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Home , Absolute magnitude, Bolometric correction, Effective temperature, Magnitude (astronomy), North Polar Sequence, Photographic magnitude

THE OF THE , THE STELLAR SCALE, AND BOLOMETRIC CORRECTIONS

G. P. KUIPER

ABSTRACT Three connected subjects are treated in this paper. The first section deals with the , the absolute photovisual and bolometric magnitudes, and the spectral type of the sun. The quantities adopted are: Te =.5713° (Unsold) ; Afpv = 4.73; Mbol = 4.62; spectrum dG2. The second section contains a discussion of Te of the O, B, and A . The result is given in Table 3 and shown in Figure 1. In the third section, tables of are constructed, as far as pos- sible independently of assumptions about the shape of the spectral energy-curves. For spectral types earlier than Fo, theoretical spectral energy-curves are used; the results are found in Tables 6 and 3. For later types, empirical bolometric corrections are de- rived from Pettit and Nicholson’s radiometric observations. The discussion is based on Table 7 and Figures 2 and 3; the result is given in Tables 9 and 10. In the fourth section, effective are derived for late-type stars with suitable data: C and the giants for which diameters were measured interfero- metrically. has been taken into account. The result for Castor C is 3550 ± iio°; for the giants, the values in Table 11. In the last section, color temperatures are collected and reduced to the system of effective temperatures. Particular attention is paid to the observations of the dwarfs (Tables 12 and 14), on which the reduction to effective temperatures depends to a large extent (sun). Table 13, last column, gives the derived effective temperatures for classes A-M; they are shown graphically in Figure 4. Directions are given for the calibration of temperatures derived from W. Becker’s Catalogue of color indices and Pettit and Nicholson’s water-cell absorptions, to effective temperatures (p. 469). Two applications are given. The result on e Aurigae reduces a difficulty previously encountered by B. Strömgren. INTRODUCTION The purpose of this article is the selection or derivation of the most probable set of effective temperatures and of bolometric cor- rections that can now be obtained. These data are required in the discussion of the mass- relation given in the following paper. Only a few new results are derived here, but the main object is to select and relate the most reliable observations and results thus far obtained. Among the recent investigations of the temperature scale, those by Pannekoek1 and Pilowski2 are of special interest. The results of Pannekoek’s paper are extensively used in our article. Pilowski’s pa- 1 Ap. 84,481, 1936. 2Zs.f.Ap., ii, 265, 1936. 429

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1938Ap J 88. . 42 9K 3 4 per isanattempttouse,formain-sequencestarsonly,thestatistical 430 G.P.KUIPER relation betweenradiusandspectraltype, found fromspectroscopicbinaries;thelast,starsingeneral. type andabsolutemagnitude,fromwhichtherelationbetweenspec- tral typeandsurfacebrightnessisderived.Thefirstrelation But asPannekoekhaspointedout,therelationbetweenspectral from whichthisrelationisderivedbyPilowski,sothatitseemspref- type andtemperatureismuchcloserthanthestatisticalrelations peratures/’ asdefined*byBrill,andnoteffectivetemperatures(“ra- erable torestrictthediscussionthosestarsforwhichalldata diation temperatures”areformaltemperaturesrelatedtothevisual are welldetermined.Furthermore,Pilowskiderived“radiationtem- on thewavelengthusedinmeasuringsurfacebrightness).It surface brightnessesbyPlanck’sformula;sincethestarsdonotex- will beouraimtoobtainthemorefundamentaleffectivetempera- actly radiateasblackbodies,the“radiationtemperatures”depend tures, definedby constant.IfLandRareexpressedintermsofthesun,we have in whichListheluminosityandRradiusofastar, be usedinpracticeonlyifatableofbolometriccorrectionsisavail- since Lisneverknowndirectlyfromobservation,butusuallyonly the absolutevisual(orphotovisual)magnitude,equation(2)can If LandRareknown,Tmaybefoundfromequation(2).But if able. Thebolometriccorrectionisdefinedby e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 3 Op.çit.jp.505. B.C. —A/ibolWbol'W'-pv, Mbol =Mhoi(sun)—2.5logL. 24 L =47rR•0T,(1) 4 E.g.Pannekoek,op.cit.,p.483. (3) (4) 1938Ap J 88. . 42 9K 5 6 portance infixingtheunitofstellarluminosities. or —26.68IPv. present paper,asinthenextone,anattemptismadetoreduce of thewholeaccessiblesolarspectrum.Unsoldhasjustpublished termination ofthisconstantinvolvesaspectrophotometricstudy North PolarSequenceof1922.Othervisualmagnitudesarereduced of bolometriccorrectionscanbeconstructedwithoutrecoursetothe to thissystembeforetheyareusedasisexplainedonp.477. discussion, whichgavera=—26.72andM+4.85.Inthe or tained. Theluminosity,expressedinstellarmagnitudes,isofim- effective temperaturescanbebroughtintoafairlysatisfactory assumption ofblack-todyradiation,andthatalsothescalestellar corrections. Thephotovisualmagnitudes(w)usedinthispaper this paper.Itinvolvesthezeropointoftablebolometric a carefulrediscussionoftheavailabledata,andweadopthisresult: since theatmosphericextensiondependsonwavelength,de- absolute determinationsofstellartemperatureswhichmaybeob- are internationalphotovisualmagnitudes(IPv)onthesystemof IPv. tabulated byRussellIPvmagnitudeswillbeused: all magnitudestotheIPvsystem.Accordingly,incomparisons state. The definitionofthequantityTfi(sun)isgivenincourse vY pv bo 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 5 PhysikderSternatmosphären,chap,ii,1938. The solareffectivetemperatureisderivedfromtheconstant; In thefollowingsectionsitwillappearthatasatisfactorytable M/>. J.,43,103,1916. b) Fabryfoundittobe26.94mag.brighterthan ,or—26.85 a) Zöllnerfoundthesun26.87±-°5mag.brighter thanCapella, 2. ThestellarmagnitudeofthesunisusuallytakenfromRussell’s i. Theeffectivetemperatureofthesunisoneveryfew THE TEMPERATUREANDLUMINOSITYOESUN o T =5713±30K(estimatedmeanerror), e THE STELLARTEMPERATURESCALE431 log T—2)-757io.002(m.e.). e 1938Ap J 88. . 42 9K 8 7 V other dependingonPettitandNicholson’sradiometric work. Accordingly, wehavem=—26.26IPgor—26.79Giving determination ofthemagnitudedifferencebetween thesunand minations, onewhichusesthemoonasanintermediate step,andthe double weighttoKing’sdetermination,weobtain—26.60IPvfrom which thecorrection—0.14mag.shouldapplyasmentionedabove. magnitudes ofthecomparisonstarsweredeterminedfromdirect the twophotographicdeterminations. as reducedbyRusselltoKing’sphotographicsystem,is—26.12, IPg, or—26.50IPv. his value=2.69,ascomparedwith2.55IPg.Applyingthe of thesedependoncomparisonswithPolaris,forwhichKingused comparisons withPolaris),King’sdeterminationbecomes—25.97 same correction,—0.14mag.,totheother4comparisons(since responding colorindexis+0.53. tioned inRussell’spaper.SincethedwarfFandGstarshavecolors been publishedsinceRussell’sdiscussion. type, asdeterminedbyMorganandthewriter,isG2.Thecor- to adoptthecolorindexcorrespondingspectraltype.This almost uniquelydeterminedbytheirspectraltype,itappearssafe IPv. which givetheaverage—26.93IP- With weights4,and1,asusedbyRussell,Ceraski’svalueis 432 G.P.KUIPER pg — 26.64IPv- © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 7 Cf.Morgan,ibid.,87,460,1938.