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Home , 104 Herculis, 14 Herculis, 17 Cygni, 29 Herculis, 56 Ursae Majoris, 68 Herculis

THE EMPIRICAL - RELATION

G. P. KUIPER

ABSTRACT After a brief historical introduction the problem is subdivided into two main parts. The first, on the temperature scale and bolometric corrections, is treated in the preceding article. The second problem, the derivation of the empirical mass-luminosity refation, is treated in three sections in this article: (i) the visual binaries, (2) some selected spectroscopic binaries, and (3) Trumpler’s massive in clusters. An at- tempt has been made to obtain the most accurate observational data for all quantities entering into the discussion, including magnitudes and spectral types. Tables 1 and 7 giye the visual binaries used at present. Table 5 shows the derived quantities for the stars of Table 1, and Table 6 shows the quantities of theoretical interest for the stars of Table 5 for which the accuracy is sufficiently great. Visual binaries in the same class as those of Table 1, but for which the data are still incomplete, are collected in Table 8. The problem of the spectroscopic binaries is only partly treated in this paper. Only some representative objects are discussed; they are found in Table 12. Theoretical values of the ellipticity and reflection constants are used in the discussion of the ob- servations. Trumpler’s massive stars are discussed in Table 13. In all three sections the results of the preceding article have been used. The data are shown graphically in Figures 1 and 2.

INTRODUCTION The discovery of the mass-luminosity relation has been of great importance to the progress of astronomy. The relation has been used in statistical astronomy and in double- astronomy and has been a central problem of theoretical astrophysics. Since for most stars no direct determination of the mass can be made, the use of the mass-luminosity relation is the only method of estimating the total mass of the known stars per volume of space—an important dy- namical quantity.1 In binary-star statistics the observable Aw and a rough knowledge of the can now be used in obtaining a statistically useful determination of the mass ratio, which is of great cosmogenetic interest.2 Investigations, such as a recent one on e Aurigae3 show the need of having the relation well established. But particularly the theoretical importance, both in ab-

1 E.g., Oort, 6, 285, Table 34, 1932. 2 Pub. A.S.P., 47, 17 ff-, 143 ff-, I935- sAp. 86, 574, 1937. 472

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1938ApJ 88 . . 472K 6 10 7 4,5 8 less explicitform{Ap.J.,42,115,n.2). laxes withabsolutemagnitudesderivedbymethodsofstellarsta- by comparingabsolutemagnitudesderivedfromhypotheticalparal- established mainlyfromdouble-linespectroscopicbinaries.There- type andofspectralwithluminosity.Thefirstrelationwas statistical relationbetweenintrinsicbrightnessandmass.Hiscon- mass-luminosity relationmusthavedelayedthediscoveryofthat that oneofthebest-knownstars,SiriusB,doesnotfollowgeneral laxes increased.Eventodaytheknowledgeofmassesistoalarge related seemstohavedevelopedgraduallyastheknowledgeofparal- stract formandinnumericalforparticularstars,isclearfrom Monograph,” chaps,viiandviii,1938.