Testing Scaling Relations for Solar-Like Oscillations from the Main Sequence to Red Giants using Kepler Data
D. Huber, T. R. Bedding, D. Stello, S. Hekker, S. Mathur, B. Mosser, G. A. Verner, A. Bonanno, D. L. Buzasi, T. L. Campante, Y. P. Elsworth, S. J. Hale, T. Kallinger, V. Silva Aguirre, W. J. Chaplin, J. De Ridder, R. A. Garcia, T. Appourchaux, S. Frandsen, G. Houdek, J. Molenda-Zakowicz, M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, R. L. Gilliland, S. D. Kawaler, H. Kjeldsen, A. M. Broomhall, E. Corsaro, D. Salabert, D. T. Sanderfer, S. E. Seader, J. C. Smith
aa r X i v : . [ a s t r o - ph . S R ] S e p accepted for publication in ApJ Preprint typeset using L A TEX style emulateapj v. 08/22/09
TESTING SCALING RELATIONS FOR SOLAR-LIKE OSCILLATIONS FROM THE MAIN SEQUENCE TORED GIANTS USING
KEPLER
DATA
D. Huber , T. R. Bedding , D. Stello , S. Hekker , S. Mathur , B. Mosser , G. A. Verner , A. Bonanno ,D. L. Buzasi , T. L. Campante , Y. P. Elsworth , S. J. Hale , T. Kallinger , V. Silva Aguirre ,W. J. Chaplin , J. De Ridder , R. A. Garc´ıa , T. Appourchaux , S. Frandsen , G. Houdek ,J. Molenda- ˙Zakowicz , M. J. P. F. G. Monteiro , J. Christensen-Dalsgaard , R. L. Gilliland ,S. D. Kawaler , H. Kjeldsen , A. M. Broomhall , E. Corsaro , D. Salabert , D. T. Sanderfer ,S. E. Seader , and J. C. Smith accepted for publication in ApJ ABSTRACTWe have analyzed solar-like oscillations in ∼ Kepler
Mission, spanningfrom the main-sequence to the red clump. Using evolutionary models, we test asteroseismic scalingrelations for the frequency of maximum power ( ν max ), the large frequency separation (∆ ν ) and os-cillation amplitudes. We show that the difference of the ∆ ν - ν max relation for unevolved and evolvedstars can be explained by different distributions in effective temperature and stellar mass, in agree-ment with what is expected from scaling relations. For oscillation amplitudes, we show that neither( L/M ) s scaling nor the revised scaling relation by Kjeldsen & Bedding (2011) is accurate for red-giantstars, and demonstrate that a revised scaling relation with a separate luminosity-mass dependencecan be used to calculate amplitudes from the main-sequence to red-giants to a precision of ∼ Subject headings: stars: oscillations — stars: late-type — techniques: photometric Sydney Institute for Astronomy (SIfA), School ofPhysics, University of Sydney, NSW 2006, Australia;[email protected] Astronomical Institute ’Anton Pannekoek’, University of Ams-terdam, Science Park 904, 1098 XH Amsterdam, The Netherlands School of Physics and Astronomy, University of Birmingham,Birmingham B15 2TT, UK High Altitude Observatory, NCAR, P.O. Box 3000, Boulder,CO 80307, USA LESIA, CNRS, Universit´e Pierre et Marie Curie, Universit´eDenis, Diderot, Observatoire de Paris, 92195 Meudon cedex,France Astronomy Unit, Queen Mary University of London, Mile EndRoad, London E1 4NS, UK INAF Osservatorio Astrofisico di Catania, Italy Eureka Scientific, 2452 Delmer Street Suite 100, Oakland, CA94602-3017, USA Danish AsteroSeismology Centre (DASC), Department ofPhysics and Astronomy, Aarhus University, DK-8000 Aarhus C,Denmark Department of Physics and Astronomy, University of BritishColumbia, Vancouver, Canada Institute of Astronomy, University of Vienna, 1180 Vienna,Austria Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany Instituut voor Sterrenkunde, K.U.Leuven, Belgium Laboratoire AIM, CEA/DSM-CNRS, Universit´e Paris 7Diderot, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-Yvette,France Institut d’Astrophysique Spatiale, UMR 8617, UniversiteParis Sud, 91405 Orsay Cedex, France Astronomical Institute of the University of Wroc law, ul.Kopernika 11, 51-622 Wroc law, Poland Space Telescope Science Institute, 3700 San Martin Drive,Baltimore, Maryland 21218, USA INTRODUCTION
Empirical relations connecting observable quantitieswith physical parameters of stars are of fundamental im-portance for many fields of stellar astrophysics, with clas-sical examples including the colour-temperature scale forlate-type stars (see, e.g., Flower 1996; Casagrande et al.2010) and the period-luminosity relation for Cepheidvariables (Leavitt 1908). The calibration of these rela-tions relies either on the direct measurement of stellarproperties (e.g. through trigonometric parallaxes andinterferometry) or the comparison of observations withstellar models. In both cases measurements spanning anextended range in parameter space (such as temperature,luminosity and metallicity) are required.One of the most powerful methods to constrain funda-mental properties of field stars is asteroseismology (see,e.g., Brown & Gilliland 1994; Christensen-Dalsgaard2004; Aerts et al. 2010). The connection of globalasteroseismic observables to fundamental stellar prop-erties was calibrated by Kjeldsen & Bedding (1995,hereafter KB95) by introducing relations that scalestellar properties from the observed values of the Department of Physics and Astronomy, Iowa State University,Ames, IA 50011 USA Universit´e de Nice Sophia-Antipolis, CNRS, Observatoire dela Cˆote d’Azur, BP 4229, 06304 Nice Cedex 4, France NASA Ames Research Center, MS 244-30, Moffett Field, CA94035, USA SETI Institute/NASA Ames Research Center, MS 244-30,Moffett Field, CA 94035, USA
D. Huber et al.Sun. The scaling relations have been used extensivelyboth for the direct inference of stellar properties (e.g.,Stello et al. 2009b; Kallinger et al. 2010a; Mosser et al.2010; Kallinger et al. 2010b; Chaplin et al. 2011a;Hekker et al. 2011b,a; Silva Aguirre et al. 2011)as well as for calculating pulsation amplitudes ofmain-sequence and subgiant stars (e.g., Michel et al.2008; Bonanno et al. 2008; Mathur et al. 2010b;Huber et al. 2011; Chaplin et al. 2011c) and red-giantstars (Edmonds & Gilliland 1996; Gilliland 2008;Stello & Gilliland 2009; Mosser et al. 2010; Huber et al.2010; Baudin et al. 2011; Stello et al. 2011). The threeglobal observables discussed in this paper are the fre-quency of maximum power ( ν max ), the large frequencyseparation (∆ ν ) and the mean oscillation amplitude perradial mode.Tests of the scaling relations for these observables overa wide range of parameter space has so far been ham-pered by the relative sparsity of detections of solar-like oscillations. The success of the Kepler mission haschanged this picture, with the detection of oscillations inthousands of stars covering a large part of the low-massregion in the H-R diagram (Gilliland et al. 2010b). Inthis paper, we aim to use the large number of detectionsby
Kepler to test scaling relations for stars ranging fromthe main-sequence to He-core burning red-giants. DATA ANALYSIS AND MODELS
The
Kepler space telescope was launched in March2009 with the primary goal of finding Earth-like planetsorbiting solar-like stars through the detection of photo-metric transits.
