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Astronomy&Astrophysicsmanuscriptno.masana February5,2008 (DOI:willbeinsertedbyhandlater) Effective temperature scale and bolometric corrections from ⋆ 2MASS photometry 6 0 E.Masana1,2,C.Jordi1,2,3,andI.Ribas3,4 0 2 1 Departamentd’AstronomiaiMeteorologia,UniversitatdeBarcelona,Avda.Diagonal647,08028Barcelona,Spain n 2 CERforAstrophysics,ParticlePhysicsandCosmology,associatedwithInstitutdeCie`nciesdel’Espai-CSIC a 3 Institutd’EstudisdeEspacialsdeCatalunya(IEEC).Edif.Nexus,C/GranCapita`2-4,08034Barcelona,Spain J 4 InstitutdeCie`nciesdel’Espai,CSIC,CampusUAB,Fac.Cie`ncies,TorreC5parell,2aplanta,08193Bellaterra,Spain 3 Received/Accepted 1 v 9 Abstract.Thispaperpresentsamethodtodetermineeffectivetemperatures,angularsemi-diametersandbolometriccorrections 4 forpopulationIandIIFGKtypestarsbasedonV and2MASSIRphotometry.Accuratecalibrationisaccomplishedbyusing 0 asampleofsolaranalogues,whoseaveragetemperatureisassumedtobeequaltothesolareffectivetemperatureof5777K. 1 Bytakingintoaccountallpossiblesourcesoferrorweestimateassociateduncertaintiesbetterthan1%ineffectivetemperature 0 and in the range 1.0–2.5% in angular semi-diameter for unreddened stars. Comparison of our new temperatures with other 6 determinationsextractedfromtheliteratureindicates,ingeneral,remarkablygoodagreement. Theseresultssuggest thatthe 0 effectivetemperaurescaleofFGKstarsiscurrentlyestablishedwithanaccuracybetterthan0.5%–1%.Theapplicationofthe / h methodtoasampleof10999dwarfsintheHipparcoscatalogueallowsustodefinetemperatureandbolometriccorrection(K p band)calibrationsasafunctionof(V K),[m/H]andlogg.BolometriccorrectionsintheVandKbandsasafunctionofT , - [m/H]andloggarealsogiven.Wepr−ovideeffectivetemperatures,angularsemi-diameters,radiiandbolometriccorrectionseiffn o theVandKbandsforthe10999FGKstarsinoursamplewiththecorrespondinguncertainties. r t s a Key words.stars:fundamental parameters–stars:late-type–stars:sub-dwarfs–infrared:stars–techniques: photometric– : methods:analytical v i X r 1. Introduction rectmethodsaremainlybasedontheuseofphotometry,spec- a troscopy,or a combinationof both.In the case of thetemper- Effectivetemperatureandluminosityaretwofundamentalstel- atures, although many of the published calibrations claim to lar parametersthat are crucial to carry out tests of theoretical haveuncertaintiesof theorderof severaltensof degrees,val- models of stellar structure and evolution by comparing them uesobtainedbydifferentauthorscaneasilyhavediscrepancies with observations.Theaccuracyin the determinationofother of 100 K or even larger. The reason for such nagging differ- stellar properties,suchasmetallicity,ageorradius,hingeson encesmustbefoundsomewhereintheingredientsofthemeth- ourabilitytoestimatetheeffectivetemperaturesandluminosi- ods: atmosphere models, absolute flux calibrations, oscillator ties. strengths,calibrationstars,etc. There are several approaches in the literature to compute Inthispaperwepresentasemi-empiricalmethodtodeter- effective temperature and/or luminosity.Except when applied mine effective temperatures (T ) and bolometric corrections eff to the Sun, very few of them are direct methods that permit (BC)from2MASS1 JHK2 photometry(Cutrietal.2003)that anempiricalmeasurementoftheseparameters.Usually,semi- is applicable to FGK type stars. As all others, our method is empirical or indirect methods are based to a certain extent susceptible to problems derived from the uncertainties in the on stellar atmosphere models. Among the direct approaches ingredientsmentionedabove.However,ourapproachbenefits we find the remarkable work by Code et al. (1976), which is from two major features: First, it provides a way to evaluate basedoninterferometricmeasurementsofstellarangularsemi- realistic individual uncertainties in T , θ and luminosity by eff diameters(θ)andtotalfluxes(Fbol)atEarth,andthemorere- considering all the involved errors; and second, as it is cali- cent works of Mozurkewich et al. (2003) and Kervella et al. bratedtousethe2MASSphotometry,itallowsthecalculation (2004),alsobasedoninterferometry.Ontheotherhand,indi- ofconsistentandhomogeneousT andBCforseveralmillion eff stars in the 2MASS catalogue. This paper also provides T , eff ⋆ Table 3 is only available in electronic form at the CDS via anonymousftptocdsarc.u-strasbg.fr (130.79.128.5)orvia 1 http://www.ipac.caltech.edu/2mass http://cdsweb.u-strasbg.fr/cgi-bin/qcat?]/A+A/ 2 Throughoutthepaper,KreferstoK band. s 2 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections angular semi-diameters, radii and BCs for 10999 dwarfs and However,theIRfluxbecomesverysensitivetometallicityand subdwarfsintheHipparcoscatalogueESA(1997).Suchlarge surfacegravityforstarshotterthan8000Ksothatsmalluncer- samplehasallowedusto constructsimpleparametriccalibra- taintiesintheseparameterstranslateintolargeuncertaintiesin tions as a function of (V K) , [m/H] and logg. Note that a theeffectivetemperature.InsuchsituationtheSEDFapproach 0 − preliminaryversionofthemethodpresentedherewasalready becomes inadequate. At the cold end, the accuracy of stellar successfullyappliedtothecharacterizationofthepropertiesof atmospheremodelslimitstheuseofthemethodtostarshotter planet-hostingstars(Ribasetal.2003). than4000K(molecularopacityplaysanimportantrolebelow The present paper is organized as follows. Section 2 thistemperature).Theselimitationsrestricttheapplicabilityof presentsthemethodandexplainsindetailtheproceduretoob- the SEDF to FGK type stars. Fortunately,these stars are very tain T and angular semi-diameters, including the fitting al- common in the Galaxy and dominate the content of most of eff gorithm,zeropointcorrectionsanderrorestimates. Thecom- surveycatalogues.Theyarecrucialforseveralkeyastrophys- parisonofourtemperatureswithseveralpreviousworks,both icaltopics, suchasthe studyofthe structureandevolutionof based on photometric and spectroscopic techniques, is de- theGalaxy,boththediskandthehalo,andthecharacterization scribed in Sect. 3. In Sect. 4 we present simple parametric ofplanet-hostingstars,amongothers. calibrations of T and BC as a function of (V K) , [m/H] eff 0 − and logg valid for dwarf and subdwarf stars. The sample of 2.1.Calculationofsyntheticphotometry 10999 stars used to build the calibrations is also described in thissectiontogetherwithadetailedexplanationofthedifferent The calculation of the synthetic photometry requires a well- contributors to the final uncertainties. Finally, the results are characterized photometric system, an accurate flux calibra- discussed in Sect. 5 and the conclusions of the present work tion and suitable synthetic spectra. The work by Cohen et al. arepresentedinSect.6. (2003a,b)providesconsistentabsolutefluxcalibrationsinboth thevisible(V)(Landoltsystem)andIR(2MASSJHK)bands. The calibration given by Cohen et al. is computed from a set 2. TheSpectralEnergyDistributionFit(SEDF) of calibrated templates, using the synthetic Kurucz spectrum method of Vegaof Cohenetal. (1992).Inthe case of theIR photom- Theuseofinfrared(IR)photometrytodetermineeffectivetem- etry, they consider the transmission of the camera and filters, peratureswasinitiallyproposedbyBlackwell&Shallis(1977). the detector properties and the Earth’s atmosphere character- Theirso-calledInfraredFluxMethod(IRFM)usestheratiobe- istics. From the comparison between observed and synthetic tween the bolometric flux of the star and the monochromatic photometryfora set of 9 A-typestars and 24 coolgiants,the flux at a given infrared wavelength, both measured at Earth, authorsinfertheneedtointroduceazeropointoffsetinthesyn- as the observable quantity. This ratio is then compared with theticphotometrytomatchtheobserved2MASSphotometry: a theoreticalestimate derivedfromstellar atmospheremodels 0.001 0.005 mag (J); 0.019 0.007 mag (H); 0.017 0.005 ± − ± ± tocarryoutthedeterminationoftheeffectivetemperature.The mag(K).Thecalculationofsuchvaluesisnotexemptofsome IRFMhasbeenwidelyusedbyanumberofauthors,beingmost difficultysincethedispersionsofthedifferencesbetweenboth noteworthytheworkbyAlonsoetal.(1995,1996a,b). photometries(syntheticandobserved)areofthesamemagni- The Spectral Energy Distribution Fit (SEDF) method that tudeasthezeropointoffsetsthemselves. we propose here follows a somewhat different approach, To computethe syntheric magnitudeswe made use of the namelythefitofthestellarspectralenergydistributionfromthe no-overshoot Kurucz atmosphere models grid (Kurucz 1979) optical(V) to the IR(JHK) usingsynthetic photometrycom- takenfromhttp://kurucz.harvard.edu/grids.html: putedfromstellaratmospheremodels.UnliketheAlonsoetal. Fi (1996a)implementationoftheIRFM,whichaveragestemper- mi (T ,logg,[m/H])=2.5log cal (1) syn eff F(T ,logg,[m/H]) aturesderivedindividuallyforeachIRband,ourmethodtakes i eff into account the four bands simultaneously (and naturally). whereFi istheabsolutefluxcalibrationgivenbyCohenetal. cal In addition, and also unlike the IRFM, the bolometric flux is (2003b)(formi = 0) and F(T ,g,[m/H])is theflux inthe cal i eff not required a priori by the SEDF method but results self- ibandcomputedfromtheintegrationofthemodelatmosphere consistently with the temperature. The fitting algorithm (see convolved with the transmission function (filter, detector and Sect.2.2)minimizesthedifferencebetweenobservedandsyn- Earth’satmosphere)fromCohenetal.