(Inpress.) is independentofspectraltype. nen shortlyafterwardemphasizedthatthemass-luminosityrelation formula forthemassratioofabinaryderivedfromAw,and tistics. was notrecognisedinRussell’spaperof1913,although1914Rus- scopic binariesareveryrareamonggiants,sothatHalmwases- sult wasthereforepartlyaccidental,becausedouble-linespectro- relation. extent limitedbytheknowledgeofaccurateparallaxes.Thefact Eddington’s workandfrommorerecentdevelopments. first formulafordynamical(nothypothetical).VanMaa- in fairagreementwithmodemdata.Healsogavethecorresponding sell foundevidenceforadefinitecorrelation,whichwasobtained sentially dealingwithmain-sequencestarsonly.Infact,therelation clusion wasessentiallybasedonthecorrelationofmasswithspectral 10 9 9 678 5 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Pub.A.S.P.,31,231,1919. Halm wasprobablythefirsttostateexplicitlyexistenceofa ÆV.,208,96,1919.Alreadyin1915Hertzspmnghadgiven thesamerelationina Hertzsprung, in1918,gavetherelation M.N.,71,638,19TI.Observatory,36,327,1913. Pop. Astr.,23,340,1914. S.Chandrasekhar,IntroductiontotheStudyofStellarStructures, “Astrophysical 4 B.Strömgren,Erg.d.exact.Naturwiss.,16,497,513,1937. i. Historical.—Theideathatthemassandluminosityarecor- THE EMPIRICALMASS-LUMINOSITYRELATION473 log m=—o.o6(M5), v 1938ApJ 88 . . 472K 1 474 G.P.KUIPER pleted. Insomewaysthetwoinvestigationsrunparallel:forin- viewed byLandmark.Ofallthese,Eddington’sresultsareun- doubtedly thebestknown,particularlybecauseEddingtonused which wasreceivedaftermostofthepresentinvestigationcom- relation inconnectionwithhiswell-knowntheoreticalinvestigations. mass ratios,andAwvaluesdoesnotusespectroscopicparallaxes. ric observationsandtheestimationofweightsindividualmass stance, intheuseofbolometriccorrectionsderivedfromradiomet- is entirelydifferentinthetwopapers:Parenagousesspectroscopic with ourTables1and5.Thetreatmentofthespectroscopicbinaries determinations. Butthepresentstudyusesseveralimprovedorbits, renago; amongtheseareTrumpler’sstars. A numberofobjectsareusedherewhichnotconsideredbyPa- absolute magnitudeswhereasweusethestellartemperaturescale. Minor discrepancieswillbefoundbycomparingParenago’sTable2 fair precision.Thederivationoftheradius,inadditiontoother which thethreefundamentalquantities—themassm,radiusR, which isneededinanycasebeforetheeclipsingbinariescanbeused. and luminosityaresimplyrelatedbytheeffectivetemperature, two quantities,doesnotrequireanyextrainformation,sinceradius and theluminosityL—maybeobtainedfromobservationswith vestigation. ment theresultsbyastatisticaltreatmentofotherdatasmaller cal work,weshallgivethemexplicitlyasthefinalresultofin- In viewofthefactthatallthreeparametersarerequiredintheoreti- for determiningthetrendofmass-luminosity relationforthe giants andtheOstars,wherefewdataofhigh individualaccuracy individual accuracy.Suchastatisticaltreatment wouldbeuseful are available. (as requiredintheoreticalwork),noattemptismadeheretosupple- 12 12 11 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Many otherinvestigationsfollowed,mostofwhichhavebeenre- The mostrecentstudyofthesubjectisthatbyP.Parenago, 2. Thepresentinvestigationislimitedtothoseinftividualstarsfor A closeco-operationbetweentheoryandobservation willcertainly Since alltheemphasisislaidonaccuratedataforindividualstars AJ.Sov.Union,14,33,1937. HandbuchderAstrophysik,5,PartII,1933. 1938ApJ 88 . . 472K 5 J preference indouble-starobservationsanddeterminationsof andmassratio.Atleastfourtosixdeterminationsforeach viously thesecondgroupofobservationsespeciallyneeded.Finally, object willbeneededinordertomakethemassdeterminationssuffi- it wouldseemthatthevisualbinariesofTables1,7,and8deserve be thequickestandleastwastefulmethodofsolvingfundamen- tal problemsofstellarstructureandevolution.Inthisconnection meridian positions;manyneworbits,ofwhichwewanttomention line spectroscopicbinariesthatarealsoeclipsingvariablesob- logue ofParallaxes;Boss’sGeneralCatalogue33342Stars,contain- of stellarmasseshaverecentlybecomeavailable:SchlesingersCata- computation ofluminositiesfromradiiandviceversa. investigations onthestellartemperaturescalearerequiredfor ciently accurate.Photometricandspectroscopicmeasuresofdouble- importance