that the relation byMosser et al. (2010) and the average relation for Kepler stars de-rived using seven different analysis methods are indistinguishablefrom the red dashed line.
Lower panel:
Residuals after subtractingthe fit indicated by the red dashed line. coefficients in Equation (9), we performed least-squaresfits to the results of seven different methods. Note thatwe have omitted all stars with ν max > µ Hz from theanalysis of the ∆ ν - ν max relation due to the difficulty ofdetermining accurate ν max values close to the Nyquistfrequency. As can be seen in Table 1, there is good agree-ment between the results of different methods, as well aswith the result by Mosser et al. (2010) that was basedon CoRoT stars with ν max < µ Hz.As already shown by Stello et al. (2009), a value of β ∼ .
75 is expected from the scaling relations for ν max and ∆ ν . Raising Equation (1) to the power of 0.75 anddividing by Equation (3) yields( ν max /µ Hz) . ∆ ν/µ Hz ∝ (cid:18) MM ⊙ (cid:19) . (cid:18) T eff T eff , ⊙ (cid:19) − . . (10)This removes the dependence on luminosity and leavesonly weak dependences on mass and effective tempera-ture, with the latter only varying by a small amount onthe red-giant branch.As noted by Mosser et al. (2010), the difference be-tween the relation for red giants and the one includingmain-sequence stars by Stello et al. (2009) is small butsignificant. To illustrate this difference we plot the rela-tions together with observations and evolutionary tracksin Figure 4. Note that ν max and ∆ ν for the models havebeen calculated using equations (1) and (3). Comparedto Figure 3, we replaced the ordinate in the upper panelby the ratio ν max /∆ ν , which scales with the radial or-der of maximum power. The lower panel shows this ra-tio with ν max raised to the power of 0.75, which betterillustrates the mass dispersion since in this plot, con-lobal oscillation parameters of red giants observed by Kepler TABLE 1Coefficients of the ∆ ν - ν max relation. Source α β ν max µ Hz) ( µ Hz)A2Z ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Kepler red giants: A2Z - Mathur et al. (2010), CAN -Kallinger et al. (2010b), COR - Mosser & Appourchaux (2009),DLB - Buzasi et al. (unpublished), OCTI & OCTII - Hekker et al.(2010d), SYD - Huber et al. (2009). CoRoT red giants, see Mosser et al. (2010). Main-sequence (MS) and red giant (RG) stars, see Stello et al.(2009) and references therein. stant masses correspond to almost horizontal lines (seeEquation (10)). The
Kepler stars are shown in bins of20 µ Hz (red diamonds). Note that the red giants used byStello et al. (2009) have been omitted for clarity, but donot differ significantly from the
Kepler sample. The un-certainties for the main-sequence sample have been col-lected from the literature where available or were set totypical values of 3% for ν max and 1% for ∆ ν .The scatter about the ∆ ν - ν max relation is considerablylarger than the measurement uncertainties (which are ∼
1% for ∆ ν in our sample) and for red giants is mainlycaused by the spread of stellar masses (Kallinger et al.2010c). While Figure 4 shows that this spread is signif-icant on the red-giant branch, it can be seen that theevolutionary tracks (solid lines) almost overlap for main-sequence stars. This effect causes a different fit to the re-lation between ∆ ν and ν max for a sample consisting onlyof red giants compared to when main-sequence stars areincluded, which is clearly reflected by the observationsand the determined fits shown in Figure 4.The dotted and dashed triple-dotted lines in Figure4 illustrate the effect of changing the metallicity forthe 1 M ⊙ track. The effect is relatively small on thegiant branch (Kallinger et al. 2010c), but more signifi-cant for less evolved stars. In fact, both main-sequencestars significantly above the Stello et al. (2009) rela-tion, η Boo and µ Her, are observed to be metal-rich(Carrier et al. 2005; Yang & Meng 2010), while the sub-giant ν Indi which falls below the relation is metal-poor(Bedding et al. 2006). Hence, these observations arequalitatively in agreement with expectations from evo-lutionary models combined with scaling relations.A detailed investigation of these effects using pulsa-tion models (which would be necessary to quantify massand metallicity changes in Figure 4 in absolute terms)is beyond the scope of this paper. However, our resultsshow that the observed difference in the ∆ ν - ν max rela-tion for red giants compared to when main-sequence starsare included is consistent with scaling relations. A moredetailed investigation of the relation for main-sequencestars using Kepler short-cadence data will be presentedin a forthcoming paper.
