The Astrophysical Journal, Vol. 151, February 1968 EFFECTIVE

The Astrophysical Journal, Vol. 151, February 1968

EFFECTIVE TEMPERATURES AND BOLOMETRIC CORRECTIONS OF EARLY-TYPE STARS Donald C. Morton and Thomas F. Adams* Princeton University Observatory Received June 29, 1967 ABSTRACT New scales of effective temperatures and bolometric corrections are presented for main-sequence stars hotter than the Sun. These scales are based on model atmospheres with ultraviolet and Balmer line blanketing included where necessary. The B-type models are in reasonable agreement with the available measures of ultraviolet stellar fluxes while at AO the visual energy distribution and YLy profile of Vega are adequately reproduced. These scales are used to transform the observed initial main sequence and mass- luminosity law into theoretical coordinates. Model interiors agree rather well with both relations though the latter one requires a He/H ratio of 0.10 or less by number for the B and A stars. I. INTRODUCTION Effective temperatures and bolometric corrections are of fundamental importance in relating the observed properties of stars to the physical parameters useful for theoretical considerations. The usual astronomical observations yield magnitudes over some band width in the visible spectrum and colors or energy distributions also in the visible. On the other hand, theoretical calculations of stellar interiors give absolute luminosities or bolometric magnitudes integrated over the whole spectral range, and some measure of the total flux such as the effective temperature. In addition to establishing a distance scale to determine absolute visual magnitudes, we must obtain the transformations from the observed coordinates Mv and B — V to the theoretical ones L or Ifboi and Te, in order to compare our theories with observed relations such as the initial main sequence and the mass-luminosity law. For the hotter stars of type A and earlier, it is necessary to correct for the significant fraction of emission in the ultraviolet which is absorbed by the Earth’s atmosphere. Originally Kuiper (1938) did this with crude theoretical model atmospheres to estimate the bolometric correction as a function of effective temperature. Temperatures were as- signed to each spectral type with a scale of ionization temperatures and the effective temperatures determined directly from observations of the two double-line eclipsing binaries pi Scorpii and ß Aurigae which have measured parallaxes. Later Keenan and Morgan (1951) revised the scale by giving some weight to color temperatures derived from six-color photometry. None of these approaches to the temperature scale has been entirely satisfying because ionization temperatures and color temperatures are not effective temperatures, and the parallaxes and photometric data for the two binaries are not well known. These problems led Popper (1959) and Harris (1963) to use model atmospheres for effective temperatures as well as bolometric corrections. The two major difficulties with this procedure are the identification of the model with a real star of par- ticular color and type, and the uncertainty as to whether the model truly represents the ultraviolet spectrum of the star. Soon it should be possible to determine bolometric corrections and effective temperatures more directly from broad-band photoelectric observations of stars in the ultraviolet with rockets and satellites. Even then models will be necessary for the temperature scale except for those few stars whose angular radii can be measured accurately. The majority of stars still will require fitting the predictions of a model to the observed energy distributions and line strengths, but use of the whole spectrum should improve * Now at Washburn Observatory, Madison, Wisconsin. 611