isPannekoek’srediscussionofthetemperaturesO van deKamp.Forthediscussionofeclipsingbinaries,special especially thosederivedwiththehelpofphotographicmeasures been published.Last,butnotleast,Trumpler’sdiscoveryofvery and Bstars,derivedfrommaximaofspectralseries.Dataforvery ing manymassratiosforvisualbinaries,basedonalltheavailable massive starsingalacticclustersshouldbementioned.Thesemany interesting systems,suchasfAurigaeandVVCephei,haverecently formula to beknown.Thisrelationisgivenbythebolometric correction. integral partofthesubjectunderdiscussion,beingneededin available. Before thisformulacanbeused,therelationbetween MiandLhas conversion ofradiiintoluminositiesandviceversa,accordingtothe advances makeitpromisingtoassembleanddiscussthedatanow stellar Tscaleandthederivationofbolometric corrections;and (Strand, vandenBos,Hertzsprung);andseveralnewmassratiosby yS e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Many importantdirectorindirectcontributionstotheknowledge We havealreadyreferredtothestellartemperaturescaleasan Hence, thesubjectfallsintotwoparts:(1) discussionofthe THE EMPIRICALMASS-LUMINOSITYRELATION475 1938ApJ 88 . . 472K 13 preceding article;thesecondnaturallyfallsintothreeparts: visual binaries,theeclipsingandTrumpler’smassivestars. 476 G.P.KUIPER known, Kepler’sthirdlaw data, andofTrumpler’sstars.Thefirstproblemistreatedinthe wdll givethetotalmass,2m,ofsystem.Iforbitiswellde- parallax equalsone-tenthoftheitself,probableerror termined, andhencetheerrorin(3loga"—2P)issmall, in log2mwillbe±0.13.Thisvaluecorrespondstoabout±1.4ab- error in2mwilldependmainlyontheparallax;we pensation oferrorsoccurswhenthedataareusedinadiagramcorre- nitude isalsoaffectedbyanerrorintheparallax,andacertaincom- then have3A(logtt")=—A(log2m).Iftheprobableerrorof (2) thediscussionofbinarysystems,withsuitableobservational larger than2mag.areveryrareindeed,itisobviousthatdatawith ered bythevisualbinariesofourlist.Thecomputedabsolutemag- solute magnitudeintheregionofmass-luminosityrelationcov- probable errorslargerthan+1.2mag.areofnoindividualvalue. Hence, allsystemsforwhichtheprobableerrorofparallaxex- lating massandluminosity;thevalue+1.4mag.isthenreducedto considered recentlybyProfessorRussell. those forwhichtheprobableerrorofparallaxstillexceeds tempt ismadeheretouseaveragesforbinariesofdifferent spectraltypes—aproblem fourth columngives,ontheinternationalphotovisual system,the may beusedinthenearfuture. servers wouldpayspecialattentiontotheseobjects,particularly great importancefortheknowledgeofstellarmassesifparallaxob- ceeds 10percentareexcludedfromourdiscussion. ±1.2 mag.Astruedeviationsfromthemeanmass-luminositycurve + o''oo4.Ofnotmuchlessimportancearethestars inTable8,which © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem If theorbitalelementsandparallaxofavisualbinaryare The remainingbinariesarecollectedinTable1.Itwouldbeof The firstthreecolumnsofTable1needno explanation.The ^ Wehave,furthermore,excludedstarswithparallaxessmaller thano''045.Noat- 3 loga"—2P2>=2m(2) THE VISUALBINARIES 1938ApJ 88 . . 472K 14 IS In computingthemass,Russell’sorbithasbeengivendoubleweight. is aparabola:n=0.0707;q2T175.Russell’sorbitrepresentsthemeasuresbetterthandoesFinsen’s. these dataarebasedonZinner’scatalogue(whichisthePotsdam apparent magnitudeofthecombinedlightbinary.Most cases thesameorbitisgiveninR.G.Aitken’srecentbook,TheBinaryStars,p.284,Table1.Thislast ADS 10075. this sectionwaswritten. table iscompletetoSeptember,1933.Baize’scatalogue(Bull.Astr.,10,273,1937)wasreceivedafter nitude datawerefirstreducedtothePDsystem withtheaidofZin- 89, 105-107,1935. -8°4352 system), correctedbytheamount—0U22+ o.nC.Othermag- HRisq..