Mon.Not.R.Astron.Soc.000,1–17(2012) Printed10January2013 (MNLATEXstylefilev2.2) A Bayesian approach to scaling relations for amplitudes of solar-like oscillations in Kepler stars E. Corsaro,1,2,3(cid:63) H.-E. Fro¨hlich,4 A. Bonanno,2 D. Huber,5 T. R. Bedding,6,7 O. Benomar,6,7 J. De Ridder3, and D. Stello6,7 3 1 1Department of Physics and Astronomy, Astrophysics Section, University of Catania, Via S. Sofia 78, I-95123 Catania, Italy 0 2I.N.A.F. - Astrophysical Observatory of Catania, Via S. Sofia 78, I-95123 Catania, Italy 2 3Instituut voor Sterrenkunde, K.U. Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium 4Leibniz Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany n 5NASA-Ames Research Center, Moffett Field, CA 94035-0001, USA a 6Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia J 7Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 8 ] R Accepted.Received; S . h ABSTRACT p We investigate different amplitude scaling relations adopted for the asteroseismology of - stars that show solar-like oscillations. Amplitudes are among the most challenging astero- o seismic quantities to handle because of the large uncertainties that arise in measuring the r t background level in the star’s power spectrum. We present results computed by means of s a a Bayesian inference on a sample of 1640 stars observed with Kepler, spanning from main [ sequence to red giant stars, for 12 models used for amplitude predictions and exploiting re- centlywell-calibratedeffectivetemperaturesfromSDSSphotometry.Wetestthecandidate 2 amplitude scaling relations by means of a Bayesian model comparison. We find the model v having a separate dependence upon the mass of the stars to be largely the most favored 6 5 one. The differences among models and the differences seen in their free parameters from 1 early to late phases of stellar evolution are also highlighted. 1 Key words: methods: data analysis – methods: statistical – stars: oscillations – stars: . 2 solar-like – stars: late-type 1 2 1 : v 1 INTRODUCTION These asteroseismic studies have recently led to the birth i X oftheensembleasteroseismology(Chaplinetal.2011a),which The study of solar-like oscillations (e.g. see Bedding & Kjeld- isshowinggreatpotentialforathoroughunderstandingofstel- r sen 2003, 2008; Bedding 2011, for summaries and reviews) has a larevolutiontheory.Thesuccess ofensemble asteroseismology experienced an enormous growth in the last five years thanks reliesmainlyonadoptingscalingrelations:generallysimpleem- to the launch of the photometric space-based missions CoRoT piricallawsthatallowforthederivationoffundamentalstellar (Baglin et al. 2006; Michel et al. 2008) and Kepler (Borucki properties for stars different from the Sun by scaling their as- et al. 2010; Koch et al. 2010). The latter in particular, is pro- teroseismic quantities from the solar values. viding a very large amount of high quality light curves, with a Among the most challenging asteroseismic quantities to very high duty cycle (see Gilliland et al. 2010b; Brown et al. measure and model one can certainly mention the oscillation 2011; Garc´ıa et al. 2011, for a general introduction to the Ke- amplitude. This is due to both the difficulty in estimating the pler asteroseismic program, for the presentation of the Kepler backgroundlevelinthepowerspectrumandtherathercompli- Input Catalog, and for a description of the preparation of Ke- catedphysicsinvolvedinthedrivinganddampingmechanisms pler light curves, respectively). These will become longer with ofthemodes(e.g.seeKjeldsenetal.2005,2008).Differentscal- the upcoming era of the Kepler extended mission (Still 2012). ing relations aimed at predicting amplitudes by scaling from the Sun’s values have been derived and discussed by several (cid:63) E-mail:[email protected]; authors,boththeoretically(Kjeldsen&Bedding1995;Houdek (cid:13)c 2012RAS 2 E. Corsaro et al. etal.1999;Houdek&Gough2002;Houdek2006;Samadietal. the stars of our sample. Unfortunately, those provided by the 2007; Belkacem et al. 2011; Kjeldsen & Bedding 2011; Samadi KIC (Brown et al. 2011) are known to suffer from significant etal.2012)andobservationally(Stelloetal.2011;Huberetal. systematic effects (see Pinsonneault et al. 2012a for a detailed 2011b;Mosseretal.2012).Avarietyofamplitudescalingrela- discussion of the problem). We used revised T derived by eff tions has been used extensively in literature in both ensemble Pinsonneault et al. (2012a) for a total of 161977 KIC stars studies(e.g.seeStelloetal.2010;Chaplinetal.2011b;Verner fromSloanDigitalSkySurveygriz filters,whichwerecorrected et al. 2011; Huber et al. 2011b; Mosser et al. 2012; Belkacem usingtemperatureestimatesfrominfraredfluxmethod(IRFM) 2012)anddetailedanalysesofsinglestarsfrommainsequence (J −K ) color index for hot stars (e.g. see Casagrande et al. s to the subgiant phase of evolution (e.g. Bonanno et al. 2008; 2010). The revised effective temperatures are available in the Samadi et al. 2010; Huber et al. 2011a; Mathur et al. 2011; online catalog (Pinsonneault et al. 2012b). Campante et al. 2011; Corsaro et al. 2012a). By cross-matching the stars of our sample with the tem- The underlying physical meaning of these various ampli- peratureestimatesprovidedbyPinsonneaultetal.