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1968ApJ. . .151. .61IM 4 for theheavierelementswerethosedeterminedfromsolarphotosphere.Thesurface spectrum. broadening intheheavyionsistentimesclassicalvalue.Forcoolerstarsfrom blanketed modelhasaneffectivetemperaturelowerby2100°Kandabolometriccorrec- The lineabsorptionhasasignificanteffectinsomecases;forexample,atB1.5Vthe portant lineblanketingisfromtheLymanlinesofhydrogenandionsheavier stars. lines intheBaimer-blanketedmodels,butwehave includedhiscorrectiontoB—Vfor equilibrium wasassumed;arecentcalculationbyMihalas(1967)indicatesthatdevia- equilibrium withthefluxconstantindepthto1percentorbetter.Localthermodynamic gravity gwasalways10torepresentmain-sequencestars.Themodelswereinradiative atmospheres. Alllineswereassumedtobeformedinpureabsorption. drogen lines.Inallmodelsthehydrogenlinebroadeningwastreatedbyaccurate B8 toF2wehaveaseriesbyMihalas(1966)withblanketingfromonlytheBalmerhy- In thesecases,wheremostofthefluxisbetweenLymanandBalmerlimits,im- B0 V,MihalasandMorton(1965)atB1.5Adams(1968)B4V. due toelectronscatteringandthebound-freeabsorptionofHi,H“,Hen. sequence starsbecauseanewseriesofmodelsfrom05toF2isnowavailablewhichin- worthwhile toredeterminethetemperaturesandbolometriccorrectionsformain- of stellarangulardiameters. stars. Acheckonthetemperaturescalecanbeobtainedfrominterferometermeasures 612 DONALDC.MORTONANDTHOMASF.ADAMSVol.151 unblanketed modelatmospheres.Weprefertotie ourscaletotheobservedrelationfor fitted hisscaletothetheoreticalbolometriccorrections giveninPopper’sFigure2for make thescaleagreewithPopper’s(1959)relation atB—F=+0.40.Mihalashad metric corrections(B.C.)ofMihalas(1966)and andMorton(1965)inorderto tions fromthisstatecanhavelittleeffectonatleastthecontinuumpropertiesof elements. Thegreatestuncertaintyistheassumptionthatdampingconstantforline tion lessnegativeby0.40magcomparedtoanunblanketedmodelwiththesamevisual cludes theblanketingofstrongestabsorptionlinesaswellcontinuousopacity best basisforobtainingtheeffectivetemperaturesandbolometriccorrectionsofhot luminosity coordinates.Atpresent,theoreticalmodelatmospheresstillseemtobethe line-blanketed onesareused.TherecentsatellitemeasurementsbySmith(1967)ina350Â Balmer discontinuities.In thesemodelstheBalmerlinesareweakandcannot havemuch 1, whered=5040.2/7VWehaveneglectedMihalas’ correctionto6fortheLyman band around1376ÂshowgoodagreementwiththeblanketedB1.5VmodelofMihalas Houck, McNall,andTaylor(1965)Stechersuggestthatlongwardof2100Âthe Griem theory,aswastheionizedheliumbroadeningwhichincludedin05andBO accuracy, orwavelengthrangetodeducethetransformationstemperatureand observed fluxesarereasonablyconsistentwiththebestmodelatmospheres,particularlyif our confidenceintheadoptedtemperatures.ReportsfromrocketflightsbyBless,Code, now availableforuppermain-sequencestars. recent B4V,B0and05models.Webelievethat themodelsinTable1arebest correction alreadyhasbeenincorporatedinthe reports tobepublishedonthemore cooler starsinPopper’sFigure1,whichkeepsB.C.o at—0.07.Theresulting+0.08 the linesofheavierelements.Asystematic0.08 maghasbeenaddedtothebolo- and Morton(1965).Still,thepublishedobservationsarenotyetsufficientinquantity, influence onD.Foreach main-sequencestarwhosediscontinuitywasmeasured by e e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem For thehotteststarswetakemodelsofHickokandMorton(1968)at05 In eachmodelthehelium-to-hydrogenabundancewas0.15bynumber,andratios The resultsfortheelevenline-blanketedmodelatmospheresaresummarizedinTable In thispaperthetransformationsarederivedentirelyfrommodelatmospheres.Itis The typesandcolorsofthe fourhottestmodelsweredeterminedfromthe calculated II. THEMODELS 1968ApJ. . .151. .61IM are fromJohnson’s(1963,1966)listsofaveragesforMKtypes. has notedthatthecalculatedDisnoteasilyrelatedtoobservations;thereforehe Johnson (1963,1966)arelationwasthenderivedbetweenDandU—Vtogivethe No. 2,1968EARLY-TYPESTARS613 preferred toidentifyhismodelsusingthecalculatedcolorindexB—V,eventhoughthis main-sequence relationswithU—F.ForthesevenBaimer-blanketedmodelsMihalas ultraviolet colorinTable1.B—VandtheclosestMKtypefollowedfromusual 0 1346 The identificationofeachmodelwithastarparticularspectraltypeorcolorremains the leastsecurestepinourprocedure. required anempiricalnormalizationandacorrectionforthelinesofheavierelements. 