-- HR 6516.. HR 6426.. HR 7162. HR 6416.. Kr. 60 44 Boo... 99 Her.... 85 Peg.... 70 Oph 26 Dra.... 15 14 03 EriBC. 7 Vir ji HerBC p Eri 7} Cas.... © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem a Cen.... 9 Pup.... a CMi.. a Gem... f UMaA. a CMa... a Aur f Her.... T Cyg.... £ Boo.... £ UMa... Ô Equ.... 4. W.J.Luyten,“InvestigationsofBinaryStars,”Pub.Minnesota,2,PartI,1934. 3. K.A.Strand,AnnalsLeidenObs.,18,PartII,1937. 9. W.JLuytenandE.Ebbighausen,ibid.,45,54,1936. 8. P.Rudnick,A.J.,43,164,1934. 6. G.P.Kuiper,unpublished.Theorbitisnearlycircularinappearance;i=0°,e0.04. 5. E.Hertzspnmg,B.A.N.,7,330,1936. 1. W.S.Finsen,“CatalogueofVisualBinaryStarOrbits,”Circ.UnionObs.,91,24,1934.Inmost F.W.Scares,Trans.I.A.U.,4,142,1932;C.PayneGaposchkin, HarvardAnn., 11. W.J.LuytenandE.Ebbighausen,A.J.,44,119,1935. 10. R.C.Buffer,Ap.J.,80,270,1934. 7. H.N.Russell,A./.,45,95,1936;alsoW.S.Finsen,CzVc.UnionObs.,No.q$,230,1936.Finsen’sorbit 2. W.H.vandenBos,Circ.UnionObs.,No.98,1938. Verojf.Bamberg,2,1926. Name THE EMPIRICALMASS-LUMINOSITYRELATION477 Visual BinarieseortheMass-LuminosityRelation hI 17 34.0 17 25.2 17 12.2 14 32.8 13 19-9 12 36.6 23 56.9 21 10.8 21 09.6 18 53-3 18 03.2 18 00.4 I? 42.5 17 II-5 16 50.1 16 37-5 16 24.5 15 00.5 14 46.8 22 24.4 a 1900 O32?2 0 43.0 09-3 36.0 47.1 34-1 40.7 10.7 28.2 EXPLANATION OFNINTHCOLUMN(REF.) +48 03 -56 42 +45 54 +57 17 +26 33 +37 37 + 936 +32 46 +27 47 +61 57 +19 31 +55 27 +32 06 + 529 +32 06 +57 12 +30 33 + 231 +31 47 +18 37 -16 35 - 749 -25 19 -34 53 - 6025 - o54 - o59 -46 32 - 809 -13 38 5 1900 -o. 20 -1.52 4-79 0.19 3-57 3- 7 0.45 9-37 4- Si 5-27 5-21 5.61 6.82 2- 43 2.98 1- 59 AB TABLE 1 Ks, ••. Ki, K4 Ko, ... Gi, G5 A2, A2 Ao, Ai Ao, A5 Go, K+ G2, ... Gi, ... Gi, ... G8, K G4, Ki G4, F4 Go, ... Bg, Mse M+,6 Fo, Fo F9, F9 F, .•- Fo, ... Fs, ... S 5 4 3 Spec. K2 G6 Gs G M M4 G2 F6 7 4 10.3 10.06 (0.82) O.I4 0.03 0.43 0.15 0.06 3-74 0.3 O. I O. I O.OI o.o± 0.7 0.86 0.0 35: 3-40 I .2 2.6 2.80 i .96 1.37 1.48 1.78 1.68 2.61 2.82 1-5 Am 34O 219.5 526.0 242 217.1 149-95 171-37 IO4.02¿ 247.9 251 42.2 34-42 46.0 80.09 87.85 43.02 59-86 4O.23 49.94 44-52 49.8 61.8 76.06 25.00 54-7 20.54¿ 23-34 26.46 5-70 1.72 .06 •94 .19 -349 . 198 .609 .884 ■ 593 .0115 .746 -5355 .69 . 26 .84 .62 •0536 .894 ■31 -534 Ü670 .82 .362 ■94 . 26 -24 •13 .558 . 287 -55 83 ZB ¡:í Ref., Q 11, A 10, A 3, A 3, A 4, C 8, B 6, AB 'r 4, AB 4, A g,A i, A i, A 7, BC i, AB i, A i, A i, A i, A i, AB 2, A i, A i,B i, A i, A '?o72± 6 1.090+ 6 .oso±4 • i32±6 -I48±4 • iio±5 . io9±6 -o66± 5 .o59±6 -147+6 -046+4 .058+5 •049+5 -196+4 • 756±7 •045 -o89± 7 -i38±6 .061 ±4 .291 ±4 •073±3 • 376±2.6 .0632 -256+4 .060 +4 .147 ±6 •o79±5 .202 ±3 • i6i±7 . 182+5 (Trig.) 1938ApJ 88 . . 472K 16 17 Table 8(exceptyCrA,whichseemstobeinerror). makes C=1.08forgKoifoAo. in Table2.ThecolorindexCisdefinedasIPg—IPv,which which seemstobewellestablished,istabulatedagainstspectraltype ner’s tables,andthentotheIPvsystem.ThemagnitudesofoEri- of thisstarwiththose61Cygni(dKs),eand 40Eridani(dK2), metrically. has askedDr.Shapley’spermissiontocompare theHarvardspectra this valueledtoanabnormallylargeradiusfor thisstar,thewriter good agreementwiththesmallvisualdifferencefoundinterfero- were usedwhichcomparedwithsimilarspectrogramsofother spectra ofthecomponentsCapella.Three-prismspectrograms two componentsseparatelyorofthecombinedlight.ForAstars A2. Ao. For ¡jiHercuhsBC,thereducedHarvardmagnitudeis9.79mag., 478 G.P.KUIPER 3, 1938),wasreceivedafterthispaperpracticallycompleted. Itwasusedfor F andGgiants.Dr.MorganestimatesAw(pg)=0.0,whichisin the writerwishestoacknowledgeDr.Morgan’sclassificationof types wereobtainedbyDr.Morganorthewriter.Inparticular, the HDclasseswereused,andformostofotherstarsspectral As. B8. Bo. o.. and myownmeasuresgive9.82mag. dani BC,BD—8°4352,andKrüger60aremyowndeterminations. FS- Fo. 2 16 17 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem For convenience,thecorrectionIPv—PD=—0.22+0.11C, ThevaluablecompilationbyC.PayneGaposchkin{Harvard Obs.Mimeograms, For aCentauriBthepublishedspectraltypeisK5;butsince The fifthcolumnofTable1givesthespectraltypeeither Cf.pp.451and463oftheprecedingarticle. Spec. IPv-PD — O.24 — O.18 . 