DRAFTVERSIONJANUARY11,2017 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 STELLARPOPULATIONSYNTHESISBASEDMODELLINGOFTHEMILKYWAYUSINGASTEROSEISMOLOGYOF DWARFSANDSUBGIANTSFROMKEPLER. SANJIBSHARMA1,DENNISSTELLO1,2,DANIELHUBER1,2,3,JOSSBLAND-HAWTHORN1,TIMOTHYR.BEDDING1,2 DraftversionJanuary11,2017 ABSTRACT EarlyattemptstoapplyasteroseismologytostudytheGalaxyhavealreadyshownunexpecteddiscrepancies for the mass distribution of stars between the Galactic models and the data; a result that is still unexplained. 7 Here, werevisittheanalysisoftheasteroseismicsampleofdwarfandsubgiantstarsobservedbyKeplerand 1 investigateindetailthepossiblecausesforthereporteddiscrepancy. WeinvestigatetwomodelsoftheMilky 0 Way based on stellar population synthesis, Galaxia and TRILEGAL. In agreement with previous results, we 2 findthatTRILEGALpredictsmoremassivestarscomparedtoGalaxia,andthatTRILEGALpredictstoomany bluestarscomparedto2MASSobservations. Bothmodelsfailtomatchthedistributionofthestellarsamplein n (logg,T )space,pointingtoinaccuraciesinthemodelsand/ortheassumedselectionfunction.Whencorrected a eff J forthismismatchin(logg,Teff)space,themassdistributioncalculatedbyGalaxiaisbroaderandthemeanis shiftedtowardlowermassescomparedtothatoftheobservedstars. Thisbehaviourissimilartowhathasbeen 0 reportedfortheKeplerredgiantsample. Theshiftbetweenthemassdistributionsisequivalenttoachangeof 1 2%inν ,whichiswithinthecurrentuncertaintyintheν scalingrelation. Applyingcorrectionstothe∆ν max max ] scaling relation predicted by the stellar models makes the observed mass distribution significantly narrower, A butthereisnochangetothemean. G Subject headings: Galaxy: disk – Galaxy: stellar content – Galaxy: structure – asteroseismology – stars: fundamentalparameters . h p - 1. INTRODUCTION we can obtain asteroseismic information that is sensitive to, o andhencecapableofmeasuring,stellarradiusandmassina r Our current understanding of the Milky Way has, to a t large extent, been informed by stellar data from large scale largelymodelindependentway. s A promising approach to take advantage of the large en- a photometric,astrometric,andspectroscopicsurveys,suchas, semblesofseismically-inferredstellarpropertiesistoinvoke [ 2MASS(Skrutskieetal.2006),SDSS(Juric´etal.2008),Hip- stellar population synthesis-based models of the Milky Way parcos(ESA1997),GCS(Nordstro¨metal.2004),RAVE(Ko- 1 (e.g. Miglio et al. 2009; Chaplin et al. 2011a; Sharma et al. rdopatisetal.2013), SEGUE(Yannyetal.2009), APOGEE v 2016). This offers a way to link stellar structure and evo- (Zasowskietal.2013),andGaia-ESO(Gilmoreetal.2012). 4 lution with Galactic structure and evolution by combining As a result, we have already come a long way from simple 6 isochrones with star-formation history, the initial mass func- empirical models of the Galaxy that fit star counts in a few 4 tion,andthespatialdistributionofstarsoftheGalaxy.Thisal- lines of sight (Bahcall & Soneira 1980b,a, 1984), to mod- 2 lowsonetopredictstellarobservablesliketemperature,pho- 0 els that aim to be dynamically self-consistent (Robin et al. tometry, asteroseismic parameters, as well as fundamental . 2003; Binney 2010, 2012b; Binney & McMillan 2011; Bin- 1 stellarpropertiessuchasradiusandmass. ney 2012a; McMillan & Binney 2012; Czekaj et al. 2014; 0 Chaplin et al. (2011a) used the first seven months of Ke- Scho¨nrich & Binney 2009a,b; Sharma et al. 2014). Some of 7 pler data of dwarfs and subgiants, to compare the distribu- these models, such as , Besanc¸on (Robin et al. 2003), TRI- 1 tions of seismically-inferred radii and masses of about 400 LEGAL (Girardi et al. 2005), and Galaxia (Sharma et al. : stars with a synthesis-based Galactic model using TRILE- v 2011) have been constructed to directly satisfy the observa- GAL.Theyfoundthattheradiusdistributionofthesynthetic i tionalconstraintsfromvariouslargescalephotometric,astro- X populationmatchedthedata,butthemassdistributionsignif- metricandspectroscopicsurveysoftheMilkyWay.However, r to understand the Milky Way’s formation history, and hence icantly under-predicted the number of low-mass stars (M < a 1.15M ),andhenceover-predictedthenumberofmoremas- tofurtherverifythemodels, itisimportanttoknowthefun- sive(y(cid:12)ounger)stars. UsingredgiantsfromKepler,wefound damental properties of the stars, including radius and mass. theoppositeeffectwhencomparingtheobservedmasseswith Untilrecently,ithasbeendifficulttoreliablydeterminethese predictions from the Galactic model Galaxia (Sharma et al. properties model-independently for large numbers of distant 2016). WeshowedthatintheKeplerregionforamagnitude stars. Fortunately,thespacemissionsCoRoT(Baglin&Frid- limitedsample,TRILEGALpredictsmorebluestarsascom- lund2006)andKepler(Boruckietal.2010),andnowalsoK2 pared to 2MASS, while Galaxia has no such problem. Be- (Howelletal.2014),providehighlyaccuratetime-seriespho- cause blue stars are young and massive, this suggests that tometry of thousands of stars across the Galaxy, from which TRILEGAL overpredicts the number of young and massive stars. Hence,itisimportanttocomparetheobservedmasses 1SydneyInstituteforAstronomy,SchoolofPhysics,UniversityofSyd- ney,NSW2006,Australia ofdwarfsandsubgiantswiththepredictionsfromGalaxia. 