(2012a),we tude scaling relations is still not properly understood (e.g. see arrived at a final sample of 1640 stars with an accurate T eff the discussions by Samadi et al. 2007; Verner et al. 2011; Hu- (1111observedinLCand529inSC),whichwillbeusedforour ber et al. 2011b; Samadi et al. 2012; Belkacem 2012). Testing investigation.Totaluncertaintiesontemperature,asderivedby them with observational data is vital for assessing the com- Pinsonneault et al. (2012a), include both random and system- petingrelationsandforimprovingourunderstandingofstellar aticcontributions.Theamplitudesofthefinalsampleareplot- oscillations,i.e.thedrivinganddampingmechanismsthatpro- tedagainstν and∆ν ininFigure1(topandmiddlepanels, max duce the observed amplitudes (see also Chaplin et al. 2011b). respectively).ThebottompanelshowsanasteroseismicHRdi- Inthiscontext,Bayesianmethodscanbeofgreatuse(seee.g. agramofoursampleofstars(amplitudesagainstT fromPin- eff Trotta2008)becausetheyallowustomeasurephysicalquanti- sonneaultetal.2012a),similartotheoneintroducedbyStello tiesofinterestinarigorousmanner.Moreover,Bayesianstatis- et al. (2010). 1σ error bars are overlaid on both quantities for ticsprovidesanefficientsolutiontotheproblemofmodelcom- each panel. The average relative uncertainty in amplitude is parison, which is the most important feature of the Bayesian (cid:104)σ /A(cid:105)=9.2% for the entire sample, and (cid:104)σ /A(cid:105) =11.7% A A SC approach(seealsoBenomaretal.2009;Handberg&Campante and (cid:104)σ /A(cid:105) =8.1%, for SC and LC targets, respectively. A LC 2011; Gruberbauer et al. 2012). In the present paper we analyze amplitude measurements of a sample of 1640 stars observed with Kepler, together with 3 AMPLITUDE SCALING RELATIONS temperature estimates derived from SDSS photometry, which we introduce in Section 2. In Section 3 we discuss the differ- Several scaling relations for oscillation amplitudes have been ent scaling relations for predicting the oscillation amplitude proposedsofar.Wewillbrieflyintroducetheminthefollowing perradialmode.TheresultsobtainedfromaBayesianparam- (see Huber et al. 2011b for further discussion). eter estimation for the different scaling relations are shown in Section 4 and the model comparison is presented in Section 5. Lastly,discussionandconclusionsabouttheresultsofouranal- 3.1 The L/M scaling relation ysis are drawn in Section 6 and 7, respectively. The first scaling relation for amplitudes was introduced by Kjeldsen & Bedding (1995) for radial velocities, based on the- oretical models by Christensen-Dalsgaard & Frandsen (1983). 2 OBSERVATIONS AND DATA It is given by We use amplitude measurements and their uncertainties, ob- (cid:18)L(cid:19)s v ∝ , (1) tainedbyHuberetal.(2011b)forasampleof1673starsspan- osc M ning from main sequence (MS) to red giant stars (RGs) ob- wherev representsthepredictionfortheamplitudeinradial servedwithKepler inshortcadence(SC;mostlyMSstarsbut osc velocity, L is the luminosity and M the mass of the star, and alsosomesubgiantsandlowluminosityRGs)andlongcadence s=1. Kjeldsen & Bedding (1995) also showed that the corre- (LC;allRGs)modes(Gillilandetal.2010a;Jenkinsetal.2010, spondingphotometricamplitudeA ,observedatawavelength respectively). Most of the 542 stars observed in SC have pho- λ λ, is related to v by tometry for one month, while the 1131 stars observed in LC osc have light curves spanning from Kepler’s observing quarters A ∝ vosc . (2) 0 to 6. All amplitudes were derived according to the method λ λTr eff describedbyKjeldsenetal.(2005,2008),whichprovidesampli- Foradiabaticoscillations,theexponentris1.5(seealsoHoudek tudesperradialmode(seeHuberetal.2009,formoredetails). 2006),butKjeldsen&Bedding(1995)foundtheobservedvalue Valuesofthefrequencyofmaximumpower,ν ,thelargefre- max for classical p-mode pulsators to be 2.0 (see also Samadi et al. quencyseparation,∆ν,andtheiruncertaintiesforallthestars 2007). By combining the two equations, one obtains were also taken from Huber et al. (2011b), who used the SYD pipeline (Huber et al. 2009). (cid:18)L(cid:19)s 1 A ∝ . (3) Itisimportanttohaveaccuratetemperatureestimatesfor λ M λTr eff (cid:13)c 2012RAS,MNRAS000,1–17 A Bayesian approach to amplitudes of solar-like oscillations 3 We are interested in a sample of stars observed with Kepler, whosebandpasshasacentralwavelengthλ=650nm.Byscal- ing Eq. (3) to our Sun, we have A (cid:18) L/L (cid:19)s(cid:18) T (cid:19)−r λ = (cid:12) eff , (4) A M/M T λ,(cid:12) (cid:12) eff,(cid:12) where A = 3.98ppm is the Sun’s photometric amplitude 650,(cid:12) observed at the Kepler wavelength (e.g. see Stello et al. 2011) and T =5777K. The exponent s has been examined both eff,(cid:12) theoretically (e.g. see Houdek et al. 1999; Houdek & Gough 2002; Samadi et al. 2007) and observationally (Gilliland 2008; Dziembowski & Soszyn´ski 2010; Stello et al. 2010; Verner et al. 2011; Baudin et al. 2011), and found to be 0.7 < s < 1.5. The exponent r has usually been chosen to be 2.0 (e.g. see the discussion by Stello et al. 2011), which implies that solar- like oscillations are not fully adiabatic. This was also shown in the case of CoRoT red giant stars by Samadi et al. (2012). Nonetheless,someauthors(e.g.seeMicheletal.2008;Mosser etal.2010)havechosentoadoptr=1.5(seealsothediscussion by Kjeldsen & Bedding 2011). We can also use the results introduced by Brown et al. (1991), which suggest that the frequency of maximum power scales with the cut-off frequency of the star. Hence, ν ∝ √ max g/ T , and by considering that L/M scales as T4 /g, one eff eff obtains L T3.5 ∝ eff . (5) M ν max Thus, Eq. (4) can be rewritten as A (cid:18) ν (cid:19)−s(cid:18) T (cid:19)3.5s−r λ = max eff , (6) A ν T λ,(cid:12) max,(cid:12) eff,(cid:12) where ν = 3100µHz. The functional form of the ampli- max,(cid:12) tude scaling relation given by Eq. (6) has the advantage of simplifying the inference problem presented in Section 4 with respecttotheoneofEq(4).ThisisbecauseEq.