20 . 21 .24 . 20 .22 •23 dMo dGo Ks K2 Ko Gs Spec. TABLE 2 IPv—PD — O.I7 —0.05 .09 .12 .14 . l6 gMo g Go Ks K2 Ko Gs Spec. IPv-PD — 0.03 -0.15 . 10 •05 .09 •13 1938ApJ 88 . . 472K 18 mined andmakesthestaranormaldwarf.Thewriterwishesto between thecomponents.Themajorityofdeterminationsare my own,buttenHarvardvalueswereavailableandthesein- and rCeti£Bootis(dG8).Theresult,Ki,isquitewelldeter- thank Dr.ShapleyandMissHoffleitfortheirkindco-operation. nearly asgoodmaybeobtainedforthepairinquestion;Bmeans data, includingthereferencesexplainedatbottomoftable, cluded. in 1934.OfparticularinterestisStrand’spublicationonsixclassi- cover theorbitonlypartly;andCmeansthatisquiteun- a moreuncertainorbit,usuallyduetothefactthatmeasures and thequalityQoforbit.=Aindicatesanorbitasgoodor tions sincetheappearanceof“CatalogueOrbits,”byFinsen certain. graphic measuresareincorporated. cal binaries(fiveofwhichareusedhere),inmanyphoto- cases thesameasinSchlesinger’sCatalogueof1935.Thefewmore recent determinationsavailablewereincluded. in Table3.AmostimportantsourceofinformationisthenewBoss General Catalogue,whichcontainsthemassratiosthatcanbederived least fiftyplates. determined severalmassratiosphotographically,eachbasedonat from meridianobservations.Inaddition,vandeKamphasrecently fects) willbethatinsuchcasesnotthebrighter starisobservedbut that arenotseparatedontheplatesused,or,ifobservedvisually, finite sizeoftheimage(causedbyimperfect seeing andopticalef- the centeroffight.Thepositionthis maybecomputedif ,HR7162,rCygni,and85Pegasi. Theresultofthe are notresolvedinthemeridiancircle.Thesebinaries arefHerculis, Ann., 105,57,1937). 18 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The sixthcolumnofTable1containsthemagnitudedifferences The seventh,eighth,andninthcolumnsgivetherelevantorbital The listofreferencesillustratestheactivityinorbitaldetermina- The lastcolumngivesthetrigonometricparallax,inallbutafew The dataonmassratiosforthebinariesofTable1arecollected One wordshouldbesaidaboutmassratiosdeterminedforbinaries Theresultisinagreementwiththatjustpublishedby MissHoffleit{Harvard THE EMPIRICALMASS-LUMINOSITYRELATION479 1938ApJ 88 . . 472K was positions. Theprobableerroris±0.025. separated bythepresentwriterfromComstock’svaluesobtainedmeridian was found,which,correctedforAm=3.00,gives0.382. 0.045 found,which,correctedforAm=1.78,givesB = 0.45. munifcation byDr.Jenkins,theweightofthisdeterminationissmall.85Peg:B(obs.) Her: B(obs.)=0.223:;Am3.40;(corr.)0.265:.Accordingtoapersonalcom- 480 0.470 ±0.012;Am=3.5;£(corr.)0.51. 0.012; fromthey-direction£(obs.)=0.A19±0.034,weighted mean£(obs.)= 0.024. CorrectedforAm=2.82,B0.328. observations publishedbyBerman.Theprobableerroris±0.028. (The differencebetweenthex-andy-solutionssuggestsasomewhat largeruncertainty.) i 7,themeridianpositionsmustrefertoprimary,andBneedsnocorrection.99 = 0.313;Am3.00;B(corrected)0.372.26Dra:sincetheseparationreaches = 0.48;Aw3.5;£(corr.)0.52. 44 Boo. 02 EriBC 7 Vir.. r UMa a Cen.. a CMa. a Aur.. 7] Cas... a CMi. a Gem. ^ Boo.. © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 4. LickObs.Bull.,i,35,1901;Pub.A.S.P.,34,179,1922.Valueisapproximate. 8. Ibid.,32,157,1920.Basedonrelativemicrometerpositionsofthreefieldstars, 6. Pub.MichiganObs.,2,101,1915. 3. B.A.N.,3,132,1926. 