(2003b): theticphotometrybytuningthevaluesoftheeffectivetemper- ature andthe angularsemi-diameter.The BC can be obtained F(T ,logg,[m/H])= ∞φ(T ,logg,[m/H],λ) (λ)dλ(2) i eff eff i Z T fromthesetwoparameters,andthen,whenthedistancetothe 0 starisknown,theluminosityiscomputedfromtheBCandthe where φ(Teff,logg,[M/H],λ) is the flux given by the stellar absolute magnitude in a given photometric band. The uncer- atmospheremodeland (λ)theeffectivetransmissionfunction i T taintiesofthederivedparameters(T ,angularsemi-diameter intheibandnormalizedtoapeakvalueofunity. eff andBC)areestimatedfromtheerrorsintheobservedandsyn- thetic photometryas well as in the assumed[m/H],logg and 2.2.Fittingalgorithm A . V From a theoreticalpointof view, the SEDF methodcould The fitting algorithm is based on the minimization of the χ2 be applied to stars of any spectral type and luminosity class. function defined from the differences between observed (cor- E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 3 rected forinterstellar extinction)and synthetic VJHK magni- theyare ofno use to calibratethe SEDFmethod.As analter- tudes,weightedwiththecorrespondingerror: native, we have used the list of photometric solar analogues compiled by CayreldeStrobel (1996).We assume that, as an V A V 2 J A J 2 χ2 = − V − syn + − J− syn + ensemble, the average of the effective temperatures of these σV ! σJ ! photometric solar analogues should be equal to the effective H A H 2 K A K 2 temperatureoftheSun(i.e.,5777K). H syn K syn + − − + − − (3) After selecting a subsample of 50 unreddened stars σ ! σ ! H K with non-saturated 2MASS photometry from table 1 of This function depends (via the synthetic photometry)on T , eff CayreldeStrobel(1996),we computedtheirtemperaturesus- logg,[m/H]andamagnitudedifference ,whichistheratio ingtheSEDFmethod.Weobtainedanaveragetemperatureof A betweenthesynthetic(star’ssurface)andtheobservedflux(at 5832 14 K, i.e., 55 K (or 1%) higherthan the solar effec- Earth)( = 2.5logF /F ). isdirectlyrelatedtothe ± ∼ star Earth tivetemperature.Exactlythesamevalueisobtainedifweuse A − A angularsemi-diameterbythefollowingexpression: the subset of solar “effectivetemperatureanalogues”fromta- θ=10 0.2 (4) ble 5 of CayreldeStrobel. Withouta profoundanalysis of all − A theingredientsinvolved–fromthestellaratmospheremodelto Althoughthesyntheticphotometrydependsimplicitlyongrav- the absoluteflux calibration,– itis verydifficultto assess the ityandmetallicity,inpractice,thespectralenergydistribution reasons for such difference. However, it seems clear that the in the optical/IRfor ourrangeof temperaturesis onlyweakly temperature scale as obtained from the synthetic photometry dependent on these quantities. This fact makes it possible to aloneneedsacorrectiontoagreewiththeaverageofthesolar obtainaccuratetemperaturesevenforstarswithpoordetermi- analogues.Fromaformalpointofview,thiscorrectioncanbe nationsofloggand[m/H]. computedfromthesyntheticphotometrythatresultsfromforc- As canbeseen,the χ2 functiondependsalsoonthe inter- ingavalueofT =5777Ktotheentiresample.Afterdoingso, eff stellarabsorptionAV (theabsorptionintheotherbandscanbe wereplacedthezeropointsgivenbyCohenetal.(2003b)(see computedusingtheextinctionlawofSchaifers&Voigt(1982): Sect.2.1)bytheaveragedifference(foreachband)betweenthe AJ = 0.30AV, AH = 0.24AV and AK = 0.15AV). In principle, observedandsyntheticphotometrycomputedforthesolarana- itispossibletoconsiderAV asafreeparameter.However,the logues.AssumingthatthereisnooffsetontheV band,theoff- strongcorrelationbetweenTeffandAV,especiallyforthehotter setsfortheotherbandsare:0.027 0.003mag(J);0.075 0.005 ± ± stars, decreasesthe precisionin the determinationof both pa- mag (H); 0.022 0.005 mag (K). It is interesting to note that ± rameters,withresultingtypicaluncertaintiesof4%inTeff and both in the case of Cohen et al. (2003a) and in our case, the 0.25maginAV.Thus,forbestperformance,AV shouldonlybe valueoftheoffsetinthe H banddifferssignificantlyfromthe consideredasafreeparameterwhenitsvalueissuspectedlarge offsetsin J andK.Itshouldbestressedthattheeffectivetem- andnoothermethodforitsestimationisavailable.Ingeneral, peraturesgivenbyCayreldeStrobel(1996)havenotbeenused thebestapproachis tofix thevalueof AV inEq.(3)fromthe here.Wehaveonlyusedthepropertyofthestarsinbeingclas- estimationbyphotometriccalibrations,forinstance. sifiedassolaranalogues,and,consequently,weassumedtheir Therefore,theonlytwoadjustableparametersbytheSEDF averagetemperaturetobeequaltothesolareffectivetempera- method in the present work are Teff and , whereas logg, ture. A [m/H] and AV are fixed parameters. To minimize Eq. (3) Inourprocedure,we are implicitlyassumingthatthe cor- with respect to these two parameters we use the Levenberg- rectionin our temperaturescale is just a zero pointoffsetand Marquardtalgorithm(Press etal. 1992),which isdesignedto that no dependence on temperature or metallicity is present. fita setofdatato anon-linearmodel.Inallourtests, conver- These assumptionsare justified a posteriori in Sect. 3, where gence towards the minimum value of χ2 was reached rapidly several comparisonsof SEDF temperatureswith other photo- andunequivocally. metricandspectroscopicdeterminationsareshown. The angular semi-diameters computed from Eq. (4) were used to check the consistency of the new zero points in our 2.3.CalibrationoftheSEDFmethodusingsolar temperature scale. These angular semi-diameters were com- analogues pared with the direct values compiled in the CHARM2 cata- The standard procedure for the calibration of an indirect logue(Richichi&Percheron2005).Werestrictedthecompar- method to determine effective temperatures is based on the isontostarswith accurateVLBIorindirect(spectrophotome- comparisonoftheresultswith accuratetemperaturesfromdi- try)measurementsofthesemi-diameter.Only10ofthesestars rectmethodsforasetofstars.Inthisway,thelistofstarswith fulfill the conditions for applicability of the SEDF method. empirical effective temperatures and angular semi-diameters Figure1showsthecomparisonofthesemi-diametersforthese from Code et al. (1976) has been widely used for calibration 10 stars. The agreement is excellent, with an average differ- purposes. This list has been recently increased with the the ence (θ θ ), weighted with the inverse of the square dir SEDF − worksofMozurkewichetal.(2003)andKervellaetal.(2004). of the error, equal to 0.3% with a s.d. of 4.6% (see Table − Otherauthorsuse well-studiedstars,suchastheSun,Vegaor 1). All the direct values used in the comparison correspond Arcturus,tocalibratetheirmethods. to an uniform stellar disk. A crude comparison of both uni- Unfortunately, the few stars with empirical values of form disk and limb darkened values for about 1600 F, G and T are too bright to have accurate 2MASS photometry and K stars in the CHARM2 catalogue indicates a 4% positive eff ∼ 4 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 2.4.Errorestimation One ofthe featuresofthe SEDFmethodisthatityieldsindi- vidualuncertaintiesofbothT andθ.Thetotaluncertaintycan 0.5 eff be calculated by combining the contributions from the spec- tral energy distribution fit and also from the uncertainties in thefixedparameters,i.e.,logg,[m/H],andA .Assumingnull V 0.4 correlationbetweenthesedifferent(and,inprinciple,indepen- M2 (mas) (dTeenfft)osrour)ceisstohfeeqrruoard,rtahteictostuamluonfcethrteaidnitffyeorefnttheerYrokrpcaoranmtriebtuer- R A A tions: H meter C0.3 (∆Yk)2 = (∆YkSEDF)2+ ∂[∂mY/kH]!2(σ[m/H])2 a di mi ∂Y 2 ∂Y 2 Se0.2 + ∂logkg! (σlogg)2+ ∂Ak ! (σAV)2 V (5) Asmentionedabove,theminimizationofχ2iscarriedoutwith HD 209458 0.1 respecttothefourmagnitudesVJHK,withtheparametersA , 0.1 0.2 0.3 0.4 0.5 V Semidiameter (SEDF) (mas) [m/H]andloggbeingheldfixed.Thefirsttermoftheequation (∆Y SEDF)istheerrorin theparameterY comingfromthe fit k k tothespectralenergydistribution,whichiscomputedfromthe covariance matrix (∆Y SEDF √C ). The derivatives in the k kk ≡ Fig.1. Comparison of angular semi-diameters computed from the otherthreetermsoftheequationaredeterminednumerically: SEDFmethod(withthenewzeropointinthetemperaturescale)and ∂Y Y (+∆[m/H]) Y ( ∆[m/H]) fromtheCHARM2catalogue.InthecaseofHD209458,thecompar- k k − k − (6) ∂[m/H] ≈ 2 isonofthesemi-diameterisbetweentheSEDFmethodandanempir- icaldeterminationfromahigh-precisiontransitlightcurve. inthecaseofthemetallicity,andinananalogouswayforthe surfacegravityandtheinterstellarabsorption. TheerrorinT isobtaineddirectlyfromEq.(5),whereas eff Table1.ComparisonofdirectandSEDFangular semi-diameters. theerrorinθmustbecalculatedfromtheerrorinthe param- θdirareVLBIandspectrophotometricvalues(Richichi&Percheron eter: A 2005).ForHD209458,θ isderivedfromaplanetarytransitlight dir σ =0.2 ln10θσ (7) curve. θ A The adopted values for the errors in the magnitudes (σ ), Star θ (mas) θ (mas) ∆θ(%) mi dir SEDF metallicity (σ ) and surface gravity (σ ) are discussed [M/H] logg HIP6702 0.190 0.050 0.189 0.004 0.5 ± ± inSect.4.1.1. HIP8433 0.225 0.050 0.286 0.005 27.1 ± ± − HIP48113 0.405 0.005 0.402 0.006 0.7 HIP50786 0.260±0.010 0.265±0.004 1.9 3. Comparisonwithothermethods ± ± − HIP51056 0.290 0.030 0.306 0.005 5.5 ± ± − Five samples of FGK stars with accurate determinations of HIP85365 0.380 0.015 0.410 0.036 7.9 ± ± − effective temperatures (both photometric and spectroscopic) HIP91237 0.210 0.015 0.251 0.004 19.5 ± ± − wereselectedfromtheliterature(Alonsoetal.1996a,Ram´ırez HIP96895 0.270 0.015 0.278 0.004 3.0 HIP96901 0.260±0.015 0.255±0.003 −1.9 & Mele´ndez 2005, Fuhrmann 1998, Santos et al. 2003 and HIP113357 0.365±0.010 0.352±0.006 3.6 Edvardsson et al. 1993) to carry out a comparison with our ± ± HD209458 0.113 0.010 0.115 0.002 1.8 results. We put special attention in correcting for the effects ± ± − of interstellar reddening, which could lead to systematic dif- ferences.FortheAlonsoetal.andRam´ırez&Mele´ndezsam- ples(the mostreddened),interstellarreddeningwas corrected using the values of E(B V) given by the authors so that − the two temperature estimations would be directly compara- ble. For the other three samples, composed of stars at closer correctionforlimbdarkening,ofthesameorderofthedisper- distances, we restricted our comparisonsto unreddenedstars. sion of the relative differencesshown in Table 1. In addition, Anyhow,thismeantrejectingveryfewstarsfromfurtheranal- we compared the radius of HD 209458 – obtained with the ysis. Among several papers in the literature, we chose these