(6)isbasedon a set of independent observables, namely ν and T , which max eff representtheinputdatausedforthiswork.Ananalogousargu- ment has been applied to the other scaling relations described in the following sections. Thus,Eq.(6)representsthefirstmodeltobeinvestigated, which we will refer to as M . For this model, both the ex- 1 ponents s and r are set to be free parameters. In parallel, an extended version of Eq. (6) given by A (cid:18) ν (cid:19)−s(cid:18) T (cid:19)3.5s−r Figure 1. Oscillation amplitudes for 1640 stars observed with Ke- λ =β max eff , (7) pler inSC(orangesquares)andLC(bluecircles)modesandplotted Aλ,(cid:12) νmax,(cid:12) Teff,(cid:12) against the frequency of maximum power νmax (top) and the large is also considered, where the factor β allows the model not to frequencyseparation∆ν(middle)ofthestarsinalog-logscale.Am- necessarily pass through the solar point, as we will discuss in plitudesagainsttheeffectivetemperatureT areshowninthebot- eff moredetailinSection4.2.Eq.(7)istreatedasaseparatemodel tompanel,representinganasteroseismicHRdiagramforoursample andwillbedenotedasM .Clearly,M dependsuponthe ofstars.1σerrorbarsareshownonbothquantitiesforalltheplots. 1,β 1,β TheSunisshownwithitsusualsymbol((cid:12)). additional free parameter, β. 3.2 The bolometric amplitude The stars considered span over a wide range of temperatures, from about 4000K to more than 7000K. Hence, following the (cid:13)c 2012RAS,MNRAS000,1–17 4 E. Corsaro et al. discussion by Huber et al. (2011b), a more valuable expres- to Huber et al. (2011b), an obvious way to modify Eq. (11) is sion for the photometric amplitude could be represented by given by the bolometric amplitude A , which is related to A by (see bol λ Ls 1 Kjeldsen & Bedding 1995): A(3) ∝ , (14) bol MtTr−1c (T ) v eff K eff A ∝λA T ∝ osc . (8) bol λ eff Tr−1 where now the mass varies with the independent exponent t. eff Forthiscase,thedependenceuponthequantitiesν andT max eff By using Eq (1) we thus have becomesslightlymorecomplicatedbecausethesimplepropor- (cid:18)L(cid:19)s 1 tionalityexpressedbyEq.(5)cannolongerbeadopted.There- A(1) ∝ , (9) fore, the first step to derive the functional form based on our bol M Tr−1 eff setofobservables(ν ,∆ν,T ),istoconsiderthescalingre- max eff which by scaling to the Sun and adopting Eq. (5) yields lationsforthelargefrequencyseparation∆ν (e.g.seeBedding et al. 2007) A(1) (cid:18) ν (cid:19)−s(cid:18) T (cid:19)3.5s−r+1 bol = max eff , (10) ∆ν (cid:18) M (cid:19)0.5(cid:18) R (cid:19)−1.5 Abol,(cid:12) νmax,(cid:12) Teff,(cid:12) = , (15) ∆ν M R (cid:12) (cid:12) (cid:12) where A = 3.6ppm represents the Sun’s bolometric am- bol,(cid:12) plitude, determined by Michel et al. (2009) and also adopted with∆ν(cid:12) =135µHz,andforthefrequencyofmaximumpower by Huber et al. (2011b). Eq. (10) is the second model, M2, νmax to be investigated in Section 4, with the exponents s and r ν (cid:18) M (cid:19)(cid:18) R (cid:19)−2(cid:18) T (cid:19)−0.5 set again to be the free parameters. We also consider the new max = eff , (16) ν M R T model M , which again includes the proportionality term β max,(cid:12) (cid:12) (cid:12) eff,(cid:12) 2,β playing the same role as in Eq. (7). both expressed in terms of the fundamental stellar properties M,R,andT .Itishoweverworthmentioningthatthesescal- eff ing relations are empirical approximations whose validity and 3.3 The Kepler bandpass-corrected amplitude limitations are not yet fully understood and currently under debate (e.g. see Miglio et al. 2012, and references therein). By Ballotetal.(2011)haverecentlyestablishedabolometriccor- combining Eq. (15) and (16), one can derive an expression for rection for amplitude of radial modes observed with Kepler, the seismic radius of a star (e.g. see Chaplin et al. 2011a), which translates into a correction factor for effective tempera- namely turesfallingwithintherange4000–7500K.Againfollowingthe approachbyHuberetal.(2011b),weconsiderarevisedversion R (cid:18) ν (cid:19)(cid:18) ∆ν (cid:19)−2(cid:18) T (cid:19)0.5 = max eff . (17) of Eq. (9), which reads R ν ∆ν T (cid:12) max,(cid:12) (cid:12) eff,(cid:12) A(2) ∝(cid:18)L(cid:19)s 1 , (11) As a second step, we express Ls/Mt in terms of R, νmax, and bol M Terff−1cK(Teff) Teff and scale to the Sun’s values, yielding where (cid:18) L (cid:19)s(cid:18) M (cid:19)−t (cid:18) R (cid:19)2s−2t(cid:18) ν (cid:19)−t = max c (T )=(cid:18) Teff (cid:19)0.8 (12) L(cid:12) M(cid:12) R(cid:12) νmax,(cid:12) (18) K eff 5934 (cid:18) T (cid:19)4s−0.5t eff . is the bolometric correction expressed as a power law of the Teff,(cid:12) effective temperature. By scaling once again to the Sun and Finally, by combining Eqs. (12), (14), (17) and (18) we arrive applying Eq. (5), we obtain at A(b2o)l =(cid:18) νmax (cid:19)−s(cid:18) Teff (cid:19)3.5s−r+0.2 , (13) A(b3o)l =(cid:18) νmax (cid:19)2s−3t(cid:18) ∆ν (cid:19)4t−4s Abol,(cid:12) νmax,(cid:12) Teff,(cid:12) Abol,(cid:12) νmax,(cid:12) ∆ν(cid:12) (19) whichwewillrefertoasM .Asfortheothermodels,weintro- (cid:18) T (cid:19)5s−1.5t−r+0.2 3 eff . ducethemodelM3,β withtheproportionalitytermβincluded. Teff,(cid:12) Thisrepresentsthemodelforthemass-dependentscalingrela- tion for amplitudes, hereafter denoted as M . In this case, we 3.4 The mass-dependent scaling relation 4 have the three free parameters s, r, and t and the set of ob- Amassdependenceoftheoscillationamplitudeswassuggested servables now includes also our measurements of ∆ν. The cor- forthefirsttimebyHuberetal.