Fig. 4.—
Upper panel:
Solar-metallicity (Y=0.283, Z=0.017)evolutionary tracks (solid lines) together with the relation byStello et al. (2009) (blue dashed-dotted line) and the relation fit-ted to the
Kepler data (red dashed line). Blue squares show themain-sequence and sub-giant stars used by Stello et al. (2009), andred diamonds the
Kepler observations as shown in Figure 3 inbins of 20 µ Hz. The black dashed triple-dotted and dotted linesshow the solar-mass tracks for chemical compositions of (Y,Z) =(0.291,0.009) and (Y,Z) = (0.265,0.035), respectively.
Lower panel:
Same as the upper panel, but with the luminosity dependence ofthe ordinate removed by raising ν max to the power of 0.75 (seeEquation (10)). Population effects in the ν max and ∆ ν distributions The large number and diversity of red giants for whichsolar-like oscillations have been detected enables us toidentify stellar populations using global oscillation pa-rameters. Due to the straightforward relationship be-tween oscillation parameters and fundamental parame-ters (see equations (1) and (3)), comparisons with mod-els could allow us to draw some conclusions about thestar-formation history in the
Kepler field. Such studieshave already been performed for red giants observed by
CoRoT (Miglio et al. 2009; Yang et al. 2010).Before such a comparison is made, it is important toconsider potential biases in the sample of
Kepler redgiants. Firstly, limitations on the bandwidth of space-craft communications restrict the number of stars forwhich data can be obtained. Hence, absolute star countsfor a given field and magnitude range will be incom-plete. Secondly, about two thirds of the
Kepler red-giantsample have been chosen as astrometric reference stars(Batalha et al. 2010; Monet et al. 2010). These havebeen selected to be distant, bright but unsaturated gi-ants ( T eff , KIC < g KIC < . R/R ⊙ , KIC > Kepler magnitude range 11.0–12.5. Through theremainder of this paper we will refer to these stars as theastrometric sample, and to the remaining stars as theasteroseismic sample. Thirdly, as was the case for the
CoRoT sample, the
Kepler sample excludes stars pul- D. Huber et al.
Fig. 5.—
Histograms of ν max (left panels) and ∆ ν (right panels) in percent comparing a synthetic stellar population for the Kepler
FOV(top panels), the observed distributions in the
Kepler sample (middle panels) and the observed distributions in the
CoRoT sample (bottompanels). The dashed and dotted lines in the middle panel separately show the distributions of the astrometric and asteroseismic sample,respectively. Note that the comparison for the
Kepler sample should be restricted to stars with ν max & µ Hz (∆ ν & . µ Hz). sating at frequencies that are too low to be resolved withthe given time base available at this stage of the mission( ν max . µ Hz).To obtain a first qualitative analysis of the observeddistributions of the
Kepler sample, we calculated a syn-thetic stellar population the same way as Miglio et al.(2009), using the stellar synthesis code
TRILEGAL (Girardi et al. 2005). The input parameters were identi-cal to the simulations performed by Miglio et al. (2009)for the
CoRoT sample, with the exception of restrictingthe simulation to a 10 deg field centered on the KeplerFOV ( α =290.7 ◦ , δ =44.5 ◦ ) in the magnitude range 8 – 13mag in the Kepler bandpass. A constant star-formationrate was assumed.Figure 5 compares the distributions of the synthetic
Kepler population with the observed one, together withthe
CoRoT sample (Mosser et al. 2010). Note that thelatter includes stars from both fields of view of
CoRoT ,one directed to the galactic center, the other to the galac-tic anti-center. As for the