Kepler monitors the brightness of starsat two sampling rates, either in 29.4 min (long cadence)or in 58.8 sec (short cadence) intervals (Gilliland et al.2010a; Jenkins et al. 2010). For asteroseismic studies ofsolar-like oscillators, the former are primarily used tostudy oscillations in red-giant stars, and short cadencedata are used to measure the more rapid oscillations inmain-sequence and subgiant stars. While red giants ob-served in long cadence have been continuously monitoredsince the launch of the mission, the number of short ca-dence slots is restricted due to bandwidth limitations.Short cadence slots have therefore so far been primarilyused to survey a large number of stars for a period of onemonth each.
Kepler observations are subdivided into quarters,starting with the initial commissioning run (10 d,Q0), followed by a short first quarter (34 d, Q1) andsubsequent full quarters of 90 d length. Our studiesare based on
Kepler data spanning from Q0 to Q6 forlong cadence data and Q0 to Q4 for short cadence data.Our final sample contains 1686 stars, of which 1144have been observed at long cadence, mostly since thelaunch of the mission ( ∼
500 days), and 542 at shortcadence for a typical length of one month. We have used
Kepler raw data, which were reduced in the mannerdescribed by Garc´ıa et al. (2011) and analyzed usingseveral automated analysis methods (Bonanno et al.2008; Campante et al. 2010; Hekker et al. 2010b;Huber et al. 2009; Kallinger et al. 2010a; Karoff et al.2010; Mathur et al. 2010a; Mosser & Appourchaux2009; Mosser et al. 2011b; Verner & Roxburgh 2011).We refer the reader to Hekker et al. (2011c) andVerner et al. (2011a) for an extensive comparison of the results provided by these methods. Unless otherwisementioned, all results presented here are based on themethod by Huber et al. (2009). We have only retainedresults for stars with at least one matching pipelineresult within 10% and 5% of the determined ν max and∆ ν value, respectively. The same outlier rejectionprocedure was repeated for all methods that returnedresults for both long cadence and short cadence data,and these datasets have then been used to validate allresults and conclusions reported in this paper. Note thatall amplitudes shown in this paper have been normalizedto recover the full sine-amplitude of an injected signal(commonly referred to as peak-scaling). All amplitudesshown have been calculated using the method describedby Kjeldsen et al. (2008) with c = 3 .
04 (Bedding et al.2010) to convert to amplitude per radial mode.Uncertainties on ν max , ∆ ν and amplitudes reportedin this paper were estimated using Monte-Carlo simu-lations by generating synthetic power spectra followinga χ distribution with two degrees of freedom with ex-pected values corresponding to the observed power den-sity levels. For each star, the method by Huber et al.(2009) was repeated on each synthetic dataset and thestandard deviation of the distribution after 500 iterationswas taken as an estimate of the uncertainty. The typi-cal relative uncertainties obtained using this method forlong cadence and short cadence data are 3% and 4% for ν max , 1% and 3% for ∆ ν , and 7% and 11% for the ampli-tude. These estimates agree with the results obtained byHekker et al. (2011c) and Verner et al. (2011a). For ouranalysis, only stars with uncertainties lower than 20% in ν max , 10% in ∆ ν and 50% in amplitude were retained.Note that oscillation amplitudes for all stars observed inlong-cadence with ν max > µ Hz have been removedfrom our analysis due to the difficulty of estimating thenoise level close to the Nyquist frequency of 283 µ Hz.The asteroseismic relations discussed in this paper relyon scaling from observed values of the Sun. To ensurethat these reference values are consistent with our analy-sis method, we used 111 30-day subsets of data collectedby the VIRGO instrument (Fr¨ohlich et al. 1997) aboardthe SOHO spacecraft spanning from 1996 to 2005 andanalyzed them in the same way as the
Kepler data. Thisyielded solar reference values of ν max , ⊙ = 3090 ± µ Hzand ∆ ν ⊙ = 135 . ± . µ Hz, which are consistent withpreviously quoted values in the literature. We deter-mined the oscillation amplitude to be A ⊙ = 4 . ± . λ = 500 nm) which, us-ing the approximation by KB95, translates into a solarbolometric amplitude of A ⊙ , bol = 3 . ± . A ⊙ , bol = 3 . Kepler stars, a test of as-teroseismic scaling relations relies on the comparisonof observations with models. In our study, we usecanonical BaSTI evolutionary models (Pietrinferni et al.2004) with a solar-scaled distribution of heavy elements(Grevesse & Noels 1993). Mass loss in BaSTI modelsis characterized according to the Reimers law (Reimers1975), and we used models with the mass loss parame-ter set to the commonly used value η = 0 . Fig. 1.—