9. Ann.LeidenObs.,18,PartII,135,1937.Valuederivedfromradial-velocity 5. A.J.,45,124,1936;59plateswereused,datingfrom1920to1933. 1. B.Boss,GeneralCatalogueof33342Stars,1,Appen.II,1937.fHer:B(observed) 7. A.J.,44,83,1935.From50platesbetween1919and1934,B=0.323+0.015 2. Ann.LeidenObs.,18,PartII,75,1937;theprobableerrorifc±0.019. 13. Personalcommunication.Frommeasuresinthe^-direction £(obs.)=0.476± 12. Ibid.,47,i,1938.From57platesbetween1916and1936, B=0.359±0.0,07. 10. A.J.,46,36,1937.From54platesbetween1919and1936,B(obs.)=0.288± 11. Ibid.,45,121,1936.From53platesbetween1916and1935, B(obs.)=0.257± Star O.405 .791 .461 •235 .326 . 282 .3B5 .446 •Soi .508 .64 •443 + •SI Æm- EXPLANATION OFFOURTHCOLUMN(REF.) Van deKamp Van denBos Boss GC Boss GC Boss GC Boss GC Hadley Boss GC Boss GC Boss GC Strand Boss GC Reese, Sanford Authority G. P.KUIPER TABLE 3 Ref. HR 7162 Kr. 60. 99 Her. 85 Peg. 70 Oph. 26 Dra. r Her. r Cyg. Star 416 45 427 478 38 372 359 328 382 51 52 265: Van deKamp Van deKamp Van deKamp Van deKamp Van deKamp Boss GC Boss GC Boss GC Boss GC Boss GC Strand Comstock Authority Ref. 13 12 11 10 9 8 7 i i i i i 1938ApJ 88 . . 472K 19 ponents, measuredinthewavelengthconsidered,and5istheirsepa- mination ofmassratiosisthereforemostimportant. lute massdeterminationseach,freefromassumptions. Thedeter- hasnotyetbeenobserved,asinthecaseof7]Cassiopeiae, been replacedbytheGCmassratios.Particularlywhenwhole have beenpublished,butonlythosethatseemedmostreliable.Sev- order thatsignificantmassratiosmaybeobtained.IfAmiswell close pairswithsmallAm. determined, thereisnodifficultyinobtainingmassratiosevenfor line connectingthecomponents,atadistance[i/(L+L^)]sfrom the accuracyinvolvessquareoftimeinterval used. than twentyyears.Thisisobviousfromthefact thatinsuchcases should giveamorereliabledeterminationofthemassratiothan eral extensiveinvestigationsbasedonmeridianpositionshavenow even thebestphotographicmeasuresextendedoveraperiodofless except iflargerthan3mag.,mustbeknownquiteaccuratelyin rection (B—B)istabulatedagainstAm.ItappearsthatAm, to givethetruevaluesof5.Thiscorrectionhasbeenappliedin ration. Itfollowsthattheobserved(orapparent)valuesoíB= the magnitudedifference,Am,isknown.Thecenterwillbeon Table 3,withthedetailsgiveninfootnotes.In4cor- tfu/(m! +m)willhavetobeincreasedbyi/(LZ,inorder the brighterstar,whereLandareluminositiesofcom- a Geminorum,7Virginis,and£Bootis,themeridianobservations 21 ohs 2x x2 19 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The binariesforwhichthemassratiosareknown yieldtwoabso- We havenotattemptedtoincludeinTable3allmassratiosthat Cf.E.Silbernagel,A.N.,233,168,1928;P.vandeKamp, AJ.,46,37,1937. 0.5 0.0. 2.0. 1-5 i .0, THE EMPIRICALMASS-LUMINOSITYRELATION481 Aw B —Bohn O.I37 O.500 . 201 .285 ■387 TABLE 4 4.0, 3-5 5-0. 3- 0 2- 5 Aw B —.Bobs 0.010 O.091 .038 .025 •059 1938ApJ 88 . . 472K as of thetwomagnitudes.ThishasbeendoneforninepairsinTable1. 482 G.P.KUIPER equal magnitudes,weshallassignhalfthetotalmasstoaverage In thesecaseseachbinaryyieldsonlyoneabsolutemassdetermina- shall usethemassratiocorrespondingtoobservedAw,adopting small, andnomassratioisavailablefromobservations.Herewe tion. the slopeofmass-luminosityrelationfoundinthispaper.We nearapastron,wherethecurvatureofrelativeorbitis van deKamp’sphotographicdeterminationextendsoverthirteen in ratioabout1:2;hence,B=0.392hasbeenadopted.ForSirius, shall usetheprimariesofthesetwosystemsonly. weight 0.7totheGCdetermination. meridian observations(forsimphcity,thecomparisonismadein tions. Theweightsofthetwovalues77Cassiopeiaeareprobably the massratio,relativeweightsmustbeassignedtothesedetermina- relative orbit).ThevalueB=0.295hbeenadopted,whichgives slightly over1".Thesizeofthewholeorbit,14",iscoveredby time thephotographicdeterminationwillbeveryaccurate. well