2StellarAstrophysicsCentre,DepartmentofPhysicsandAstronomy, Besides inaccuracies in the Galactic model, there are a AarhusUniversity,DK-8000AarhusC,Denmark numberofotherfactorsthatcouldcontributetothemismatch 3SETI Institute, 189 Bernardo Avenue, Mountain View, CA 94043, in the mass distributions seen by Chaplin et al. (2011a). (i) USA 2 The observed asteroseismic information is in the form of 25 the average seismic parameters ∆ν (average frequency spac- 20 (a) ingbetweenovertoneoscillationmodes)andν (frequency max at maximum oscillation power) that are extracted from time 15 mean=1.0006 series photometry using a specific algorithm/method. Prior stddev=0.02 p 10 tothelaunchofKepler, about2000starswithKp<12mag were selected as potential asteroseismic dwarf and subgiant 5 targets based on their parameters in the Kepler Input Cata- log (KIC) (Brown et al. 2011). They were all observed with 0 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 shortcadenceforonemontheachduringaninitial10-month ∆νC14/∆νC11 seismic survey phase. Four hundred stars showed detectable 25 oscillationsafterthefirstsevenmonthsandwerepresentedby Chaplinetal.(2011a)(hereafterdenotedasChaplin-11sam- 20 (b) ple).Followingthecompletionofthe10-monthsurvey,anup- dateddwarf/subgiantsample(518stars)showingoscillations 15 mean=1.0051 stddev=0.03 was presented by Chaplin et al. (2014) (here after denoted p 10 as Chaplin-14 sample), where a method different from that ofChaplinetal.(2011a)wasusedforextractingseismicpa- 5 rameters. IntheChaplin-14sample467starshavemeasured values of ∆ν, ν and T . Of these, 290 stars are in com- 0 max eff 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 mon with the previous Chaplin-11 sample. In Figure 1, we νC14/νC11 max max comparethe∆νandν valuesofChaplin-2014withthose max ofChaplin-2011,forstarscommontobothsamples. Itcanbe FIG.1.—(a)Ratioof∆νforthe290starsincommonbetweentheChaplin- seen that there are no systematic shift between the two data 2014andChaplin-2011samples. (b)Sameaspanel(a)butforνmax. The dashedlineinpanel(b)showstheratioofνmax, adoptedinChaplin-2014to sets. This means there is no method-specific differences in thatadoptedinChaplin-2011. Thereisnosyst(cid:12)ematicshiftbetweenthetwo the two data sets. In this paper we mainly make use of the methods. Thestandarddeviationisoftheorderoftheuncertaintiesonthe Chaplin-2014sample,becauseithasmorestars. estimatedvaluesofνmax(4%)and∆ν(2%). Inaccuraciesintheselectionfunctioncanleadtoamismatch 2.2. Scalingrelationsandsolarreferencevalues and need to be checked. There could be systematics associ- Thestellarmassandradiuscanbeestimatedfromtheseis- atedwiththealgorithmusedtoestimateaverageseismicpa- mic parameters ∆ν and ν , and the effective temperature max rameters. (ii) The probability to detect oscillations can dif- T usingthefollowingscalingrelations: eff ferfromonealgorithmtoanotherandthiscanleadtodiffer- encesintheselectionfunction. SincetheanalysisbyChaplin M (cid:18) νmax (cid:19)3(cid:18) ∆ν (cid:19)−4(cid:18) Teff (cid:19)1.5 etal.(2011a),anewdatasetofdwarfsandsubgiantshasbeen = (1) M f ν f ∆ν T publishedbyChaplinetal.(2014),whichusedadifferental- (cid:12) νmax max,(cid:12) ∆ν (cid:12) eff,(cid:12) gorithm to estimate the seismic parameters. Additionally, it R (cid:18) νmax (cid:19)(cid:18) ∆ν (cid:19)−2(cid:18) Teff (cid:19)0.5 = . (2) contains more stars and extends to slightly lower gravities. R f ν f ∆ν T Hence, it is necessary to investigate the mass distributions (cid:12) νmax max,(cid:12) ∆ν (cid:12) eff,(cid:12) with the new data set as well. (iii) Theoretical modelling of These relations are based on the relations ∆ν ∝ ρ1/2, and thestellaroscillationspredictdeparturesfromthe∆νscaling ν ∝g/T1/2(Brownetal.1991;Kjeldsen&Bedding1995; relation (Stello et al. 2009; White et al. 2011; Miglio et al. max eff Belkacemetal.2011),whichinturnarebasedontheassump- 2013) and the effect of this needs to be taken into account. tion that the structure of any given star is homologous with (iv) To estimate mass from average seismic parameters, one respecttotheSun. Thisassumptionisnotstrictlycorrectand has to adopt certain solar reference values. Currently, there canleadtodeparturesfromthescalingrelations. Toaccom- is no consensus on the choice of these with systematic dif- modatethesedepartureswehaveintroducedthefactors f ferences ranging from 1% to 2%. Hence, one needs to in- νmax and f . Thereisalsoconsiderableuncertaintyregardingthe vestigate whether the discrepancy between observations and ∆ν choiceofsolarreferencevaluesandthisleadstouncertainties predictions is less or greater than the current diversity in the in f and f , when we adopt a specific set of canonical solarreferencevalues. νmax ∆ν solarreferencevalues. Below,wediscussthisindetail. Inthispaperwerevisitthedwarf/subgiantanalysisofChap- It is clear from the scaling relations that