Upper panel: ∆ ν versus ν max for the entire sample of Kepler stars. Red triangles show stars observed in long cadence,while black diamonds are stars observed in short cadence. Lowerpanel: Same as upper panel, but with the luminosity dependenceremoved by raising ν max to the power of 0.75. Green lines showpower law fits to the ∆ ν - ν max relation for two different intervalsof ν max (see text). The blue dashed line shows the relation byStello et al. (2009a). Fusi-Pecci & Renzini 1976). Note that mass loss forthese models is only significant in the red giant phaseof stellar evolution. SCALING RELATIONS FOR ν max AND ∆ ν Brown et al. (1991) first argued that the frequency ofmaximum power ( ν max ) for Sun-like stars should scalewith the acoustic cut-off frequency. KB95 used this as-sumption to relate ν max to stellar properties as follows: ν max ≈ M/M ⊙ ( T eff /T eff , ⊙ ) . L/L ⊙ ν max , ⊙ . (1)This scaling relation has since been found to workwell both observationally (see, e.g., Bedding & Kjeldsen2003; Stello et al. 2008; Bedding 2011) as well as theo-retically (see, e.g., Chaplin et al. 2008; Belkacem et al.2011).The mean large frequency separation (∆ ν ) betweenmodes of consecutive radial overtone and equal spher-ical degree is directly related to the sound travel timeacross the stellar diameter, and is therefore sensitive tothe mean stellar density (Ulrich 1986). This is expressedin the following scaling relation:∆ ν ≈ ( M/M ⊙ ) . ( T eff /T eff , ⊙ ) ( L/L ⊙ ) . ∆ ν ⊙ . (2)Note that in our analysis ∆ ν is measured as the meanspacing of all detectable modes around the value of ν max in the power spectrum.It has been well established for both main-sequenceand red-giant stars that ν max and ∆ ν follow a power TABLE 1Coefficients of the ∆ ν - ν max relation. Method α, β α, β ν max < µ Hz ν max > µ HzA2Z 0.259(3),0.765(2) 0.25(1),0.779(7) 919,257COR 0.267(3),0.761(2) 0.23(1),0.789(6) 1150,415OCT 0.263(3),0.763(2) 0.20(1),0.811(7) 1082,281SYD 0.267(2),0.760(2) 0.22(1),0.797(5) 1228,458A2Z - Mathur et al. (2010a), COR - Mosser & Appourchaux(2009); Mosser et al. (2011b), OCT - Hekker et al. (2010b), SYD -Huber et al. (2009). law relation (Stello et al. 2009a; Hekker et al. 2009;Mosser et al. 2010; Hekker et al. 2011b,a):∆ ν = α ( ν max /µ Hz) β . (3)Figure 1 shows this relation for the entire Kepler samplein our analysis. Although the relation appears to beconstant over several orders of magnitude, Mosser et al.(2010) and Huber et al. (2010) noted that the slope isdifferent for red-giant and main-sequence stars. This canbe illustrated more clearly by removing the luminositydependence by raising ν max to the power of 0.75, yielding( ν max /µ Hz) . ∆ ν/µ Hz ∝ (cid:18) MM ⊙ (cid:19) . (cid:18) T eff T eff , ⊙ (cid:19) − . . (4)The lower panel of Figure 1 displays this ratio as func-tion of ν max . The distribution shows a prominent di-agonal structure around ν max ∼ − µ Hz, which weidentify as the red clump, comprising He-core burningred giant stars (see Bedding et al. 2011; Mosser et al.2011a,c). It is evident that the power law becomessteeper as ν max increases. To measure this effect, we fit-ted Equation (3) to the sample in two groups subdividedat ν max = 300 µ Hz, which roughly marks the transitionfrom low-luminosity red giants to subgiants. The best-fitting power laws for the SYD pipeline are shown assolid green lines in the lower panel of Figure 1, and theresults for each pipeline with values for both long- andshort cadence data are listed in Table 1. The differencein the coefficients between red giant and main-sequencestars is significant, and should be noted when using therelation for determining ∆ ν from ν max . For ν max closeto the solar value, for example, the use of a power-lawrelation calibrated to red-giant stars would lead to anunderestimation in ∆ ν by ∼ ν max and ∆ ν com-pare with evolutionary models? The upper panelof Figure 2 compares the observed distributions withsolar-metallicity models with masses 1.0 M ⊙ , 1.3 M ⊙ and 2.0 M ⊙ (solid lines), which roughly correspond tothe lower bound, median and upper bound of themass distribution derived by Kallinger et al. (2010a)and Chaplin et al. (2011a). The model values for ν max and ∆ ν have been calculated using Equations (1) and(2). Additionally, we have color-coded the mass ofeach star calculated using Equations (1) and (2) with D. Huber et al. Fig. 2.—
Upper panel: Observed values of ν . /∆ ν versus ν max (symbols) compared to solar-metallicity ( Y = 0 . Z = 0 . M ⊙ , 1.3 M ⊙ and 2.0 M ⊙ (solid lines). Asteroseis-mic masses are color-coded as indicated in the plot. Typical errorbars for different ranges of ν max are indicated near the bottom ofthe plot. Lower panel: Same as top panel but with the effectivetemperature dependence removed (see Equation (4)) and omittingmodel tracks. Symbol types and colors are the same as in Figure1. The green solid lines show linear fits to the linear-log plot forthe same intervals of ν max as in Figure 1. the observed values of ν max and ∆ ν adopting the ef-fective temperature listed in the Kepler
Input Catalog(KIC, Brown et al. 2011) . The observed and theoreticalmasses in this plot essentially correspond to a compari-son of the so-called direct method and grid-based methodof estimating asteroseismic masses (see, e.g., Gai et al.2011), assuming solar metallicity.As noted by Kallinger et al. (2010b) and Huber et al.(2010), the spread in stellar mass is pronounced on thered giant branch, while for less-evolved stars the spreadin observations is weaker and the models almost over-lap. The overall agreement between the models and thedata is very good, and we do not observe any signifi-cant offset of the models with respect to the observa-tions. It is also remarkable how well the models of dif-ferent masses track the He-core-burning red clump. Al-though the spread of data points about the models formain-sequence stars can be explained by measurementuncertainties, we note that stellar population models pre-sented by Silva Aguirre et al. (2011) yield evidence thatthe Kepler sample is on average metal-poor, which canhave a some impact on the ∆ ν - ν max relation of main-sequence stars (see Huber et al. 2010). As noted bySilva Aguirre et al. (2011), however, the measurement Note that for 39 stars in our list, no effective temperatures wereavailable in the KIC, and we omitted those stars for the remainderof our analysis. uncertainties are currently too large to test a metallicityinfluence on the scaling relations.What causes the different ∆ ν - ν max power-law relationsfor evolved and unevolved stars? The scaling relation inEquation (4) suggests that the change in slope for theunevolved stars must be partially due to a variation ineffective temperature. To test this hypothesis, we againused the effective temperatures listed in the Kepler