established. of themass-luminosityrelationwouldgiveB=0.61for44Bootis. mate weights2:1:1,makingtheaverage0.450;thisvalueisvery weight. Theobservationsastheystandwould indicatethatthepri- mary isofabnormallylowmassandthesecondary ofabnormally termination wasdescribedinnote13ofTable 3,andisofhigh the massratioisbasedon0.400from^-direction (weight0.71) In viewoftherelativelysmallpartorbitcoveredbyob- gas!. Takingtheduplicityofcompanionintoaccount,slope contradictory tothevalueB=0.61. servations, theobservedvalueofBcanprobablynotbeconsidered and 0.669fromthey-direction(weight0.29). VandeKamp’sde- © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem If themassratioisnotknown,andcomponentshavenearly Only twopairs,HR6416and6426,areleftforwhichAwisnot For thesystemsinTable3withmorethanonedeterminationof For fHerculistheagreementisbetterthancouldbeexpected.In Three determinationsareavailablefor70Ophiuchi,withapproxi- Table 3containstwoanomalousvalues:for44Bootisand85Pe- The caseof85Pegasiisperhapsmorepuzzling. TheGCvalueof 1938ApJ 88 . . 472K 20 21 pages 480and482.Theweightsincolumn8willbeupperlimitsfor neath thetable. high mass;butiftheparallaxofsystemshouldbetoolargeby ly equal.Forthenineobjectswhereaveragecomponentisused tion arefoundinTable5.Explanationofthecolumnsisgivenbe- to bedouble. the largemassofsecondarymightbeexplainedbyassumingit twice itsprobableerror,theprimarywouldbenearlynormaland of Table5havingweight3ormore;aGeminorumand£UrsaeMa- in column10,theweightremainsidenticallysameasgiven the massesintenthcolumn;secondarieswillbeoflower certainties ofthemassratiosused.Qualitativedataaregiven relative accuracythantheprimariesunlesscomponentsarenear- joris areomittedbecausethecomponentsspectroscopicbinaries, nosities followfromthebolometricmagnitudesofTable5,incon- known, whereasforKrüger60BnoreliableestimateofTcanbe and HR6426isomittedbecausethespectraltypereferstoablendof metric measuresareavailable.Forthesestars weshalladoptthe nection withthebolometricmagnitudeofsun,4.62.Thevalue system, buttheevidencehasneverbeenentirelyconvincing,and made. 70Ophiuchihasoccasionallybeensuspectedtobeatriple as thatofProcyon,cannotbeusedbecausenospectraltypeis the twocomponents.Furthermore,faintandclosecompanions,such column 8,ofcourse. dication ofthepresenceathirdbody. A logT=/Ofollowsfromthetemperaturescaleofpre- nearly thesameforgM2anddM2;colorindices arealsonearly siderations. temperature scaleforthegiants,onbasis ofthefollowingcon- Strand’s discussionofaccuratephotographicmeasuresgivesnoin- ceding article,exceptfordwarfslaterthanM2, forwhichnoradio- e e 2021 We havenotattemptedtogivenumericalestimatesoftheun- © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The dataofinterestinconnectionwiththemass-luminosityrela- Table 6givesthequantitiesoftheoreticalinterestforstars As wehaveseen,theheatindices(orbolometric corrections)are The massesofTable6aretakendirectlyfrom5;thelumi- Cf.p.438oftheprecedingpaper.Table13, p.464. THE EMPIRICALMASS-LUMINOSITYRELATION483 CM r" «

Visual Binaries for the Mass-Luminosity Relation © American Astronomical Society £ a O 0303 | +1 Ov ®.T-OH>Upqpq SK£v3o^fíWoW U vopq I H-j-COVOCO-|-HTl-00«N|OCN(N.h|O'^'H-j-COVO¿0 ^ VO OTf-M 00 CO !>.10O I>- OvOO- Tf- IOCM^ co htoi^vO co Of-lO O CO CO HTfIO o tocooo“H ^ OM H VO(N to vocmmw.: OOOHOOOOOOO m Ttr^HiocotoTf-ovoovo Provided bytheNASA Astrophysics DataSystem vo r?oo Tf 00 CO CM VO vo VO vo (N Tf- H Tf- Il +I+++1I I + . od vo VO Tf" co 12 S3 H CO w u co I Tf" Tf ^ O u M a. oovq¿¿o vO H to coTf-CMOv