(2010),andlateronstudiedin respondingmodelM hasthelargestnumberoffreeparam- 4,β detailbyStelloetal.(2011)forclusterRGswiththeintroduc- eters among those investigated in this work. Note that models tion of a new scaling relation. It was also tested afterwards by M and M reduce to models M and M , respectively, 4 4,β 3 3,β Huberetal.(2011b)forawidersampleoffieldstars.According for t=s. (cid:13)c 2012RAS,MNRAS000,1–17 A Bayesian approach to amplitudes of solar-like oscillations 5 3.5 The lifetime-dependent scaling relation themassdependenceoftheamplitudes,weintroduceaslightly modified version of the amplitude relation given by Eq. (21), Kjeldsen&Bedding(2011)haverecentlyprovidedphysicalar- where we set the mass to vary with an independent exponent guments to propose a new scaling relation for predicting the t, thus yielding amplitudes of solar-like oscillations observed in radial veloci- ties. Their relation arises by postulating that the amplitudes A(5) ∝ Lτo0s.c5 1 . (25) depend on both the stochastic excitation (given by the gran- bol Mt T1.25+rc (T ) eff K eff ulation power, see Kjeldsen & Bedding 2011, for details) and By adopting again Eq. (18) and rearranging, we finally obtain the damping rate (given by the mode lifetime). It reads A(5) (cid:18) ν (cid:19)2−3t(cid:18) ∆ν (cid:19)4t−4(cid:18) τ (cid:19)0.5 Lτ0.5 bol = max osc vosc ∝ M1.5oTsc2.25 , (20) Abol,(cid:12) νmax,(cid:12) ∆ν(cid:12) τosc,(cid:12) (26) eff (cid:18) T (cid:19)4.55−r−1.5t where τ is the average mode lifetime of radial modes. By eff , osc T meansofEq.(8),andwiththebolometriccorrectionintroduced eff,(cid:12) in Section 3.3, the corresponding relation for the bolometric hereafter marked as model M6. As done for the other scaling amplitude is given by (see also Huber et al. 2011b) relations, the model M6,β is also included in our inference. Clearly,modelsM andM reducetomodelsM andM , Lτ0.5 1 6 6,β 5 5,β A(4) ∝ osc . (21) respectively, for t=1.5. bol M1.5 T1.25+rc (T ) eff K eff In order to obtain the expression for the model M , we use 5 similar arguments to those adopted in Section 3.4, arriving at 4 BAYESIAN INFERENCE A(4) (cid:18) ν (cid:19)−2.5(cid:18) ∆ν (cid:19)2(cid:18) τ (cid:19)0.5 We now use Bayesian inference for the free parameters of the bol = max osc models described above. The Bayes’ theorem tells us that A ν ∆ν τ bol,(cid:12) max,(cid:12) (cid:12) osc,(cid:12) (22) (cid:18) Teff (cid:19)2.3−r , p(ξ|A,M)= p(A|pξ(,AM|)Mπ(ξ)|M), (27) T eff,(cid:12) where ξ = ξ ,ξ ,...,ξ is the vector of the k free parameters where τ = 2.88d, as adopted by Kjeldsen & Bedding 1 2 k osc,(cid:12) that formalize the hypotheses of the model M, considered for (2011). For our computations we assume that the mode life- theinference,andAisthesetofamplitudemeasurements.The timeisafunctionoftheeffectivetemperatureofthestaralone term p(A|ξ,M) is now identified with the likelihood L(ξ) of (e.g.seeChaplinetal.2009;Baudinetal.2011;Appourchaux the parameters ξ given the measured oscillation amplitudes: et al. 2012; Belkacem et al. 2012; Corsaro et al. 2012b). We i used the empirical law found by Corsaro et al. (2012b), which p(A|ξ,M)=L(ξ|A,M). (28) relatesthemodelinewidthsΓoftheradialmodes((cid:96)=0)tothe Thus, the left-hand side of Eq. (27) is the posterior probabil- effectivetemperatureofthestarwithintherange4000–7000K. ity density function (PDF), while the right-hand side is the In particular, they found that product of the likelihood function L(ξ), which represents our (cid:18)T −T (cid:19) mannerofcomparingthedatatothepredictionsbythemodel, Γ=Γ exp eff eff,(cid:12) , (23) 0 T andthepriorPDFπ(ξ|M),whichrepresentsourknowledgeof 0 theinferredparametersbeforeanyinformationfromthedatais where Γ =1.39±0.10µHz and T =601±3K. This relation 0 0 available.Thetermp(A|M)isanormalizationfactor,known wascalibratedusingKepler RGsintheopenclustersNGC6791 as the Bayesian evidence, which we do not consider for the in- andNGC 6819,and asampleof MSand subgiant Kepler field ference problem because it is a constant for a model alone. As stars. Given τ =(πΓ)−1, we obtain we will argue in Section 5, the Bayesian evidence is essential (cid:18)T −T (cid:19) for solving the problem of model comparison. τ =τ exp eff,(cid:12) eff , (24) osc 0 T Forourinferenceproblem,weadoptthecommonGaussian 0 likelihood function, which presumes that the residuals arising with τ = 2.65±0.19d. The mode lifetimes were computed 0 fromthedifferencebetweenobservedandpredictedlogarithms for all the stars of our sample by means of Eq. (24), together oftheamplitudesareGaussiandistributed,i.e.theamplitudes withtheircorrespondinguncertaintiesfromtheerrorpropaga- themselvesarepresumedtobelog-normaldistributed(seealso tion.Asfortheotherscalingrelations,wealsointroducemodel Appourchaux et al. 1998). Therefore, we have M . 