CoRoT sample, we observea maximum at ν max ∼ µ Hz, corresponding to the redclump. These are low-mass ( < M ⊙ ) stars, for which He-core burning occurs at similar luminosities (and hencesimilar ν max ), forming a dominant population on thegiant branch. Miglio et al. (2009) noted that for the CoRoT stars, the maximum appears at a lower valueof ν max (higher luminosity) than for the model. This isnot the case for the Kepler field.The dashed and dotted lines in the middle panel showthe distributions of the astrometric and asteroseismicsample separately, to illustrate potential biases intro-duced in the overall distribution by combining these dis-tinctly selected sets of stars. We see that the red clumpmaximum is predominantly formed by the astrometricsample, while the asteroseismic sample contributes moreto slightly higher ν max values. This suggests that the astrometric sample is largely unbiased, while the broad-ening of the red clump peak compared to the model ispotentially caused by selection bias in the asteroseismicsample.Apart from the red clump, a second and much broadercomponent in the synthetic population can be identifiedin the interval ν max ≃ − µ Hz and ∆ ν ≃ − µ Hz.This consists of more massive stars that, compared to thered clump, occupy lower luminosities (and hence higher ν max ) over a wider range when they settle as He-coreburning stars. This component is referred to as the sec-ondary clump by Girardi (1999). A comparison with thehistogram of the Kepler sample shows that the observa-tions qualitatively reproduce the distributions for bothhigher ν max and ∆ ν values.To investigate this further, Figure 6 shows the ratio ν max /∆ ν as a function of ν max for the synthetic popula-tion and for the Kepler observations. The solid lines areevolutionary tracks of different masses. Figure 7 showsthe same plot but with the luminosity dependence on theordinate removed by raising ν max to the power of 0.75.In both figures, the separation of the red clump and thesecondary clump is obvious in the population synthesis(upper panels). For the observations (lower panels) wecan identify a group of stars with M > M ⊙ extend-ing up to ν max ∼ µ Hz, as expected for the secondaryclump population.The qualitative agreement between the distributionsis encouraging and the excess of stars with ν max =40 − µ Hz and
M > M ⊙ suggests that the Kepler sample includes stars belonging to the secondary clumppopulation. As pointed out by Girardi (1999), the de-tection of these stars should allow us to probe importantphysics, such as convective-core overshooting, and helpto put tight constraints on the recent star-formation his-tory in the galaxy. The detailed modelling required forlobal oscillation parameters of red giants observed by
Kepler Fig. 6.—
Ratio of ν max /∆ ν as a function of ν max for the syntheticpopulation (upper panel, 1530 stars) and the Kepler red-giantsample (lower panel, 801 stars). Red lines show solar-metallicity(Y=0.283, Z=0.017)
ASTEC evolutionary tracks with masses asindicated in the plots. The red clump and secondary clump popu-lations are indicated for the synthetic population. these inferences is beyond the scope of this paper.
Small frequency separations
As discussed in Section 3.2, the small frequency sep-arations of main-sequence stars depend on the sound-speed gradient in the stellar core and are sensitive to theevolutionary state of a star. The excellent quality of the
Kepler data allows us to measure the small separations inan unprecedented number of stars covering a wide rangeof evolutionary states on the giant branch.To measure the small separations, we first examinedthe ´echelle diagram of each star using the power spec-trum in the frequency range ν max ± ν and then man-ually fine-tuned the large separation to make the l = 0ridge vertical. The adjustment to ∆ ν was typically a fewtenths of a microhertz. Stars for which no unambiguousidentification of l = 0 and 2 could be found were dis-carded from this analysis. The adjusted ´echelle diagramwas then collapsed, and a Gaussian function was fittedto each mode ridge that was identified. The center of thefitted Gaussian was taken as the position of that ridge.Using this technique we were able to measure the l = 0and 2 ridges for 470 stars. Of these, we measured the l = 1 ridge for 400 stars and the l = 3 ridge for 45 stars.The uncertainties were determined from extensive simu-lations of artificial Kepler data that were analyzed usingthe same method. Before discussing the separations ofthe ridges, we first consider their absolute positions.