0 70 (1953) toclassifynormalmain-sequencestars.TheintrinsiccolorsU—VandBF Chalonge andDivan(1952),theMKtypewasobtainedfromBrightStarCatalogue polating 1/Tbetweenthemodels,with£/—FasindependentvariableforO (Hoffleit 1964).WiththeintrinsicU—Vcolorforeachspectraltypeaccordingto The Sunisincludedtocompletethelowerendofscalebecauseitspropertiesareso given toMihalas’successfulfitofboththeH7profileandOke’sabsoluteenergydis- and Bmodels—Fforthecoolertypes.AroundA0Vconsiderableweightwas models showsthatthenewcalibrationiscoolerby 1000°KormoreforB1-A0.Inthis well determined.Itstemperatureof5800°Kcomes fromAllen’s(1963)valueof3.90X predictions oftheunblanketedmodelsbyMihalas (1964),butasSmithhimselfnotes, dependence isapproximately linearandisentirelytheoretical,independent of theidenti- the temperaturescalesobtainedbyHarris(1963)and Underhill(1966)fromunblanketed B0; thisisaconsequenceofthechangeinslope¿7 —Fatthispoint.Comparisonwith tribution ofVega(A0V,B—F=0.0)toamodelwithlogg4.0andT9600°K. range ourscaleissurprisinglyclosetothetemperatures ofKeenanandMorgan(1951), 1033 g-iforthetotalluminosity. fication ofthemodelswith realstars,exceptforthezeropoint,whichwas chosento the useofblanketedmodelsshouldraisehisscale closetoours. cooler temperaturescalebycomparinghisobserved ratioB(1376)/B(5475)withthe though ourAstarsaresomewhatcooler.Smith (1967)hasobtainedasignificantly e e erS sec .23 65 60 45 40 30 55 50 176 The effectivetemperaturesforeachspectraltypeinTable2wereobtainedbyinter- Table 2listsalltheMKspectraltypesG2andearlierusedbyJohnsonMorgan An abruptchangeinthetemperaturedifferencesbetween spectraltypesoccursaround For thehotterstarsbolometric correctionsusuallyarerelatedtologTbecause this e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 37450 21910 28640 10080 11200 12600 16800 9160 8400 7200 III. EFFECTIVETEMPERATURESANDBOLOMETRICCORRECTIONS 7750 T e Properties ofLine-blanketedModelswithlogg=40 0 241 0 020 075 D 130 -0 82 -1 17 -1 34 -1 47 FOR EACHSPECTRALTYPE U—V TABLE 1 + 27 +0 37 + 16 + 06 - 02 - 07 - 10 - 18 - 25 - 29 -0 32 B—V -0 01 -0 03 -0 11 -0 26 -0 59 -0 77 -1 -1 -2 -3 B C 0 00 92 34 57 23 Mihalas—Balmer lines Mihalas—Balmer lines Adams andMortonB4V Hickok andMortonB0V Mihalas—Balmer lines Mihalas—Balmer lines Mihalas—Balmer lines Mihalas—Balmer lines Mihalas—Balmer lines Mihalas andMortonB1.5V Hickok andMorton05 Remarks 1968ApJ. . .151. .61IM pendent ontheradiometricmeasuresofPettitandNicholson(1928)madefortyyears from thebolometriccorrectionsofUnderhill(1965)andMihalas(1966)basedon star forwhichwehaveagoodmeasureoftheabsoluteluminosityintegratedoverall it isonlythedifference(B.C.—B.C.o)thatmatters.SinceSunstill 08 . 07.. 06 . 05 . wavelengths, thisstarshouldbethebasisofluminosityscale. vations sincemostoftheirlightpenetratesouratmosphere.Inthesecaseswearede- the Vmagnitudeband. B0.5 BO 09.5 09 . 4.35. Atthehighertemperaturesblanketedmodelsconsistentlyhavelessnegative best unblanketedmodels.TheHarris(1963)scalehasgreaterdeviationsforlogT> metric magnitudesforF2Vandlater.Atearliertypesournewscaleislessnegative 614 DONALDC.MORTONANDTHOMASF.ADAMSVol.151 B2 B1 ago. HerewehaveadoptedPopper’sscalewhichreproducesKuiper’sintheregion corrections forthesameTsincesomeofultravioletfluxhasbeenredistributedinto B7 B6 B5 B3 than Kuiper’sscalebyasmuch0.5mag,butneverdiffersmore0.3mag B9 B8. (5) V-G8VandleavesaG2starliketheSunwithB.C.