5,β L(ξ)=(cid:89)N √ 1 exp(cid:34)−1(cid:18)∆i(ξ)(cid:19)2(cid:35), (29) 3.6 A new scaling relation i=1 2πσ(cid:101)i 2 σ(cid:101)i where N is the total number of data points (the number of Following similar arguments to those adopted by Stello et al. stars, in our case), while (2011) for introducing a new scaling relation for amplitudes of cluster RGs, and the discussion by Huber et al. (2011b) about ∆ (ξ)=lnAobs−lnAth(ξ) (30) i i i (cid:13)c 2012RAS,MNRAS000,1–17 6 E. Corsaro et al. is the difference between the observed logarithmic amplitude Table 1. Maximum and minimum values of the free parameters, for the i-th star and the predicted one (which depends on the adoptedforallthemodelsandsamples. adopted model, i.e. on the parameters vector ξ). The term (cid:104) (cid:105) σ(cid:101)i appearing in the leading exponential term of Eq. (29) is ξj ξjmin,ξjmax the total uncertainty in the predicted logarithmic amplitude, s [0.2,1.2] namely the relative uncertainty of the amplitude enlarged by r [−6.5,11.0] the relative errors of the independent variables νmax, ∆ν, and t [1.0,2.0] Teff.Thismeansthatwearenotconsideringerror-freevariables lnβ [−1.0,1.0] inourmodels(seee.g.D’Agostini2005;Andreon&Hurn2012, formoredetails).Forsimplifyingthecomputationsamodified then performed by integrating (marginalizing) the posterior version of the likelihood function, known as the log-likelihood, distribution function p(ξ |d,M) over the remaining k−1 pa- is preferred. The log-likelihood function is defined as Λ(ξ) ≡ rameters ξ ,ξ ,...,ξ . We obtain the corresponding marginal lnL(ξ), which yields 2 3 k PDF of the parameter ξ 1 Λ(ξ)=Λ0− 12(cid:88)N (cid:20)∆σi(ξ)(cid:21)2 , (31) p(ξ1 |A,M)=(cid:90) p(ξ|A,M)dξ2,dξ3,...,dξk, (34) (cid:101)i i=1 where whosestatisticalmomentsandcredibleintervals(i.e.Bayesian confidenceintervals)arethequantitiesofinterestforourwork. Λ =−(cid:88)N ln√2πσ . (32) Since the dimensionality of our problem is not higher than 0 (cid:101)i k=4,alltheintegrationscanbecomputedbydirectnumerical i=1 summationoftheposteriordistributionovertheremainingpa- The choice of reliable priors is important in the Bayesian ap- rameters(onlyincasetheobservables,likeT ,wereerror-free proach. For our purpose, uniform priors represent a sensible eff all integrations could be computed analytically). choice for most of the free parameters. This means one has no The results presented in the coming sections are derived assumptions about the inferred parameters before any knowl- inthreecases:fortheentiresampleofstars,andforSC(domi- edge coming from the data, with equal weight being given to natedbyMSstars)andLC(RGs)targetsseparately.Analyzing all values of each of the parameters considered. In particular, the two subsets separately allows us to test whether the fitted we use standard uniform priors for the exponents s, r and t of parameters of the scaling relations are sensitive to the evolu- themodelsdescribedabove,lettingtheparametersvarywithin tionary stage of the stars (see also Huber et al. 2011b). alimitedrangeinordertomakethepriorsproper,i.e.normal- Themeanvalues(orexpectationvalues)ofthefreeparam- izable to unity. For the proportionality term β introduced in eters of the models, together with their corresponding 68.3% Eq. (7), we adopt the Jeffreys’ prior ∝ β−1 (Kass & Wasser- Bayesiancredibleintervals,arelistedinTable2forthecaseof man1996),aclassofuninformativepriorthatresultsinauni- the entire sample, and in Tables 3 and 4 for the subsets of SC form prior for the natural logarithm of the parameter. In this and LC targets. We also computed a weighted Gaussian rms, manner, the parameter of interest is represented by the offset σw , of the residuals ∆2(ξ), where we adopted the weights lnβ (see below), whose prior is uniform distributed and also rms i w = σ−2, σ being the total uncertainty used in Eq. (29). limited in range. Hence, uniform priors are included in the in- i (cid:101)i (cid:101)i The maximum of the log-likelihood function, Λ , and σw , ference problem as a simple constant term depending on the max rms used as an estimate of the fit quality, are also listed in the intervals adopted for the inferred parameters sametables.Inaddition,wederivedcorrelationcoefficientsfor π(ξ|M)=(cid:89)4 (cid:104)ξjmax−ξjmin(cid:105)−1 (33) eaancahlypsiasiruosifnfgreSeinpgauralamreVtearlsuebyDmeceoamnpsoosfitpiorinnc(iSpValDc)omfropmontehnet j=1 posterior PDFs. The results are shown in Tables 5, 6, 7 for with ξ = s, ξ = r, ξ = t and ξ = lnβ, and ξmin,ξmax the entire sample, and SC and LC targets, respectively, with 1 2 3 4 j j −1 meaning total anti-correlation and 1 total correlation. The theminimumandmaximumvaluesdefiningtheintervalofthe effects of the correlations will be discussed in Section 6. j-th parameter. The intervals that we adopt are listed in Ta- ble 1. These ranges are used for both the Bayesian parameter estimation and the model comparison. 