to estimate mass lin et al. (2011a) and analyze each of the above mentioned and radius one has to adopt some solar reference values, factors. In Section 2, we discuss the observational data and ∆ν andν . Unfortunately,thereisnoconsensusonthe theGalacticmodels.Systematicsassociatedwiththedifferent max, cho(cid:12)ice of sola(cid:12)r reference values. Ideally, to estimate them datasetsandGalacticmodelsarediscussedhere. InSection we would require high quality data of the Sun in the Kepler 3,weanalyzetheasteroseismicinformationfordifferentdata bandpass,which,unfortunatelyisnotavailable. Thedatafor setsanddifferentGalacticmodelsanddiscusstheroleofthe theSunisavailableinotherbandpassesandthishasbeenan- selectionfunction. Wealsodoaquantitativestudyofthedif- alyzed. The results using the SOHO/VIRGO green channel ference between observed and predicted mass distributions. data (Frohlich et al. 1997) are shown in Table 1 for various Finally,inSection4wediscussandconcludeourfindings. methods. Whiletheestimatesof∆ν agree(differenceabout 0.2%),theestimatesofν differ(cid:12)significantly(difference 2. DATA,SCALINGRELATIONSANDGALACTICMODELS max, about2.5%)andsofarthisdi(cid:12)sagreementhasnotbeenexplic- 2.1. Observationaldata itlyexplained. Themostlikelycauseforthedifferencesisthe 3 ThemainGalacticstellar-population-synthesismodelused TABLE1 SOLARREFERENCEVALUESFORDIFFERENTMETHODSOFCOMPUTING inthispaperisfromtheGalaxia4 code(Sharmaetal.2011). AVERAGESEISMICPARAMETERS. ItusesaGalacticmodelthatisbasedontheBesanc¸onmodel byRobinetal.(2003)butwithsomemodifications. Galaxia Method ∆ν (µHz) νmax, (µHz) SYDa 135(cid:12).10 0.01 3090(cid:12) 3e has a 3D extinction scheme that is based on Schlegel et al. CANb 134.88±0.04 3120±5 (1998)dustmaps. Wealsoapplyalowlatitudecorrectionto CORc 134.90±0.1 3060±10 thedustmapsasinSharmaetal.(2014). Theisochronesused OCTd 135.03±0.1 3140±10 topredictthestellarpropertiesarefromthePadovadatabase aHuberetal.(2009±,2011,2013)± (Bertellietal.1994;Marigoetal.2008). Theuniquefeature bKallingeretal.(2010) of Galaxia is its novel star-spawning scheme which, unlike cMosseretal.(2012);Hekkeretal.(2013) previouscodes,doesnotdiscretizethespatialdimensionsinto dHekkeretal.(2013) multiplelinesofsight;insteaditgeneratesacontinuousthree- eHuberetal.(2011)reportanuncertaintyof30µHzforνmax, and10µHz dimensionaldistributionofstars. for∆ν . However,thisisforone30daysolartimeseriessubs(cid:12)etoutof111 analyz(cid:12)edbythemintotal. For comparison, we also used the TRILEGAL5 Galac- tic stellar-population-synthesis model (Girardi et al. 2005). TRILEAGLasacodeisveryflexibleandoffersmultipleop- tions to the user to change various aspects of the Galactic 3.0 KIC model,i.e.,IMF,SFR,agescaleheight,localsurfacedensity 2.5 Galaxia ofstarsandsoon. However,thereisadefaultversionofthe TRILEGAL 2.0 Galactic model that is advocated in the (Girardi et al. 2005) p1.5 paperandiscommonlyused. Weherewishtoinvestigatethe discrepanciesreportedbyChaplinetal.(2011a),whodonot 1.0 mention any specific changes to the default settings. So we 0.5 choosetousethedefaultmodelofTRILEGALforouranaly- 0.0 sis. 0.5 0.0 0.5 1.0 − Unlike Galaxia, TRILEGAL cannot generate stars over a J-Ks(mag) wide angular area, so we generated stars along 21 lines of FIG.2.—(a)J Kscolordistributionofstarswithr<14intheKIC(black) sight pointing towards the centers of the 21 Kepler CCD- comparedwithp−redictionsfromGalaxia(red)andTRILEGAL(green). (b) modules. We used the default settings but with binary stars aspanel(a),butforg rcolor. Theintegratedprobabilitydistributionsare turned off. To model the dust, we used the 3D extinction scaledtounity. − modelofGalaxia.OnenoticeabledifferencebetweenGalaxia andTRILEGAListhatGalaxiausesaconstantstar-formation rate, whilethedefaultsettinginTRILEGALusesatwo-step method-specificsystematicsassociatedwiththeestimationof star-formation rate, which is twice as large between 1-4 Gyr ν . This would argue for the use of “method-specific” so- max asatanyothertime. larvalues,meaningthevaluesreturnedfromsolardatawhen using the same method (pipeline) as used for the rest of the stellarsample. However, thereisnostrongevidencetoback 2.3.1. ComparingGalacticmodelswiththeKeplerInputCatalog uptheuseofmethod-specificvalues.Onthecontrary,Hekker Before comparing the Galactic models with the data, it is et al. (2013) found that for red giants, method-specific solar important to check that they reproduce the stellar photom- reference values introduce biases. In other words, method- etry in the KIC. This is because the KIC formed the basis specific systematics in νmax for the Sun are not necessarily fortheselectionoftheobservedsampleofstars. Hence,any representativeofthesystematicsforotherstars. significantmismatchbetweenthemodelsandtheKICwould Similar to giants, for dwarfs and subgiants it is not clear indicate a fundamental problem with the models, making it whetheroneshouldadoptmethodspecificsolarreferenceval- difficulttoperformmeaningfulcomparisonswiththeseismic ues. The Chaplin-14 sample adopted the SYD method (Hu- data. beretal.2009,2011)forcomputingtheseismicparameters. InSharmaetal.