=—0.07.Inallapplications agree withPopper’s(1959)rediscussionofKuiper’s(1938)scaleobtainedfromradio- Al. AO. B9.5 A2.. F2. FO A4 A3 A7. A5 F5 F3. F7. F6. e F8 e Sun G2. G1 GO mk © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem For starscoolerthanF2itisbettertodeterminethebolometriccorrectionsbyobser- * Magnitudesandluminositiesinthe lastthreecolumnsareappropriateforstarsonthezero-agemain sequence -1 40 -1 43 -1 44 -1 45 -1 46 -1 47 -1 -1 -1 29 -1 38 -0 72 -0 54 -0 63 -0 91 -0 39 +0 06 -0 25 40 17 +0 12 -0 13 40 21 40 25 4-0 39 40 30 4-0 75 40 37 40 73 40 79 +0 70 -fO 60 4-0 54 40 48 40 43 40 41 U —V 0 00 19 10 Colors, EffectiveTemperatures,andBolometric - 30 - 28 - 30 - 31 - 31 - 32 - 32 -0 32 - 24 - 26 - 20 - 16 - 12 - 14 4 03 - 06 - 09 4 06 - 03 + 12 4 09 4 14 4 19 -f 63 4 40 4 36 4 31 40 63 4 61 4 .59 4 54 4 51 4 47 4 43 B—V 00 Corrections ofMain-SequenceStars 32100 34300 35000 37500 30900 35700 36500 20500 22600 26200 15600 17900 14600 12000 13600 10000 10700 9070 9320 9600 8840 8500 8630 6810 8200 6370 6580 6210 7000 7240 7520 5800 5890 5980 5800 Te TABLE 2 log T 3.763 e 418 354 490 506 544 553 562 574 312 535 079 000 029 253 982 958 969 936 946 193 929 134 164 914 833 845 860 818 876 804 777 763 770 793 -3 -3 23 -2 36 -2 -2 -3 -3 -3 -1 -2 -0 91 -1 -1 -0 47 -0 69 -1 05 -0 25 -0 13 -0 17 -0 10 -0 05 -0 07 -0 04 -0 02 -0 01 -0 07 -0 04 -0 02 -0 07 -0 07 -0 06 B C 0 00 0 00 0 00 0 00 01 07 00 48 85 76 77 12 19 17 40 72 40 30 -2.78 -3 30 41 33 41 07 -1 10 -1 90 -2 31 -0 05 -0 38 41 55 41 70 41 95 41 83 42 10 42 44 42 20 43 00 43 29 44 00 43 75 43 55 44 84 44 99 44 75 44 44 87 44 26 M* v ! 40 60 40 03 41 08 -0 61 -1 -2 -3 67 -4 31 -5 14 -6 06 41 38 -1 41 73 41 57 41 88 +2 16 42 05 42 +3 75 43 55 43 29 43 00 44 24 43 99 44 92 44 80 44 40 44 77 44 69 ^bol 57 58 10 log L/Lq* +3 43 43 44 33 41 +2 35 42 54 42 94 42 15 +1 41 41 41 41 +1 40 94 41 41 40 40 40 41 40 49 40 03 40 15 40 21 40 31 -0 06 -0 01 0 63 96 90 38 48 67 36 09 04 22 28 59 16 71 1968ApJ. . .151. .61IM 1 ? r r lata quotedbyPopper(1959) andthebolometriccorrectionappropriatefor theA2V I Aurandjj,Scowithour scale. ForßAurwetakethetrigonometricparallax andorbital liminated thepossibilityitisaclosedoublestar seenpole-on,sothatitmayhave 3 1.0magabovetheage-zeromainsequence.The interferometerobservationshave f VegaisabettervaluefortypicalA0Vstar,though wemustrememberthatVega tar (Conti1965)whileVegaseemstobenormal.We thereforebelievethetemperature Irown etal.found10380°±180°KforSirius(Al V) thoughbothstarshaveB—F^ ures whichareevenclosertoourscaleformoststars. Theinterferometertemperature or theBstarsacorrectiontofluxforabsorption resultsinslightlyhighertempera- elation ofthemodelstoobservedparameters isentirelydifferentineachcase. ourse thetwoscalesdependonsameblanketedmodelstosomeextent,though tirough instrinsiccolorsorspectraltypes,wefindratherencouragingagreement.Of sction totheflux,butcompareourscalewithtemperaturesquotedbyBrownetal. olors inestablishingtherelationbetweenB—VandT.Ifweneglectsmallcor- ive temperatures,and,moreimportant,wemustuseintrinsicratherthanobserved iat somecorrectionsarerequiredinthecalculatedsurfacefluxesandderivedeffec- i validforgiantsandsupergiants.AfewofthestarsinSydneylistarereddenedso osity classesIVorV,butwecannotbesurethatthe(F,T)relationofmodels comparisonwithourtemperaturescaleismeaningfulforthetenofthesestarslumi- ion ofthespectraltypemodel.TheSydneygrouphasestimatedeffectivetem- tep intheuseofmodelsforobtainingatemperaturescale,namelydetermina- firomatic fluxandeffectivetemperature.Thisprocedureeliminatesthemostuncertain dll requiredtogiveabolometriccorrectionorsomeotherrelationbetweenthemono- irface ofthestar,withoutknowledgedistance.However,modelatmospheresare lagnitude atsomevisualwavelengthtogiveanabsolutemonochromaticfluxthe ngular diametersofstarsbyBrown,Davis,Allen,andRome(1967).Anangular sro pointusedforthescaleinTable2. lated sensitivityfunctionsandthereforeweprefertheempiricaldeterminationof rere alwayshavebeendifficultiesincalculatingmagnitudesandcolorsfromthetabu- le scalesaretiedtotheSun,butanerrorof0.29magishardunderstand.However, iis shouldbeunimportantinearlyBstars.Thedifferenceapparentlyarisesfromhow 4 VmodelhasB.C.