HubbleSpaceTelescopefromahighprecisionplanetarytran- five samples because they have a minimum of 25 stars with sitlightcurve(Brownetal.2001)–withourestimationfrom non-saturated2MASSphotometryandthevaluesof[m/H]and SEDF, obtainingvery goodagreement:1.146 0.050R and logg – needed for a consistent comparison– are providedby ± ⊙ 1.160 0.058R ,respectively. theauthors. ± ⊙ E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 5 3.1.MethodsbasedonIRphotometry:Alonsoetal. (1996a)andRam´ırez&Mele´ndez(2005) 8000 Asmentionedabove,theIRFMisthemostpopularmethodto computeeffectivetemperaturesfromIRphotometry.Thework 7000 by Alonso et al. (1996a) is undoubtedly the widest applica- K) tionoftheIRFMtoFGKstars.Theauthorscomputedeffective M) ( temperaturesfor 462starswith knowninterstellarabsorption, RF 6000 surfacegravityandmetallicity.Afterselecting thestars inthe T(Ieff Alonso et al. sample in the range 4000 < Teff < 8000 K and 5000 witherrorsinthe2MASSphotometrybelow0.05mag,weob- tainedeffectivetemperaturesfromtheSEDFmethodforasub- 4000 setof315stars.Thecomparisonbetweenbothdeterminations ofT isshowninFig.2.Theaveragedifference∆T (IRFM 300 eff eff K) −SEDF)wasfoundtobe−67K,withastandarddeviationof EDF( 150 81K.Thedependenceofthisdifferenceonthetemperatureis STeff 0 nbootttsoimgnpifiacnaenlto:fTFeIRffigF.M2=,th1e.0re30isTneSoffEDdFe−pe2n3d9enKc.eAosfsthheowtenmipnetrhae- RMF - -150 turedifferencewithmetallicity. ITeff-300 4000 5000 6000 7000 8000 In a recent work, Ram´ırez & Mele´ndez (2005) have re- T (SEDF) (K) eff computed the IRFM temperatures of almost all the stars in Alonso et al. (1996a) using updated input data. According to K) 300 the authors, the difference between the old and new temper- EDF ( 150 ature scales is not significant. They also compare their effec- ST eff 0 tivetemperatureswithsomedirectdeterminations.Theauthors MF - -150 csoolnacrlutedmeptheraatttuhreer(einisthaesysesntesmeaIRticFMdifferenc–ethoefirabvoaulute4s0).KThaet IRT eff-300 Alonso -4 -3 -2 -1 0 comparisonbetweenthetemperaturesofRam´ırez&Mele´ndez [m/H] andourdeterminationsisshowninFig.3.For385starsincom- Fig.2.ComparisonoftheeffectivetemperaturesfromtheIRFMand mon we find ∆Teff (IRFM −SEDF) equal to −58 K (σTeff=67 theSEDFmethodfor315starsinthesampleofAlonsoetal.(1996a). K), and TIRFM = 1.061 TSEDF 403 K. Unlike in the case Thebottom panel shows the temperature difference asafunction of eff eff − ofAlonso etal. (1996a),thedependenceof∆Teff in [m/H]is themetallicity. relevant (Fig. 3, bottom panel). For [m/H] < 2.0 the tem- − peratures from Ram´ırez & Mele´ndez are clearly hotter than ourtemperatures.ThesametrendwasfoundbyCharbonnel& Primas (2005)when comparingtheir temperaturesof 32 halo dwarfs( 3.5< [Fe/H]< 1.0)withthevaluesofRam´ırez& − − Mele´ndez. halo. Effective temperatures were determined from fits to the wings of the Balmer lines. Of those stars, 24 have accurate 3.2.Othermethods 2MASSphotometrysothatreliableSEDFtemperaturescanbe derived. The comparison is shown in Fig. 4. The mean aver- Besides the IRFM, which uses IR photometry, we have also age difference ∆T (Fuhrmann SEDF) is 12 K, (σ =45 comparedtheeffectivetemperaturesobtainedusingtheSEDF K), with a slight deeffpendenceon−the temperature:TFuhrTmeaffnn = eff method with other determinations. Two of these (Fuhrmann 0.895TSEDF+618K.Nodependencewasfoundbetween∆T eff eff 1998 and Santos et al. 2003) are spectroscopic works, while and[m/H](Fig.4,bottompanel). in another case (Edvardssonet al. 1993) the temperaturesare basedonuvby βphotometry.Thechiefprobleminthecaseof − spectroscopicdeterminationsisthat,ingeneral,theyaremostly 3.2.2. Santosetal.(2003) applied to bright stars, which have poor 2MASS photometry (the 2MASS detectors saturate for stars brighter than K 4 Tostudythecorrelationbetweenthemetallicityandtheprob- ≈ mag). This fact reduces the number of stars in the Fuhrmann ability of a star to hosta planet,Santoset al. (2003)obtained (1998)andSantosetal.(2003)samplesthatcanbecompared spectroscopic temperatures for 139 stars based on the analy- withSEDFmethod. sis of several iron lines. Effective temperatures for a total of 101 stars in the sample of Santos et al. can be obtained us- ing the SEDF method.In this case, ∆T (Santos SEDF) is 3.2.1. Fuhrmann(1998) eff − 28 K, with σ = 68 K, and practically independent of the Teff This sample is composed of about 50 F and G nearby stars, temperature:TSantos = 1.053TSEDF 270K(Fig.5).Thereis eff eff − both main sequence and subgiants, of the Galactic disk and nodependenceof∆T with[m/H](Fig.5,bottompanel). eff 6 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 6500 8000 6250 7000 6000 K) K) MF) ( 6000 ec.) ( 5750 T(IReff T(Speff 5500 5000 5250 4000 SEDFT (K) eff 1350000 SEDFT(K)eff 50100000 IRMF - T eff--310500 Spec.T - eff --210000 4000 5000 6000 7000 8000 5000 5250 5500 5750 6000 6250 6500 T (SEDF) (K) T (SEDF) (K) eff eff K) 300 K) 200 SEDF T( eff 1500 SEDF T( eff 1000 IRMF - T eff--310500 Spec. T - eff--210000 -4 -3 -2 -1 0 1 -2.5 -2 -1.5 -1 -0.5 0 0.5 [m/H] [m/H] Fig.3.ComparisonoftheeffectivetemperaturesfromtheIRFMand Fig.4.Comparisonof theeffectivetemperaturesfromfitstoBalmer theSEDFmethodfor386starsinthesampleofRam´ırez&Mele´ndez linesandtheSEDFmethodfor24starsincommonwiththesampleof (2005).Thebottompanelshowsthetemperaturedifferenceasafunc- Fuhrmann(1998).Thebottompanelshowsthetemperaturedifference tionofthemetallicity. asafunctionofthemetallicity. 3.2.3. Edvardssonetal.(1993) tobepreferredoverthebestpossibleaccuracy.Inthissection ThesampleofEdvardssonetal.iscomposedby189nearbyF we present calibrations for both Teff and BC as a function of andGtypestars.Incontrastwiththeprevioustwo,theeffective (V K)0, [m/H]and logg. To calculate the calibrations, the − temperatureisnotderivedfromspectroscopybutfromuvby β SEDFmethodwasappliedtoasampleofstarsintheHipparcos − photometry. To do so, the authors built a grid of synthetic catalogue,asdescribedbelow.Notethatthesecalibrationsare photometryusing the atmospheremodels of Gustafsson et al. subjecttotwolimitationswithrespecttothefullSEDFmethod: (1975) and further improvedit by adding several new atomic First, theyare simplificationssince notallthe available infor- and molecular lines. Effective temperatures for 115 stars in mationisused,andsecond,individualuncertaintiescannotbe their sample could be derived using the SEDF method. The determined. average difference ∆T (Edvardsson SEDF) is 10 K, with eff − a dispersion of 70 K and no dependence on the temperature: 4.1.Thestellarsample TEdvardsson = 1.006TSEDF 27K(Fig.6).Asinthetwoprevi- eff eff − ouscases,thebottompanelofFig.6showsthatthetemperature We collected a sample of FGK dwarfs and subdwarfs in the differenceisnotcorrelatedwith[m/H]. Hipparcos catalogue, and therefore with measured trigono- metric parallaxes. Their V magnitudescome mainly from the Hauck & Mermilliod (1998) catalogue, except for those stars 4. Parametriccalibrations withlessthantwoobservations,whereweusedtheHipparcos ThepracticaluseoftheSEDFmethodasithasbeendescribed catalogue. The entire sample has complete and non-saturated in Sect. 2 is not straightforward since it requires the calcula- JHK photometry in the 2MASS catalogue. The metallicity tion of synthetic photometry from stellar atmosphere models wasextractedfromthecompilationofCayreldeStrobeletal. and then use a numerical algorithm to minimize the χ2 func- (2001) or computed from uvby β photometry– either mea- − tion.Parametriccalibrations(asafunctionofoneormorepa- sured fromour own observationsor obtainedfrom the Hauck rameters) may offer a suitable means to estimate reliable ef- &Mermilliod(1998)catalogue–,usingaslightlyrevisedver- fective temperatures in cases where simplicity and speed are sionoftheSchuster&Nissen(1989)calibration.Therangeof E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 7 6500 7000 6000 6500 K) K) pec.) ( 5500 hot.) ( S P T(eff T(eff 6000 5000 5500 K)4500 K) 300 DF ( 150 DF ( 150 SETeff 0 SET eff 0 Spec.- Teff --310500 Phot.T - eff--310500 4500 5000 5500 6000 6500 5500 6000 6500 7000 T (SEDF) (K) T (SEDF) (K) eff eff K) 300 K) 200 SEDF T( eff 1500 SEDF T( eff 1000 Spec. - T eff--310500 Phot. T - eff--210000 -1 -0.5 0 0.5 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 [m/H] [m/H] Fig.5.Comparisonoftheeffectivetemperaturesfromironlinefitsand Fig.6. Comparison of the effective temperatures from photometry theSEDFmethodfor101starsincommonwiththesampleofSantos and the SEDF method for 115 stars in common with the sample of etal.(2003).Thebottompanelshowsthetemperaturedifferenceasa Edvardsson et al. (1993). The bottom panel shows the temperature functionofthemetallicity. differenceasafunctionofthemetallicity.Thestandarddeviationfor asinglestarinEdvardssonetal.(1993)is81K. metallicities covered by the sample is 3.0 . [m/H] . 