InputCatalog to model this dependency using Equation (4).The result is shown in the lower panel of Figure 2. Asexpected, correcting for the higher average effective tem-peratures of main-sequence stars compared to red giantsremoves the gradient in the distribution. Subdividing thesample again at ν max = 300 µ Hz, we find a positive slopefor the red giant sample and a negative slope for main-sequence stars (solid green lines). These variations arequalitatively in agreement with different mass distribu-tions in the sample: while for red giants ν max is correlatedto stellar mass for He-core burning stars (Mosser et al.2011a,c), main-sequence stars with higher ν max are gen-erally low-mass stars (see, e.g., Chaplin et al. 2011a).Without mass estimates independent of asteroseismicscaling relations, however, it is not possible to make fur-ther quantitative conclusions about the distribution. AMPLITUDES
The L/M Scaling Relation
KB95 suggested that model predictions byChristensen-Dalsgaard & Frandsen (1983) implied ascaling for velocity amplitudes of A vel ∝ (cid:18) LM (cid:19) s , (5)with s = 1. They further argued that the oscillationamplitude A λ observed in photometry at a wavelength λ is related to the velocity amplitude: A λ ∝ v osc λT r eff . (6)The exponent s has since been revised in the range ofroughly s = 0 . − . r has so farbeen chosen to be either r = 1 . r = 2 . Kepler bandpass to compare predictions made withEquation (6) to our observations. To do so, we used theexpression for the bolometric amplitude given by KB95and converted these values to amplitudes observed in the
Kepler bandpass, as follows: A Kp ∝ (cid:18) LM (cid:19) s T r − c K ( T eff ) , (7)with c K being the bolometric correction factor as aesting Asteroseismic Scaling Relations 5function of effective temperature, given by Ballot et al.(2011): c K ( T eff ) = (cid:18) T eff (cid:19) . . (8)The reason for choosing to correct the model amplitudesrather than the observed values is that effective tempera-tures for most of the stars in our sample are rather uncer-tain, and we hence prefer not to perform the correctionon the observed amplitudes.Figure 4.1(a) shows the observed amplitudes for thefull Kepler sample as a function of ν max . The spreadis much larger than the typical measurement uncertain-ties. As first noted by Huber et al. (2010) and later con-firmed by Mosser et al. (2011a), Mosser et al. (2011c)and Stello et al. (2011), this spread in the amplitude- ν max relation for red giants is related to a spread inmass. To demonstrate this, Figure 4.1(b) shows the sameplot but color-coded by stellar masses, as calculated inthe previous section. We observe that, particularly forlow-luminosity red giants, the higher-mass stars showlower amplitudes than lower-mass stars for a given ν max .For unevolved stars we can tentatively identify the sametrend, although the separation is less clear.The observation of a mass dependence for a given ν max indicates that Equation (7) should be revised to includean additional mass dependence. To demonstrate this,solid lines in Figure 4.1(b) show model tracks with dif-ferent masses scaled using typical values of r = 2 and s = 0 . L/M ) s scaling clearlyfails to reproduce the observed spread on the red-giantbranch, but predicts a strong mass dependence for un-evolved stars, contrary to what is observed.An obvious way to account for an additional mass de-pendence is to rearrange Equation (7) as follows: A Kp ∝ L s M t T r − c K ( T eff ) . (9)Note that such a formulation has been introduced byKjeldsen & Bedding (2011) (hereafter KB11) and hasalso been used by Stello et al. (2011) in the Kepler studyof cluster red giants. To evaluate the coefficients thatbest reproduce our observations, we again used modelswith masses 1.0 M ⊙ , 1.3 M ⊙ and 2.0 M ⊙ and calculatedtheir expected amplitudes according to Equation (9) fora given set of s , t and r . The agreement of models andobservations was evaluated as follows: For each observedamplitude, we interpolated each model track to obtainthe model amplitude at the observed ν max value. Wethen calculated the minimum squared deviation of theobserved amplitude to the three model amplitudes, nor-malized by the measurement uncertainty. The parame-ters s , t and r were then optimized using a least-squaresfit. Note that for simplicity we have only used modelvalues below the tip of the red-giant branch, since aninclusion of later evolutionary stages would require theidentification of all red-clump stars in our sample whichis beyond the scope of this paper.First tests quickly showed that the strong correla-tion between effective temperature and luminosity madean independent determination of s , t and r impossible.Following Stello et al. (2011), we have therefore fixed Fig. 3.— (a) Oscillation amplitude versus ν max for the entire Kepler sample. Symbol types and colors are the same as in Figure1. The position of the Sun is also marked. (b) Same as panel (a)but with asteroseismic masses color-coded. Error bars have beenomitted for clarity. Solid lines show the 1.0, 1.3 and 2.0 M ⊙ solar-metallicity ( Y = 0 . Z = 0 . r = 2 and s = 0 .