(2016),weanalyzedthe(J K )colordis- s The solar reference values corresponding to this method are tributionofamagnitudelimitedsample(r<1−4mag)ofstars ∆ν = 135.1 µHz, νmax, = 3090 µHz. For the Chaplin- in the Kepler field and below we summarize the results for 11(cid:12)sample, however, a di(cid:12)fferent method was used, namely the completeness of this paper. We generated two synthetic the OCT pipeline (Hekker et al. 2013) available at that time catalogsoftheMilkyWay,usingeachofGalaxiaandTRILE- whose method specific solar reference values were ∆ν = GAL.StarsfromthesyntheticcatalogsandfromtheKICthat 134.9 µHz and νmax, =3150 µHz. These were adopted(cid:12)by laywithin8degreesofthecenteroftheKeplerfieldandwith Chaplinetal.(2011a)(cid:12)forcomputingthestellarmassesintheir magnituder<14wereselectedforcomparison. IntheKIC, sample. As discussed earlier (Figure 1), there is no system- therepeatabilityofphotometryforstarsbrighterthanmagni- atic shift in seismic parameters between the Chaplin-11 and tude 14 is about 2%, so it is very likely to be complete for Chaplin-14datasets.Hence,νmax, =3090µHzisanequally r<14. WeshowtheresultingdistributionsofJ Ks colorin validchoiceforestimatingmasses.(cid:12)Forclaritythroughout,we Figure2. ThepredictionofGalaxiaisinexcelle−ntagreement adopt∆ν =135.1µHz,andνmax, =3090µHz. withthestarsfromtheKIC,butthatfromTRILEGALshows Basedo(cid:12)nthediscussioninthepr(cid:12)evioustwoparagraphs,we asignificantdifference. TRILEGALcorrectlyreproducesthe conclude that fνmax is uncertain by at least 2%, and we will region around the red peak of the color distribution, but not usethisfactlater. 4http://galaxia.sourceforge.net 2.3. Galacticmodels 5http://stev.oapd.inaf.it/cgi-bin/trilegal 4 0.35 12 (a) (b) 0.30 TRILEGAL 10 0.25 Galaxia 8 Galaxia-matchedtoTRILEGAL 0.20 6 0.15 4 0.10 0.05 2 0.00 0 0 2 4 6 8 10 12 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Age(Gyr) J-Ks 0.0018 2.5 0.0016 (c) (d) 0.0014 2.0 0.0012 1.5 0.0010 0.0008 1.0 0.0006 0.0004 0.5 0.0002 0.0000 0.0 7000 6500 6000 5500 5000 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Teff (K) M(M ) (cid:12) FIG.3.—ComparisonofstellarpropertiesofTRILEGAL(black)andGalaxia(blue). Shownaredistributionsofage(panela),color(panelb),temperature (panelc),andmass(paneld). ThecurvesinredshowthepredictionsofGalaxiawhenitsstarsarereweightedtomatchtheagedistributionoftheTRILEGAL stars.Inpanela,theredandblacklinesareontopofeachother. aroundthebluepeak. Specifically,itoverestimatesthenum- encesbetweenGalaxiaandTRILEGALaremainlyrelatedto ber of stars to the left of the blue peak and underestimates ageand/ormass,andnotduetodifferencesinisochrones. If the number of stars on the right side of the blue peak. Stars the difference were due to isochrones, then even after forc- leftwardofthebluepeakaretypicallyyoungmain-sequence ing a match on mass, age and metallicity, the two models stars,suggestingthatTRILEGALoverpredictsthenumberof would have shown differences in the color distribution. The youngstarsintheKeplerfield. two main factors that control the age distribution of a stellar Now, to better understand the above mentioned difference samplearethestar-formationrateandtheagescaleheightre- betweenTRILEGALandGalaxia,weselectedstarsfromboth lation.BothofthesefactorsaredifferentbetweenGalaxiaand modelssatisfying4000K<T <6600K,3.7<logg<4.2 TRILEGALandcouldberesponsibleforthemismatchinthe eff and r<12 mag (we call this selection criteria S ). This J K colordistribution. dwarf s − was designed to select dwarfs and subgiants, which are the To conclude, we find that TRILEGAL cannot fit the color main focus of this paper. The distributions of stellar prop- distributionofthestars,probablybecauseitpredictstoomany erties are shown in Figure 3. The same color difference as youngandmassivestars.Unlesswecanexplainthemismatch seeninFigure2canbeseenhere. Adifferenceinage, tem- in the color distribution by some other means (e.g., system- peratureandmassdistributionscanalsobeseen. Mass, age, atics in the isochrones), a color or temperature-limited sam- andmetallicityarethethreeintrinsicpropertiesofastarthat pleofdwarfsandsubgiantsselectedfromTRILEGALisex- largely define its observable properties. The metallicity dis- pectedtobebiasedtowardshighermasses. tributions (not shown here) did not show any significant dif- ference. Hence,thecolordifferenceismostlikelyduetodif- 3. ANALYSISOFASTEROSEISMICINFORMATION ferences in mass and age distributions. Next, we therefore TocomparethepredictionsoftheGalacticmodelwiththe investigate if the color, temperature, and mass distributions asteroseismic information from Kepler, we generated a new would match (between Galaxia and TRILEGAL) if we were syntheticpopulationofstarsusingbothGalaxiaandTRILE- toaltertheGalacticmodelofGalaxiainsuchawaythatthe GAL.Thesyntheticstarswerethenselectedtomatchthese- age distribution of stars obeying the subgiant and dwarf se- lectioncriteriaoftheobservedstars. Themainselectioncri- lectioncriteriamatchestheTRILEGALprediction. However teria was based on apparent magnitude and a lower limit on we cannot easily alter the model in Galaxia, instead we use ν . However, not alltargeted starsshowedoscillations. A max theideaofimportancesampling. Weassignaweighttoeach scheme to compute the detection probability was presented Galaxia star such that the weighted age distribution matches inChaplinetal.