Variation of ǫ The measured mode ridge centroids allow us to in-vestigate the parameter ǫ in Equation (2). Guided bythe Sun, for which ǫ ≃ . Fig. 7.—
Same as Figure 6, but with the luminosity dependenceof the ordinate removed by raising ν max to the power of 0.75 (seeEquation (10)). Kjeldsen & Bedding 1995), we have plotted the centroidpositions in Figure 8, which shows that ǫ is a function of∆ ν .Figure 8 also shows a considerable spread in ǫ for agiven ∆ ν , in particular for stars with ∆ ν < µ Hz. Whilethe spread is of the same order as the estimated uncer-tainties, we have tested whether some of it might be re-lated to physical properties of the stars. We have in-vestigated this by correlating ǫ with ν max /∆ ν which, asshown in Figure 6, is sensitive to stellar mass. No clearcorrelation could be found.The relation between ǫ and ∆ ν (and hence also ν max ) implies that ǫ is a function of fundamental pa-rameters. If this relation can be quantified for less-evolved stars, clear mode ridge identifications as pre-sented in this paper could potentially be used to predict ǫ for other stars. As suggested by Bedding & Kjeldsen(2010), such comparisons can be of considerable helpin cases where the mode ridge identification is diffi-cult due to short mode lifetimes or rotational splitting(see, e.g., Appourchaux et al. 2008; Benomar et al. 2009;Garc´ıa et al. 2009; Kallinger et al. 2010a; Bedding et al.2010a).Meanwhile, we can already use the ensemble resultsin Figure 8 to suggest ridge identifications for the four CoRoT red giants discussed by Hekker et al. (2010a).These stars have ∆ ν in the range 3.1 to 5.3 µ Hz and so weexpect ǫ to be 1.0 or slightly less. Looking at the ´echellediagrams of these stars (see Figures 8, 10, 12 and 14 inHekker et al. 2010a), we indeed see in all cases that oneof the two ridges falls there. In each of these diagrams,this indicates it is the left-hand ridge that correspondsto l = 1 and the other to l = 0. C-D diagrams
D. Huber et al.
Fig. 8.—
Mode ridge centroids as measured from the folded andcollapsed power spectrum for l = 0 (black squares), l = 1 (greydiamonds) and l = 2 (grey triangles). Note that the upper abscissalabel shows ǫ for l = 0 modes. Error bars are only shown for l = 0for clarity. Using the measured mode ridge centroids, we wereable to derive the small separations δν , δν and δν for 470, 400 and 45 stars, respectively. The results areshown in so-called C-D diagrams (Christensen-Dalsgaard1988) in Figure 9. We also plot the frequency separa-tion ratios δν l /∆ ν which, according to models, are ex-pected to be largely insensitive to surface layer effects(Roxburgh & Vorontsov 2003; Ot´ı Floranes et al. 2005;Mazumdar 2005; Chaplin et al. 2005).We can see in Figure 9a that δν is an almostfixed fraction of ∆ ν , which confirms the findings byBedding et al. (2010b) for low-luminosity red giants. Weobtain the following relation for the full range of ν max : δν = (0 . ± . ν + (0 . ± . , (11)in agreement with the relation found by Bedding et al.(2010b). We see a deviation from this linear relationshipfor stars with ∆ ν . µ Hz ( ν max . µ Hz), which canbe seen in Figure 9b as an increase of the ratio δν / ∆ ν .Additionally, we observe an increased spread of δν , par-ticularly for stars with ∆ ν ∼ µ Hz. While most of thisspread in the red clump is due to the much larger numberof stars (see Figure 5) and the larger uncertainty in de-termining δν , we tested whether it could also be related Fig. 9.—