4.1 Statistically independent models A note of caution concerns the treatment of the uncer- tainties. In fact, by using the natural logarithm of the equa- Beforegettingtoadescriptiononhowtocorrectlyincludethe tionsthatdescribethemodels(seealsoSection4.2),weensure models in the inference problem, it is useful to highlight that thatwearenotfavoringforexamplefrequenciesuponperiods, the models M , M , and M on the one hand, and the 1,β 2,β 3,β amplitudesuponpower,temperaturesuponsurfacebrightness, modelsM ,andM theotherhand,arestatisticallyidentical 2 3 etc.,whichhastheadvantageofmakingtheerrorpropagation to one another (but not identical in the general sense, since law fully correct. Thus, the corresponding uncertainties to be their underlying physical assumptions are different). In case considered in Eq. (29) will be the relative uncertainties. the intervals of their free parameters are the same for all the The inference problem for a given parameter, e.g. ξ , is modelswhenperformingtheBayesianparameterestimation(as 1 (cid:13)c 2012RAS,MNRAS000,1–17 A Bayesian approach to amplitudes of solar-like oscillations 7 itisfortheanalysispresentedinthiswork)thisimpliesthata Eq. (39) only holds in case of uncorrelated uncertainties and statisticalinferenceforthesemodelswouldleadtoidentical(or linear relations (see also D’Agostini 2005; Andreon & Hurn directly related) values of these free parameters. In particular, 2012).Intuitively,Eq.(39)isthequadraticsumoftherelative according to Eqs. (7), (10), and (13) we have that uncertaintiesoverthephysicalquantitiesconsidered,according toEq.(38).AvariationofEq.(38)isrepresentedbythemodel s(M )=s(M )=s(M ) 1,β 2,β 3,β (35) M1,β given by Eq. (7), whose natural logarithm reads s(M )=s(M ) 2 3 (cid:18) (cid:19) (cid:18) (cid:19) A ν for the exponent s, ln λ =−sln max A ν λ,(cid:12) max,(cid:12) r(M1,β)=r(M2,β)−1=r(M3,β)−0.2 (36) +(3.5s−r)ln(cid:18) Teff (cid:19)+lnβ. (40) r(M2)=r(M3)−0.8, Teff,(cid:12) for the exponent r, and which differs from Eq. (38) by the additional term lnβ. As al- (cid:18) (cid:19) A ready argued before, the offset lnβ allows the model not to lnβ(M )=lnβ(M )+ln bol,(cid:12) 1,β 2,β Aλ,(cid:12) necessarily pass through the solar point (A(cid:12),νmax,(cid:12),Teff,(cid:12)). (cid:18) (cid:19) Its introduction in the inference is of importance if one wants =lnβ(M3,β)+ln AAbol,(cid:12) (37) to assess whether or not the Sun is a good reference star for λ,(cid:12) the sample considered. This choice is also motivated by the fact that the Sun is falling at the edge of the sample of stars fortheoffsetlnβ,wherethetermln(Abol,(cid:12)/Aλ,(cid:12))arisesfrom when plotting amplitudes against νmax and ∆ν (Figure 1, top the difference in considering the amplitudes to be either ob- andmiddlepanels).Thispeculiarpositionisalsoevidentfrom servedatλ=650nm(M )orbolometric(M andM ). our asteroseismic HR diagram (Figure 1, bottom panel), and 1,β 2,β 3,β Asaconsequence,fromnowontheentireanalysisforthemod- is caused by the lack of solar twins in our sample of stars (see elsM ,M ,andM ononeside,andforthemodelsM the discussion by Chaplin et al. 2011a). In fact, in the case of 1,β 2,β 3,β 2 and M on the other side, will be reduced to that of the two lnβ (cid:54)=0, by replacing the solar values in Eq. (40) (or alterna- 3 models M and M , respectively. The reader can derive the tively Eq. (7)), the predicted amplitude for the Sun would be 1,β 2 correspondingparametersfortheotherdependentmodelsusing resizedbyafactorβ.Thismeansthatthebestreferenceampli- Eqs. (35), (36), and (37). tudeforscalingtheamplitudesofoursampleofstarswouldbe represented by βA . According to Eq. (40), the total uncer- λ,(cid:12) tainty to be considered in building the likelihood function for the model M is given again by Eq. (39) because the offset 1,β 4.2 Models M1 and M1,β doesnotplayanyroleinthetotalcontributeoftheuncertain- ties. ThefirstmodeltobeinvestigatedisgivenbyEq.(6).Asargued A representative sample of the resulting marginal PDFs above, we need to consider the natural logarithm in order to is plotted in Figure 2 for the case of the entire sample, where treat the observables independently of the function adopted 68.3% Bayesian credible regions (shaded bands) and expec- (see the discussion in Section 4). Hence the model reads tation values (dashed lines) are also marked. The comparison (cid:18) (cid:19) (cid:18) (cid:19) A ν ln λ =−sln max betweenthepredictedandtheobservedamplitudesisshownin Aλ,(cid:12) νmax,(cid:12) Figure 3 for the three cases considered (top panels for model (38) +(3.5s−r)ln(cid:18) Teff (cid:19) . M1, bottom panels for model M1,β), together with a plot of T the residuals arising from the difference between the models eff,(cid:12) and the observations. At this stage we briefly describe how the uncertainties have been included in our analysis. The new uncertainties on the scaledamplitudetobeconsideredinEq.(29)areclearlygiven 4.3 Model M byσ /A ,hereafterσ forsimplicity,whereA istheobserved 2 Ai i (cid:101)Ai i amplitude for the i-th star and σAi its corresponding uncer- ThemodelM2 givenbyEq.