(2011b)andthiswasusedbyChaplinetal. thatofTRILEGAL.Usingtheseweightswethencomputethe (2011a)toselectstarsfromaGalacticmodel. Inthisscheme, weighted distriutions of other quantities like color, tempera- mass,radius,andeffectivetemperatureofeachsyntheticstar ture,andmass,andcomparethemwiththoseofTRILEGAL. wereusedtopredictitstotalmeanoscillationpowerandthe We found that the color, temperature and mass distributions granulationnoiseinthepowerspectrum.Theapparentmagni- of the reweighted Galaxia sample now matched the TRILE- tudewasusedtocomputetheinstrumentalnoiseinthepower GAL sample. Reweighting the Galaxia sample to match the spectrum which, combined with granulation noise, gave the massdistributionoftheTRILEGALsampleproducedsimilar total noise. The mean oscillation power and the total noise results. Thisisexpectedbecausemassandagearecorrelated were then used to derive the probability of detecting oscil- forthetypesofstarsthatweanalyze. Thisshowsthatdiffer- lations, p , with less than 1% possibility of false alarm. detect 5 1.0 ∆ν =135.1µHz Kepler-C14 ∆ν =135.1µHz Kepler-C14 2.0 νma(cid:12)x,(cid:12)=3090µHz Galaxia 1.0νma(cid:12)x,(cid:12)=3090µHz Galaxia 0.8 1.5 0.8 p p0.6 3.5 1.0 0.4 0.5 (a) 0.2 (b) 0.6 02..00 ∆νmνa(cid:12)x,(cid:12)==13350.910µµHHzz KTRepILleErG-CA1L4 01..00∆νmνa(cid:12)x,(cid:12)==13350.910µµHHzz KTRepILleErG-CA1L4 logg4.0 0.4pMin()detect 0.8 1.5 p p0.6 1.0 4.5 0.4 0.2 0.5 0.2 (c) (d) 0.0 0.0 2.0 ∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L4 1.0∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L4 7000 6500 T6eff00(0K) 5500 5000 0.0 0.8 1.5 FIG.6.—Minimumprobabilityofdetectingoscillationsasafunctionof p1.0 p00..46 lboagsegdaonndpTreeffd.icTtihoensmfarpomistfhoercstoadreswGiatlhaxaipap.aDreenttecmtiaognniptruodbeabri=lity11isacnodmis- putedusingtheschemeofChaplinetal.(2011b). Asharptransitionfrom 0.5 (e) 0.2 (f) hightolowdetectionprobabilitycanbeseen.Therectangularboxmarksthe 0.0 0.0 regionwherethesampleisapproximatelycomplete. 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5 M/M R/R the TRILEGAL model, which was used by Chaplin et al. (cid:12) (cid:12) (2011a). Here, the predicted mass distribution is shifted FIG.4.— MassandradiusdistributionsforChaplin-14sampleobserved slightly to the right. Figure 4e,f also show the predictions byKepler(black)andpredictedbyGalaxia(red)andTRILEGAL(green). of TRILEGAL alongside observed stars, but with masses (a-b): ResultusingGalaxiaandourchoiceofsolarreferencevaluesfor∆ν andνmax. (c-d): ResultusingTRILEGALandourchoiceofsolarreference of observed stars computed using the solar reference values values. (e-f): ResultusingTRILEGALandthesolarreferencevaluesused adopted by Chaplin et al. (2011a) in their analysis. As ex- byChaplinetal.(2011a)(ourreproductionoftheChaplin-11result). The pected, this result is the same as that presented by Chaplin integratedprobabilitydistributionsarescaledtounity. etal.(2011a),withasignificantshiftofthemassdistribution from TRILEGAL compared to the observed distribution. In Figure 5, we show the same analysis but using the Chaplin- ∆ν =135.1µHz Kepler-C11 ∆ν =135.1µHz Kepler-C11 2.0 νma(cid:12)x,(cid:12)=3090µHz Galaxia 1.0νma(cid:12)x,(cid:12)=3090µHz Galaxia 11sample,whosesizeisslightlysmallerthantheChaplin-14 0.8 sample. Asexpected,weseethesametrendsasseeninFig- 1.5 p p0.6 ure4. 1.0 Thevalueofν usedbyChaplinetal.(2011a)isabout 0.4 max, 2%higherthanthev(cid:12)alueusedbyus(Figure4a-d)andthisre- 0.5 0.2 (a) (b) duces the masses of the observed stars by about 8%, which 0.0 0.0 exacerbates the mass discrepancy between the TRILEGAL ∆ν =135.1µHz Kepler-C11 ∆ν =135.1µHz Kepler-C11 2.0 νma(cid:12)x,(cid:12)=3090µHz TRILEGAL 01..80νma(cid:12)x,(cid:12)=3090µHz TRILEGAL phraesdnicotisoingnainfidctahnetoebffseecrtv.atTioonsc.oTnchleudsme,atlhledimffeisrmenactechinin∆νth(cid:12)e, 1.5 stellar mass distribution found by Chaplin et al. (2011a) can p p0.6 1.0 0.4 bealleviatedif:(i)oneadoptsavalueofνmax, thatisslightly smallerand(ii)weuseGalaxia(withdefaults(cid:12)ettings)instead 0.5 (c) 0.2 (d) ofTRILEGAL(withdefaultsettings)astheGalacticmodel. 0.0 0.0 ThisdoesnotnecessarilymeanthattheGalaxiamodeliscor- 2.0 ∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L1 01..80∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L1 rleecctti.oTnhfeunmcatisosndaisntdribaubtiioansiinssmenasssitidvueetotothaencinhcooicrereocfttchheosicee- 1.5 oftheselectionfunctioncancanceloutabiasduetoanincor- p p0.6 1.0 rectgalacticmodel. Hence,inthenextsubsection,weinves- 0.4 tigatetheaccuracyoftheselectionfunction. 