0.5. − Values of logg were computed from uvby β photometry a few measurements(and with no evaluationof systemat- − (Masana 1994; Jordi et al. 1996). Originally, the sample was ics),wehavesetaminimumerrorinV equalto0.015mag. builttostudythestructureandkinematicsofthediskandhalo Theuncertaintiesintheabsolutefluxcalibrationaregiven oftheGalaxy(Masana(2004))andafulldescriptionincluding by Cohen et al. (2003a,b) and are in the range 1.5–1.7% thephotometryandacompletesetofphysicalparameterswill (0.016–0.019mag),dependingontheband.Forthosestars beprovidedinaforthcomingpaper(Masanaetal.2006). affectedbyinterstellarreddening,theuncertaintyin A as V Inspiteoftheproximityofthestars(90%ofthemarecloser derived from photometric calibrations based on uvby β − than 200 pc), we computedindividualinterstellar absorptions photometry (Jordi et al. 1996) is expected to be of about fromuvby β photometryandcorrectedthe observedmagni- 0.05mag,or 1.5%inT . eff − ∼ tudes.Asdiscussedbelow,interstellarabsorptionisoneofthe – Errors in [m/H] and logg: As mentioned above, mostimportantsourcesofuncertaintyintheT determination. [m/H] was obtained, whenever possible, from spectro- eff scopicmeasurements,andotherwise we used photometric calibrations, with assigned uncertainties of 0.10 dex and 4.1.1. Errors 0.15 dex, respectively. We assigned uncertainties of 0.18 For our sample, the errors in the magnitudes, metallicity and dex to logg values determined from photometric calibra- surfacegravitywereestimatedinthefollowingmanner: tions. The effecton the final effectivetemperaturesdueto the uncertainties of both [m/H] and logg is very small: – Errors in the VJHK magnitudes: The total error in each an error of 0.5 dex in [m/H] has an effect in T of less eff magnitude was computed as the quadratic sum of the ob- than 0.5%, whereas the same error in logg has an effect servational error, the error in the absolute flux calibration thatrangesbetween0%and1%,dependingonthevalueof and the error in the determination of the interstellar ex- T andlogg. eff tinction. The first one comes from the photometric cata- logues.However,topreventtheunderestimationoftheer- No error was attributed to the flux in the stellar atmosphere ror in the V band, usually computed from the average of models. Comparisonscarried out by using other stellar atmo- 8 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections Fig.7.Relativeerror(%)ineffectivetemperatureassumingσ = 0.015mag,σ = 0.2,σ = 0.2andtheerroroftheabsoluteflux VJHK [m/H] logg calibration.Topleft:[m/H]=0.0.Topright:logg=4.5.Bottomleft:[m/H]= 2.0.Bottomright:logg=2.0. − sphere models such as those by Castelli et al. (1997) and the keptinmindthatmostofthestarsinoursampleareessentially NextGenmodelsbyHauschildtetal.(1999)showresultingdif- unreddened,thusyieldingthebestpossibleaccuracy. ferencesintemperaturebelow 0.3%inallcases(Ribasetal. Table 3 lists effective temperatures, angular semi- ∼ 2003). diameters, radii and bolometric corrections in the V and K AnestimationofthefinalerrorsinT asfunctionofT , (2MASS) bands with the corresponding uncertainties for the eff eff [m/H] and logg is shown in Fig. 7. As can be seen, the fi- entire sample. Using these values, we calculated simple para- nalerrorisalmostindependentof[m/H]andlogg,butnotof metriccalibrationsofeffectivetemperatureandbolometriccor- T . Hotter stars have greater uncertainties (slightly >1% for rectionasdescribedbelow. eff T = 7500 K) than cooler stars (0.6% for T = 5000 K). eff eff It is important to note that, in the case of reddened stars, an 4.2.Effectivetemperaturecalibration uncertaintyof0.05maginA candoubletheerrorinT com- V eff pared to the values in Fig. 7. For the angular semi-diameter Although the effective temperature for FGK type stars is the behaviour of the errors is very similar to those of the ef- strongly correlated with the (V K) index (see for instance 0 − fectivetemperature,withvaluesforunreddenedstarsofabout Alonsoetal.(1996a)),italsodependsweaklyonthemetallicity 1.0–2.5%.ThismeansthatforHipparcosstarswithgoodparal- andsurfacegravity,aswementionedinSect.2.Therefore,an laxes,weareabletodeterminethestellarradiiwithremarkable empiricalcalibrationofT shouldincludetermsinall(V K) , eff 0 − uncertaintiesofabout1.5–5.0%. [m/H]andlogg.Furthermore,inourcasethecalibrationswere Figure 8 shows the cumulative histograms of the relative constructedseparatelyintwo(V K)0intervals.Starsdeparting − errorsin effectiVetemperature,angularsemi-diameterandra- morethan3σfromthefitwererejected.Theresultingexpres- dius for the 10999stars of the sample. As can be seen, about sionsare: 85%ofthestarshavedeterminationsofT betterthan1.1%. eff The relative error in the angular semi-diameter is also better – 0.35<(V K)0 <1.15(4954stars): − than1.5%for about85%ofthe stars. In thecase ofthe radii, θ = 0.5961+0.1567(V K) +0.0309(V K)2+ themaincontributortotheerroristheuncertaintyintheparal- eff − 0 − 0 lax.Evenso,50%ofthestarshaveradiusdeterminationsbetter + 0.009[m/H]+0.0022[m/H]2 than 10%, and 85%of the stars better than 25%. It shouldbe + 0.0021(V K) [m/H] 0.0067logg 0 − − E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 9 11 nn ctioctio00..7755 aa ar’s frar’s fr00..0022..5555 StSt 00 00 0.25 00..55 0.75 11 1.25 11..55 1.75 22 2.25 22..55 2.75 33 Relative error in T (%) eff 11 nn ctioctio00..7755 aa ar’s frar’s fr00..0022..5555 StSt 00 00 0.25 00..55 0.75 11 1.25 11..55 1.75 22 2.25 22..55 2.75 33 Relative error in θ (%) 11 nn ctioctio00..7755 aa ar’s frar’s fr00..0022..5555 StSt 00 00 5 1100 15 2200 25 3300 35 4400 45 5500 Relative error R (%) Fig.8. Cumulative histograms of the relative error in effective tem- perature,angularsemi-diameterandradiusforthe10999starsinthe sample. Fig.9. Residuals of the T fit as function of effective temperature, eff metallicityandsurfacegravity. σ = 0.0028 (8) θeff getfromMKclassification.Theerrorineffectivetemperature – 1.15 (V K) <3.0(5820stars): 0 ≤ − causedbyanerrorinloggwillbe: θ = 0.5135+0.2687(V K) 0.0174(V K)2+ eff + 0.0298[m/H] 0.00−09[m0/−H]2 − 0 ∆Teff = 50a40Te2ff∆logg (11) − 0.0184(V K) [m/H] 0.0028logg 0 whereaisthecoefficientoftheloggtermsinEqs.(8)and(9). − − − σθeff = 0.0026 (9) In the worst case, that of the hotter stars, ∆Teff = 85 ∆logg. Thus,eveniftheuncertaintyinloggisasmuchas0.5dex,the whereθ = 5040.ThestandarddeviationofEqs.(8)and(9)is eff Teff errorinducedinTeff isjust40K. about20 K and 25 K, respectively.As shown in Fig. 9, there The fits for four different metallicities and logg = 4.5 to- isnoresidualtrendasafunctionof(V K) ,[m/H]orlogg. 0 getherwiththestellarsampleareshowninFig.10.Figure11 − Equation(8)isaplicableintherange3.25 . logg . 4.75and showsthe empiricalT -(V K) relationshipsasa function eff 0 eq.9intherange3.75.logg.4.75.Furthermorethecalibra- − ofthemetallicity. tionsarevalidintherangesofcoloursandmetallicitiesofthe sample: 4.3.Bolometriccorrectioncalibration 3.0<[m/H]< 1.5 for 1.0<(V K) <2.9 0 − − − Since the SEDF method provides both effective temperature 1.5 [m/H]< 0.5 for 0.5<(V K) <2.9 − ≤ − − 0 andangularsemi-diameter,italsonaturallyallowsforthede- 0.5 [m/H]<0.0 for 0.4<(V K)0 <3.0 termination of the bolometric correction in a specific band. − ≤ − 0.5 [m/H]<0.5 for 0.35<(V K) <2.8 (10) From this, if the distance is known, one can compute the lu- 0 ≤ − minosityofthestar.Thebolometriccorrectioninagivenband While(V K) isanobservationalquantityand[m/H]can 0 isdefinedasthedifferencebetweenthebolometricmagnitude − be obtained from photometric and/or spectroscopic measure- andthemagnitudeinthatband: ments, a good determination of logg is usually unavailable for the most of the stars. This could severely restrict the ap- BC = M M =m m (12) i bol i bol i − − plicability of the above calibrations. However, some photo- where m and m are assumed to be corrected of interstellar metricindexes,astheStro¨mgrenδc (see Crawford(1975)or bol i 1 reddening. M can be easily expressed as a function of the Olsen(1988)),aregoodsurfacegravityindicatorsand,ifavail- bol radiusandeffectivetemperature: able,canhelptoestimatelogg.Ontheotherhand,catalogues ofspectroscopicmetallicitiesusuallyprovideanestimationof R T M = 5log 10log eff +4.74 (13) thesurfacegravity.Finally,acrudeestimationofloggcanbe bol − R − Teff ⊙ ⊙ 10 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections riccorrection: BC = M M = x bol x − θ T = 5log 10log 0.26 m (14) x − KR !− T − − ⊙ ⊙ where is the factor correspondingto the transformation of K units.Oncethebolometriccorrectionforabandiisknown,the bolometriccorrectionforanyband jcanbedeterminedfrom: BC =(m m )+BC (15) j i j i − The error in the bolometric correction can be expressed as a functionoftheuncertaintiesinT ,θ(or )andm,asinSect. eff i A 2.4: 5 σ 2 10 σ 2 (σ )2 = θ + Teff +(σ )2 = BCi ln10 θ ! ln10 Teff ! mi 10 σ 2 = (σ )2+ Teff +(σ )2 (16) A ln10 Teff ! mi The procedure described here was used to compute the Fig.10. T -(V K) fits for four groups of stars with different bolometric correction in the K (2MASS) band for the stars metallicitieeffs.The−empir0icalrelationshipscorrespondtologg=4.5and in our sample. In the same way as for the effective tempera- [m/H]= 2.0, 1.0, 0.25and+0.25. ture, we calibrated BC as a function of (V K) , [m/H] and 0 − − − − loggwiththefollowingresults: – 0.35<(V K) <1.15(4906stars): 0 − BC = 0.1275+0.9907(V K) 0.0395(V K)2+ K − 0− − 0 +0.0693[m/H]+0.0140[m/H]2 8000 +0.0120(V K) [m/H] 0.0253logg [m/H] = -3.0 − 0 − [m/H] = -2.0 σ = 0.007mag (17) BC [m/H] = -1.0 7000 [m/H] = 0.0 – 1.15 (V K)0 <3.0(5783stars): ≤ − BC = 0.1041+1.2600(V K) 0.1570(V K)2+ ) K − − 0− − 0 K +0.1460[m/H]+0.0010[m/H]2 T (eff6000 −0.0631(V−K)0[m/H]−0.0079logg σ = 0.005mag (18) BC 5000 Therangeofvalidityofthesecalibrationsisthesameasinthe caseoftheeffectivetemperature.Thebolometriccorrectionin 4000 anybandcanbeobtainedfromBCK viaEq.(15). Figure12showsthefitsforfourdifferentmetallicities,to- 0 1 2 3 4 gether with the stars in the sample used to obtain the calibra- (V-K) tions. The BC (V K) relationships as a function of the 0 K − − 0 metallicity are shown in Fig. 13. The calibration is tabulated inTable2andcomparedwiththecalibrationsbyAlonsoetal. (1995)andFlower(1996)inFig.14,showinggoodagreement. Fig.11.T -(V K) relationshipsforlogg=4.5andfourdifferent eff − 0 metallicities. 5. Discussion whereR =6.95508108mandT =5777K.FortheSunwe Theproceduredescribedinthispaperyieldsthreebasicstellar eff adopt V(⊙ ) = 26.75 mag and m⊙ ( ) = 26.83 mag, and parameters: the best-fitting effective temperature and angular bol ⊙ − ⊙ − thereforeBC ( )= 0.08(Cox2000). semi-diameterand,fromthem,thebolometriccorrection.Ifthe V ⊙ − Using the definition of the absolute magnitude at a given distance is known, θ can be transformed into the true stellar band (M = m + 5logπ + 5) and expressing the radius as radius. The accuracies of the parameters for the stars in our x x function of the parallax (π) and the angular semi-diameter sampleare0.5–1.3%inT ,1.0–2.5%inθand0.04–0.08mag eff (R = θ/π), we obtain the following formula for the bolomet- fortheBC.