=—1.63.Bothschemesneglectlineblanketingintheband,but here FoisthesmoothedsolarfluxgivenbyAllen(1963).Inthiscomputation m berelateddirectlytotheSunbyexpression lodel weobtainedB.C.=—1.34.Asanalternativeprocedurethebolometriccorrection s quotedbyMatthewsandSandage(1963)fornoatmosphere.FortheB4Vblanketed 3 theadoptedvalueforSun.HereVissensitivityfunctionmagnitude lodels andradiometricobservationsaroundF2Vinordertorelatethetheoreticalscale ith acorrectionforlineblanketing,andzeropointdeterminedfromcomparisonof [consistency. ThescalepresentedinthispaperwasbasedonMihalas’evaluationof b. 2,1968EARLY-TYPESTARS615 iameter, aftercorrectionforlimbdarkening,canbecombinedwiththeapparent eratures offifteenstarsthiswayusingtheblanketedmodelatmospheresTable1. f 9500°±300°forVega(A0V)isentirelyconsistent withouradopted9600°K,but volved asmuch0.1maginB—F. .00. DetailedspectralanalysishasshownthatSirius hasabundanceslikeametallic-line e ve t v In thecalculationofbolometriccorrectionsfromhot-starmodelsthereisapuzzling Effective temperaturescanbeobtainedfromtheinterferometermeasuresof It isofinteresttocompare theeffectivetemperaturesobtainedfromtwo binaries © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem B.C. =-0.07+2.5log/TrFyVdv-log./VF.o-10log(T/To), e +2.5 logíirFVdv-10T ve 1968ApJ. . .151. .61IM 1415 1 0.10 orlessbynumber.Therefore, weshallconsideronlytheirsetsofmodels with metric parallaxesarecorrectedfortheirultravioletcolorexcessesthesestarsagreewith because EggenandSandage(1962)haveshownthatifthefieldstarswithgoodtrigono- binary-star data(Morton1968)suggeststhatthe helium proportionshouldbeHe/H^ used avarietyofcompositions,butcomparison theirmass-luminosityrelationswith increases withevolution,andconsequentlythetotalincreaseinbrightnessispredictedto star onthezero-agemainsequence.(Actuallytheoryshowsthattemperaturealso done byoneofus(Morton1959)itanattempttodecidewhethertheN(ÿ,7)0reac- interiors canbecheckedbyacarefulcomparisonwiththeobservations. been upheldbyindirectlaboratoryevidence,butotheraspectsofthetheorystellar useful toreviewthecomparisonbetweenpredictionsofmodelstellarinteriorsand for bolometriccorrectionsandeffectivetemperaturesontheuppermainsequence. from anyintrinsicpropertyoftheinitialmainsequence. be 0.33mag.)TherelationsinTable2thengivelogL/LoasafunctionofT.This although heusedKeller-Meyerottopacitieswhich omitlineabsorption.At30Mo it isnecessarytoextendtherange1and15Mo withinteriorsbyIben(1966,1967), curve isthesolidlineplottedasaHertzsprung-RusselldiagraminFigure1. the Hyadesmainsequence.ForSunweadoptM=+4.84,consistentwithV their chemicalcompositionmaybeasignificantlydifferentonefromtheotherclusters, represents ameanlinethroughtheunevolvedstarsofeachcluster,notlowerenvelope. shall adoptBlaauw’srelationhere.ItisgivenintheseventhcolumnofTable2and cluster, NGC2362,IIICephei,and6611ontheassumptionthattheyallhave sequence byfittingtheunevolvedsectionsofHyades,Pleiades,aPersei lent, butprobablysomewhatfortuitous.Nevertheless,wefeelthistemperatureismore because itsparallaxofO''0056isobtainedfrommembershipintheScorpio-Centaurus Stothers (1963)hasamodelwithjustelectron-scattering opacity.KelsallandStrömgren ture scaleofKeenanandMorgan.Theslightwaves inourcurveprobablyresultfrom tained bySchwarzschild(1958)fromKuiper'sbolometriccorrectionsandthetempera- the