C-D diagram of δν versus ∆ ν for l = 2 ( a ), l = 1 ( c )and l = 3 ( e ) and frequency separation ratios of δν /∆ ν for l = 2( b ), l = 1 ( d ) and l = 3 ( f ). Dashed lines in panels ( a ) and ( e )show linear fits to the data. to physical properties of the stars. C-D diagrams for redgiants calculated from stellar models (T. R. White et al.,in preparation) show that the expected range in δν / ∆ ν for M = 1 − M ⊙ and solar metallicity in non He-coreburning models is about 0.02, which is roughly compara-ble to the range of values observed in the data outside thered clump. Indeed, a comparison of stars with different δν in the lower panel of Figure 6 has shown the spreadcan be partially explained by a spread in stellar masses,lobal oscillation parameters of red giants observed by Kepler δν values. Morequantitative conclusions about the mass spread will bepossible once the measurement uncertainties are reducedby the collection of more data.Figure 9c shows that δν is negative for almost all redgiants, confirming the findings by Bedding et al. (2010b).As for l = 2, we observe a trend (but with oppositesign) in the frequency separation ratio δν /∆ ν with ∆ ν ,which can be seen in Figure 9d. The decrease of δν /∆ ν and the increase of δν /∆ ν appear to affect stars inthe same range of ∆ ν and by a similar amount ( ∼ − .
1. While this indicates thatthe observed trends are statistically almost uncorrelated,we cannot at this point exclude that this is due to thelarge uncertainty of measuring δν .As shown by Bedding et al. (2010b), the Kepler dataallow the detection of l = 3 modes in red giants. Wewere able to measure the small separation δν for 45stars and a linear fit to the relation with ∆ ν , shown inFigure 9e, yields: δν = (0 . ± . ν + (0 . ± . . (12)Figure 9f shows the ratio δν /∆ ν , which shows some ev-idence for an increase of δν with decreasing ∆ ν . Whilethe error bars appear to be too large to make any firmstatements on this variation, the observed scatter sug-gests that the uncertainties are overestimated. Giventhat the relative amplitude of l = 3 modes is not wellknown, it seems likely that the strength of l = 3 modeshas been considerably underestimated in the simulationsused to derive the uncertainties. Ensemble collapsed ´echelle diagram
In order to display the separations of the mode ridgesfor all stars, we use the measured values of ǫ to shiftand align the folded and collapsed power spectra in anensemble collapsed ´echelle diagram. The result is shownin the upper panel of Figure 10. Note that this diagramis slightly different from the scaled ´echelle diagram pre-sented by Bedding et al. (2010b) in that we have shifted the folded and collapsed power spectra rather than scal-ing frequencies. The thick solid line in the lower panelshows the upper panel summed along the full rangeof ν max . It clearly reveals the presence of ridges with l = 0 , ,
2, and also l = 3. The power for l = 3 is weakerthan found by Bedding et al. (2010b), who considered asmaller sample of low-luminosity giants with high signal-to-noise. We attribute this to the larger (and less biased)sample of stars in this paper, including stars with unde-tectable l = 3 modes, as well as to the fact that the l = 3ridge has a slight tilt, making its power spread out in thelower panel of Figure 10.The trends of the frequency separation ratios observedin Figure 9 are clearly visible in Figure 10. As ν max (andhence ∆ ν ) decreases, the l = 2 ridge slopes to the leftwhile the l = 1 ridge slopes to the right, corresponding toan increase of δν /∆ ν and a decrease of δν /∆ ν . Thesefeatures are also clearly shown by the thin red and bluelines in the lower panel of Figure 10, which correspondto subsets with ν max < µ Hz and ν max > µ Hz,respectively. The red and blue lines also show evidence