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Astronomy&Astrophysicsmanuscriptno.masana February5,2008 (DOI:willbeinsertedbyhandlater) Effective temperature scale and bolometric corrections from ⋆ 2MASS photometry 6 0 E.Masana1,2,C.Jordi1,2,3,andI.Ribas3,4 0 2 1 Departamentd’AstronomiaiMeteorologia,UniversitatdeBarcelona,Avda.Diagonal647,08028Barcelona,Spain n 2 CERforAstrophysics,ParticlePhysicsandCosmology,associatedwithInstitutdeCie`nciesdel’Espai-CSIC a 3 Institutd’EstudisdeEspacialsdeCatalunya(IEEC).Edif.Nexus,C/GranCapita`2-4,08034Barcelona,Spain J 4 InstitutdeCie`nciesdel’Espai,CSIC,CampusUAB,Fac.Cie`ncies,TorreC5parell,2aplanta,08193Bellaterra,Spain 3 Received/Accepted 1 v 9 Abstract.Thispaperpresentsamethodtodetermineeffectivetemperatures,angularsemi-diametersandbolometriccorrections 4 forpopulationIandIIFGKtypestarsbasedonV and2MASSIRphotometry.Accuratecalibrationisaccomplishedbyusing 0 asampleofsolaranalogues,whoseaveragetemperatureisassumedtobeequaltothesolareffectivetemperatureof5777K. 1 Bytakingintoaccountallpossiblesourcesoferrorweestimateassociateduncertaintiesbetterthan1%ineffectivetemperature 0 and in the range 1.0–2.5% in angular semi-diameter for unreddened stars. Comparison of our new temperatures with other 6 determinationsextractedfromtheliteratureindicates,ingeneral,remarkablygoodagreement. Theseresultssuggest thatthe 0 effectivetemperaurescaleofFGKstarsiscurrentlyestablishedwithanaccuracybetterthan0.5%–1%.Theapplicationofthe / h methodtoasampleof10999dwarfsintheHipparcoscatalogueallowsustodefinetemperatureandbolometriccorrection(K p band)calibrationsasafunctionof(V K),[m/H]andlogg.BolometriccorrectionsintheVandKbandsasafunctionofT , - [m/H]andloggarealsogiven.Wepr−ovideeffectivetemperatures,angularsemi-diameters,radiiandbolometriccorrectionseiffn o theVandKbandsforthe10999FGKstarsinoursamplewiththecorrespondinguncertainties. r t s a Key words.stars:fundamental parameters–stars:late-type–stars:sub-dwarfs–infrared:stars–techniques: photometric– : methods:analytical v i X r 1. Introduction rectmethodsaremainlybasedontheuseofphotometry,spec- a troscopy,or a combinationof both.In the case of thetemper- Effectivetemperatureandluminosityaretwofundamentalstel- atures, although many of the published calibrations claim to lar parametersthat are crucial to carry out tests of theoretical haveuncertaintiesof theorderof severaltensof degrees,val- models of stellar structure and evolution by comparing them uesobtainedbydifferentauthorscaneasilyhavediscrepancies with observations.Theaccuracyin the determinationofother of 100 K or even larger. The reason for such nagging differ- stellar properties,suchasmetallicity,ageorradius,hingeson encesmustbefoundsomewhereintheingredientsofthemeth- ourabilitytoestimatetheeffectivetemperaturesandluminosi- ods: atmosphere models, absolute flux calibrations, oscillator ties. strengths,calibrationstars,etc. There are several approaches in the literature to compute Inthispaperwepresentasemi-empiricalmethodtodeter- effective temperature and/or luminosity.Except when applied mine effective temperatures (T ) and bolometric corrections eff to the Sun, very few of them are direct methods that permit (BC)from2MASS1 JHK2 photometry(Cutrietal.2003)that anempiricalmeasurementoftheseparameters.Usually,semi- is applicable to FGK type stars. As all others, our method is empirical or indirect methods are based to a certain extent susceptible to problems derived from the uncertainties in the on stellar atmosphere models. Among the direct approaches ingredientsmentionedabove.However,ourapproachbenefits we find the remarkable work by Code et al. (1976), which is from two major features: First, it provides a way to evaluate basedoninterferometricmeasurementsofstellarangularsemi- realistic individual uncertainties in T , θ and luminosity by eff diameters(θ)andtotalfluxes(Fbol)atEarth,andthemorere- considering all the involved errors; and second, as it is cali- cent works of Mozurkewich et al. (2003) and Kervella et al. bratedtousethe2MASSphotometry,itallowsthecalculation (2004),alsobasedoninterferometry.Ontheotherhand,indi- ofconsistentandhomogeneousT andBCforseveralmillion eff stars in the 2MASS catalogue. This paper also provides T , eff ⋆ Table 3 is only available in electronic form at the CDS via anonymousftptocdsarc.u-strasbg.fr (130.79.128.5)orvia 1 http://www.ipac.caltech.edu/2mass http://cdsweb.u-strasbg.fr/cgi-bin/qcat?]/A+A/ 2 Throughoutthepaper,KreferstoK band. s 2 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections angular semi-diameters, radii and BCs for 10999 dwarfs and However,theIRfluxbecomesverysensitivetometallicityand subdwarfsintheHipparcoscatalogueESA(1997).Suchlarge surfacegravityforstarshotterthan8000Ksothatsmalluncer- samplehasallowedusto constructsimpleparametriccalibra- taintiesintheseparameterstranslateintolargeuncertaintiesin tions as a function of (V K) , [m/H] and logg. Note that a theeffectivetemperature.InsuchsituationtheSEDFapproach 0 − preliminaryversionofthemethodpresentedherewasalready becomes inadequate. At the cold end, the accuracy of stellar successfullyappliedtothecharacterizationofthepropertiesof atmospheremodelslimitstheuseofthemethodtostarshotter planet-hostingstars(Ribasetal.2003). than4000K(molecularopacityplaysanimportantrolebelow The present paper is organized as follows. Section 2 thistemperature).Theselimitationsrestricttheapplicabilityof presentsthemethodandexplainsindetailtheproceduretoob- the SEDF to FGK type stars. Fortunately,these stars are very tain T and angular semi-diameters, including the fitting al- common in the Galaxy and dominate the content of most of eff gorithm,zeropointcorrectionsanderrorestimates. Thecom- surveycatalogues.Theyarecrucialforseveralkeyastrophys- parisonofourtemperatureswithseveralpreviousworks,both icaltopics, suchasthe studyofthe structureandevolutionof based on photometric and spectroscopic techniques, is de- theGalaxy,boththediskandthehalo,andthecharacterization scribed in Sect. 3. In Sect. 4 we present simple parametric ofplanet-hostingstars,amongothers. calibrations of T and BC as a function of (V K) , [m/H] eff 0 − and logg valid for dwarf and subdwarf stars. The sample of 2.1.Calculationofsyntheticphotometry 10999 stars used to build the calibrations is also described in thissectiontogetherwithadetailedexplanationofthedifferent The calculation of the synthetic photometry requires a well- contributors to the final uncertainties. Finally, the results are characterized photometric system, an accurate flux calibra- discussed in Sect. 5 and the conclusions of the present work tion and suitable synthetic spectra. The work by Cohen et al. arepresentedinSect.6. (2003a,b)providesconsistentabsolutefluxcalibrationsinboth thevisible(V)(Landoltsystem)andIR(2MASSJHK)bands. The calibration given by Cohen et al. is computed from a set 2. TheSpectralEnergyDistributionFit(SEDF) of calibrated templates, using the synthetic Kurucz spectrum method of Vegaof Cohenetal. (1992).Inthe case of theIR photom- Theuseofinfrared(IR)photometrytodetermineeffectivetem- etry, they consider the transmission of the camera and filters, peratureswasinitiallyproposedbyBlackwell&Shallis(1977). the detector properties and the Earth’s atmosphere character- Theirso-calledInfraredFluxMethod(IRFM)usestheratiobe- istics. From the comparison between observed and synthetic tween the bolometric flux of the star and the monochromatic photometryfora set of 9 A-typestars and 24 coolgiants,the flux at a given infrared wavelength, both measured at Earth, authorsinfertheneedtointroduceazeropointoffsetinthesyn- as the observable quantity. This ratio is then compared with theticphotometrytomatchtheobserved2MASSphotometry: a theoreticalestimate derivedfromstellar atmospheremodels 0.001 0.005 mag (J); 0.019 0.007 mag (H); 0.017 0.005 ± − ± ± tocarryoutthedeterminationoftheeffectivetemperature.The mag(K).Thecalculationofsuchvaluesisnotexemptofsome IRFMhasbeenwidelyusedbyanumberofauthors,beingmost difficultysincethedispersionsofthedifferencesbetweenboth noteworthytheworkbyAlonsoetal.(1995,1996a,b). photometries(syntheticandobserved)areofthesamemagni- The Spectral Energy Distribution Fit (SEDF) method that tudeasthezeropointoffsetsthemselves. we propose here follows a somewhat different approach, To computethe syntheric magnitudeswe made use of the namelythefitofthestellarspectralenergydistributionfromthe no-overshoot Kurucz atmosphere models grid (Kurucz 1979) optical(V) to the IR(JHK) usingsynthetic photometrycom- takenfromhttp://kurucz.harvard.edu/grids.html: putedfromstellaratmospheremodels.UnliketheAlonsoetal. Fi (1996a)implementationoftheIRFM,whichaveragestemper- mi (T ,logg,[m/H])=2.5log cal (1) syn eff F(T ,logg,[m/H]) aturesderivedindividuallyforeachIRband,ourmethodtakes i eff into account the four bands simultaneously (and naturally). whereFi istheabsolutefluxcalibrationgivenbyCohenetal. cal In addition, and also unlike the IRFM, the bolometric flux is (2003b)(formi = 0) and F(T ,g,[m/H])is theflux inthe cal i eff not required a priori by the SEDF method but results self- ibandcomputedfromtheintegrationofthemodelatmosphere consistently with the temperature. The fitting algorithm (see convolved with the transmission function (filter, detector and Sect.2.2)minimizesthedifferencebetweenobservedandsyn- Earth’satmosphere)fromCohenetal.