*Ibid.,86,180, 1937, The useofthemoonasanintermediatestep involves,first,the The precedingvaluesmaybesupplementedby twootherdeter- The secondphotographicdeterminationisbyBirck.result, No directvisualdeterminationsofthemagnitudesunhave E. S.King,from10determinations,found=—25.83.Six 3. Twophotographicdeterminationsareavailable,alreadymen- d) W.H.Pickeringmade10determinationsusingdifferentstars, The straightmeanofthesefourvisualdeterminationsis—26.77 c) Ceraskifoundittobe 22.93 —0.66+3.22mag.brighterthanSirius. 22.93—°-66 +6.30mag.brighterthanPolaris,and 22.93—°-66 +4.86mag.brighterthanProcyon, 1938Ap J 88. . 42 9K 9 10 . moon, andsecond,thedeterminationofstellarmagnitude phase coveredbyRougier. visional zeropointofthemagnitudescale.Withthispoint, of theabsorptionbyobjectives.Theuncertaintyinreduc- reduction wasobtainedbyplacingtwomicroscopeobjectivesand sun andmoonhasrecentlybeenpublishedbyRougier.Heusedfor the intensitiestobemeasuredofsameordermagnitude.The vidual comparisonswithPolaris.Inprinciple,therefore,oneob- well established,thecalibrationtostellarmagnitudesbymeansof moon, —3.04;andofthemoonatotherphases,valuesgivenin magnitude ofthesunwasfoundtobe—17.33+0.03;full moon werecomparedwithastandardlamp,whichprovidedpro- tion constantwasestimatedtobe+0.01mag.Boththesunand these measurementsaphotoelectric(potassium)cell.Heobserved Rougier’s Table30. lowed fromgeometricalopticsandanempiricaldetermination the sunthroughanopticaldevicewhichreducedbrightnessby both authorscouldbemadeatgreateraltitudes (63observations), between thefirstandlastquarterisused,when theobservationsby we find—11.58+0.05(p.e.)onKing’sscale. Adoptingthelatter King’s workshouldgiveagooddeterminationifnoserioussystem- servation shouldsufficetofixRougier’szeropoint,andthewholeof some diaphragmsatproperdistances.Thereductionconstantfol- atic errorsarepresent. the mooncanbemadefrommagnitudedeterminationsatanylunar tude ofthefullmoonis—11.64onKing’sscale. Ifonlythepart tude determinationsofthemoon. 14.87 mag.,andobservedthemoonwithoutthatdevice.Thismade 12 1 10 1112 9 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 4. Averyfinedeterminationofthemagnitudedifferencebetween Rougier’s magnitudedifferencemaybecombinedwith5magni- If allofKing’sobservationsareused,itisfoundthatthemagni- a) King’sphotographicmeasures,83inall,weremadebyindi- Since therelativemagnitudesinRougier’ssystemseemtobevery /fo'd., 2,319. HarvardAnn.,59,63-94,1912.Rougier,op. At.,p.323. Ann.Strasbourg,2,Part3,1933;3>5,1937. THE STELLARTEMPERATURESCALE433 1938Ap J 88. . 42 9K 15 13 16 on moon areavailable,thefirstofwhichisbyZöllner.(Twostillolder value, andtheIPgmagnitudeforPolaris(2.55insteadofKing’s 434 G.P.KUIPER inverse-square law.Reducingtheobservationstofullmoonwith determinations willnotbeusedhere.)Zöllnerfoundthefullmoon value 2.62),weobtain made betweenfirstandlastquarter,wefindforthefullmoon own curve,wefind12.48mag.forthedifference.Hence,have poor. GivingPickering’sresultdoubletheweightofZöllner’s, solution actingasafilter.Thescalewasexclusivelybasedonthe full moonwiththeaidofRougier’slight-curveinsteadZöllner’s to be12.31mag.brighterthanCapella.Reducinghisobservations . Theinternalprobableerrorofthemeanobserva- the aidofRougier’slight-curve,andusingonlyobservations color ofwhichwasmadeequaltotheseobjectsbymeansablue compared boththemoonandArcturuswithapentanelamp, tions ofArcturusaloneis±0.11mag.ButPickering’sobservations This resultisbasedon18observationsofthemoonbutonly4 IPv magnitudeofArcturuswefind+0.04,sothat are ofinterestbecausetheirextremesimplicity. on them. combined visualresultis — 12.63Pickering’sscale(withArcturus0.24mag.).Forthe 14 1415 16 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem c) ThesecondvisualdeterminationisbyW.H.Pickering.He b) Twovisualdeterminationsofthestellarmagnitude HarvardAnn.,62,62-69,1908.Ibid.,TableXI, p.65. Twoobservations,Nos.9and10,wereomittedbecause of Pickering’scomments ^ PhotometrischeUntersuchungen,pp.109and124,1865. The agreementbetweenthetwovisualdeterminationsisrather y m(full moon)=—12.83IP(Pickering). V w(full moon)=—12.29IP(Zöllner). w(full moon)=—11.65IPg• w(full moon)=—12.65IPv. 1938Ap J 88. . 42 9K 19 18 22 21 observations, inviewoftheyellowcolorreflectedmoonlight. full moon.Althoughcolordeterminationsofthemoonshowno tion ofthelight-curvetofullmoon,whichisnecessarilysomewhat flected lightarenotinverygoodagreement.Itseemsprobablethat lower limittothebrightnessoffullmoonascomparedwithRou- arbitrary, particularlywhenfewobservationsaremadeverynear necting thesolarmagnitudewithalllunarobservationsbetween at leastpartofthediscrepancycanbeattributedtoextrapola- lower limitinthesystemusedhere,weshalladopt—12.75IPvas first andlastquarter.Hence,itdoesnotconcernuswhetheror we useRougier’sdefinitionofthefullmoononlyasaconstantcon- gier’s system,whichwehaveusedinthereductions.(Itisclearthat change duringthelunation,onlyasmallvariationcouldappreciably cidentally adeterminationofitsvisualmagnitude. find thephotoelectricmagnitudestobe+0.75, +0.09,and1.39. magnitudes. Withthedirectionsofpage478wefindforIPvmag- to obtainthephotoelectricmagnitudeoflamponinterna- Vega, andDeneb,withastandardlampasintermediary.Inorder as derivedbyPettitfromhisowncurve.Inourpreviousdiscussion this factarisesonlybecauseweusethemagnitudeoffullmoon affect thedifferencebetweenphotoelectricandradiometric respectively; andhence,withthecolorindices givenelsewherewe nitudes ofCapella,Vega,andDeneb,+0.19,0.09,+1.34, tional scale,weapply0.8ofthecolorindex(IPg—IPv)toIPv Rougier’s definitionisexactlycorrect.) These values,inconnectionwiththedifferences foundbyCalder, a probablevalue. — 12.63mag.,whichisontheHarvardscale.Sincethisvaluea 17 21 22 18 20 11 1920 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Unfortunately, Rougier’slight-curveandPettit’scurveofthere- d) Pettit’sradiometricworkonthemoonasawholegivesin- E.g.,StebbinsandWhitford,Ap.J.,87,252,1938. Ibid.,86,180,1937. Pettit’s computedvalueforthebrightnessoffullmoonis Ap.J.,81,33-34- e) CaldercomparedthemoonphotoelectricallywithCapella, Ap.J.,81,17,1935;Contr.Mt.W.Obs.,No.504. On accountofthisdifference,Pettit’sdeterminationwillgivea Ibid.,Fig.4,p.28.HarvardAnn.,105,445,1937. THE STELLARTEMPERATURESCALE435 1938Ap J 88. . 