(10)deservesasimilardescription tainty as derived by Huber et al. (2011b). However, Eq. (38) to that presented in Section (4.2) for model M , where the 1 suggeststhattheuncertaintyonamplitudeisnottheonlyone natural logarithm is now given by affecting the predicted amplitude A . In fact, uncertainties on both νmax (derived by Huber et al. 2λ011b) and Teff (from Pin- ln(cid:32) A(b1o)l (cid:33)=−sln(cid:18) νmax (cid:19) sonneault et al. 2012a) have to be included in our computa- A ν bol,(cid:12) max,(cid:12) tions. The total uncertainty to be used in Eq. (29) is given by (41) (cid:18) (cid:19) T the Gaussian error propagation law, which gives +(3.5s−r+1)ln eff , T σ2(s,r)=σ2 +s2σ2 +(3.5s−r)2σ2 (39) eff,(cid:12) (cid:101)i (cid:101)Ai (cid:101)νmax,i (cid:101)Teff,i with a total uncertainty for the i-th star expressed as where we defined σ ≡ σ /ν and σ ≡ (cid:101)νmax,i νmax,i max,i (cid:101)Teff,i σ /T , similarly to what was done for the amplitudes. σ2(s,r)=σ2 +s2σ2 +(3.5s−r+1)2σ2 (42) Teff,i eff,i (cid:101)i (cid:101)Ai (cid:101)νmax,i (cid:101)Teff,i (cid:13)c 2012RAS,MNRAS000,1–17 8 E. Corsaro et al. Table2.ExpectationvaluesoftheinferredparametersforallthemodelsdescribedinSection3inthecaseoftheentiresample(bothLCand SC targets), having N =1640 stars. 68.3% Bayesian credible intervals are added. The maximum value for the log-likelihood function Λmax andaweightedGaussianrmsoftheresiduals,σw ,arealsoreported. rms Model s r t lnβ Λmax σrwms M1 0.680±0.002 4.31±0.04 – – −3533.1 0.23 M 0.524±0.004 5.51±0.05 – 0.400±0.010 −2491.3 0.24 1,β M2 0.722±0.002 4.96±0.04 – – −4159.9 0.24 M4 0.822±0.003 3.93±0.06 1.58±0.02 – −948.5 0.23 M 0.643±0.005 4.46±0.06 1.36±0.02 0.528±0.012 163.4 0.18 4,β M5 – −5.71±0.03 – – −925.5 0.31 M – −5.07±0.05 – −0.122±0.007 −786.9 0.29 5,β M6 – −5.04±0.05 1.80±0.02 – −727.1 0.30 M – −4.97±0.06 1.75±0.02 −0.035±0.012 −722.8 0.29 6,β Table 3.SamedescriptionasforTable2butinthecaseofSCtargets,havingN =529stars. Model s r t lnβ Λmax σrwms M1 0.775±0.003 3.23±0.05 – – −321.4 0.20 M 0.624±0.010 3.68±0.06 – 0.241±0.016 −199.3 0.19 1,β M2 0.838±0.003 4.05±0.05 – – −443.8 0.22 M4 0.984+−00..000190 2.79±0.09 1.66+−00..0045 – −94.9 0.23 M 0.748±0.015 3.47±0.09 1.27±0.04 0.321±0.020 18.0 0.20 4,β M5 – −2.78±0.09 – – −83.1 0.24 M – −2.75±0.09 – 0.020±0.008 −80.2 0.24 5,β M6 – −2.80+−00..1009 1.56±0.02 – −79.6 0.24 M – −2.75±0.10 1.72±0.04 0.087+0.014 −59.4 0.24 6,β −0.015 Table 4.SamedescriptionasforTable2butinthecaseofLCtargets,havingN =1111stars. Model s r t lnβ Λmax σrwms M1 0.464+−00..000076 9.53+−00..1156 – – −799.2 0.28 M 0.548±0.009 9.67+0.15 – −0.35±0.03 −737.4 0.27 1,β −0.16 M2 0.491+−00..000076 10.51+−00..1156 – – −769.0 0.28 M4 0.666+−00..000065 6.99±0.12 1.28±0.02 – 207.4 0.18 M 0.602±0.008 5.87±0.14 1.31±0.02 0.45±0.03 301.0 0.16 4,β M5 – −6.08±0.04 – – −357.9 0.24 M – −4.38±0.16 – −0.27+0.02 −309.2 0.24 5,β −0.03 M6 – −5.94±0.08 1.55±0.03 – −356.0 0.24 M – −4.39±0.16 1.45+0.02 −0.30±0.03 −307.1 0.24 6,β −0.03 Table 5.Correlationcoefficientsforpairsoffreeparametersforeachmodelinthecaseoftheentiresample. Model svsr svsb svst r vsb r vst bvst M1 −0.90 – – – – – M −0.85 −0.92 – 0.66 – – 1,β M2 −0.94 – – – – – M4 −0.74 – 0.59 – 0.01 – M −0.74 −0.76 0.44 0.31 −0.25 0.04 4,β M5 – – – – – – M – – – −0.74 – – 5,β M6 – – – – 0.74 – M – – – −0.41 0.18 0.71 6,β (cid:13)c 2012RAS,MNRAS000,1–17 A Bayesian approach to amplitudes of solar-like oscillations 9 Table 6.SamedescriptionasforTable5butinthecaseofSCtargets. Model svsr svsb svst r vsb r vst bvst M1 0.22 – – – – – M −0.40 −0.94 – 0.49 – – 1,β M2 0.20 – – – – – M4 −0.29 – 0.88 – −0.27 – M −0.48 −0.85 0.71 0.44 −0.37 −0.31 4,β M5 – – – – – – M – – – 0.12 – – 5,β M6 – – – – −0.07 – M – – – 0.03 −0.01 0.76 6,β Table 7.SamedescriptionasforTable5butinthecaseofLCtargets. Model svsr svsb svst r vsb r vst bvst M1 −0.98 – – – – – M −0.49 −0.71 – −0.25 – – 1,β M2 −0.98 – – – – – M4 −0.92 – 0.50 – −0.19 – M −0.31 −0.66 0.30 −0.46 −0.30 0.18 4,β M5 – – – – – – M – – – −0.98 – – 5,β M6 – – – – 0.88 – M – – – −0.88 0.04 0.37 6,β to be included in Eq. (29). The resulting models are shown in According to Eq. (43), the total uncertainty to be considered Figure 4, with similar descriptions as those adopted for Fig- in Eq. (29) reads ure 3. σ2(s,r,t)=σ2 +(2s−3t)2σ2 +(4t−4s)2σ2 (cid:101)i (cid:101)Ai (cid:101)νmax,i (cid:101)∆νi (44) +(5s−1.5t−r+0.2)2σ2 , (cid:101)Teff,i 4.4 Models M and M 4 4,β with σ ≡ σ /∆ν , as done for the other quantities. As The models M and M (see Section 3.4) are clearly the (cid:101)∆νi ∆νi i 4 4,β one can intuitively expect, the new total uncertainty depends most complex ones among those investigated in this work be- onthethreefreeparametersofthemodel.Theresultingmodels cause the largest number of free parameters is involved, and are shown in Figure 5. measurements of ∆ν are also needed. We note that, although a tight correlation between ν and ∆ν has been found in max previousstudies(e.g.seeStelloetal.2009a),wechoosenotto adopt the νmax–∆ν relation to express model M4 in terms of 4.5 Models M5 and M5,β ν only (or alternatively ∆ν) because additional uncertain- max The models described in Section 3.5 are derived with a quite ties arising from the scatter around this relation would affect differentapproach,whichrequiresanestimateofthemodelife- the results of our inference. This is also motivated by recent timeforeachstarconsideredinoursample.Wenotethatmodel results by Huber et al. (2011b) who found that the ν -∆ν max M wasalsoinvestigatedbyHuberetal.(2011b),whohowever relation changes as a function of T between dwarf and giant 5 eff did not take into account mode lifetimes. stars. The natural logarithm of model M lead us to Therefore,byconsideringthenaturallogarithmofEq.