0.5 0.2 (e) (f) 0.0 0.0 3.1. Theroleoftheselectionfunction 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5 M/M R/R The accuracy of a selection function that is based on de- (cid:12) (cid:12) tectabilityofoscillationshingesuponourabilitytoaccurately FIG.5.—SameasFigure4,butfortheChaplin-11sample. predict the mean oscillation power. The mean oscillation power is computed from the maximum mode amplitude and Starswith p >0.9wereassumedtobedetectable. Here- forthisanempiricalrelationisused, whichispronetoinac- detect afterwerefertothisselectionfunctionasS0. curaciesandcanpotentiallybiastheselectionfunction. We applied the S0 selection function to the synthetic stars Wenowstudywhetheralteringtheassumptionsbehindthe and calculated the mass and radius distributions (Figure 4). selectionfunctionhaveanyeffectonthedistributionofstars Figures 4a,b show the predictions of Galaxia, which match in the (logg,T ) space. In Figure 6, we show p as a eff detect well with the observations. Figure 4c,d show predictions of function of logg and T for stars with r=11 mag, selected eff 6 ingtheassumednoiseintheselectionfunctionorincreasing 2.5 r would have a similar effect. On average, the stellar mass increaseasonemovesdiagonallyfromthelowerrighttothe 2.0 upperleft(Figure7). Hence, abiasin(logg,Teff)spacewill 3.5 alsoleadtoabiasinthemassesofthestars. In Figure 8, we show the distribution of the Kepler sam- 1.5 ple in (T ,logg) space alongside predictions from Galactic eff logg4.0 M(M)(cid:12) manoddeplrsedbiacsteedd doinstrtihbeutSio0nssemleacttciohnwfeulnlcitniotnh.e cTehnetraolbsreegrvioedn 1.0 butdifferencescanbeseenalongtheboundary. Bothmodels overpredictthenumberofstarsataround(6400,3.7). Addi- 4.5 tionally,TRILEGALpredictsfewerstarsaround(5600,4.1), 0.5 whileGalaxiapredictsmorecoolerstars(T<5000K).Atthe low-temperatureendforevolvedstarsweexpectthedetection 0.0 probability to be high. Hence, the differences seen here are 7000 6500 6000 5500 5000 most likely due to inaccuracies in the models. At high tem- Teff(K) perature, the detection probability is low and is sensitive to FIG.7.—MeanmassofastarasafunctionofloggandTeff,aspredictedby the assumptions made in the selection function. Here, stars Galaxia,forstarswithapparentmagnituder=11.Themeanmassincreases are close to the instability strip where convection zones are asonemovesdiagonallyfromthelowerrighttotheupperleft. thin, whichmakesitdifficulttomodelthemodedrivingand damping mechanisms. Hence, the differences seen here are 3.2 mostlikelyduetoinaccuraciesintherelationusedtopredict TRILEGAL modeamplitudes. 3.4 Kepler-C14 The fact that the observed and the predicted distributions 3.6 of stars do not match in (T ,logg) space means the mass eff distributions will also not match. To eliminate this bias, we 3.8 g creatednewselectionfunctionsbyresamplingthemodelstars g4.0 to satisfy the observed distribution of stars in (T ,logg,r) o eff l4.2 space. Thedisadvantageofsuchresamplingisthatwereduce our sensitivity to model-based differences, because the first 4.4 (a) order differences are already taken out. However, they are 4.6 stillusefultounderstandsystematicsrelatedtoasteroseismic 4.8 analysis. Belowaretwowaysfordevisingsuchnewselection 7500 7000 6500 6000 5500 5000 4500 functions. T eff S1: Hereweresamplethemodelstarstomatchthedis- 3.2 • tributionofobservedstarsin(T ,logg,r)space. This Galaxia eff 3.4 isdonebydividingthe(T ,logg,r)spaceintobinsand Kepler-C14 eff thenmakingsurethateachbinhasthesamenumberof 3.6 model and observed stars. Because the number of ob- 3.8 g servedstarsislow,onehastoadoptlargebinsandthis g4.0 canaffecttheaccuracyoftheselectionfunction. o l4.2 S2: Inthisweselectaboxin(T ,logg,r)spacewhere eff • 4.4 weexpectthedetectionprobabilitytobecloseto1and (b) wherethedistributionoftheobserveddatamatchesthat 4.6 of the Galactic model. Compared to the case S1, the 4.8 case S2 leads to fewer stars, but has a more accurate 7500 7000 6500 6000 5500 5000 4500 T selectionfunction. Theselectedbox, eff  1 if(5800<T <6600)& shFoIwGn.8a.s—blSuetarpsoiinnt(slowghgi,leTesffy)nsthpeatciec.sTtahresKweipthlerpddewtecatr>fsa0n.9dasruebgsihaonwtsnaares p(ST ,logg,r)= (3.8<logge<ff4.1)& (3) rdeedxpinoilnotgs.g,TThheeosbimseurlvaetdeddsattaarshwaseraencuonncveorltvaeindtywiothf1u5n0ceKrtaiinntTieeseftfyapnicdal0o.2f | eff  (r<11) thatintheobserveddata,150KinTeff and0.2dexinlogg. BothGalactic 0 otherwise, modelsoverpredictthenumberofstarsataround(6500,3.7). Inaddition TRILEGALunderpredictsthenumberofstarsataround(5700,4.1)while isshowninFigure6,anditcanbeseenthattheboxis Galaxiapredictsmorestarsatthelowtemperatureend. mainly inside the region of high detection probability (red). from Galaxia. Stars with high p are confined to the re- detect gion in (logg,T ) space shown in red. For the red region, In Figure 9, we compare the predicted mass distributions eff therightboundaryisbecausetherearenostarstotherightof with observations, for the three different selection functions, thatboundary. Theleftandlowerboundariesarebecausethe S0, S1 and S2. Compared to the observed distributions, the oscillationamplitudedecreaseswithincreasingloggandT . predicted