42 9K 24 / e 23 lead to—4.33,—4-35,and—4.33mag.forthelamp,or—4.34 of themoontoonesystem,forwhichweadoptIPvsystem.To visual determinationhalf-weight. the sunhasbeenassumedtobe+0.53mag.Thedifferencebe- results. Thephaseangleduringtheseobservationswas+5°5o,so somewhat differentdefinitionoffullmoon,obtained—0.18mag.). 436 G.P.KUIPER measures (themoonistheredder): that endthecplorindexofmoonisneeded.Thecolor standard lamp,or—12.09IP- that thecorrectiontofullmoonis—0.12mag.(Calder,witha in themean. Pettit’s determinationhasbeengivendoubleweightandDanjon’s tween thecolorindicesofsunandmoonisfoundfromfollowing sponding todKo,inagreementwithPettit’sresult. determinations. ItshouldberemarkedthatPettit’s ownreduction, Calder’s datathenmakethefullmoon7.75mag.brighterthan In formingthemean,doubleweightwasgiven tothetwomodern 224 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We mustnowreducethefivedifferentvaluesformagnitude 3 Rougier,op.cit.,Table30.Cf.p.43- Accordingly, thecolorindexofmoonis+0.75mag.,corre- Calder measuredthemoonontwonights,oneofwhichledtofinal The fivemagnitudedeterminationsofthefullmoonare,therefore, — 12.09IPe=12.69IPv(Calder) — 11.65IPg=12.40IPv(King) to beequalthatofCapella,andhence0.17mag.largerthan Wilsing, Schemer,Russell*0.25mag. for thesun. Danjonf 3 Pettitf 20 Calder §0.17 Mean =—12.66IPv (7) * Ap.J.,43,126,1916. f Pub.A.S.P.,38,243,1926. § HarvardAnn.,105,452,1937.Thecolorindexwasfound %Ann. Strasbourg,3,146,274,1936-37. Weighted mean0.22mag.(6) 12.65 (Zöllner,W.H.Pickering) 12.75 (Pettit) 1938Ap J 88. . 42 9K 25 26 rection wasneededtoallowforthedifferencebetweenRougier’sand what greaterrelativebrightnessofthefullmoonasdefinedbyRou- Pettit’s light-curve. gier’s light-curve. Russell’s result—12.55HS;thedifferenceisaboutequaltosome- of energy,fromwhichtheauthorsobtainedradiometricmag- Rougier’s photoelectricdifference,14.29mag.,gives ments” containsacalibrationofradiometricmagnitudesinterms magnitude isfoundtobe—27.18—Am+0.62=—27.01mag. nitude ofthesun,—27.18mag.Fromthisfigurebolometric magnitude ofthesunis—26.94IPv. in thispaper.TheaveragebolometriccorrectionforaG2dwarfis, then, 0.07,asfoundbelow(Table9).Hence,theresultingvisual The constant0.62mag.isexplainedonpage451;itreducesthe quantity (m—Am)tothezeropointofbolometricmagnitudesused —12.63 HS,isveryclosetotheaverage;butwefeltthatasmallcor- heat indicesformain-sequencestars,whichalsoentersintheempiri- mag. But,asweshallseelater,theremaybesomescatterinthe from PettitandNicholson’smeasuresisprobablylessthan0.05 loss inthetelescope.(PettitandNicholson). may beinerrorbyasmallamount. Two photographicdeterminations 26.60 | cally determinedbolometriccorrections.Hence,thequantity0.07 Four visualdeterminations —26.77 weight1 r Sun-moon, plus5determinationsofmoon 26.77 2 sun thusfardiscussedare: Radiometric determination —26.94 3 r 26 25 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The value—12.66IPvhereobtaineddoesnotappreciablymodify This resultforthebrightnessoffullmoon,togetherwith 5. PettitandNicholson’spaperon“StellarRadiationMeasure- The uncertaintyofthebolometricmagnitudesunderived ThequantityAmincludesthezenithabsorptionofatmosphere aswellthe The fourgroupsofdeterminationstheIPvmagnitude Ap.68,279,1928;Contr.MLW.Obs.,No.369. r V Weighted mean—26.84+0.04 (p.e)IPv m(sun) =—12.66—(14.290.8X0.22)—26.77IP• THE STELLARTEMPERATURESCALE437 1938Ap J 88. . 42 9K 20 438 G.P.KUIPER lamp, which,aswehaveseen,wasinturncomparedwiththree mag. brighterthanthestandardlamp,forwhichweobtained—4.34 stars. Thephotoelectricmagnitudeofthesunwasfoundtobe21.46 value beinerrorbyanamountmuchlargerthan0.1or0.2mag.We yielding veryaccurateresults.Thedifferencecannot,therefore,be cussed. Caldercomparedthesunonthreedayswithhisstandard The photoelectricmethodusedbyCalderis,ofcourse,capable IPe. Hencew(sun)=—25.80IPe—26.22IPv. be differentfromthoseassignedinformingthemeanIPvvalue.We mean, —26.75IPv,ofthefirstthreegroupsobservationsgiven magnitude ofthesun,wecombineresultobtainedfromPettit conclude thatCalder’sobservationmaybeaffectedbyasystematic ascribed toanaccidentalerrorinCalder’swork,norcanourmean whereas lower thetotalweightofotherdeterminationsfrom3!to2^. raise theweightofradiometricdeterminationfrom3to5and error andthatitseemsbesttoomitatthepresenttime. The weightedmeanthenbecomes certainty inthebolometriccorrection,relativeweightshaveto and Nicholson'sradiometricwork,—27.01mag.,withtheweighted low-dispersion slitspectraDr.Morganandthewriterindependently above. Thismeanleadstomi=—26.82.Onaccountoftheun- found G2ontheconventionalscale. needed forfurnishingreliabledeterminations of effectivetempera- have radiiandluminositiesbeendeterminedwith suchaccuracyasis ture. Forthatreasonitisfortunatethesun providesasecond ho © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem One observationofthemagnitudesunremainstobedis- This valuedeviates0.6mag.fromthemeanobtainedabove. In ordertoobtainthemostprobablevalueofbolometric As mentionedbelow,onlyforthreestarsinaddition tothesun The thirdquantityneededforthesunisitsspectralclass.From = Wbol =“26.95,-^bol+4-62,(8) Wpv 26.84,-^pv+4*73•(9) SUNSPOTS 1938Ap J 88. . 42 9K 3 o29 27 32 /o28 30o Wanders haveshownthattheumbraofsunspotsisinradiative nearly thesameforGandKdwarfs,relationfoundwillbevalid determination foratypemuchlaterthanG2.SinceMinnaertand perature determination,wederivethefollowingconclusions. With thelimitsindicated,valuehasbeenplottedinFigure4. 4450 fromthesurfacebrightness,and4700degreeof and spectraltypewillbethesameasforordinarystarsinradiative equilibrium, itfollowsthattherelationbetweensurfacebrightness for dwarfs. equilibrium. Since,further,thevalueofsurfacegravityisvery present time. koek andfromPannekoek’srecentreviewofthemethodstem- of numerouslinesandbandsfortheearly-typestarsbyspace ory ofthecontinuousabsorptioncoefficient.Therelationisfurther An accuratecomparisonwithdwarfK-typespectrawouldbeof spots is5713'Vo.42=4600asdeterminedfromthetotalradiation, tory valuesbetweencertainpointsinthespectralsequenceforwhich great interest. sume thatthismeansthetypeiswithinlimitsG8andK2. ionization. Weadopt4600. show certainanomalieswhichmakethismethod uncertainatthe effective temperaturesmaybederived. reddening. Nevertheless,colorswillbeusefulforgivinginterpola- complicated forthelate-typestarsbypresenceinspectrum stellar colors;therelationisnotyetfullyaccountedforbythe- 28 27 2 31 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Ibid., p.361.vZs.f.Ap.,8,32,Í933;225, 1934. PhysikderSternatmosphären,p.363,1938. The effectivetemperatureoftheumbramediumtolarge-sized ^ Ibid.,p.363.95,S9)ïQSS- The spectraltypeofthesespotsisusuallygivenasKo.Weas- 3° Ibid.,p.366.33Ap.J.,84,481,1936. Temperature determinationsbasedontheBoltzmannfunction Effective temperaturesdependinasomewhatirregularwayon 2. Forthe0andBstarsbestdetermination maybemade 1. PartlyonthebasisoftheoreticalworkbyUnsoldandPanne- THE STELLARTEMPERATURESCALE439 TEMPERATURES OFEARLY-TYPESTARS 1938Ap J 88. . 