(2003b): theticphotometrybytuningthevaluesoftheeffectivetemper- ature andthe angularsemi-diameter.The BC can be obtained F(T ,logg,[m/H])= ∞φ(T ,logg,[m/H],λ) (λ)dλ(2) i eff eff i Z T fromthesetwoparameters,andthen,whenthedistancetothe 0 starisknown,theluminosityiscomputedfromtheBCandthe where φ(Teff,logg,[M/H],λ) is the flux given by the stellar absolute magnitude in a given photometric band. The uncer- atmospheremodeland (λ)theeffectivetransmissionfunction i T taintiesofthederivedparameters(T ,angularsemi-diameter intheibandnormalizedtoapeakvalueofunity. eff andBC)areestimatedfromtheerrorsintheobservedandsyn- thetic photometryas well as in the assumed[m/H],logg and 2.2.Fittingalgorithm A . V From a theoreticalpointof view, the SEDF methodcould The fitting algorithm is based on the minimization of the χ2 be applied to stars of any spectral type and luminosity class. function defined from the differences between observed (cor- E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 3 rected forinterstellar extinction)and synthetic VJHK magni- theyare ofno use to calibratethe SEDFmethod.As analter- tudes,weightedwiththecorrespondingerror: native, we have used the list of photometric solar analogues compiled by CayreldeStrobel (1996).We assume that, as an V A V 2 J A J 2 χ2 = − V − syn + − J− syn + ensemble, the average of the effective temperatures of these σV ! σJ ! photometric solar analogues should be equal to the effective H A H 2 K A K 2 temperatureoftheSun(i.e.,5777K). H syn K syn + − − + − − (3) After selecting a subsample of 50 unreddened stars σ ! σ ! H K with non-saturated 2MASS photometry from table 1 of This function depends (via the synthetic photometry)on T , eff CayreldeStrobel(1996),we computedtheirtemperaturesus- logg,[m/H]andamagnitudedifference ,whichistheratio ingtheSEDFmethod.Weobtainedanaveragetemperatureof A betweenthesynthetic(star’ssurface)andtheobservedflux(at 5832 14 K, i.e., 55 K (or 1%) higherthan the solar effec- Earth)( = 2.5logF /F ). isdirectlyrelatedtothe ± ∼ star Earth tivetemperature.Exactlythesamevalueisobtainedifweuse A − A angularsemi-diameterbythefollowingexpression: the subset of solar “effectivetemperatureanalogues”fromta- θ=10 0.2 (4) ble 5 of CayreldeStrobel. Withouta profoundanalysis of all − A theingredientsinvolved–fromthestellaratmospheremodelto Althoughthesyntheticphotometrydependsimplicitlyongrav- the absoluteflux calibration,– itis verydifficultto assess the ityandmetallicity,inpractice,thespectralenergydistribution reasons for such difference. However, it seems clear that the in the optical/IRfor ourrangeof temperaturesis onlyweakly temperature scale as obtained from the synthetic photometry dependent on these quantities. This fact makes it possible to aloneneedsacorrectiontoagreewiththeaverageofthesolar obtainaccuratetemperaturesevenforstarswithpoordetermi- analogues.Fromaformalpointofview,thiscorrectioncanbe nationsofloggand[m/H]. computedfromthesyntheticphotometrythatresultsfromforc- As canbeseen,the χ2 functiondependsalsoonthe inter- ingavalueofT =5777Ktotheentiresample.Afterdoingso, eff stellarabsorptionAV (theabsorptionintheotherbandscanbe wereplacedthezeropointsgivenbyCohenetal.(2003b)(see computedusingtheextinctionlawofSchaifers&Voigt(1982): Sect.2.1)bytheaveragedifference(foreachband)betweenthe AJ = 0.30AV, AH = 0.24AV and AK = 0.15AV). In principle, observedandsyntheticphotometrycomputedforthesolarana- itispossibletoconsiderAV asafreeparameter.However,the logues.AssumingthatthereisnooffsetontheV band,theoff- strongcorrelationbetweenTeffandAV,especiallyforthehotter setsfortheotherbandsare:0.027 0.003mag(J);0.075 0.005 ± ± stars, decreasesthe precisionin the determinationof both pa- mag (H); 0.022 0.005 mag (K). It is interesting to note that ± rameters,withresultingtypicaluncertaintiesof4%inTeff and both in the case of Cohen et al. (2003a) and in our case, the 0.25maginAV.Thus,forbestperformance,AV shouldonlybe valueoftheoffsetinthe H banddifferssignificantlyfromthe consideredasafreeparameterwhenitsvalueissuspectedlarge offsetsin J andK.Itshouldbestressedthattheeffectivetem- andnoothermethodforitsestimationisavailable.Ingeneral, peraturesgivenbyCayreldeStrobel(1996)havenotbeenused thebestapproachis tofix thevalueof AV inEq.(3)fromthe here.Wehaveonlyusedthepropertyofthestarsinbeingclas- estimationbyphotometriccalibrations,forinstance. sifiedassolaranalogues,and,consequently,weassumedtheir Therefore,theonlytwoadjustableparametersbytheSEDF averagetemperaturetobeequaltothesolareffectivetempera- method in the present work are Teff and , whereas logg, ture. A [m/H] and AV are fixed parameters. To minimize Eq. (3) Inourprocedure,we are implicitlyassumingthatthe cor- with respect to these two parameters we use the Levenberg- rectionin our temperaturescale is just a zero pointoffsetand Marquardtalgorithm(Press etal. 1992),which isdesignedto that no dependence on temperature or metallicity is present. fita setofdatato anon-linearmodel.Inallourtests, conver- These assumptionsare justified a posteriori in Sect. 3, where gence towards the minimum value of χ2 was reached rapidly several comparisonsof SEDF temperatureswith other photo- andunequivocally. metricandspectroscopicdeterminationsareshown. The angular semi-diameters computed from Eq. (4) were used to check the consistency of the new zero points in our 2.3.CalibrationoftheSEDFmethodusingsolar temperature scale. These angular semi-diameters were com- analogues pared with the direct values compiled in the CHARM2 cata- The standard procedure for the calibration of an indirect logue(Richichi&Percheron2005).Werestrictedthecompar- method to determine effective temperatures is based on the isontostarswith accurateVLBIorindirect(spectrophotome- comparisonoftheresultswith accuratetemperaturesfromdi- try)measurementsofthesemi-diameter.Only10ofthesestars rectmethodsforasetofstars.Inthisway,thelistofstarswith fulfill the conditions for applicability of the SEDF method. empirical effective temperatures and angular semi-diameters Figure1showsthecomparisonofthesemi-diametersforthese from Code et al. (1976) has been widely used for calibration 10 stars. The agreement is excellent, with an average differ- purposes. This list has been recently increased with the the ence (θ θ ), weighted with the inverse of the square dir SEDF − worksofMozurkewichetal.(2003)andKervellaetal.(2004). of the error, equal to 0.3% with a s.d. of 4.6% (see Table − Otherauthorsuse well-studiedstars,suchastheSun,Vegaor 1). All the direct values used in the comparison correspond Arcturus,tocalibratetheirmethods. to an uniform stellar disk. A crude comparison of both uni- Unfortunately, the few stars with empirical values of form disk and limb darkened values for about 1600 F, G and T are too bright to have accurate 2MASS photometry and K stars in the CHARM2 catalogue indicates a 4% positive eff ∼ 4 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 2.4.Errorestimation One ofthe featuresofthe SEDFmethodisthatityieldsindi- vidualuncertaintiesofbothT andθ.Thetotaluncertaintycan 0.5 eff be calculated by combining the contributions from the spec- tral energy distribution fit and also from the uncertainties in thefixedparameters,i.e.,logg,[m/H],andA .Assumingnull V 0.4 correlationbetweenthesedifferent(and,inprinciple,indepen- M2 (mas) (dTeenfft)osrour)ceisstohfeeqrruoard,rtahteictostuamluonfcethrteaidnitffyeorefnttheerYrokrpcaoranmtriebtuer- R A A tions: H meter C0.3 (∆Yk)2 = (∆YkSEDF)2+ ∂[∂mY/kH]!2(σ[m/H])2 a di mi ∂Y 2 ∂Y 2 Se0.2 + ∂logkg! (σlogg)2+ ∂Ak ! (σAV)2 V (5) Asmentionedabove,theminimizationofχ2iscarriedoutwith HD 209458 0.1 respecttothefourmagnitudesVJHK,withtheparametersA , 0.1 0.2 0.3 0.4 0.5 V Semidiameter (SEDF) (mas) [m/H]andloggbeingheldfixed.Thefirsttermoftheequation (∆Y SEDF)istheerrorin theparameterY comingfromthe fit k k tothespectralenergydistribution,whichiscomputedfromthe covariance matrix (∆Y SEDF √C ). The derivatives in the k kk ≡ Fig.1. Comparison of angular semi-diameters computed from the otherthreetermsoftheequationaredeterminednumerically: SEDFmethod(withthenewzeropointinthetemperaturescale)and ∂Y Y (+∆[m/H]) Y ( ∆[m/H]) fromtheCHARM2catalogue.InthecaseofHD209458,thecompar- k k − k − (6) ∂[m/H] ≈ 2 isonofthesemi-diameterisbetweentheSEDFmethodandanempir- icaldeterminationfromahigh-precisiontransitlightcurve. inthecaseofthemetallicity,andinananalogouswayforthe surfacegravityandtheinterstellarabsorption. TheerrorinT isobtaineddirectlyfromEq.(5),whereas eff Table1.ComparisonofdirectandSEDFangular semi-diameters. theerrorinθmustbecalculatedfromtheerrorinthe param- θdirareVLBIandspectrophotometricvalues(Richichi&Percheron eter: A 2005).ForHD209458,θ isderivedfromaplanetarytransitlight dir σ =0.2 ln10θσ (7) curve. θ A The adopted values for the errors in the magnitudes (σ ), Star θ (mas) θ (mas) ∆θ(%) mi dir SEDF metallicity (σ ) and surface gravity (σ ) are discussed [M/H] logg HIP6702 0.190 0.050 0.189 0.004 0.5 ± ± inSect.4.1.1. HIP8433 0.225 0.050 0.286 0.005 27.1 ± ± − HIP48113 0.405 0.005 0.402 0.006 0.7 HIP50786 0.260±0.010 0.265±0.004 1.9 3. Comparisonwithothermethods ± ± − HIP51056 0.290 0.030 0.306 0.005 5.5 ± ± − Five samples of FGK stars with accurate determinations of HIP85365 0.380 0.015 0.410 0.036 7.9 ± ± − effective temperatures (both photometric and spectroscopic) HIP91237 0.210 0.015 0.251 0.004 19.5 ± ± − wereselectedfromtheliterature(Alonsoetal.1996a,Ram´ırez HIP96895 0.270 0.015 0.278 0.004 3.0 HIP96901 0.260±0.015 0.255±0.003 −1.9 & Mele´ndez 2005, Fuhrmann 1998, Santos et al. 2003 and HIP113357 0.365±0.010 0.352±0.006 3.6 Edvardsson et al. 1993) to carry out a comparison with our ± ± HD209458 0.113 0.010 0.115 0.002 1.8 results. We put special attention in correcting for the effects ± ± − of interstellar reddening, which could lead to systematic dif- ferences.FortheAlonsoetal.andRam´ırez&Mele´ndezsam- ples(the mostreddened),interstellarreddeningwas corrected using the values of E(B V) given by the authors so that − the two temperature estimations would be directly compara- ble. For the other three samples, composed of stars at closer correctionforlimbdarkening,ofthesameorderofthedisper- distances, we restricted our comparisonsto unreddenedstars. sion of the relative differencesshown in Table 1. In addition, Anyhow,thismeantrejectingveryfewstarsfromfurtheranal- we compared the radius of HD 209458 – obtained with the ysis. Among several papers in the literature, we chose these HubbleSpaceTelescopefromahighprecisionplanetarytran- five