samechemicalcomposition.SimilarmainsequenceshavebeenpublishedbySandage tion inthecarboncyclewasresonant.Thenegativeconclusionofthatpaperhassince the observedinitialmainsequenceandmass-luminosityrelation.Sometimeagothiswas correction forB1.5VweobtainT=21300°K.Theagreementwithourscaleisexcel- spectrum. Ifweassumethetwocomponentscontributeequallytolight,T= absorption. Unfortunatelytheirmodelscoveronly massesfrom1.58to7.08Mgsothat the mostaccurateformulaeforenergygeneration andopacityincludingbound-bound the inaccuraciesinidentificationofmodelswith particularspectraltypesratherthan reliable thanthe16600°KfoundbyKopal(1955). association (Bertiau1958).WiththeorbitaldataofStibbs(1948)andbolometric give noweighttothisdetermination.Thetemperaturefrom/xScoisevenlessreliable of theuncertaintiesinbothparallaxandphotometricorbit,consequentlywe 616 DONALDC.MORTONANDTHOMASF.ADAMSVol.151 (1957) andbyJohnsonIriarte(1958)whichdiffernomorethan0.1magbutwe 10300° K,whichishotterthanourscaleby1200°K.Popperhasdiscussedtheeffects — 27.63(StebbinsandKron1957);theSunapparentlyhasevolved0.15magaboveaG2 e v e e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem All togetherwebelievethatTable2representsthebestcalibrationpresentlypossible Blaauw (1963)hasderivedtherelationbetweenMandB—Vonzero-agemain We ignorethesuggestionsthatparallaxesofHyadesmaybewrongor Now thatwehaveasetoftransformationsinwhichthereissomeconfidence,it The bestzero-ageinteriormodelsarethoseofKelsall andStrömgren(1965),whoused For logT<4.3,i.e.,B2.5andlater,thecurveisremarkablyclosetooneob- v e X =0.70,Y0.28,Z 0.02,He/H=0.10, IV. THEINITIALMAINSEQUENCE 1968ApJ. . .151. .61IM 3-1 The dashedlinesrepresenthomogeneous interiormodels. from observationsofgalacticclusters andthetemperaturebolometric-correctionscales ofthispaper. for thetwosetsofmodelswithsameZarenearlyidenticalexceptM<2Mo for allhismodels,whileStothersadopted where Ibentendstobelessluminousbyasmuch0.1inlogL/Lo. cause headoptedLo=3.86X10ergssec;otherwisethemass-luminosityrelations with KelsallandStrömgrenforZ=0.02itisseenthataddinglineabsorptioninthe No. 2,1968EARLY-TYPESTARS617 tween themodels.Asmallcorrectionof—0.004shouldbeappliedtoIben’slogL/Lqbe- interior requiresacorrection—0.02inlogTtoIben’smodels.Thisissomewhatless Iben used than the—0.03correctionIben(1963)found,sothereremainsadiscrepancyof0.01be- e Fig. 1.—Hertzsprung-Russell diagram withzero-agemainsequences.Thesolid,wavyline isderived These homogeneousmodelsareplottedinFigure1.IntheregionofoverlapIben © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem X =0.70,F0.27,Z0.03,He/H0.0965. X =0.708,F0.272,Z0.02,He/H0.0960 X =0.70,F0.27,Z0.03,He/H0.0965. 1968ApJ. . .151. .61IM 1415 9 opacity orthenuclearreactionrates.Morton(1959)showedthatifN(^,y)0 0.02 ontheuppermainsequence,itmaybethatsmallcorrectionsarerequiredin prepared alistofthemostreliableorbits.HisTable1givestenpairsfromG2to09 been madebyMorton(1968)inordertodetermineheliumabundancesfromF2V rate wereresonantthehomogeneousmodelswouldbeshiftedbyabout—0.06inlogT, log TV but thisseemstobemorethancantoleratedbythedeviationsinFigure1. 0.03 models,butneithertheobservationsnorourtransformationsareaccurateenough Hya (F5),WZOph(F8),andUVLeo(G2)areplotted ascrossesinFigure2. luminosities werecalculatedusingtheobservedradii.ThefourpairsZZBoo(F2),VZ wish toextendthecomparisonhotterandcoolertemperatures. B5 V,whereinteriormodelsareavailableforarangeofcompositions.Inthispaperwe comparison betweenobservationandtheoryinFigure1isveryencouraging.IfZ=0.02 Analysis ofsolarcosmic-raydatabyGaustad(1964)favorsZ=0.02.Onthewhole lent widthsareproportionaltothesquarerootofnumberabsorbingions. thermore, increasingZto0.03shouldnothaveamajoreffectonthelineblanketingbe- the differenceinlogTisatmost0.03,andonly0.015forZ=0.03.Ifweprefer is excellent,buthereachangeinmixinglengthcancompensateforZ. cause