42 9K 34 + log T e 440 G.P.KUIPER from astudyofthemaximacertainspectralseries,rediscussedon of abundancewithspectraltypeenterasaccidentalerrors.Ofcourse, independent temperaturedeterminationinwhichpossiblevariations tion, whichseemsparticularlyvaluablesinceeachelementgivesan Pannekoek hasdone.HisTableIVcontainsthenecessaryinforma- the basisoftheorycontinuousabsorptioncoefficient,as plotted inourFigure1.Theaccordancebetweenthedifferentele- ments isverysatisfactory.Evenifinthefuture itshouldbeneces- sary torevisethetheoryofcontinuousabsorption coefficient,it the early-typestars. the methodisindependentofspaceabsorption,whichaffectsmost tions couldbesolargeastoaffectappreciably thepositionsof placed atBgratherthanAoiftheclassificationnear isbasedonthestrengthof In additionDr.Morganhasinformedthewriterthat maximum ofSishouldbe seems unlikelythatthevariationswithtemperature ofthecorrec- the KlineandHe4471.Cf.Morgan,Pub.YerkesObs.,7, Part 3,1935. © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Pannekoek’s valuesforlogg=4.4(main-sequencestars)are 34 Ibid.,p.496.Doubleweightwasgiventothecolumnheaded “E.G.Williams.” Fig. i.—Thestellartemperaturescalefortypes0-G 1938Ap J 88. . 42 9K o ß AurigaeatmaximumontheIPvscale,2.10 +0.04andavalue P.D. Fromthesedata,andfromTable2,page 478,wefindfor ß Tauriis1.98+0.04(m.e.)P.D.andthatof Aurigae2.26+0.06 ß Tauri.AccordingtoZinner’scompilation, themagnitudeof our rediscussionleadstoasomewhatlowervalueofthetemperature other threesystemsarediscussedbelow,CastorConpages455- proximity ofA.TheparallaxRSCVnisstillveryuncertain. yet, andthelight-curveisstillsomewhatprovisional,owingto and asomewhatgreateruncertaintyoftheresult obtained. have ameasurableparallax.TheyarecollectedinTable1. mon toallfourpoints. to Pannekoek,thereisapossibilityofsmallsystematicerrorcom- maxima. Theinfluenceontheobservedmaximaofagradualtransi- 458. stars byPannekoek,basedontheintensityofKline.According are systematically3000lower. great differencesbetweentheionizationandexcitationpotentialsof temperatures foundaresomewhattoohigh. the seriesusedbetweentypesBoand08.Ifeffectispresent, lines shouldbeconsidered,butitisprobablysmallbecauseofthe tion withincreasingtemperatureofabsorptionlinestoemission 35 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 3s Ap.84,506,1936. Stebbins foundßAurigaeatmaximum0.35 mag. fainterthan 3. Fiveeclipsingbinariesareknownwhichshowtwospectraand Pannekoek hasalreadydiscussedthesystemofßAurigae,but For 44BootisBnospectroscopicorbithasbeendeterminedas In Figure1arealsoplottedfourtemperaturedeterminationsofA Pannekoek’s temperaturesforlogg=2.4(giantBand0stars) 44 BooB. Pi Sco.. ß Aur a GemC RS CVn... THE STELLARTEMPERATURESCALE441 TABLE 1 dK6+, dK6+ dCLp Ai, Ai B3, B3 F, G8 4 Spec. 48 ±12 37+4 73+3 79+5 7?4± o?8 1938Ap J 88. . 42 9K o 36 of (Ao)andisnearlyequaltothata Geminorum(Ai). This valueisplottedinFigure1. ness.) or ±0.04inlogT.(Obviously,theuncertaintyislarger preciably. of ^=0.5for10,000inthevisualregionspectrum(selenium creases moreslowlywithincreasingTthanthebolometricbright- than ±0.24/10,sinceforthistemperaturethevisualbrightnessin- koek’s theoreticalworkonlimbdarkening,whichindicatesavalue The lastlinefollowsfromequation(2),page430,inconnectionwith the intermediatesolutionaprobableerrorof±960°inTisfound, cell). Aslightlydifferentvalueofuwouldnotchangetheresultap- as wellanintermediatecase,wehavefortt=o''037: the tableofbolometriccorrections(Table6). by Pannekoek.Thetrigonometricparallaxbeingo''o37±o''oo4 slightly brighterthan2.04+0.06,respectively.Weshalladopt2.07 tainty intheabsolutemagnitude,owingtoparallax. 0.24 (p.e.). IPv, whichhappenstobeidenticalwiththeHDmagnitudeused 442 G.P.KUIPER ture: (1)theunknownvalueoflimbdarkening;(2)uncer- 2 (p.e.), theabsolutemagnitudeofaveragecomponentis+0.66 e U (uniform)andD(fullydarkened).Consideringtheseextremes, e e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We have,accordingly,forAi: According toBaker,thespectraltypeismoreadvanced thanthat The secondsourceofuncertaintyismoreimportance.With The intermediatesolutionwillbeadoptedonthebasisofPanne- Shapley hasgiventwophotometricsolutionsofStebbins’data, There aretwomainsourcesofuncertaintyinthederivedtempera- 95, 733,1935- log R/RO COS Í (h. Te. log T=3.99±0.04(p.e.).(10) e 0.414 0.218 0.147 10150 u \{U +D) 0.431 0.223 0153 9800 0.448 9500 0.228 0.159 1938Ap J 88. . 42 9K 38 37 o o 40 39 ponents, —1.82IPv,hasaprobableerrorofatleast±0.24mag. lax dependslargelyupontheprobableerrorofPCGpropermo- nate. Theprobableerroroftheparallaxis,therefore,atleast11per mag. brighterthan/¿.Themagnitudeofthelatterstaris3.92PD method usedinthenextarticlethatR/a=0.346,andadjusted 0.6 or0.7ofA.SinceJf/Jb=0.613(RudnickandElvey),wemay nick andElvey.Atmaximum,^Scorpiiwasfoundtobe0.60 value oftheprobableerrorthisresult(owingtouncertainty uncertain byprobablynotmorethan0.1mag.,wefind16,000asthe value oíai/a=0.370.WiththisofRandMissMaury’s assume thatthecomponentshaveequalradii.Incasea/a= tion, whichisiipercentofthepropermotionitselfineachco-ordi- ter parallaxis0^0074(Kapteyn).Theprobableerrorofthisparal- of theparallaxalone)is±1900.Inlogarithms: of asini=9,534,000km,togetherwith62?o,wefindRR/ cent ofitsvalue,andtheaverageabsolutemagnitudecom- ratio havebeendetermined,isSZCamelopardalis,forwhichWes- average effectivetemperatureofthecomponents.Theminimum a//a =0.380andbi/a0.336.Fromthesedatawefindbythe mary is—4.8,ifAm=1.0assumed. distance modulusof12.0,theabsoluteIPvmagnitude ofthepri- cording toMissMaury. This valueisshowninFigure1atB3,theaveragespectraltypeac- determination oftheabsolutemagnitudewill bepossible.Witha tem beingamemberoftheclusterNGC1502, afairlyaccurate selink publishedanexceptionallyaccuratelight-curve. Thesys- come ofgreatinterestwhenthespectroscopicorbitandmass (Zinner), or3.68IPv,sothatatmaximum^is3.08IPv.Theclus- 2 b b = 5-370. 3839 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 4. Thesystemof^ScorpiihasrecentlybeendiscussedbyRud- Cf.thetreatmentofVPuppis,p.500.Dissertation, Leiden,1938. Adopting thebolometriccorrectionsofTable6below,whichare Miss MauryfoundthephotographicintensityofBtobeabout 4oTrampler, LickObs.Bull.,14,156,1930.SeealsoAp.J86, 187,1937. vAp. 87,553,1938. 5. Aneclipsingsystemofstillearliertype(Bo±),whichwillbe- THE STELLARTEMPERATURESCALE443 log r=4.20±0.05(minimump.e.).(11) e 1938Ap J 88. . 