samples because they have a minimum of 25 stars with sitlightcurve(Brownetal.2001)–withourestimationfrom non-saturated2MASSphotometryandthevaluesof[m/H]and SEDF, obtainingvery goodagreement:1.146 0.050R and logg – needed for a consistent comparison– are providedby ± ⊙ 1.160 0.058R ,respectively. theauthors. ± ⊙ E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 5 3.1.MethodsbasedonIRphotometry:Alonsoetal. (1996a)andRam´ırez&Mele´ndez(2005) 8000 Asmentionedabove,theIRFMisthemostpopularmethodto computeeffectivetemperaturesfromIRphotometry.Thework 7000 by Alonso et al. (1996a) is undoubtedly the widest applica- K) tionoftheIRFMtoFGKstars.Theauthorscomputedeffective M) ( temperaturesfor 462starswith knowninterstellarabsorption, RF 6000 surfacegravityandmetallicity.Afterselecting thestars inthe T(Ieff Alonso et al. sample in the range 4000 < Teff < 8000 K and 5000 witherrorsinthe2MASSphotometrybelow0.05mag,weob- tainedeffectivetemperaturesfromtheSEDFmethodforasub- 4000 setof315stars.Thecomparisonbetweenbothdeterminations ofT isshowninFig.2.Theaveragedifference∆T (IRFM 300 eff eff K) −SEDF)wasfoundtobe−67K,withastandarddeviationof EDF( 150 81K.Thedependenceofthisdifferenceonthetemperatureis STeff 0 nbootttsoimgnpifiacnaenlto:fTFeIRffigF.M2=,th1e.0re30isTneSoffEDdFe−pe2n3d9enKc.eAosfsthheowtenmipnetrhae- RMF - -150 turedifferencewithmetallicity. ITeff-300 4000 5000 6000 7000 8000 In a recent work, Ram´ırez & Mele´ndez (2005) have re- T (SEDF) (K) eff computed the IRFM temperatures of almost all the stars in Alonso et al. (1996a) using updated input data. According to K) 300 the authors, the difference between the old and new temper- EDF ( 150 ature scales is not significant. They also compare their effec- ST eff 0 tivetemperatureswithsomedirectdeterminations.Theauthors MF - -150 csoolnacrlutedmeptheraatttuhreer(einisthaesysesntesmeaIRticFMdifferenc–ethoefirabvoaulute4s0).KThaet IRT eff-300 Alonso -4 -3 -2 -1 0 comparisonbetweenthetemperaturesofRam´ırez&Mele´ndez [m/H] andourdeterminationsisshowninFig.3.For385starsincom- Fig.2.ComparisonoftheeffectivetemperaturesfromtheIRFMand mon we find ∆Teff (IRFM −SEDF) equal to −58 K (σTeff=67 theSEDFmethodfor315starsinthesampleofAlonsoetal.(1996a). K), and TIRFM = 1.061 TSEDF 403 K. Unlike in the case Thebottom panel shows the temperature difference asafunction of eff eff − ofAlonso etal. (1996a),thedependenceof∆Teff in [m/H]is themetallicity. relevant (Fig. 3, bottom panel). For [m/H] < 2.0 the tem- − peratures from Ram´ırez & Mele´ndez are clearly hotter than ourtemperatures.ThesametrendwasfoundbyCharbonnel& Primas (2005)when comparingtheir temperaturesof 32 halo dwarfs( 3.5< [Fe/H]< 1.0)withthevaluesofRam´ırez& − − Mele´ndez. halo. Effective temperatures were determined from fits to the wings of the Balmer lines. Of those stars, 24 have accurate 3.2.Othermethods 2MASSphotometrysothatreliableSEDFtemperaturescanbe derived. The comparison is shown in Fig. 4. The mean aver- Besides the IRFM, which uses IR photometry, we have also age difference ∆T (Fuhrmann SEDF) is 12 K, (σ =45 comparedtheeffectivetemperaturesobtainedusingtheSEDF K), with a slight deeffpendenceon−the temperature:TFuhrTmeaffnn = eff method with other determinations. Two of these (Fuhrmann 0.895TSEDF+618K.Nodependencewasfoundbetween∆T eff eff 1998 and Santos et al. 2003) are spectroscopic works, while and[m/H](Fig.4,bottompanel). in another case (Edvardssonet al. 1993) the temperaturesare basedonuvby βphotometry.Thechiefprobleminthecaseof − spectroscopicdeterminationsisthat,ingeneral,theyaremostly 3.2.2. Santosetal.(2003) applied to bright stars, which have poor 2MASS photometry (the 2MASS detectors saturate for stars brighter than K 4 Tostudythecorrelationbetweenthemetallicityandtheprob- ≈ mag). This fact reduces the number of stars in the Fuhrmann ability of a star to hosta planet,Santoset al. (2003)obtained (1998)andSantosetal.(2003)samplesthatcanbecompared spectroscopic temperatures for 139 stars based on the analy- withSEDFmethod. sis of several iron lines. Effective temperatures for a total of 101 stars in the sample of Santos et al. can be obtained us- ing the SEDF method.In this case, ∆T (Santos SEDF) is 3.2.1. Fuhrmann(1998) eff − 28 K, with σ = 68 K, and practically independent of the Teff This sample is composed of about 50 F and G nearby stars, temperature:TSantos = 1.053TSEDF 270K(Fig.5).Thereis eff eff − both main sequence and subgiants, of the Galactic disk and nodependenceof∆T with[m/H](Fig.5,bottompanel). eff 6 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 6500 8000 6250 7000 6000 K) K) MF) ( 6000 ec.) ( 5750 T(IReff T(Speff 5500 5000 5250 4000 SEDFT (K) eff 1350000 SEDFT(K)eff 50100000 IRMF - T eff--310500 Spec.T - eff --210000 4000 5000 6000 7000 8000 5000 5250 5500 5750 6000 6250 6500 T (SEDF) (K) T (SEDF) (K) eff eff K) 300 K) 200 SEDF T( eff 1500 SEDF T( eff 1000 IRMF - T eff--310500 Spec. T - eff--210000 -4 -3 -2 -1 0 1 -2.5 -2 -1.5 -1 -0.5 0 0.5 [m/H] [m/H] Fig.3.ComparisonoftheeffectivetemperaturesfromtheIRFMand Fig.4.Comparisonof theeffectivetemperaturesfromfitstoBalmer theSEDFmethodfor386starsinthesampleofRam´ırez&Mele´ndez linesandtheSEDFmethodfor24starsincommonwiththesampleof (2005).Thebottompanelshowsthetemperaturedifferenceasafunc- Fuhrmann(1998).Thebottompanelshowsthetemperaturedifference tionofthemetallicity. asafunctionofthemetallicity. 3.2.3. Edvardssonetal.(1993) tobepreferredoverthebestpossibleaccuracy.Inthissection ThesampleofEdvardssonetal.iscomposedby189nearbyF we present calibrations for both Teff and BC as a function of andGtypestars.Incontrastwiththeprevioustwo,theeffective (V K)0, [m/H]and logg. To calculate the calibrations, the − temperatureisnotderivedfromspectroscopybutfromuvby β SEDFmethodwasappliedtoasampleofstarsintheHipparcos − photometry. To do so, the authors built a grid of synthetic catalogue,asdescribedbelow.Notethatthesecalibrationsare photometryusing the atmospheremodels of Gustafsson et al. subjecttotwolimitationswithrespecttothefullSEDFmethod: (1975) and further improvedit by adding several new atomic First, theyare simplificationssince notallthe available infor- and molecular lines. Effective temperatures for 115 stars in mationisused,andsecond,individualuncertaintiescannotbe their sample could be derived using the SEDF method. The determined. average difference ∆T (Edvardsson SEDF) is 10 K, with eff − a dispersion of 70 K and no dependence on the temperature: 4.1.Thestellarsample TEdvardsson = 1.006TSEDF 27K(Fig.6).Asinthetwoprevi- eff eff − ouscases,thebottompanelofFig.6showsthatthetemperature We collected a sample of FGK dwarfs and subdwarfs in the differenceisnotcorrelatedwith[m/H]. Hipparcos catalogue, and therefore with measured trigono- metric parallaxes. Their V magnitudescome mainly from the Hauck & Mermilliod (1998) catalogue, except for those stars 4. Parametriccalibrations withlessthantwoobservations,whereweusedtheHipparcos ThepracticaluseoftheSEDFmethodasithasbeendescribed catalogue. The entire sample has complete and non-saturated in Sect. 2 is not straightforward since it requires the calcula- JHK photometry in the 2MASS catalogue. The metallicity tion of synthetic photometry from stellar atmosphere models wasextractedfromthecompilationofCayreldeStrobeletal. and then use a numerical algorithm to minimize the χ2 func- (2001) or computed from uvby β photometry– either mea- − tion.Parametriccalibrations(asafunctionofoneormorepa- sured fromour own observationsor obtainedfrom the Hauck rameters) may offer a suitable means to estimate reliable ef- &Mermilliod(1998)catalogue–,usingaslightlyrevisedver- fective temperatures in cases where simplicity and speed are sionoftheSchuster&Nissen(1989)calibration.Therangeof E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 7 6500 7000 6000 6500 K) K) pec.) ( 5500 hot.) ( S P T(eff T(eff 6000 5000 5500 K)4500 K) 300 DF ( 150 DF ( 150 SETeff 0 SET eff 0 Spec.- Teff --310500 Phot.T - eff--310500 4500 5000 5500 6000 6500 5500 6000 6500 7000 T (SEDF) (K) T (SEDF) (K) eff eff K) 300 K) 200 SEDF T( eff 1500 SEDF T( eff 1000 Spec. - T eff--310500 Phot. T - eff--210000 -1 -0.5 0 0.5 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 [m/H] [m/H] Fig.5.Comparisonoftheeffectivetemperaturesfromironlinefitsand Fig.6. Comparison of the effective temperatures from photometry theSEDFmethodfor101starsincommonwiththesampleofSantos and the SEDF method for 115 stars in common with the sample of etal.(2003).Thebottompanelshowsthetemperaturedifferenceasa Edvardsson et al. (1993). The bottom panel shows the temperature functionofthemetallicity. differenceasafunctionofthemetallicity.Thestandarddeviationfor asinglestarinEdvardssonetal.(1993)is81K. metallicities covered by the sample is 3.0 . [m/H] . 