dampingbroadeningdominatesinallthestronglinesandconsequentlyequiva- tions ofdouble-lineeclipsingbinaries.Popper(1967)hasreviewedtheavailabledataand with bothcomponentsdefinitelyonthemainsequence.Effectivetemperaturescor- to eliminatetheZ=0.02case.At1MoagreementwithWeymann-Searsmodel the heliumabundancewere0.10orlesstobeconsistentwithinteriormodels.Fur- which assumedHe/H=0.15andZ0.02,butwedonotexpectlargedifferencesif Z =0.02,He/H0.0865agreeswithIbeninluminosity,butiscoolerby0.007 responding tothespectraltypeswereobtainedfromournewscale,andthenabsolute magnitudes, trigonometricparallaxes,andthebolometric correctionsappropriateforthe radius andluminosityin4.5X10yr.Theirinitialmodelwith=0.728,Y0.252, Iben’s modelsmaybeinerrorbecauseofuncertaintiesaboutthemixinglength Mo modelsshouldnotrequirelargecorrections.Below1.5thetemperaturesof scattering becomesmoreimportant,sothatthedifferencescausedbychoiceofZand models from1to15MoandStother’smodelat30 Mo.Forthesamecompositionthis spectral types.ExceptforaCMa,whichliesoffthe diagram,thesestarsareshownin binaries. TheeightmostreliablepairsinalistbyHarris, Strand,andWorley(1963)are and heliumabundancesuchthatahomogeneousmodelwillevolvetotheobservedsolar convective envelope.At1MoWeymannandSears(1965)havederivedamixinglength the inclusionofbound-boundopacityoughttobelessandconsequently1530 618 DONALDC.MORTONANDTHOMASF.ADAMSVol.151 Figure 2asdots. e relation differsverylittlefromthatofKelsalland Strömgren from2to7Mo,butbelow given inTable3.Heretheabsoluteluminosities must beobtainedfromtheapparent 0.08 ±0.02.At17Mothe 09.5VbinaryYCygshowsexcellentagreement withthe*. below thetheoreticalrelations foreitherZ=0.02or0.03andthusrequire He/H= this to1.6Mothelattermodelsaresomewhatmore luminous. e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem A comparisonoftheobservedandtheoreticalmass-luminosityrelationsalreadyhas Our temperatureandbolometric-correctionscalesdependonmodelatmospheres On theuppermainsequencestellarmassesaredeterminedprimarilyfromobserva- Figure 1showsthattheobservedzero-agemainsequenceliesveryclosetoZ= For massesinexcessofthemodelscomputedbyKelsallandStrömgrenelectron We wishtocompareallthesebinarieswiththemass-luminosity lawgivenbyIben^ Masses alsoareknownforafewA,F,andGmain-sequence starsinnearbyvisual As notedinthepaperby Morton(1968)thestarsfromB5toF2tend lie alittle: V. THEMASS-LUMINOSITYLAW 1968ApJ. . .151. .61IM binaries (X)andvisual().ThelineisIben’stheoreticalrelationfortheindicatedcomposition. for ZZBooandaCom(rjCrBliessofartotheleftthatitmustbeanomalous).Increas- evolved tofittheSun.Theeclipsingbinaries(crosses)areconsistentwithvisualones opacity inthelattermodelsresultsaslightlylowerheliumabundancewhentheyare identical withthoseofWeymannandSearshere,thoughtheadditionbound-bound in themass-luminositylawordertocheckourtemperaturescale. in thedataandourtemperaturescale.Thetheoreticalmass-radiusrelationwouldpro- (dots) andtheobservedSun(O),butallthesepointsliewellaboveIben’scurveexcept vide amoredirectcomparisonwiththeeclipsingbinarydata,buthereweareinterested two mostmassivemodels,althoughprobablybetterthanwarrantedbytheuncertainties No. 2,1968EARLY-TYPESTARS619 # Fig. 2.—Mass-luminosityrelationforthepresentSun(O),andmain-sequenceFGeclipsing Figure 2showsthemass-luminosityrelationfrom1toAfo.Iben’smodelsarenearly © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 99 HerA 26 DraA 7} CrBA ri CasA.. 7 VirB. y VirA a ComA a CMaA a CenA Visual BinarieswithWell-determinedOrbitsand Parallaxes Star F7 V F5 V FO V FO V Al V G2 V G2 V Gl V GOV MK log M/MO + 03 + .15 + 05 + 07 +0.02 + .07 +0 33 - 18 - .06 TABLE 3 M v 49 69 67 04 41 36 52 70 50 -0.07 -0 13 B.C .06 .00 .00 07 07 02 00 log L/LO 0 06 0.14 0 19 0 06 0 30 0 43 0.50 0.51 1 40 1968ApJ. . .151. .61IM 620 DONALDC.MORTONANDTHOMASF.ADAMSVol.151 ing theheliumabundanceinmodelstoHe/H=0.14bynumberwouldgivebest diffuse nebulae.Itseemsmorelikelythatthebinariesabovelinemustbesomewhat fit, butthenthesestarswouldhavemoreheliumthantheyoungerAandB Therefore, theB.C.