42 9K o 41 + + 444 G.P.KUIPER the probableerrorofTandloghasbeencomputedunder assumption thattheprobableerrorofparallaxis10percent tain errorintheparallax.InTable2,forsixdifferentvaluesofT solute Tdetermination,itisofinteresttoknowtheeffectacer- for veryhightemperaturesfollowssimplyfromtheRayleigh-Jeans sorption). Itmaybehopedthatforgalacticclusterstheprobable which fordistantobjectsisreallythequantitydesired(spaceab- its value.Thiscorrespondsto+0.22mag.inthedistancemodulus, It followsfromTable2thatfortherange10,000°-40,000(A0-O8) error maybereducedbelowthisvalue.TheprobableoflogT the percentageerrorinTisaboutsameasthatparallax. law: of theparallax,whereasfortemperaturesGandKstarsper- was extensivelyusedbyPannekoek,wemaytentatively deriveone by Pannekoek.FromthepublicationE. G. Williams,which more maximum,thatforHe.AplotofWilliams’ measuredinten- centage errorwillbelessthanhalfthatoftheparallax. For muchhighertemperaturestheerrorinTwouldbedoublethat e e e ably lessagain.Themaximumisprobablynear O7.Accordingto sities showsasteadyincreaseinstrengthofHe towardtheearher types uptoO7,whilefortheonly06star strengthisconsider- e e e 41 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Since, ingeneral,theparallaxismainsourceoferroranab- 6. Starsearlierthan08arenotcoveredbythemaximadiscussed Ap.J.j83,296,TableVI,1936. Probable Error,ifP.E.ofttIs10PerCent Mp =C V 2.5 logr(r-+00). e TABLE 2 1938Ap J 88. . 42 9K o o43o4 42 45 ++++ +4-4- 1 46 spectrum butalsoanabsorptionO7spectrum.Thesestars,HD 44,000 forlogg=4.4. by BealsusingtheZanstramethod)ofabout59,000and65,000. stars. TwostarsarementionedwhichhavenotonlyaWolf-Rayet Pannekoek, thetemperaturecorrespondingtothismaximumis peratures. Theaverageofthetwominimumtemperaturesfoundby As Gordeladsestates,thetemperaturesderivedbystrictapplica- levels contributeappreciably.AsDr.Wurmmentionedtothewrit- absorptions fromthegroundstateandthathigher shells (Wolf-Rayetstars,Beetc.,contrarytoplanetarynebu- strong, whichwillcausemanyatomstofallintointermediatelevels. lae), wasdiscussedtheoreticallybyGordeladse,whopointedout that theobservedemissionsinsuchobjectsareonlypartlydueto peratures derivedforBestarsbytheunmodifiedZanstramethod, the Zanstramethod,62,000°,maythereforebesomewhattoohigh. tion oftheZanstramethodwillinthiscaseleadtotoo-hightem- er, thisfactwillbeofimportancewhentheobservedemissionsare obtained byadetailedtreatmentofthebehaviordifferentatoms probably accountsforthesmallscatterin temperaturesfound tures fortheabsorptionlines.Moredefiniteresultscouldonlybe and CinHD193793,WVI;He,V++,N from thecircumstancethatcertainemissionsarepresent(He,C 937 andHD211856,havemimimumtemperatures(determined as givenbyO.Mohler,maybeinerror.Inaddition,aselection effect enters;onlystarshavingemissionsabove acertainstrength 5o,ooo°-55,ooo°, foundbycomparisonwithPannekoek’stempera- in dilutedradiation. are discussed,whateverthecauseoftheseemissions maybe.This 211856, W6orW7).Thiswouldleadtotemperaturesoftheorder 44 42 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Another sourceofinformationisBeals’sstudytheWolf-Rayet The Zanstramethodasappliedtostarshavingrelativelysmall Another estimatemayeasilybemade,asDr.Wurmsuggested, Op.cit.,Table8andp.143. 43 Op.cit.,Table8,p.139. Gordeladse’s remarksalsoapplytotheemissionBstars.Tem- Pub.Dom.Ap.Obs.Victoria,6,95,1934. Zs.f. Ap.,13,48,,1936.^Pub.U.Mich.Obs.,5, 43, 1933. THE STELLARTEMPERATURESCALE445 1938Ap J 88. . 42 9K 49 47 48 by Mohler(whichaverage18,000°)andtheabsenceofanotable derived inthenextsection,areadded. extrapolations. Thebolometriccorrectionsforthesetemperatures, 446 G.P.KUIPER photovisual magnitudesareverysimilartothose forvisualmagni- rections, B.C.=m—w.Threecomputationsofthisquantity in Table3.Thevaluesfor06andO5,givenparentheses,aremere for theO,B,andAstars,meanvaluesmayberead,whicharefound dependence onspectraltype(B0-B8). bolometric correctionforphotovisualinsteadofmagnitudes. tion. Hertzsprungmadethecomputationforlowertempera- spectral classesitisadvantageoustoconsiderthebolometriccor- B2 Bi P- 139- Bo O9 08 07 06 05 as afunctionofThavebeenmade,allassumingblack-bodyradia- matic. we denotewithAw,bolometriccorrectionscomputedontheassump- tures, usingthesensitivitycurveofeye;Pikedidsamefor tion ofblack-bodyradiation.Itisseenthatthesecorrectionsto From hisdatawederivethebolometriccorrectionsgiveninTable4; the highertemperatures.FinallyHertzsprunghascomputed tudes, althoughphotovisualmagnitudesaremore nearlymonochro- bolpv e 8 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem v Zs.f.Wiss.Phot.,4,49,1906;Eddington,TheInternal Constitution oftheStars, 4 M.N.,89,539,1929. 7. FromFigure1,whichshowsallthetemperaturedatacollected w Zs.f.Wiss.Phot.,30,173,1931. i. Beforeattemptingtofixthetemperaturescaleoflater Spec. (4.8) 4.9 (4.9) -5.6 log T 4-31 e 4-36 4.40 4-50 4.60 4-7 BOLOMETRIC CORRECTIONS — 2.22 B.C. 4-3 3-24 3-8 2.48 2.70 TABLE 3 As- A2. Ao. B8. Bs B6. b b 4 3 Spec. log T t 93 986 03 09 16 IQ 23 27 -0.31 — 2.01 B.C. 0.52 0.72 1.58 i .04 1.79 i .42 1938Ap J 88. . 42 9K 51 53 52 54 o values Awcanthereforenotbeused;theyneedacorrection which wedenotebybm.Thiscorrectioncanbeevaluatedonlyif available tothisidealcase:(1)theoreticalspectralenergy-curves, the wholespectralenergy-curveisknown.Twoapproximationsare computed onthebasisofacertainassumedcompositionstellar the inaccessibleultraviolet). , and(2)radiometricobservationsoflow-temperature Pannekoek. Unsoldadoptedforthestellaratmospheressame stars (forhightemperaturestoomuchoftheenergyisradiatedin stellar interiors(B.Strömgren,Eddington).Thisratiocorresponds abundance ofhydrogen,one-thirdbyweight,ashadbeenfoundin to 14hydrogenatomsforeachmetalatom.Burkhardt,indiscussing 0.62 mag.,againstPannekoek’s1.20mag.Theobservationsgive used Pannekoek’scomputationsinpreferencetoUnsöld’sbecause the intensityjumpatheadofBalmerseries,laterconsidered used byPannekoek,inviewofnumericaluncertainties inthetheory about 1.2mag.(AlthoughthisagreementwithPannekoek’spredic- tions doesnotnecessarilyconfirmthehighabundance ofhydrogen Unsold, PhysikderSternatmosphären,p.134,Fig.536. for T=10,080Unsoldpredictstheintensityjumpat3647Atobe 50:1 forthisratio,whereasPannekoekadopted1000:1.Wehave 50 1 50 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem It iswellknownthatthestarsdonotradiateasblackbodies.The 2. Spectralenergy-curveshavebeencomputedbyUnsoldand s Ap.J.,84,484,1936.52Zs.f.Ap.,13,56, Zs.f.Ap.,8,241ff.,1934;PhysikderSternatmosphären,pp. 131-138,1938. Handb. d.Ap.,7,460.SeealsoKienle,Erg.exakt.Naturwiss., 16,462,1937; Unsold, op.cit.,pp.135-136. 4.000. 4.500. 3.500. 3.000. 5.000. 2.500. 2, ooo° T THE STELLARTEMPERATURESCALE447 -4.69 —0.18 Am 0.36 0.64 1-74 2.84 i .07 TABLE 4 12.000. 10.000. 15.000. 20.000. 6, ooo° 8,000. T — 0.02 — I.01 Am O.06 O.28 O.55 O.97 1938Ap J 88. . 42 9K o o 57 o 55,56 0 o of thecontinuousabsorptioncoeflScientagreementisimportant nounced amechanismofcontinuousabsorptionthatmay remove thedifficulty. values varyingbetween0.08mag.