Fig. 10.—
Upper panel:
Ensemble collapsed ´echelle diagram of470 red giants. Note that the contour threshold has been lowered to40% of the maximum value and that each row shows the folded andcollapsed power spectrum of one star. The right ordinate marks ν max values for selected rows. Lower panel:
Upper panel collapsedalong the entire range of ν max (thick black line) as well as in subsets ν max > µ Hz (thin blue line) and ν max < µ Hz (thin red line).Ridge identification and definitions of small separations used inthis paper are indicated. In both panels, the dotted lines mark thecenters of the l = 0, 2 and 3 ridge and the dashed line shows themidpoint of adjacent l = 0 modes. for a slope of the l = 3 ridge, corresponding to an increaseof δν as observed in Figure 9f.We note that the relative width of the l = 0 ridgeshown in Figure 10 (measured in terms of ∆ ν ) decreaseswith increasing ν max . In fact, the absolute width of theridge (in µ Hz), which is determined by the frequency res-olution, mode lifetime and curvature, remains roughlyconstant over the range of stars considered. As foundby Bedding et al. (2010b) for low-luminosity red giants,the l = 1 ridge is significantly broader than the others.This was interpreted as a direct confirmation of theo-retical results by Dupret et al. (2009), which predictedcomplicated power spectra due to less efficient trappingof mixed modes in the cores of low-luminosity red giants.0 D. Huber et al. Fig. 11.—
Absolute width of the l = 0 (dashed line), l = 1 (solidline) and l = 2 (dashed-dotted line) ridge as a function of ν max .The dotted line marks the formal frequency resolution of the data(0.08 µ Hz).
The models by Dupret et al. (2009) also predicted thatthis effect should become less pronounced for higher lu-minosity red giants. Indeed, we observe that the relativewidth of the l = 1 ridge shown in Figure 10 remainsroughly constant, indicating that its absolute width issignificantly lower for high-luminosity stars than for low-luminosity stars.To quantify this, we measured the absolute widths ofthe l = 0, 1 and 2 ridges in several bins of ν max , and showthe result in Figure 11. Note that we have accounted forpossible artificial broadening of the ridges due to Nyquisteffects by excluding the two stars in our sample with ν max > µ Hz from this calculation. As can be seen inFigure 11, the widths of the l = 0 and 2 ridges remainroughly constant over the range of ν max , while the widthof the l = 1 ridge increases significantly. Our observationof a much narrower l = 1 ridge for high-luminosity starstherefore provides further confirmation of the results byDupret et al. (2009). Pulsation amplitudes
Amplitudes of solar-like oscillations provide valuableinformation to test models of convection (see Section3.3). Equations (1), (7) and (8), in combination with ef-fective temperatures, can be used to test theoretical scal-ing relations for amplitudes (see, e.g., Stello et al. 2007),as follows: A λ ∝ T . s − r eff ν − s max λ . (13)As shown by Mosser et al. (2010) for the CoRoT red gi-ants, the relation between ν max and the pulsation ampli-tudes indeed follows a power law and this is shown forour Kepler sample in Figure 12. We have investigatedthe slope of the power law using amplitudes from fourdifferent methods (A2Z, COR, OCT and SYD, see Table1 for references) and find on average: A ∝ ν − . . (14)As can be seen from Figure 12, the measured ampli-tudes show considerable structure, in particular a lack Fig. 12.—
Pulsation amplitude versus ν max observed in the Kepler bandpass ( λ = 650 nm). The red solid line shows a powerlaw using a slope of − .