(19) 5 one obtains (cid:32) A(4) (cid:33) (cid:18) ν (cid:19) ln(cid:32) A(b3o)l (cid:33)= (2s−3t)ln(cid:18) νmax (cid:19)+(4t−4s)ln(cid:18) ∆ν (cid:19) ln Abbolo,l(cid:12) =−2.5ln νmmaxa,x(cid:12) Abol,(cid:12) νmax,(cid:12) ∆ν(cid:12) +2ln(cid:18) ∆ν (cid:19)+0.5ln(cid:18) τosc (cid:19) (45) +(5s−1.5t−r+0.2)ln(cid:18) Teff (cid:19) , ∆ν(cid:12) (cid:18) (cid:19) τosc,(cid:12) Teff,(cid:12) +(2.3−r)ln Teff . (43) T eff,(cid:12) for model M , and with the additional term lnβ for M . Thus, the total uncertainty for the i-th star of the sample is 4 4,β (cid:13)c 2012RAS,MNRAS000,1–17 10 E. Corsaro et al. given by Table 11. Bayesian Information Criterion (BIC) computed for all σ2(r)=σ2 +6.25σ2 +4σ2 themodelsinthreecasesconsidered:alltargets(secondcolumn),SC (cid:101)i (cid:101)Ai (cid:101)νmax,i (cid:101)∆νi (46) targets(thirdcolumn),LCtargets(fourthcolumn). +0.25σ2 +(2.3−r)2σ2 , (cid:101)τosc,i (cid:101)Teff,i Model BIC BIC(SC) BIC(LC) withσ2 ≡σ /τ ,andisthesameforboththemodels (cid:101)τosc,i τosc,i osc,i M1 7081 655 1612 here considered, as model M differs only by the additional 5,β M 5004 417 1497 1,β term lnβ. The resulting models are shown in Figure 6, with M2 8335 901 1552 similar descriptions as those adopted for Figures 2 and 3. M4 1920 209 −393 M −296 −11 −574 4,β M5 1859 172 723 4.6 Models M6 and M6,β M5,β 1589 173 632 Following the same arguments used for the other models, we M6 1469 173 726 M 1468 137 635 obtain 6,β (cid:32) A(5) (cid:33) (cid:18) ν (cid:19) ln bol = (2−3t)ln max usefultotakeintoaccounttheratios(orodds)oftheevidences, Abol,(cid:12) νmax,(cid:12) namely the so-called Bayes factor, which is given as (cid:18) (cid:19) (cid:18) (cid:19) +(4t−4)ln ∆ν +0.5ln τosc (47) B = p(A|Mi) = EMi . (50) ∆ν(cid:12) τosc,(cid:12) ij p(A|Mj) EMj (cid:18) (cid:19) +(4.55−r−1.5t)ln Teff . In case Bij > 1 the model Mi is the favored one, while con- Teff,(cid:12) versely if Bij < 1 the model Mj ought to be preferred. The resultingnaturallogarithmsoftheBayesfactor,whicharecom- forthemodelM .Thenewuncertaintiestobeconsideredwill 6 puted according to Eq. (50) are listed in Tables 8, 9, and 10, depend on the free parameters r and t, thus we have for the cases of the entire sample, and SC and LC targets, re- σ(cid:101)i2(r,t)=σ(cid:101)A2i +(2−3t)2σ(cid:101)ν2max,i +(4t−4)2σ(cid:101)∆2νi (48) spectively. Therefore, if lnBij > 0 the model Mi is preferred +0.25σ2 +(4.55−r−1.5t)2σ2 . over Mj and vice versa if lnBij <0. (cid:101)τosc,i (cid:101)Teff,i It is sometimes useful to consider so-called Information with the same quantities adopted in Eq. (46). An analogous Criteria,whichmayofferasimpleralternativetotheBayesian discussion to that used for other scaling relations has to be evidence, whose numerical computation in some cases can be applied for model M6,β. The results of the inference for both very time demanding. In particular, we adopted the Bayesian models are plotted in Figure 7. Information Criterion (BIC), also known as Schwarz Informa- tion Criterion (Schwarz 1978), which follows from a Gaussian approximation to the Bayesian evidence in the limit of a large 5 MODEL COMPARISON sample size, as it can be represented by our sample of stars (N (cid:29)1). Thus, the BIC reads As mentioned in Section 4, the term p(A | M) appearing in Eq. (27) (Bayesian evidence) is the one of interest for solving BIC≡−2Λmax+klnN, (51) the problem of model comparison in the context of Bayesian where k is the number of free parameters of the model consid- statistics(e.g.seeTrotta2008;Benomaretal.2009;Handberg ered(i.e.thedimensionofthecorrespondingparameterspace), & Campante 2011; Gruberbauer et al. 2012). The Bayesian andN thenumberofdatapoints.SinceΛ isknown,theBIC max evidence is given by integrating the numerator appearing in canbecomputedstraightforwardly.Theresultingvaluesofthe the right-hand side of Eq. (27) over all the possible values of BIC are listed in Table 11 for the cases of the entire sample the free parameters ξ ,ξ ,...,ξ . Thus we have 1 2 k (second column) and of SC and LC targets separately (third (cid:90) and fourth columns, respectively). According to the Occam’s E ≡p(A|M)= L(ξ)π(ξ|M)dξ, (49) M razorprincipleonwhichBayesianmodelcomparisonrelies,the ΩM most eligible model is the one that minimizes the BIC. where Ω represents the parameter space, defined by the in- M Ashighlightedbytheshadedrowsandcolumns,themodel tervalsofvariationofthefreeparametersthatformalizethehy- M is largely the favored one for all the samples considered 4,β pothesesofthemodelM,andhavingvolumegivenbyEq.(33). because its evidence is always greater than those of any other The Bayesian evidence given by Eq. (49) basically represents modelinvestigatedinthiswork.Inaddition,theBICconfirms the integral of the likelihood function averaged by the prior the result computed through the evidences. distribution. As the prior π(ξ | M) has to be normalized, theevidencedependsontheparameterspace.Thus,theinter- vals [ξmin,ξmax] of the free parameters ξ used for computing j j j 6 DISCUSSION Eq. (49) are those listed in Table 1. SinceameasureofE alonedoesnotcarryanymeaningful The analysis described in Section 4 and in Section 5 lead us M information, to solve the problem of model comparison it is to interesting results about the use of the amplitude scaling (cid:13)c 2012RAS,MNRAS000,1–17