{T=5040)and0.25 from integrationsoverthecomputedspectra,forthreetempera- longer exactlyholds.Pannekoekgivesthenecessarycorrections, lation betweensurfaceandeffectivetemperatures,7^=27^,no fined bytheslopeofspectralenergy-curve)andeffectivetemper- for thederivationofbolometriccorrections). for T=6600whenAwiszero.Inordertoavoidbolometriccor- sities ofTablei,wefind,forX=5500A,thebutvaluesgivenin thus obtainedaregiveninTable6fortemperatures upto50,000. Table 5.Thetruebolometriccorrectiontophotovisualmagnitudes these correctionsistoincreasethetotalbolometriccorrectionby terpolatory correctionsfortheothertemperatures.Theresultof tures stated.Intheabsenceofdetailedcomputationsweadoptin- Pannekoek hascomputedspectralintensitiesformostofthespec- which hasnotbeenclearedup. for thesolartemperaturedifferencebetweentwotempera- be consideredasfinal.Itistruethatfortemperatureshigherthan bm valuesbeforeaddingthemtoAw.Thebolometric corrections rections withdifferentsigns,wesubtract+0.10 mag.fromallthe 5000, and6000A.Acomplicationarisesfromthefactthatre- trum; forothertemperatures,onlyatthreewavelengths,4000, ature isinfairagreementwithobservationforsuchwidelydifferent 448 G.P.KUIPER tures ispredictedwiththewrongsignbybothauthors,adifficulty compositions ashavebeenadoptedbyUnsoldandPannekoek;but 65oo-7ooo° thetheoreticalrelationbetweencolortemperature(de- tinuum, thecorrectionshmobtainedfromPannekoek’sdatacannot 25,200, thehighesttemperatureconsideredbyPannekoek). is, then,B.C.=Aw+bm. e e 56 55 57 o © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem DuringasymposiumattheYerkesObservatory(June, 1938)Dr.Wildtan- Cf.alsoMinnaert,Zs.f.Ap.,12,260,1936. Op.cit..p.486. Applying thesesmallcorrectionstothecomputedspectralinten- It sohappensthatbmreachesthemaximumvalueof+0.10mag. For threeeffectivetemperatures—10,080,5040and3150— In spiteofthisagreementonthestrengthBahnercon- 1938Ap J 88. . 42 9K o vide extremelyvaluableinformationaboutboth thebolometriccor- tions isimportant. that thesecondsourceofinformationregarding bolometriccorrec- color indexispredicted.Fortunately,itfor the lowtemperatures absorption bandsandlinesthatcanbeonlyroughlycorrect;also, than 25,000,assumingthatitslowlydecreases. The correctionbmhasbeenextrapolatedfortemperatureshigher as wehaveseen,forthesolartemperaturean incorrect valueofthe Furthermore, thevaluesgivenarebasedonasmoothingprocessfor 58 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 3. TheradiometricobservationsbyPettitand Nicholsonpro- For lowtemperatures,bmisnotwelldeterminedbyTable5. Ap. 68,279^1928;Contr.Mt.W.Obs.,No.369. 6300. 3150° 5600. 5040. 7200. 10 .. 12.. 11.. 4- 5 4- 0 9.0 8.0 6- 5 6.0 3- 5 3-0 5- 5 7- 5 7.0 5- 0 Theoretical ValuesofBolometricCorrections T e THE STELLARTEMPERATURESCALE449 (Temperatures iniooo°) +O.O9 +O.06 + 0.01 — O.98 -3“2: — O.06 — I.2Ó B.C. 8vt O.78 0.12 O.57 O.4O 0.22 0.01 0.00 0.17 0-34 0.63 0.06 2.1: I-3: TABLE 6 TABLE 5 45 40 35 30 0 50 252OO. I260O. 25 22 20 IQ 17 16 1680O. IO080. 18 15 14 IS 84OO Te —o. IO -0.44 -4-3 -i^iS 1 B.C. hm 3-8 4.1 3-5 3.12 2.69 2.40 2.18 2.06 i .80 1.66 •5 1.94 1-35 .46 •37 . 21 1938Ap J 88. . 42 9K 59 450 G.P.KUIPER rections andthetemperaturescale.Thetheoreticalworkdiscussed may beaboutioor15percent.Ifthisresultiscorrect,the stars aredisplacedtowardtheredwithrespecttoPlanckcurves, low; butthebolometricmagnitudesobtainedwouldstillbecorrect, that correspondtotheireffectivetemperatures.Thedisplacement above indicatesthattheobservedenergy-curvesoflow-temperature dices, beingessentiallycolortemperatures,should,ofcourse,betoo temperatures derivedfromwater-celltransmissionsandheatin- perature hardlyaffectsthederivedbolometricmagnitude. losses) varybetween0.4and0.6mag.forspectraltypesF5 Furthermore, thecorrectionsLm(whichincludeinstrumental since thereductionsto“noatmosphere,”Am,werecomputedwith a Planckcurvecorrespondingtotheobservedenergydistribution. in Table7.Theradiometricmagnitudes,ra,werecorrectedby strument (sothatdeviationsfromaPlanckcurvecannothaveen- and M6,indicating,first,thatnearlyalltheenergyreachedin- rived forallstarsofclassesA0-M6.ForFo-Mtheresultsaregiven tered seriously)and,second,thatasmallerrorintheassumedtem- bridge 34myownmeasureswereused,andforB4342thevaluewas ner’s compilationorPotsdamPublications,Volume17;forGroom- ranges notlessthan0.2mag.areindicatedbyparentheses. taken fromMrs.Gaposchkin’scompilation.Variablestarswith both thetheoreticalcomputationandradiometricdetermination methods forstarsoftypesdFo-Go.Thisistheonlyintervalwhere of Table6byacomparisonthecorrectionsderivedtwo tions apartfromthezeropoint.Thelatterwasmadeequaltothat r r r is asmalldiscrepancyfortheAstars,probably becausethedevia- should beapproximatelycorrect.ThecomparisonisshowninTable 8; thetemperaturescaleofFigure1wasadopted. Thedifference tions fromthePlanckcurvewereaccountedfor inthefirstcomputa- F2 —Goisnearlythesameasthatfoundby both methods.There tion butnotinthesecond;differenceis therightdirection. — Aw;thenmagnitudesontheIPvscalewereobtainedfromZin- r 59 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem From PettitandNicholson’sdata,bolometriccorrectionswerede- The differences,(m—Aw,)w,givethebolometriccorrec- ThecorrectionsaregiveninTable2,p.478. rpv 1938Ap J 88. . 42 9K o We shalladoptTable6forstarsearlierthanFoand7 determined bolometriccorrectionzeroat6600)theconstantcorrec- stars laterthanFo. has recentlyobtainedaveryhomogeneousand accuratesetofspec- tion totheMountWilsonvaluesofm—Am is+0.62mag.This same zeropointhasbeenadoptedinTable7. 70 OphA.. B.. r r 02 EriA.. 7 Cyg y Tau ß Cas ß CVn.... ß Cet ß Com.... ß Her a Tri \p UMa.. 0 Aql r] Psc 1 Peg X Ser a Per a CMi.. ß Gem a Aur. a Aqu 77 Boo y AndA.. a UMaAB r Cet a Ser a Ari a Boo £Peg o Tau r Psc 5 Eri e BooA... € Leo « Vir e Eri i Per © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem With thezeropointjustdefined(whichmakestheoretically The spectraltypesinTable7aremostlydueto Dr.Morgan,who a AB Star 2 413 0-45 3- 47 4.27 4.08 4.40 4.26 o-94 3.18 3 09 3.78 3.80 3-75 3-53 309 2- 37 2.82 4.20 4-55 0.04 3- 57 3-72 2.29 1.90 3.12 4- 38 2.90 2.18 2.80 369 2.66 1.96 2-33 5.90 1.23 2.76 2.03 IPv THE STELLARTEMPERATURESCALE451 dGo dGi dGo dG6 dK dG gG6 gG6 gG6 gGi Sp. dK dKi dK gC9 gG6 gG6 gG2 gC9 gG8 gK2 gKi gK2 gK gK2 gK2 dF dF dF dF6 2 9 dF 4 2 3 CF9 CF4 CF7 3 4 2 9 Gi g G Fq 9 9 + 1.6 +3-1 + 2.6 + 2.8 +2.0 -po. 8 +4-9 +4-3 +4-4 +0.9 + 5-9 +0.7 +3-7 -po. 8 + 5-9 + 1.2 +3-8 +3*2 -4.4 + 1.0 +7-4 +6.1 + 1 — 2. +5-7 +0.4 -0.5 -0.7 -1 + — o.i -5- — 2. -0.4 -3- -1.4 -0.3 M o. + —o B.C. 0.02 47 02 07 07 01 09 18 23 16 12 13 Gr. 34.... TABLE 7 45 Ari... 30 Psc... 61 CygB 61 CygA 53 EriAB 55 Peg... Ci 1218.. 