8. Red symbols show all stars with ν max > µ Hz and
M > M ⊙ identified in Figures 6 and 7. of low amplitudes for ν max & µ Hz. This feature isfound consistently in all methods and therefore appearsto be intrinsic. Some low-amplitude stars do appear at ν max & µ Hz, but we note that these amplitude esti-mates are rather uncertain due to the difficulty of esti-mating the background noise for stars oscillating near theNyquist frequency. To investigate this lack of low ampli-tudes, we plot all stars that we have tentatively identifiedas secondary clump stars (using Figures 6 and 7, setting ν max > µ Hz and
M > M ⊙ ) in Figure 12 as red sym-bols. We observe that these stars have systematically lowamplitudes. We conclude that, for a given ν max in therange between 40 − µ Hz, stars with lower ∆ ν (andtherefore higher mass, see Equation (10)) oscillate withlower amplitudes than other stars with the same ν max .Assuming that this also holds for low-luminosity red gi-ants, the lack of low amplitude stars for ν max & µ Hzcould then possibly be explained by the relatively fastevolution of more massive stars near the bottom of thered-giant branch compared to ∼ M ⊙ stars.Both the observation of low amplitudes for stars withlower ∆ ν in the range ν max ≃ − µ Hz and the lackof low amplitude stars for ν max & µ Hz indicate asignificant dependence of the amplitude- ν max relation onthe stellar mass. Since such a dependence is not seenin Equation (13), this suggests a revision of the scalingrelation in Equation (7).The overall distribution of the measured amplitudes inFigure 12 shows considerable additional scatter, whichis possibly related to physical effects such as metallic-ity (Samadi et al. 2010a,b) in combination with measure-ment uncertainties. Tests have shown that the measure-ment error for amplitudes is dominated by the specificmethod and model used to subtract the background sig-nal due to granulation and stellar activity. Further workis needed to identify possible systematic effects for theentire range of ν max , which will then allow us to improvethe relation between the oscillation amplitude and ν max using a large sample of stars on the giant branch. SUMMARY AND CONCLUSIONS lobal oscillation parameters of red giants observed by
Kepler
Kepler mission we have studied various aspects of solar-like os-cillations in ∼
800 red giants. Our conclusions can besummarized as follows:1. The ∆ ν - ν max relation using red giants is signif-icantly different from the relation when main-sequence stars are included. This difference canbe explained by evolutionary models and scalingrelations, which show that the ∆ ν - ν max relation ismainly sensitive to stellar masses for red giants andto metallicity for main-sequence stars.2. The ν max and ∆ ν distributions are in qualitativeagreement with a simple model of the stellar pop-ulation. We observed that ν max for the red clumpstars in the Kepler field is slightly higher than forthe CoRoT fields, possibly due to the differentstellar populations that are observed. We identi-fied several stars with ν max = 40 − µ Hz and
M > M ⊙ , in agreement with a secondary clumppopulation.3. The quantity ǫ in the asymptotic relation is a func-tion of ∆ ν . If the relation between ǫ and fundamen-tal parameters can be quantified and is also validfor main-sequence stars, it could be used to studysurface layer effects or for mode identification instars where ridges cannot be clearly identified dueto short mode lifetimes or rotational splitting. Wedemonstrated the potential of the observed rela-tion by providing mode identifications for four redgiants observed by CoRoT .4. We presented C-D diagrams for l = 1, 2 and 3 andshow that frequency separation ratios of δν /∆ ν and δν /∆ ν reveal opposite trends as a function of∆ ν . We observed a spread in the small separations,in particular for stars in the red clump, and find evidence that this is partially due to the spread inmass. We also measured the small separation δν and tentatively identify a similar variation of thefrequency separation ratio δν /∆ ν as observed for δν /∆ ν as a function of ∆ ν .5. The absolute width of the l = 1 ridge for stars ofhigher luminosity is significantly narrower than forstars with low luminosity. This is the first quanti-tative confirmation of more efficient mode trappingas predicted by theory.6. We presented a first estimate for the relation be-tween pulsation amplitude and ν max for Kepler red giants. We observed a distinct lack of low-amplitude stars for ν max & µ Hz and found that,for a given ν max , stars with lower ∆ ν (and thereforehigher mass, see Equation (10)) tend to show loweramplitudes than stars with higher ∆ ν . Both obser-vations can be explained with a mass dependence ofthe amplitude- ν max relation, and therefore suggestthat the scaling relation for luminosity amplitudesof red giants needs to be revised.The authors gratefully acknowledge the Kepler
ScienceTeam and all those who have contributed to the
Kepler mission for their tireless efforts which have made theseresults possible. We are also thankful to A. Miglio forhis kind help with the stellar population synthesis andto our anonymous referee for his/her helpful comments.Funding for the
Kepler
Mission is provided by NASA’sScience Mission Directorate. DH acknowledges supportby the Astronomical Society of Australia (ASA). DS andTRB acknowledge support by the Australian ResearchCouncil. The National Center for Atmospheric Researchis a federally funded research and development centersponsored by the U.S. National Science Foundation. SH,YPE and WJC acknowledge support by the UK Scienceand Technology Facilities Council.