8. (c) Same as panel (b)but using Equation (9) with coefficients determined by fitting themodel tracks to observations. (d) Same as panel (c) but with bestfitting coefficients determined by comparing observed amplitudeswith calculated amplitudes using asteroseismic masses and radii(see text for details).
D. Huber et al.
Fig. 4.—
Observed versus calculated amplitudes for the best-fitting coefficients in Equation (9), using radii and masses deter-mined using the scaling relations for ν max and ∆ ν . The dashedblue line shows the 1:1 relation. The position of the Sun is alsoshown. the value to r = 2. The best-fitting parameters are s = 0 . ± .
002 and t = 1 . ± .
01, with uncertaintiesestimated by repeating the fitting procedure 1000 timesusing amplitudes drawn from a random distribution witha scatter corresponding to the 1 σ measurement uncer-tainties (scaled so that reduced χ = 1 for the originaldata). The parameter uncertainties were then estimatedby calculating the standard deviation of the resulting dis-tribution for each coefficient.The scaled model tracks in Figure 4.1(c) using thesecoefficients reproduce a mass spread in amplitude forred giants, with higher-mass stars showing lower am-plitudes for a given ν max , as observed by Huber et al.(2010), Mosser et al. (2011a), Stello et al. (2011) andMosser et al. (2011c). For unevolved stars, the modelamplitudes now show less dispersion but appear to sys-tematically underestimate amplitudes for both subgiantand main-sequence stars.The method presented above assumes that the excessspread of amplitudes at a given ν max is entirely due to aspread in stellar mass. A more sophisticated approach isto account for the actual mass distribution by directly us-ing masses and radii of the sample estimated using ν max and ∆ ν in Equations (1) and (2). We again used theeffective temperatures from KIC and evaluated Equa-tion (9) for each value of s and t directly using theseasteroseismic masses and radii. The corresponding best-fitting parameters in this case were s = 0 . ± .
002 and t = 1 . ± .
02, and the scaled model tracks with theseparameters are shown in Figure 4.1(d). The spread ofthe model tracks on the red giant branch is now consid-erably reduced, better matching the observations, andthe overall fit for less evolved stars is also improved.Figure 4 compares the observed and calculated ampli-tudes using s = 0 .
838 and t = 1 .
32, showing good agree-ment along the 1:1 line over the entire range of ν max .The residuals between observed and calculated ampli-tudes over the full range of ν max show a standard devi-ation of about 25%, which is consistent with the typicaluncertainties in the adopted stellar properties (estimatedfrom propagating the uncertainties in ν max and ∆ ν and assuming an uncertainty of 200 K in T eff ) and the mea-surement uncertainties in the observed amplitudes. Wedo see an overall bias of about 9%, with calculated am-plitudes being systematically underestimated when scal-ing from the solar value. This bias is stronger for un-evolved stars with ν max > µ Hz (15%) than for evolvedstars ν max < µ Hz (7%). An explanation for this biasmight be additional physical differences between the Sunand our
Kepler sample, which influence oscillation am-plitudes but have not yet been taken into account.Our best-fitting values for s and t using asteroseismicmasses and radii are significantly different to Stello et al.(2011), who found s = 0 . ± .
02 and t = 1 . ± . Kepler field. This difference is presumably due tothe fact that our sample includes a much wider range ofevolutionary states. Indeed, repeating our analysis usingonly red giants in a ν max range similar to that used byStello et al. (2011) yields coefficients which are in bet-ter agreement. Remaining differences could potentiallybe due to metallicity effects which have been suggestedto have a significant influence on oscillation amplitudes(Samadi et al. 2010) or systematic errors when estimat-ing stellar properties through the direct method of using ν max and ∆ ν . Nevertheless, all methods confirm thatscaling relations with a separate mass and luminosity de-pendence better reproduce the observed amplitudes fromthe main sequence to red giants. The KB11 Scaling Relation
Kjeldsen & Bedding (2011) recently argued that am-plitudes of solar-like oscillations should scale in propor-tion to fluctuations due to granulation. They proposeda revised scaling relation for velocity amplitudes: A vel ∝ Lτ . M . T . , (10)where τ osc is the mode lifetime.Using the same arguments as in the previous section,the KB11 relation predicts photometric amplitudes inthe Kepler bandpass as follows: A Kp ∝ Lτ . M . T . r eff c k ( T eff ) . (11)They also suggested the following relation for the gran-ulation power in intensity measured at ν max : P int ( ν max ) ∝ L M T . . (12)First results on granulation properties of Kepler red-giant stars by Mathur et al. (2011) have shown promis-ing agreement. More recently, Mosser et al. (2011c) con-firmed the proportionality between mode amplitudes andthe granulation power at ν max for a large sample of redgiants, but found that the absolute values calculated us-ing Equations (11) and (12) were offset from the observedvalues. Here we extend this comparison to include alsomain-sequence stars.To test the relation, we used the solar granulationpower at ν max determined by our analysis of VIRGOdata, which yielded a mean value of P int , ⊙ = 0 . ± esting Asteroseismic Scaling Relations 7 Fig. 5.—
Upper panel: Observed versus calculated granulationpower at ν max using Equation (12). Lower panel: Observed versuscalculated amplitudes using Equation (11) and assuming a solarmode lifetime for all stars. The dashed blue line shows the 1:1relation in both panels, and the position of the Sun is marked. .