62 Sgr... 51 Gern.. 56 Leo... 15 Tri... 13 Sge... m Vir... R Lyr... B 2935.. B 4342.. B 3100.. y Dra.. Tr Leo... ß Cnc.. ß And.. X Aqu.. a Hya.. p Gem.. ß Peg... a Tau.. a Cet... a Sco... a Ori.. ô And.. f Cep.. P Per... a HerA £ Cyg.. ô Vir... Star 10 38 (2.62) (0.96) (6.O2) (4-3±) (3-52) (5-88) (5-52) (5 •18) (3-65) 4.78 6.52 6.09 3.82 3- 5 3-67 3-94 3-42 4.60 4.62 8.09 4- 5 7 3. •3 2.22 5-43 2 .l8 3.62 2.41 1.02 5-39 5 13 2.81 1.30 7-57 + 5-58 7-50 IPv *dK+ *dMi-p *gM2 *dM2 + *gM+ *gM2 + *gMi *gM+ *gM *gM *gM *gM *gM5+ 4 3 4 5 4 4 4 gM2 gMo gM3 gM3 gM4H- gM4-P gM3 cMo cMi + dK6 dK6 dKs gK5 gK4 gK4 gKi gK3 gK-P gKs gMi CK4 cKi 5 M5 Sp. + 1-3 + 0.5 + 8 + 8 + 7 + 7 + o - 0.7 - 1+ - 0.4 + 0+ + 10 - 4 - 0.6 - 4+ + 10 + 1: — 3 — i — 1+ — o — 4+ — 0+ — I - 1+ — 2 — I Jfpv O o o o o o o o o (-4.28) —0.87 1I (3-7±) (427) (2.99) (2.68) (2 -79) (2:92) (2.25) (2.55) B.C. 0.50 0.94 0.94+ I.46 I .21 1- 50 •3 I .12 1.04 1.63 1.06 I-3I 2.54 I .96 2.08 2.83 2.31 2.36 2.06 i .89 i .98 1.98 i .66 2- 9 5 1.48 1938Ap J 88. . 42 9K 60 452 G.P.KUIPER my own.ForaOrionis,MorganobtainedCM3;butsinceitispossi- moved, butfortheKandMdwarfstypesareleftunchanged. sent thedwarfs,giants(M=o.o),andsupergiants—4.0),respectively. The systemforthegiantsispracticallyidenticalwiththatusedat Mount Wilson.Afewdeterminations,indicatedbyanasterisk,are the GstarswithrespecttoMountWilsonsystemhasbeenre- tral typesforthebrighterstars.Asmallsystematicdeviation nometric determinationsandpartlyonspectroscopic data,someof ble thatthespectraltypevariesslightly,aswellmagnitude, spectral type,takingintoaccounttheabsolute magnitude.Three which werekindlysuppliedbyDr.Morgan. lines aredrawninthediagram,indicating meanpositionsfor the average,Mi+,ofalldeterminationsavailablehasbeenadopted. v 60 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 2.—EmpiricalbolometriccorrectionsfortypesF-M.Thethreecurvesrepre- Mostlyunpublished. Figure 2showstherelationbetweenbolometric correctionand The absolutemagnitudesarebasedpartlyon up-to-datetrigo- FO F5GOG5KOgKSgMOgM3gM6 1938Ap J 88. . 42 9K Mpv =—4.0(supergiants),respectively.ThecurveforM0.0is main-sequence stars,starswith=0.0(giants),and pendent fromeachother,areseentofitthecurveverywellindeed. well determined;thenormalpoints,indicatedbycrossesandinde- rediscussion oftheaccurateradiometricobservationswithotherim- in PettitandNicholson'soriginaldiagram.Thisresultjustifiedthe proved data.Theabsolute-magnitudeeffectisinagreementwith K2. Ko. be usedforstarswhichnoradiometricdata areavailable.For magnitude groups,readfromthecurvesofFigure 2.Thistablemay k. that shownbycolors;itdisappearsintheearlyFstars. should bemadeforallstarswherethebolometric magnitudeisneed- ever, thatadirectdeterminationoftheradiometric magnitude the Mstarsincreaseinbolometriccorrection issorapid,how- G2. Go. G8. G5 ed. This,ofcourse,isalsotruefortheKsupergiants, particularly F5. F8. F 2. 3 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The absolute-magnitudeeffectisveryclearlyshown,moresothan Table 9givesthebolometriccorrectionsfor the threeabsolute Sp. dF2. dA. dGo. 2 Empirical BolometricCorrectionsforDifferentMv P Main Se- quence ■O.31 ■0.4 THE STELLARTEMPERATURESCALE453 •IS .06 .05 .04 . II . IO . IO .07 Spec. M =0.0 —o 0.04 ,08 47 SI 39 89 54 25 72 I? M =—4.0 •1-35 •0.04 0.42 0.12 0.52 0.28 0. 9 3: o. 80: 0.65: 1. 20 TABLE 8 TABLE 9 No. ii 4 5 Mo. Mi. m. m. m. M2. K. K6. K5. 5 4 3 4 Sp. Theoretical -0-53 —o. 10 .01 -(3.1O •. 1.43 Main Se- -©■S quence (2.7) (2.35) 0.85 1.14 2.03 1.70 Radiometric —0.42 — 0.06 M=0.0 : .04 '3 -4 I. II 1.72 1-55 1-35 2.72 2.26 1-95 M =—4.0 ■3.6: 3.0: 2.2 2.6 1.86: 1.56 1938Ap J 88. . 42 9K 61 mate valueoftheabsolutemagnitude,isknown.Theuncertainty when theabsolutemagnitudeisuncertain.Butforearliertypes 454 G.P.KUIPER which isgoodenoughformostpurposes. in thebolometriccorrectionshouldthennotexceedabouto.imag., color indexincreasescontinuouslyforthelaterdwarfsto2.0mag. Table gshouldbeusefulifagoodspectraltype,aswellanapproxi- (M8),andtherecanbenodoubtthatthebolometriccor- Mp. Figure3showstheresult,whichisalsotabulated inTable10. that thevaluesobtainedarenotinconsistentwithobservationsof rection valuesfortypeslaterthandM2;weshallseeonpage487 rection willalsoincrease.Weshallextrapolatethebolometriccor- Unfortunately, therelationisnotyetvery strongly determined. sequence, thebolometriccorrectionmaybeplotted directlyagainst colors andspectra. V 61 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem No radiometricobservationsareavailablebeyonddM2.The Morgan,Ap.J.,87,589,1938;Kuiper,ibid.,p.592. Because oftheverysmallscatteronlower partofthemain Fig. 3.—Empiricalbolometriccorrectionsformain-sequencestars i 2345678910iiMpv 1938Ap J 88. . 42 9K 6 64 6263 65 water-cell absorptions)thanbycolorindices.Thelatterdefineonly better determinedbyradiometricobservations(heatindicesand solved binarieswill,ofcourse,affectTable10butnot9. Table 10willbeusefulwhenthespectraltypeisuncertain.Unre- served spectralenergy-curve(uncorrectedforabsorptionlines)may which, moreover,iscrowdedwithabsorptionlinesandbands. a smallpartoftheshort-waveslopespectralenergy-curve, be systematicallytoolow,byperhaps10or15percent(Pannekoek). We coulddeterminethispercentageempiricallyifforsomelate- type starsabsolutetemperaturedeterminationswereavailable. few visualobservationsmadein1923werepublished byHaas. Van Gentalsorediscussedthedimensionsof thesystem.Latera ma. TheorbitalelementswerederivedbyJoyandSanford, found thatthesystemisaneclipsingvariablewithtwoequalmini- Joy tobeaspectroscopicbinary,showingtwospectra.VanGent diameter. an improvedfight-curvewassubsequentlypublished byvanGent. Castor Cmaybeused;forthegiants,measurementsofangular 665 62 666 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Yet thereareindicationsthattemperaturesderivedfromtheob- 3 B.A.V.,3,121,1926.B.A.iV.,6,97,1931. i. Forlow-temperaturestars,thespectralenergy-curveisfar Pub.A.S.P.,32,158,1920. This, fortunately,isthecase.Fordwarfseclipsingbinary Van Gentinhissecondpaperused1166exposures madeontwen- *Ap. 64,250,1926.A.N.,242,119,1931. 2. CastorC.—ThisimportantstarwasdiscoveredbyAdamsand 4.0. 6.0. 3.0. 5-0. Empirical BolometricCorrectionsforMainSequence Afpv THE STELLARTEMPERATURESCALE455 TEMPERATURES OEKANDMSTARS — 0.05 — 0.12 B.C. .06 •OS TABLE 10 10.o. 9.0. 8.0. 7.0. jWpv -0.51 -1.85 B.C. 0-95 1.40 1938Ap J 88. . 