0.5. − Values of logg were computed from uvby β photometry a few measurements(and with no evaluationof systemat- − (Masana 1994; Jordi et al. 1996). Originally, the sample was ics),wehavesetaminimumerrorinV equalto0.015mag. builttostudythestructureandkinematicsofthediskandhalo Theuncertaintiesintheabsolutefluxcalibrationaregiven oftheGalaxy(Masana(2004))andafulldescriptionincluding by Cohen et al. (2003a,b) and are in the range 1.5–1.7% thephotometryandacompletesetofphysicalparameterswill (0.016–0.019mag),dependingontheband.Forthosestars beprovidedinaforthcomingpaper(Masanaetal.2006). affectedbyinterstellarreddening,theuncertaintyin A as V Inspiteoftheproximityofthestars(90%ofthemarecloser derived from photometric calibrations based on uvby β − than 200 pc), we computedindividualinterstellar absorptions photometry (Jordi et al. 1996) is expected to be of about fromuvby β photometryandcorrectedthe observedmagni- 0.05mag,or 1.5%inT . eff − ∼ tudes.Asdiscussedbelow,interstellarabsorptionisoneofthe – Errors in [m/H] and logg: As mentioned above, mostimportantsourcesofuncertaintyintheT determination. [m/H] was obtained, whenever possible, from spectro- eff scopicmeasurements,andotherwise we used photometric calibrations, with assigned uncertainties of 0.10 dex and 4.1.1. Errors 0.15 dex, respectively. We assigned uncertainties of 0.18 For our sample, the errors in the magnitudes, metallicity and dex to logg values determined from photometric calibra- surfacegravitywereestimatedinthefollowingmanner: tions. The effecton the final effectivetemperaturesdueto the uncertainties of both [m/H] and logg is very small: – Errors in the VJHK magnitudes: The total error in each an error of 0.5 dex in [m/H] has an effect in T of less eff magnitude was computed as the quadratic sum of the ob- than 0.5%, whereas the same error in logg has an effect servational error, the error in the absolute flux calibration thatrangesbetween0%and1%,dependingonthevalueof and the error in the determination of the interstellar ex- T andlogg. eff tinction. The first one comes from the photometric cata- logues.However,topreventtheunderestimationoftheer- No error was attributed to the flux in the stellar atmosphere ror in the V band, usually computed from the average of models. Comparisonscarried out by using other stellar atmo- 8 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections Fig.7.Relativeerror(%)ineffectivetemperatureassumingσ = 0.015mag,σ = 0.2,σ = 0.2andtheerroroftheabsoluteflux VJHK [m/H] logg calibration.Topleft:[m/H]=0.0.Topright:logg=4.5.Bottomleft:[m/H]= 2.0.Bottomright:logg=2.0. − sphere models such as those by Castelli et al. (1997) and the keptinmindthatmostofthestarsinoursampleareessentially NextGenmodelsbyHauschildtetal.(1999)showresultingdif- unreddened,thusyieldingthebestpossibleaccuracy. ferencesintemperaturebelow 0.3%inallcases(Ribasetal. Table 3 lists effective temperatures, angular semi- ∼ 2003). diameters, radii and bolometric corrections in the V and K AnestimationofthefinalerrorsinT asfunctionofT , (2MASS) bands with the corresponding uncertainties for the eff eff [m/H] and logg is shown in Fig. 7. As can be seen, the fi- entire sample. Using these values, we calculated simple para- nalerrorisalmostindependentof[m/H]andlogg,butnotof metriccalibrationsofeffectivetemperatureandbolometriccor- T . Hotter stars have greater uncertainties (slightly >1% for rectionasdescribedbelow. eff T = 7500 K) than cooler stars (0.6% for T = 5000 K). eff eff It is important to note that, in the case of reddened stars, an 4.2.Effectivetemperaturecalibration uncertaintyof0.05maginA candoubletheerrorinT com- V eff pared to the values in Fig. 7. For the angular semi-diameter Although the effective temperature for FGK type stars is the behaviour of the errors is very similar to those of the ef- strongly correlated with the (V K) index (see for instance 0 − fectivetemperature,withvaluesforunreddenedstarsofabout Alonsoetal.(1996a)),italsodependsweaklyonthemetallicity 1.0–2.5%.ThismeansthatforHipparcosstarswithgoodparal- andsurfacegravity,aswementionedinSect.2.Therefore,an laxes,weareabletodeterminethestellarradiiwithremarkable empiricalcalibrationofT shouldincludetermsinall(V K) , eff 0 − uncertaintiesofabout1.5–5.0%. [m/H]andlogg.Furthermore,inourcasethecalibrationswere Figure 8 shows the cumulative histograms of the relative constructedseparatelyintwo(V K)0intervals.Starsdeparting − errorsin effectiVetemperature,angularsemi-diameterandra- morethan3σfromthefitwererejected.Theresultingexpres- dius for the 10999stars of the sample. As can be seen, about sionsare: 85%ofthestarshavedeterminationsofT betterthan1.1%. eff The relative error in the angular semi-diameter is also better – 0.35<(V K)0 <1.15(4954stars): − than1.5%for about85%ofthe stars. In thecase ofthe radii, θ = 0.5961+0.1567(V K) +0.0309(V K)2+ themaincontributortotheerroristheuncertaintyintheparal- eff − 0 − 0 lax.Evenso,50%ofthestarshaveradiusdeterminationsbetter + 0.009[m/H]+0.0022[m/H]2 than 10%, and 85%of the stars better than 25%. It shouldbe + 0.0021(V K) [m/H] 0.0067logg 0 − − E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections 9 11 nn ctioctio00..7755 aa ar’s frar’s fr00..0022..5555 StSt 00 00 0.25 00..55 0.75 11 1.25 11..55 1.75 22 2.25 22..55 2.75 33 Relative error in T (%) eff 11 nn ctioctio00..7755 aa ar’s frar’s fr00..0022..5555 StSt 00 00 0.25 00..55 0.75 11 1.25 11..55 1.75 22 2.25 22..55 2.75 33 Relative error in θ (%) 11 nn ctioctio00..7755 aa ar’s frar’s fr00..0022..5555 StSt 00 00 5 1100 15 2200 25 3300 35 4400 45 5500 Relative error R (%) Fig.8. Cumulative histograms of the relative error in effective tem- perature,angularsemi-diameterandradiusforthe10999starsinthe sample. Fig.9. Residuals of the T fit as function of effective temperature, eff metallicityandsurfacegravity. σ = 0.0028 (8) θeff getfromMKclassification.Theerrorineffectivetemperature – 1.15 (V K) <3.0(5820stars): 0 ≤ − causedbyanerrorinloggwillbe: θ = 0.5135+0.2687(V K) 0.0174(V K)2+ eff + 0.0298[m/H] 0.00−09[m0/−H]2 − 0 ∆Teff = 50a40Te2ff∆logg (11) − 0.0184(V K) [m/H] 0.0028logg 0 whereaisthecoefficientoftheloggtermsinEqs.(8)and(9). − − − σθeff = 0.0026 (9) In the worst case, that of the hotter stars, ∆Teff = 85 ∆logg. Thus,eveniftheuncertaintyinloggisasmuchas0.5dex,the whereθ = 5040.ThestandarddeviationofEqs.(8)and(9)is eff Teff errorinducedinTeff isjust40K. about20 K and 25 K, respectively.As shown in Fig. 9, there The fits for four different metallicities and logg = 4.5 to- isnoresidualtrendasafunctionof(V K) ,[m/H]orlogg. 0 getherwiththestellarsampleareshowninFig.10.Figure11 − Equation(8)isaplicableintherange3.25 . logg . 4.75and showsthe empiricalT -(V K) relationshipsasa function eff 0 eq.9intherange3.75.logg.4.75.Furthermorethecalibra- − ofthemetallicity. tionsarevalidintherangesofcoloursandmetallicitiesofthe sample: 4.3.Bolometriccorrectioncalibration 3.0<[m/H]< 1.5 for 1.0<(V K) <2.9 0 − − − Since the SEDF method provides both effective temperature 1.5 [m/H]< 0.5 for 0.5<(V K) <2.9 − ≤ − − 0 andangularsemi-diameter,italsonaturallyallowsforthede- 0.5 [m/H]<0.0 for 0.4<(V K)0 <3.0 termination of the bolometric correction in a specific band. − ≤ − 0.5 [m/H]<0.5 for 0.35<(V K) <2.8 (10) From this, if the distance is known, one can compute the lu- 0 ≤ − minosityofthestar.Thebolometriccorrectioninagivenband While(V K) isanobservationalquantityand[m/H]can 0 isdefinedasthedifferencebetweenthebolometricmagnitude − be obtained from photometric and/or spectroscopic measure- andthemagnitudeinthatband: ments, a good determination of logg is usually unavailable for the most of the stars. This could severely restrict the ap- BC = M M =m m (12) i bol i bol i − − plicability of the above calibrations. However, some photo- where m and m are assumed to be corrected of interstellar metricindexes,astheStro¨mgrenδc (see Crawford(1975)or bol i 1 reddening. M can be easily expressed as a function of the Olsen(1988)),aregoodsurfacegravityindicatorsand,ifavail- bol radiusandeffectivetemperature: able,canhelptoestimatelogg.Ontheotherhand,catalogues ofspectroscopicmetallicitiesusuallyprovideanestimationof R T M = 5log 10log eff +4.74 (13) thesurfacegravity.Finally,acrudeestimationofloggcanbe bol − R − Teff ⊙ ⊙ 10 E.Masanaetal.:Effectivetemperaturescaleandbolometriccorrections riccorrection: BC = M M = x bol x − θ T = 5log 10log 0.26 m (14) x − KR !− T − − ⊙ ⊙ where is the factor correspondingto the transformation of K units.Oncethebolometriccorrectionforabandiisknown,the bolometriccorrectionforanyband jcanbedeterminedfrom: BC =(m m )+BC (15) j i j i − The error in the bolometric correction can be expressed as a functionoftheuncertaintiesinT ,θ(or )andm,asinSect. eff i A 2.4: 5 σ 2 10 σ 2 (σ )2 = θ + Teff +(σ )2 = BCi ln10 θ ! ln10 Teff ! mi 10 σ 2 = (σ )2+ Teff +(σ )2 (16) A ln10 Teff ! mi The procedure described here was used to compute the Fig.10. T -(V K) fits for four groups of stars with different bolometric correction in the K (2MASS) band for the stars metallicitieeffs.The−empir0icalrelationshipscorrespondtologg=4.5and in our sample. In the same way as for the effective tempera- [m/H]= 2.0, 1.0, 0.25and+0.25. ture, we calibrated BC as a function of (V K) , [m/H] and 0 − − − − loggwiththefollowingresults: – 0.35<(V K) <1.15(4906stars): 0 − BC = 0.1275+0.9907(V K) 0.0395(V K)2+ K − 0− − 0 +0.0693[m/H]+0.0140[m/H]2 8000 +0.0120(V K) [m/H] 0.0253logg [m/H] = -3.0 − 0 − [m/H] = -2.0 σ = 0.007mag (17) BC [m/H] = -1.0 7000 [m/H] = 0.0 – 1.15 (V K)0 <3.0(5783stars): ≤ − BC = 0.1041+1.2600(V K) 0.1570(V K)2+ ) K − − 0− − 0 K +0.1460[m/H]+0.0010[m/H]2 T (eff6000 −0.0631(V−K)0[m/H]−0.0079logg σ = 0.005mag (18) BC 5000 Therangeofvalidityofthesecalibrationsisthesameasinthe caseoftheeffectivetemperature.Thebolometriccorrectionin 4000 anybandcanbeobtainedfromBCK viaEq.(15). Figure12showsthefitsforfourdifferentmetallicities,to- 0 1 2 3 4 gether with the stars in the sample used to obtain the calibra- (V-K) tions. The BC (V K) relationships as a function of the 0 K − − 0 metallicity are shown in Fig. 13. The calibration is tabulated inTable2andcomparedwiththecalibrationsbyAlonsoetal. (1995)andFlower(1996)inFig.14,showinggoodagreement. Fig.11.T -(V K) relationshipsforlogg=4.5andfourdifferent eff − 0 metallicities. 5. Discussion whereR =6.95508108mandT =5777K.FortheSunwe Theproceduredescribedinthispaperyieldsthreebasicstellar eff adopt V(⊙ ) = 26.75 mag and m⊙ ( ) = 26.83 mag, and parameters: the best-fitting effective temperature and angular bol ⊙ − ⊙ − thereforeBC ( )= 0.08(Cox2000). semi-diameterand,fromthem,thebolometriccorrection.Ifthe V ⊙ − Using the definition of the absolute magnitude at a given distance is known, θ can be transformed into the true stellar band (M = m + 5logπ + 5) and expressing the radius as radius. The accuracies of the parameters for the stars in our x x function of the parallax (π) and the angular semi-diameter sampleare0.5–1.3%inT ,1.0–2.5%inθand0.04–0.08mag eff (R = θ/π), we obtain the following formula for the bolomet- fortheBC.

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