distributions are shifted toward lower masses and eff Ifwelowertheassumedamplitudesintheselectionfunction, are slightly broader. For S0 the shift is minimal, but for S1 theredregionwouldshiftupwardsandtotheright. Increas- and S2 it is significant. We note that Figure 9a (S0) is the 7 available code ASFGrid6. ASFGrid uses the code MESA TABLE2 (v6950)(Paxtonetal.2011,2013,2015)forstellarevolution VAMLAUTECOHFESfνBmaExSFTOWRIWTHHITCHHETMHAESDSIESSTPRRIBEUDTICIOTNEDOBFYOBTHSEERGVAEDLAMCATSICSES andthecodeGYRE(Townsend&Teitler2013)forderiving MODELS.THEVALUESINSQUAREBRACKETAREFORTHECASEWHEN oscillation frequencies. The correction factors are computed THE f∆νCORRECTIONFACTORISAPPLIEDTOTHESYNTHETICSTARS. asafunctionofmetallicity, mass, andage, seeSharmaetal. Selection fνmaxGalaxia fνmaxTRILEGAL (2016) for further details. Applying these corrections makes Function the observed mass distribution narrower (blue lines in Fig- S0 1.006[1.004] 0.002 0.982[0.981] 0.002 ure 9). This is expected because for the stars that we study, S1 1.018[1.015]±0.002 1.002[1.001]±0.002 the mean mass of a star increases with temperature and the ± ± S2 1.019[1.011] 0.004 1.004[0.994] 0.004 correction factor decreases with temperature. The combined ± ± effectisthatforhigh-massstars,themassdecreases,andfor low-mass stars, the mass increases. This leads to narrowing 3.0 of the overall distribution. Computing corrections requires ∆ν=135.1µHz Kepler-C14 metallicity,andweadoptedthespectroscopicmetallicitiesre- 2.5 νmax, =3090µHz Kepler-C14,with∆νcorr ported by Buchhave & Latham (2015). Instead of applying 2.0 (cid:12) Galaxia thecorrectiontotheobservedstars,wecanalsoapplythere- p1.5 ciprocalcorrectiontothesyntheticstarswhosemetallicityis known exactly. Doing so had negligible effect on the values 1.0 quotedinTable2(seeresultsinsquarebrackets). S0 (a) 0.5 4. DISCUSSIONANDCONCLUSIONS 03..00 We have compared the asteroseismic properties of dwarfs ∆ν=135.1µHz Kepler-C14 2.5 and subgiants observed by Kepler against predictions of two 2.0 νmax,(cid:12)=3090µHz KGeaplalexria-C14,with∆νcorr population synthesis models of the Galaxy, TRILEGAL and Galaxia. The previous study by Chaplin et al. (2011a) us- p1.5 ingTRILEGALfoundthatstellarpopulationsynthesisbased 1.0 models overestimated the number of high-mass stars, which S1 (b) we are able to reproduce. We identified three potential fac- 0.5 torsthatcanshiftthemodelmassdistributionstowardhigher 03..00 masses relative to the observed masses. First, TRILEGAL 2.5 ∆ν=135.1µHz Kepler-C14 most likely overpredicts the number of young massive stars νmax, =3090µHz Kepler-C14,with∆νcorr as it fails to match the J Ks color distribution of stars in 2.0 (cid:12) Galaxia − 2MASS(Sharmaetal.2016). Second,wefoundthatachoice p1.5 ofν thatis2%lowerthanthatadoptedbyChaplinetal. max, (2011a)(cid:12),whichisequallyvalidgiventheuncertaintyintheac- 1.0 tualvalue,canincreasetheobservedmassesbyabout6%. Fi- S2 (c) 0.5 nally,wefoundthatifaselectionfunctionbasedonoscillation 0.0 amplitudesisused,theGalacticmodelscannotreproducethe 0.5 1.0 1.5 2.0 distributionoftheobservedsamplein(logg,T )space. This M/M eff mightbeduetoinaccuraciesinthemodel, butcouldalsobe (cid:12) duetoinaccuraciesintheassumedselectionfunction. Select- FIG.9.—Massdistributionofobservedstarsalongsidepredictionsfrom ingthesyntheticstarstosatisfythedistributionin(logg,Teff) Galaxia,usingthreedifferentselectionfunctions.Weusethesamesolarref- spaceremovedthisbiasbutshifted themodelmassdistribu- erencevaluesforallthreecases.Thebluelineisforthecasewheretheoreti- tiontolowermasses(Figure9b,c). callypredictedcorrectionsto∆νscalingrelationareapplied.Thecorrections werecomputedassumingsolarmetallicity(Z=0.019)foralltheobserved The bias due to the mismatch of the color distribution can stars. be corrected by using a model such as Galaxia, which does not show such a mismatch. The bias due to inaccuracies in theselectionfunctionbasedonoscillationamplitudescanbe same as Figure 4a. To quantify the shift between the mass reducedbyusingaselectionfunctionbasedonloggandT eff distributions,wedeterminedthevalueof f thatminimizes oftheobservedsample. Doingso,wefindthatthemassdis- νmax the Kolmogorov-Smirnov statistic between the observed and tributionofGalaxiaisshiftedtowardlowermassesandisalso predicted distributions. The uncertainty on the estimate was slightly broader compared to the observed distribution. A computed using bootstrapping. The results are shown in Ta- similar effect was also seen for the Kepler red giant sample ble2. Thetwoselectionfunctions(S1andS2)thatarebased Sharma et al. (2016), so the underlying cause might be the onobservableslikegravityandtemperature,givesimilarval- same. Applying corrections to the ∆ν scaling relation pre- uesfor f ,butthevalueisdifferentforS0. Itisclearthat dictedbystellarmodelsmakestheobservedmassdistribution νmax the selection function plays a crucial role and can bias f narrower than observed but does not change the mean. The νmax by 2%. The difference between TRILEGAL and Galaxia is disagreement in the mass