REFERENCESAppourchaux, T., et al. 2008, A&A, 488, 705Barban, C., et al. 2007, A&A, 468, 1033Batalha, N. M., et al. 2010, ApJ, 713, L109Bedding, T. R., & Kjeldsen, H. 2010, Communications inAsteroseismology, 161, 3Bedding, T. R., et al. 2006, ApJ, 647, 558—. 2010a, ApJ, 713, 935—. 2010b, ApJ, 713, L176Belmonte, J. A., Jones, A. R., Palle, P. L., & Roca Cortes, T.1990, Ap&SS, 169, 77Benomar, O., et al. 2009, A&A, 507, L13Brown, T. M., & Gilliland, R. L. 1994, ARA&A, 32, 37Brown, T. M., Gilliland, R. L., Noyes, R. W., & Ramsey, L. W.1991, ApJ, 368, 599Buzasi, D., Catanzarite, J., Laher, R., Conrow, T., Shupe, D.,Gautier, III, T. N., Kreidl, T., & Everett, D. 2000, ApJ, 532,L133Carrier, F., Eggenberger, P., & Bouchy, F. 2005, A&A, 434, 1085Carrier, F., et al. 2010, A&A, 509, A73Chaplin, W. J., Elsworth, Y., Miller, B. A., New, R., & Verner,G. A. 2005, ApJ, 635, L105Christensen-Dalsgaard, J. 1988, in Proc. IAU Symp. 123,Advances in Helio- and Asteroseismology, ed.J. Christensen-Dalsgaard & S. Frandsen (Dordrecht: Kluwer),295Christensen-Dalsgaard, J. 2004, Sol. Phys., 220, 137 Christensen-Dalsgaard, J. 2008, Ap&SS, 316, 13Christensen-Dalsgaard, J., & Frandsen, S. 1983, Sol. Phys., 82,469Christensen-Dalsgaard, J., & Perez Hernandez, F. 1992, MNRAS,257, 62Cochran, W. D. 1988, ApJ, 334, 349De Ridder, J., Barban, C., Carrier, F., Mazumdar, A.,Eggenberger, P., Aerts, C., Deruyter, S., & Vanautgaerden, J.2006, A&A, 448, 689De Ridder, J., et al. 2009, Nature, 459, 398Deheuvels, S., et al. 2010, A&A, 515, A87Dupret, M., et al. 2009, A&A, 506, 57Edmonds, P. D., & Gilliland, R. L. 1996, ApJ, 464, L157Frandsen, S., et al. 2002, A&A, 394, L5Garc´ıa, R. A., et al. 2009, A&A, 506, 41Gilliland, R. L. 2008, AJ, 136, 566Gilliland, R. L., et al. 2010, PASP, 122, 131Girardi, L. 1999, MNRAS, 308, 818Girardi, L., Groenewegen, M. A. T., Hatziminaoglou, E., & daCosta, L. 2005, A&A, 436, 895Gough, D. O. 1986, in Hydrodynamic and MagnetodynamicProblems in the Sun and Stars, ed. Y. Osaki, 117Hekker, S., Barban, C., Baudin, F., De Ridder, J., Kallinger, T.,Morel, T., Chaplin, W. J., & Elsworth, Y. 2010a, A&A, inpress (arXiv:1006.4284Hekker, S., et al. 2009, A&A, 506, 4652 D. Huber et al.