42 9K © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 67 68 456 G.P.KUIPER part ofthelight-curvenearminimumthathasactuallybeen of mattertowardthecenterwillnotbeverydifferentfromEdding- van Gentassumedtwoidenticalsphericalstars. have equaldepth,whichagreeswiththeequalityoftwospectral so thatthelight-curvehasconsiderableweight.Thetwominima ty-one nights.Theprobableerrorofoneexposureis±0.046mag., which theweightsofequationsconditionwerecarefullyde- 0.0068 mag.Only0.23ofthisvalue,or0.0016mag.,entersinthe ton’s standardmodel(p/p=54);(2)thevalueofr/aissmall(0.16). two reasons:(1)forstarsofnottoolargeamasstheconcentration types asobservedatMountWilson.Inhisphotometricsolution spectra tobeabout5:4.Thesurfacebrightnessesbeingthesame, used. we find,withrja=0.16,thattheratiobetweentwoequatorial Roche’s modelincomputingtheellipticityofcomponents,for for theaveragecomponenteveniftworadiiareslightlyunequal. termined. Weshallexaminewhetherhissolutionmayberetained the radiiareaboutinratioV5:. Mount Wilsonobserversfoundtheratioofintensities Taking theperiodsofrotationtobeequalperiodrevolution, each withradiusr,thefractionoflightlostis The resultswillhavealsoamoregeneralinterest. axes is1.0063.Hencethecoefficientofellipticitycorrection D =i(firstorlastcontact). where Distheprojecteddistancebetween centers,expressedin terms of2r.Dvaries,therefore,between= o (eclipsecentral)and here explicitlyareinagreementwithRussell’sstatements,Ap. /.,35,330,1912. c tude, thecaseofnearlyequalradiiisspecialimportance. Theconclusionsreached 68 6 Van Gentobtainedr¡aandsinifromaleast-squaressolutionin The assumptionofsphericityisverynearlycorrect.Wemayuse The assumptionofequahtyinsizeisnotquitecorrect,sincethe Sincealongthemainsequenceradiusvariesveryslowly withabsolutemagni- For twocirculardisksofequalanduniformsurfacebrightness, 7 Cf.p.496ofthenextpaper. 2 - (arccosD—V1),(12) 7T 1938Ap J 88. . 42 9K 69 2 2 of lightlost,neglectinghigherthansecondpowers€,is sum unchanged:ti—Ki+e)andr=r(i—e).Thenthefraction from thesymmetryofproblem.Thatmeansthatsolutions tial eclipses. This expressionisvalidforallvaluesofDexceptverysmallones unity (beginningandendofeclipse).Seriesdevelopmentgives evident fromtheequationsthatthisalsoholdsforDvaluescloseto ously changedbutthatiincreasesbyo?2oro?3. value ofthegratingconstantisused.Itappearsthatr¡anotseri- in thegratingconstantused,amountingto0.005mag.permagni- in whicha=-ï-ï-—-0.06. The smallcorrection(14)isapproximatelyequivalenttoaerror mean component,exceptthattheinclinationshould beincreasedby therefore examinedwhatchangeiscausedinthesolutioniflarger adopts 0.923mag.andOosterhoff,morerecently,0.95Ihave tude. Thisgratingconstanthasaslightuncertainty.VanGent has notyetbeenstudiedempirically;thiswould beparticularly about o?2.Thissolution,madeforuniformdisks, isr/a=0.16005; (D>e). Hence,(13)and(12)shouldbecomparedonlyforpar- (12) and(13)differonlyine,^ 2 i =86?. S 2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Suppose, now,thattheradiiareslightlydifferentbutwiththeir The firstpowerofeislackingin(13),asmighthavebeenexpected For CastorC,r\lr\isabout5/4;hencee=0.056and0.003. It ispermissible,therefore,tousevanGent’s solutionforthe The amountofdarkening,anditsdependence onwavelength, 7, ii51933. x X X (l3)(l2) =-Mi+a(iD),(14) (l2) THE STELLARTEMPERATURESCALE457 128 384 arc cos D- dVi-& (12) 2 I -Z>]3/ 2 e. Itisnotatonce D (13) 1938Ap J 88. . 42 9K 7071 71 promising inviewofthefactthatPannekoek’stheoreticalwork’ 4SS ponent ofCastorC,logg=4.61(c.g.s.units);fromPannekoek’s predicts astrongdependenceonwavelength.Fortheaveragecom- A fullydarkenedsolution,kindlymadebyDr.A.J.Wesselink,gave meantime wemayadoptthetheoreticalvaluefor4200A,u=0.3. Verification ofthistheoreticalresultwillbemostinteresting.Inthe r/a =0.163andi86?o. Table 26awefind by thewriterontwonightsandfoundtobe8.93IPv.Theprobable r/a =0.169andi85?2.Foru0.3wemaythereforeadopt ble 9,thebolometriccorrectionis—1.28mag.Table10givesfor writer, isK6+;thestardoesnotshowTiObandsinvisual error isestimatedtobe±0.05mag.Theparallaxbeing0^073+3, region (

Greenwich 1932* 0 = 1.07 + 0.05 (p.e) Greenwich 1933 t 0.91 + 0.05 KienleJ 1.15^:0.05

o Mean <¿>0 = 1.04, or Tc = 17,000 * M.N., 94, 488, 1934. ^ Ibid., p. 505. %Erg. d. Exakt. Naturwiss. 16, 446, 1937.

The fourth is the photoelectric determination by W. Becker.78 His value is 16,400o. The spectrophotometric values may be expected to be somewhat higher than those based on colors, since the absorp- tion lines, which crowd toward the violet, are avoided in the spectro- . That the high color temperatures for Ao stars are, in fact, to be expected theoretically was shown by Unsold79 and, more explicitly, by Pannekoek.80 For the classes earlier than Mi we shall base our further discussion on two sources of color temperatures: (i) W. Becker’s recent deriva- tion,78 which is of considerable weight for two reasons: {a) the photoelectric colors are accurate and homogeneous, and (b) a large number of stars (738) is included; (2) Bfill’s temperature scale,81 which is based on most of the important series of determinations of published previous to 1932.

The two scales of color temperature, expressed as c2/T, are given in the second and third columns of Table 13. The spectral types in the first column are on the same system as those in Table 7: the giants are on the Mount Wilson scale, and also the dwarfs of type earlier than K2. For the later dwarfs the scale adopted by Dr. Morgan and the writer61 is used because of its applicability to very faint stars.82 In this system the M stars (giants and dwarfs) are

78 Veröjf. Berlin-Babelsberg, 10, Heft 6, p. 8 (Table 5), 1935.

^Zs.f. Ap.} 8, 246, Fig. 4, 1934. 80 M.N., 95, 532, 1935- 81 Handb. d. Ap., 5, 193 and 199, 1932. 82 An M star of 15.0 can be obtained in less than one hour with the spectrograph used by the writer on the 40-inch telescope.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1938Ap J 88. . 42 9K 83 Whereas theclassesthusobtainedforgiantsappeartobeinex- dwarfs differsystematicallyby—0.1classatMo.AtM4thereis classified bythestrengthofTiObandsinredandorange. cellent systematicagreementwiththeMountWilsonclasses, cording totheprecedingparagraph,adopttheirtemperaturesas to benearlyontheMountWilsonsystem.Wemaytherefore,ac- agreement again. HR511.. HR 483.. HR 753-. here. ThedataarecollectedinTable12;asweshallseelater,they for theGandKdwarfs,usingspectraltypesinsystemadopted color indicesfortheindividualstars,weshallmakeanewreduction they stand,exceptfordwarfslaterthanKo.SinceBeckergivesthe ed meanspectraltypeofthecomponentsis given. Thewriteris For binariesobservedasonestar(70Oph.,61Cyg.etc.)theweight- are ofgreatimportanceinacomparisontobemadewiththesun. ticular carewastakentoomitallsubgiantsfrom Table12,sincethe dwarf stars. again indebtedtoDr.Morganforthespectral types usedhere.Par- are morefundamental. colors oftheseobjectsaredecidedlyredderthan thoseofordinary 107 Psc.. 5 1 47 UMa 39 Tau. 15 LMi. nLMi. \f/ Aur.. X Ori.. fi Cas.. 83 7] Cas.. © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem X Aur.. ô Tri.. k Cet.. The spectraltypesusedbyBeckerandBrillseem,ontheaverage, i Per.. Theoretically,thesebandshavetheadvantageoverthose inthebluebecausethey Star Becker’s ColorIndicesforGandKDwarfs THE STELLARTEMPERATURE‘SCALE463 Spec. Ko Ko K Go G6 Gi Go Gi GS G2 Go Go Go G2 Go+ G8+ 3 0.105 O.162 e .304 •143 .183 .130 .124 .300 .197 .381 •US •IS? .124 . 276 .074 . 112 'P TABLE 12 HR 7162. HR 8832. 43 Com. 61 UMa 59 Vir.. 61 Cyg. 70 Oph. 26 Dra.. 85 Peg.. 51 Peg-•