02 ppm µ Hz − . The upper panel of Figure 5 comparesthe proposed relation in Equation (12), again using as-teroseismic masses and radii, as well as effective temper-atures from KIC. We observe that the calculated valuesare systematically too high for main-sequence stars, andsystematically too low for red giants, the latter being inagreement with the results by Mosser et al. (2011c).In order to compare observed with calculated am-plitudes using Equation (11), estimates of the modelifetimes are required. While it is well establishedthat mode lifetimes in giants are significantly longerthan in main-sequence stars (De Ridder et al. 2009;Chaplin et al. 2009; Hekker et al. 2010a; Huber et al.2010; Baudin et al. 2011), τ osc is not very well con-strained for stars spanning such a large range in evolutionas considered in our sample. We have therefore decidedto neglect the influence of τ osc in Equation (11) in ourstudy. The lower panel of Figure 5 shows the comparisonbetween observed and calculated amplitudes using Equa-tion (11) and assuming solar mode lifetime for all stars.We observe that the revised scaling relation works wellfor main-sequence and subgiant stars, but predicts am-plitudes that are systematically too high for red giants,again in agreement with Mosser et al. (2011c).We note that the upper panel of Figure 5 displays anapparent systematic difference in the granulation powerat ν max in the overlapping region of the long-cadence andshort-cadence sample. This effect is due to a systematicdifference of the modelled background noise level for starsoscillating close to the long-cadence Nyquist frequency(283 µ Hz). We have confirmed that this has a negligibleeffect on the measured amplitudes, as is evident by the consistent overlap of datapoints in the bottom panel ofFigure 5.As noted by Kjeldsen & Bedding (2011), the observeddifferences are not unexpected since the relation does nottake into account potential differences in the intensitycontrast between dark and bright regions on the stellarsurface. While our study confirms the relation betweenmode amplitudes and granulation power for stars rang-ing from the main-sequence to the red-giant branch, itdemonstrates that the link between velocity and intensitymeasurements is not yet fully understood for amplitudescaling relations.
Stellar Activity & Amplitudes
The offset between observed and calculated amplitudesfor main-sequence stars noted in Section 4.1 suggests thatthere might be an additional physical dependency whichis not yet taken into account. Increased stellar activ-ity and magnetic fields are known to decrease pulsationamplitudes in the Sun (Chaplin et al. 2000; Komm et al.2000), and have been suggested as a possible cause forthe unexpected low amplitudes of oscillations in activestars (Mosser et al. 2009; Dall et al. 2010). The first di-rect evidence for an influence of stellar activity on oscil-lation amplitudes for a star other than the Sun has beenfound in the F-star HD 49933 (Garc´ıa et al. 2010). Asdiscussed by KB11, these amplitude changes may be dueto changes in the mode lifetimes.Recently, Chaplin et al. (2011b) reported the discov-ery for main-sequence and subgiant stars observed with
Kepler that oscillations are less likely to be detected instars with higher activity. This led them to concludethat, just like for the Sun, increased stellar activity sup-presses mode amplitudes.
Kepler results have further-more shown that the detection rate seems to be consid-erably lower for subgiant stars with ν max ∼ − µ Hz(Chaplin et al. 2011a,c; Silva Aguirre et al. 2011). Thiswas tentatively explained by increased magnetic activityin this stage of stellar evolution, as predicted by Gilliland(1985). While it is known that some detection bias due toinstrumental artefacts might be present in the frequencyrange from 200-500 µ Hz (Garc´ıa et al. 2011), both dis-coveries strongly suggest that activity must be consid-ered when calculating amplitudes.While Chaplin et al. (2011b) used a simple detectioncriterion, we attempt here to confirm and quantify theirdiscovery by correlating measured pulsation amplitudeswith stellar activity. Several studies have been devotedto stellar activity and variability in the
Kepler field(Basri et al. 2010; Debosscher et al. 2011; Ciardi et al.2011) with various definitions of activity measures. Sinceour study covers a large range of oscillation timescalesand amplitudes, we need to define an activity measurewhich separates variability due to oscillations from vari-ations due to activity, and at the same time remains sen-sitive to local minima and maxima in the light curve.To do this, we first smoothed each light-curve with aquadratic Savitzky-Golay filter (Savitzky & Golay 1964)with a full-width corresponding to 20 times the deter-mined oscillation period, and then measured the abso-lute maximum deviation of the smoothed curve fromits mean. We denote this measure of activity as r sg .Note that we have repeated the analysis with smoothingwidths ranging between 10-50 times the oscillation pe- D. Huber et al. Fig. 6.—
Amplitude versus ν max showing only stars observedin short cadence, with the logarithm of the activity range colorcoded. Red and blue points correspond to high and low activity,respectively. The position of the Sun with a color correspondingto the mean solar activity is also shown. riod, but found not significant difference in the results.When evaluating r sg , it must be kept in mind that oursample includes light curves with two very different time-bases ( ∼
30 d for short cadence, and ∼
500 d for long ca-dence). The samples therefore probe different activitytime scales, and have hence been separated in the subse-quent analysis.To correct our activity measure for the influence ofshot noise, we performed the following simulations. Foreach target light curve, we produced synthetic time-series including a sinusoidal variation with an ampli-tude corresponding to the measured r sg value. We thenadded white noise with a standard deviation correspond-ing to the apparent magnitude of the target, whichwas estimated from the minimal noise levels given inGilliland et al. (2010a) and Jenkins et al. (2010). Wethen measured r sg for the synthetic light curve as de-scribed above, and calculated the difference between thisvalue and the value of r sg measured from the noise-freetime series. This process was repeated 500 times for eachstar, and the median of the resulting distribution wastaken as the correction value for the observed r sg . Insummary, it was found that the typical correction val-ues were about 8% for short-cadence data, and <
1% forlong-cadence data.Figure 6 shows the amplitude- ν max relation for theshort cadence sample, with the logarithm of the activitymeasure color-coded. We see that the most active stars(shown in red) have generally lower amplitudes than lessactive stars (shown in blue). As found by Chaplin et al.(2011b), the Kepler sample is generally less active thanthe average Sun ( r sg , ⊙ = 0 . ν max , roughly di- Fig. 7.—
Upper panel: Residuals of observed amplitude dividedby the calculated amplitudes using Equation 9 versus the logarithmof the activity measure for all stars with ν max > µ Hz. Thered solid line shows an unweighted linear fit. Middle panel: Sameas the upper panel but for all stars with ν max = 300 − µ Hz.Lower panel: Same as upper panel but for all long-cadence stars( ν max < µ Hz). viding stars on the main-sequence (top panel), subgiants(middle panel) and red giants (bottom panel). The influ-ence of activity on amplitudes appears the strongest forsubgiant stars, followed by a weak correlation for main-sequence stars and no visible correlation for red giants.Unweighted linear fits to the linear-log plots yield slopesof − . ± . − . ± .