distributions reported here, trans- alsoabout2%. latestoabout2%changeinν ,whichiscomparabletothe max Detailed theoretical modelling of oscillations shows that currentuncertaintyintheν scalingrelation. Infuture,we max therearedeparturesfromthe∆νscalingrelationthatdepend need to verify the scaling relations to better than 2% to put uponmetallicity,mass,andage. AsmentionedinSection2.2, betterconstraintsontheGalacticmodels. weaccommodatedthesedeparturesusingthecorrectionfac- tor f . We computed these corrections using the publicly 6http://www.physics.usyd.edu.au/k2gap/Asfgrid ∆ν 8 However, Galaxia failed to match the distribution of ob- Institute for Theoretical Physics (National Science Founda- servedstarsinloggandT space. Thisalsoneedstobein- tion Grant No. NSF PHY11-25915) for facilitating helpful eff vestigated in future. The mismatch at high T could be due discussionsofresultsinthispaper.WethankWilliamChaplin eff to inaccuracies in predicting oscillation amplitudes because forallowingustousehiscodeforcomputingtheprobability thedetectionprobabilityofastarinthisregionissensitiveto ofdetectingoscillations. itsassumedamplitude. However,themismatchatlowT is S.S. is funded through ARC DP grant 120104562 (PI eff mostlikelyduetoinaccuraciesinthemodel,becauseherewe Bland-Hawthorn)whichsupportstheHERMESproject. D.S. expect the detection probability to be close to 1. Parallaxes, isfundedthroughFutureFellowshipfromtheAustralianRe- andhenceluminosities,fromGaiawillhelpresolvethisissue search Council (ARC). J.B.H. is funded through Laureate becauseluminositycorrelateswithgravity. Fellowship from the Australian Research Council (ARC). D.H.acknowledgessupportbytheAustralianResearchCoun- cil’s Discovery Projects funding scheme (project number ACKNOWLEDGEMENTS DE140101364)andsupportbytheNationalAeronauticsand WeacknowledgethesupportofGalacticArchaeologyand Space Administration under Grant NNX14AB92G issued Precision Stellar Astrophysics program organized by Kavli throughtheKeplerParticipatingScientistProgram. REFERENCES Baglin,A.,&Fridlund,M.2006,inESASpecialPublication,Vol.1306, Huber,D.,Chaplin,W.J.,Christensen-Dalsgaard,J.,etal.2013,ApJ,767, ESASpecialPublication,ed.M.Fridlund,A.Baglin,J.Lochard,& 127 L.Conroy,11 Juric´,M.,Ivezic´,Zˇ.,Brooks,A.,etal.2008,ApJ,673,864 Bahcall,J.N.,&Soneira,R.M.1980a,ApJ,238,L17 Kallinger,T.,Mosser,B.,Hekker,S.,etal.2010,A&A,522,A1 —.1980b,ApJS,44,73 Kjeldsen,H.,&Bedding,T.R.1995,A&A,293,87 —.1984,ApJS,55,67 Kordopatis,G.,Gilmore,G.,Steinmetz,M.,etal.2013,AJ,146,134 Belkacem,K.,Goupil,M.J.,Dupret,M.A.,etal.2011,A&A,530,A142 Marigo,P.,Girardi,L.,Bressan,A.,etal.2008,A&A,482,883 Bertelli,G.,Bressan,A.,Chiosi,C.,Fagotto,F.,&Nasi,E.1994,A&AS, McMillan,P.J.,&Binney,J.2012,MNRAS,419,2251 106,275 Miglio,A.,Montalba´n,J.,Baudin,F.,etal.2009,A&A,503,L21 Binney,J.2010,MNRAS,401,2318 Miglio,A.,Chiappini,C.,Morel,T.,etal.2013,inEuropeanPhysical —.2012a,MNRAS,426,1324 JournalWebofConferences,Vol.43,EuropeanPhysicalJournalWebof —.2012b,MNRAS,426,1328 Conferences,3004 Binney,J.,&McMillan,P.2011,MNRAS,413,1889 Mosser,B.,Elsworth,Y.,Hekker,S.,etal.2012,A&A,537,A30 Borucki,W.J.,Koch,D.,Basri,G.,etal.2010,Science,327,977 Nordstro¨m,B.,Mayor,M.,Andersen,J.,etal.2004,A&A,418,989 Brown,T.M.,Gilliland,R.L.,Noyes,R.W.,&Ramsey,L.W.1991,ApJ, Paxton,B.,Bildsten,L.,Dotter,A.,etal.2011,ApJS,192,3 368,599 Paxton,B.,Cantiello,M.,Arras,P.,etal.2013,ApJS,208,4 Brown,T.M.,Latham,D.W.,Everett,M.E.,&Esquerdo,G.A.2011,AJ, Paxton,B.,Marchant,P.,Schwab,J.,etal.2015,ApJS,220,15 142,112 Robin,A.C.,Reyle´,C.,Derrie`re,S.,&Picaud,S.2003,A&A,409,523 Buchhave,L.A.,&Latham,D.W.2015,ApJ,808,187 Schlegel,D.J.,Finkbeiner,D.P.,&Davis,M.1998,ApJ,500,525 Chaplin,W.J.,Kjeldsen,H.,Christensen-Dalsgaard,J.,etal.2011a, Scho¨nrich,R.,&Binney,J.2009a,MNRAS,396,203 Science,332,213 —.2009b,MNRAS,399,1145 Chaplin,W.J.,Kjeldsen,H.,Bedding,T.R.,etal.2011b,ApJ,732,54 Sharma,S.,Bland-Hawthorn,J.,Johnston,K.V.,&Binney,J.2011,ApJ, Chaplin,W.J.,Basu,S.,Huber,D.,etal.2014,ApJS,210,1 730,3 Czekaj,M.A.,Robin,A.C.,Figueras,F.,Luri,X.,&Haywood,M.2014, Sharma,S.,Stello,D.,Bland-Hawthorn,J.,Huber,D.,&Bedding,T.R. A&A,564,A102 2016,ApJ,822,15 ESA.1997,VizieROnlineDataCatalog,1239,0 Sharma,S.,Bland-Hawthorn,J.,Binney,J.,etal.2014,ApJ,793,51 Frohlich,C.,Andersen,B.N.,Appourchaux,T.,etal.1997,Sol.Phys.,170, Skrutskie,M.F.,Cutri,R.M.,Stiening,R.,etal.2006,AJ,131,1163 1 Stello,D.,Chaplin,W.J.,Basu,S.,Elsworth,Y.,&Bedding,T.R.2009, Gilmore,G.,Randich,S.,Asplund,M.,etal.2012,TheMessenger,147,25 MNRAS,400,L80 Girardi,L.,Groenewegen,M.A.T.,Hatziminaoglou,E.,&daCosta,L. Townsend,R.H.D.,&Teitler,S.A.2013,MNRAS,435,3406 2005,A&A,436,895 White,T.R.,Bedding,T.R.,Stello,D.,etal.2011,ApJ,743,161 Hekker,S.,Elsworth,Y.,Mosser,B.,etal.2013,A&A,556,A59 Yanny,B.,Rockosi,C.,Newberg,H.J.,etal.2009,AJ,137,4377 Howell,S.B.,Sobeck,C.,Haas,M.,etal.2014,PASP,126,398 Zasowski,G.,Johnson,J.A.,Frinchaboy,P.M.,etal.2013,AJ,146,81 Huber,D.,Stello,D.,Bedding,T.R.,etal.2009,Communicationsin Asteroseismology,160,74 Huber,D.,Bedding,T.R.,Stello,D.,etal.2011,ApJ,743,143