-T«,relationshouldbebetterdeterminedthanthatforTandB—V, ing. Webelievethemodelstobemostaccurateavailable,butsomeuncertainties metric correctionsofmain-sequencestarshotterthantheSun.Thesescaleshavebeen evolved, andmostofthembymorethanthe0.33magderivedforSun. scale. to derivetheserelationsdirectlyfromtheobservations,butourmodelsarereasonably Rocket andsatellitedataonultravioletstellarfluxesarestillnotcompleteenough still remain,especiallyintheidentificationofeachmodelwithaparticularspectraltype. based onmodelatmosphereswhichhaveincludedultravioletandBaimerlineblanket- predicted relationsbetweenmass,luminosity,andtemperatureagreeratherwellwith luminosity relationintheoreticalcoordinatesforcomparisonwithinteriormodels.The consistent withmostofthepublishedmeasurements.Thetemperaturesderivedfrom although thereisstillaquestionregardingthezeropointforbolometriccorrections. possible. Thesescalescannotbeconsideredfinal,buttheydoprovideabasisforfurther Allen, C.W.1963,AstrophysicalQuantities(2ded.;London: AthlonePress). Adams, T.F.,andMorton,D.C.1968,Ap.J.,Vol.152(in press). allowing ustousehisnewdataoneclipsingbinaries inadvanceofpublication. in thePhysicsDepartmentofUniversityCollege, London,Theauthorswishtothank over allinclinationstheconsequencescouldbeimportantforstarsearlierthanF0,par- possible discrepancies:(1)thebinariesfromF5VtoG2arenoticeablymoreluminous the interferometermeasuresofstellarangulardiametersshowgoodagreementwithour Bless, RC.,Code,AD,Houck,TE.,McNall,JFand Taylor, DJ1965,A,70,667. Blaauw, A.1963,BasicAstronomicalData,ed.K.AaStrand (Chicago:UniversityofChicagoPress), T. Kelsallforprovidingusdetailedresultsofhis interior modelsandD.M.Popperfor can beastartingpointforincludingtheeffectsofstellarrotation. cluster mainsequencesandbinary-stardataintothezero-agesequencemass- Chalonge, D.,andDivan,L1952, Annd’ap,15,201. Bertiau, F.C.1958,Ap.128,533. comparisons withultravioletfluxobservationsandinterferometermeasures,they accurate modelatmospherespresentlyavailable,butmanyimprovementsarestill the radiusandluminosityofinteriormodels(RoxburghStrittmatter1965). affect boththeenergydistributioncalculatedforamodelatmosphere(Collins1966)and Ostriker (1967).Sofaronlyuniformrotationhasbeenconsideredindetail,butitcan ticularly iftheangularvelocityincreasestowardcenterofstarassuggestedby and Astarsseemstobealittlehotterthantheobservedone. than thehomogeneousmodels,and(2)theoreticalinitialmainsequenceforB the observationsifweacceptaHe/Hratioofabout0.08bynumber.Thereremaintwo Conti, P.S.1965,Ap.J.,142,1594. Collins, G.W1966,Ap./.,146, 914. Brown, RHanbury,Davis,J,Allen,L.andRome,M. 1967,M.NR.AS.,137,393 e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem In thispaperwehavepresentednewscalesfortheeffectivetemperaturesandbolo- p. 383 The temperatureandbolometric-correctionscalespermitthetransformationof Much ofthispaperwaswrittenwhileoneus(D.C.M.)aguestM.J.Seaton Our newscalesoftemperaturesandbolometriccorrectionsarebasedonthemost In thispapertheeffectsofstellarrotationhavebeenneglected.Evenifweaverage REFERENCES VI. SUMMARY No. 2, 1968 EARLY-TYPE STARS 621 Eggen, O. J , and Sandage, A R. 1962, Ap. 136, 735. Gaustad, J. E. 1964, Ap J , 139, 406 Harris, D. L. 1963, Basic Astronomical Data, ed. K. Aa. Strand (Chicago: University of Chicago Press), p. 263. Harris, D L., Strand, K. Aa., and Worley, C. E. 1963, Basic Astronomical Data, ed. K. Aa. Strand (Chi- cago: University of Chicago Press), p. 273. Hickok, F , and Morton, D. C. 1968, Ap J., Vol. 152 (in press). 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