07 and 0 . ± .
02, respectively.While the correlations for unevolved stars are formallysignificant, a thorough comparison with other methodshas shown that the correlation for subgiant stars is onlyconfirmed in three out of six methods, while the resultsfor main-sequence and red-giant stars are consistent withzero within 3 σ for all methods. Although this compar-ison cautions us not to draw any definite conclusions,we note that the exponential decrease of amplitude withincreasing stellar activity for subgiants would be in-linewith both observational and theoretical evidence for en-esting Asteroseismic Scaling Relations 9 Fig. 8.— ν max versus effective temperature for all stars in oursample. Color codes refer to the logarithm of the pulsation am-plitude, with blue and red colors marking the highest and lowestamplitudes, respectively. The dashed line shows the empiricallydetermined cool edge of the instability strip from observations of δ Scuti stars. The theoretical cool edge for the fundamental radialmode of δ Scuti pulsations is shown for a 1.7 M ⊙ model by Houdek(2000) (solid circle) and a range of masses by Dupret et al. (2005)(dashed-dotted line). Black solid lines are solar-metallicity BaSTIevolutionary tracks with masses as indicated in the plot. The starsymbol indicates the first detection of hybrid solar-like and δ Scutioscillations by Antoci et al. (2011). hanced magnetic activity in these stars (Gilliland 1985;Chaplin et al. 2011a). A confirmation of the results pre-sented here will have to await longer timeseries (in par-ticular for short cadence data) which will allow a betterestimate of the activity measure and reduce the uncer-tainty in the adopted stellar properties through betterconstraints on ν max and ∆ ν . This will then allow a morein-depth investigation of additional physical influences onoscillation amplitudes such as stellar activity and metal-licity (Samadi et al. 2010). THE COOL EDGE OF THE INSTABILITY STRIP
The cool edge of the instability strip is widely be-lieved to be the dividing line between coherent pulsationsdriven by the opacity ( κ ) mechanism and solar-like os-cillations driven by convection. Considerable work hasbeen devoted to establishing this boundary both empiri-cally using δ Scuti and γ Doradus stars (see, e.g., Breger1979; Pamyatnykh 2000) as well as theoretically (see,e.g., Houdek et al. 1999b; Houdek 2000; Xiong & Deng2001; Dupret et al. 2004, 2005). The large number ofstars for which
Kepler has detected oscillations now al-lows the first test of this boundary from “the cool side”,using solar-like oscillations.Figure 8 shows all stars of our sample in a modi-fied H-R diagram, in which we have replaced luminosity with 1 /ν max . We also show the empirically determinedcool edge of the instability strip taken from Pamyatnykh(2000) as a dashed line, as well as the theoretical cooledge for the fundamental radial mode of δ Scuti pul-sations for a 1.7 M ⊙ model taken from Houdek (2000)(filled circle) and for a range of masses by Dupret et al.(2005) (dashed-dotted line). To convert the lines fromthe log L − log T eff to the log ν max − log T eff plane, we de-termined the closest matching grid point of evolutionarytracks with different masses to the log L − log T eff rela-tion, and then fitted a straight line to the resulting pointsin the log ν max − log T eff plane. This has been done bothfor BaSTI and a set of ASTEC (Christensen-Dalsgaard2008) evolutionary models, and both sets of modelsyielded consistent results. The empirically determinedcool edges taken from Pamyatnykh (2000) in Figure 8are: log( T eff /K ) = − .
045 log(
L/L ⊙ ) + 3 .
893 (13)log( T eff /K ) = 0 .
048 log( ν max /µ Hz) + 3 . . (14)Equation (14) appears in good agreement with thehottest stars for which solar-like oscillations are observed.However, it is important to note that the recent work byPinsonneault et al. (2011) and Molenda- ˙Zakowicz et al.(2011) suggest that many stars are significantly hotterthan the effective temperatures given in KIC, with dif-ferences of up to 250 K. With this in mind, the fact thatwe observe stars close to the theoretical limit is evidencethat the separation between the classical and stochasticpulsations is probably not a sharp dividing boundary.Indeed, we indeed do not observe a gradual decrease inamplitude towards the cool edge of the instability stripand the recent discovery of solar-like and δ Scuti oscilla-tions in HD 187547 by Antoci et al. (2011) using
Kepler data has confirmed the existence of hybrid pulsators (seestar symbol in Figure 8). Interestingly, the observedamplitude for this star is roughly a factor four higherthan expected from the traditional scaling relations dis-cussed in this paper. This was tentatively explained bythe increased mode lifetimes which are roughly 4-5 timeshigher than expected from scaling relations based on ef-fective temperature (Chaplin et al. 2009; Baudin et al.2011).To test the cool edge of the instability strip more quan-titatively, we must account for selection effects. Figure9 shows a close-up of the main-sequence region of theHRD, this time plotting luminosity on the ordinate andincluding stars for which we did not find evidence for os-cillations. Luminosities for stars without detections havebeen calculated based on values in the KIC assuming anuncertainty on the radius of 40% (Verner et al. 2011b),while for stars with detections Equations (1) and (2) wereused. Although the error bars are large, we observe anexcess of hot stars for which no oscillations have beendetected, indicating that the observed cut-off in detec-tions may not entirely due be due to selection bias. Wetherefore conclude that the empirical red-edge given inEquations (13) and (14) gives a good approximation forthe transition between opacity driven and solar-like oscil-lations. Further work will be needed to quantify this ob-0 D. Huber et al.
Fig. 9.—
H-R diagram of the low-luminosity region shown inFigure 8, comparing stars for which no oscillations have been de-tected (open black diamonds) with our sample of detections (redfilled diamonds). A typical error bar for both samples is shown inthe top region of the plot. The dashed line, dashed-dotted line andfilled circle are the same as in Figure 8. The star symbol indicatesthe first detection of hybrid solar-like and δ Scuti oscillations byAntoci et al. (2011). servation and compare the results with theoretical mod-els of excitation and damping of solar-like oscillations,as well as possible further detections of hybrid solar-likeoscillations and classical pulsation in hot stars. CONCLUSIONS
We have studied global oscillation properties in ∼ Kepler to test asteroseismic scaling re-lations. Our main findings can be summarized as follows: • By comparing evolutionary models with observa-tions, we have shown that the scaling relations for ν max and ∆ ν are in qualitative agreement with ob-servations for evolutionary stages spanning fromthe main-sequence to the He-core burning phaseof red giants. The difference in the ∆ ν - ν max re-lation between evolved and unevolved stars can beexplained by different distributions of effective tem-perature and stellar mass, in agreement with whatis expected from the scaling relations. A morequantitative test of scaling relations for ν max and∆ ν will have to await the determination of funda-mental properties from independent methods suchas spectroscopy and long-baseline interferometry. • We have shown that (
L/M ) s scaling for oscillationamplitudes fails to reproduce the amplitude- ν max relation for red giants. We have verified that arevised scaling relation using a separate mass andluminosity dependence reproduces the observationsbetter, and a relation with L s and M t coefficients of s = 0 . ± .
002 and t = 1 . ± .
02 matches ampli-tudes for field stars ranging from the main-sequenceto the red clump to a precision of 25% when adopt-ing stellar properties derived from scaling relations.The calculated amplitudes for main-sequence andsubgiant stars in our sample are systematically un-derestimated by up to 15%, indicating that theremight be an additional physical dependence whichis not yet taken into account. • We have investigated the connection of stellar ac-tivity with the suppression of oscillation ampli-tudes in main-sequence, subgiant and red-giantstars. We find evidence that the effect is strongestfor subgiant stars, but caution that these resultswill have to await confirmation with longer timeseries providing a better estimate of the activitymeasure and reduced uncertainties on stellar prop-erties used to calculated amplitudes. The presentdata do not yield strong evidence that stellar activ-ity contributes significantly to the underestimationof calculated amplitudes for main-sequence stars. • We have investigated the cool edge of the instabilitystrip using detections of solar-like oscillations. Wefind good agreement with the empirically and the-oretically determined cool edge using δ Scuti stars,but note that many stars showing solar-like oscil-lations may overlap with cooler δ Scuti stars, inagreement with the recent first discovery of hybridsolar-like and δ Scuti oscillations by Antoci et al.(2011).The authors gratefully acknowledge the
Kepler
ScienceTeam and everyone involved in the
Kepler mission fortheir tireless efforts which have made this paper possible.Funding for the
Kepler
Mission is provided by NASA’sScience Mission Directorate. We thank V. Antoci forhelpful comments on the manuscript and discussions onHD 187547. DS and TRB acknowledge support by theAustralian Research Council. SH acknowledges financialsupport from the Netherlands Organisation for ScientificResearch (NWO). HG acknowledges support by the Aus-trian FWF Project P21205-N16. JM- ˙Z acknowledges thepolish Minstry grant N N203 405139.
REFERENCESAerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010,AsteroseismologyAntoci, V., et al. 2011, Nature, in pressBallot, J., Barban, C., & van’t Veer-Menneret, C. 2011, A&A,531, A124Basri, G., et al. 2010, ApJ, 713, L155Baudin, F., et al. 2011, A&A, 529, A84Bedding, T. R. 2011, arXiv:1107.1723 Bedding, T. R., & Kjeldsen, H. 2003, PASA, 20, 203Bedding, T. R., et al. 2010, ApJ, 713, 935—. 2011, Nature, 471, 608Belkacem, K., Goupil, M. J., Dupret, M. A., Samadi, R., Baudin,F., Noels, A., & Mosser, B. 2011, A&A, 530, A142Bonanno, A., Benatti, S., Claudi, R., Desidera, S., Gratton, R.,Leccia, S., & Patern`o, L. 2008, ApJ, 676, 1248Breger, M. 1979, PASP, 91, 5 esting Asteroseismic Scaling Relations 11esting Asteroseismic Scaling Relations 11