American Astronomical Society • Provided by the NASA Astrophysics Data System 1938ApJ 88 . . 472K 2 486 G.P.KUIPER green, andblueareconsiderablyweaker.Dr.Morganhasfoundthe the classificationisbasedonthem),TiObandsinyellow, that althoughtheredTiObandscorrespond(astheyshould,since the same,aswehaveassumedinprecedingpaper. the same.Hence,effectivetemperaturesareprobablynearly Kr. 60 were aboutthesame,spectraldifferencesmentioned wouldtend sorption featurenear6^4227,presentindwarfs butabsentin spectra ofgiantsanddwarfstypeMis,course,thebroadab- as comparedwithM5-M6giants.Anotherdifferencebetweenthe same phenomenonevenmorestrikinglyinBarnard’sstar(dM6), known forthelaterMdwarfs,butcolor indicesare,approxi- to givethedwarfslargercolorindex(for two reasons)butthe giants. IftheeffectivetemperaturesofMgiants andMdwarfs smaller bolometriccorrection.The correctionsarenot -8°4352 — 8°43s2(dM4)wascomparedwithspectraofM4giants,itfound 70 Oph 02 22 ß Her a CMi_ a THer a CenA T UMaA r} Cas £ Boo When aspectrum(extendingfrom4000Ato6600A)ofBD © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Scares,Pub.Nat.Acad.Sei.,5,232,1919. Eri Aur CMa Star A.. BC A.. B. . A. . AB B. . A. . B.. + -52 + .62 + -17 + -37 + -04 + -4i - .70 - -35 - -33 - 0.60 - -45 - .01 —o. 14 - -13 - -05 - -35 - .02 - .12 - .06 - .06 log 0.00 +0.76 + 2.08 +0.59 +0.10 + 1.48 + 1-59 + 1.90 — 2.26 — i.16 —0.09 -1.77 -0.86 -0.38 — 1.40 -0.83 —0.32 -0.43 -2-59 — i.96: -1-37 log L 0.00 TABLE 6 A logTe + -303 +0.021 + .010 + .229 + .077 + -175 + .273 + .066 — .305 — .166 — .082 —0.298 — .136 — .078 — .290 — .290 “ -157 — .051 — .078 — .018 0.000 +0.25 +0.82 + 1.20 +0.29 +0.09 +0.28 +0.23 -0-37 -1.74 -0.25 —0.09 -1.65 —0.29 —o. 16 — O.O3 — O.IO — O.12 —o. 10 — 0.06 —0.06 log R 0.00 +0.56 +5-01 +0.56 +0.27 +5 -oS +0.42 +0.49 +0.20 +0.12 +0.06 +0.34 +0.25 +0.27 — 2.84 —0.23 +0.15 -1.79 -0.74 —0.07 —0.28 -0.36 log p 4.60 4.48 4-47 4- 3 1 4- 2 3 4-52 4-49 4-50 4-30 4.29 4.16 3- 3 2 4.42 4.62 4.46 4-30 3- 8 4 4-44 2.65 7-56 7.72 log g 27-5 15-5 km/sec 0-53 0.56 0.86 0.56 0.84 0.32 0.30 0.36 0.30 0.31 0.61 0.63 o. 64 0.57 0.17 o. 64 0.31 0.68 o. 61 E 1938ApJ 88 . . 472K 23 2 24 25 22 21 peratures ofgiantsanddwarfsareequal)wouldbedifficulttode- mately, fromSeares’sdeterminationforBarnard'sstar(dM6, would betoestimatethedistortionofcolorindexBarnard’s measured bytheordinarycolorindex)withadvancingtype.The star bytheTiOabsorption.Dr.Morganestimatesthatcor- would beaccountedfor(whichmeanthattheeffectivetem- C =2.0).TheMgiantsareknowntobecomeslightlybluer(as C =1.76)andfromtheapproximatevalueforWolf359(dM8, the appearanceofspectra.Whetherornotfulldifference color differenceis,therefore,inthedirectiontobeexpectedfrom rected valuemaybeabout2.0mag.,withaconsiderableuncertainty. termine. AnotherapproachtothetemperaturesofMdwarfs being bluer),ifenotcorrect,althoughtheerrorsintwo assumptionstendtocom- and whoadoptaconstantdifferencebetweenthecolorindices (0.3mag.,thedwarfs only radiometricobservationsofBarnard’sstarwillsolvetheprob- by Table13.Muchsignificancecannotbeattachedtothisresult; the meanofSeares’saverageforcomponents61Cygni,1.18, lem oftheeffectivetemperatureslaterMdwarfs. tem maybecomputedifcolorindicesfortwootherspectraltypes from 0.5to0.6forMizarAandSiriusabout3KM pensate. Cf.Pub.Tartu,28,Nos.3and5,1935-1936;30, No.1,1938. Gabovits, whoassumeequalTiOabsorptioninthevisualregion forgiantsanddwarfs, main sequenceisseentobequitesmall.Themeandensitiesvary will beusedalsoforthedwarfs. tional system,1.25).WiththeaidofTable13wethenfindc/T= and W.Becker’sphotoelectriccolorindexreducedtotheinterna- are adopted.WetakeC=0.00forAo,and1.22dKs-)-(being dwarfs. given inTable6arec.g.s.units.Thevariationofgalongthe the sameforM6aswellM2,temperaturescaleofgiants 5.19, whichhappenstobeverynearthevalue5.22forgM6given 2 25 24 2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem With this(uncertain)value,theeffectivetemperatureinoursys- Forthereasonsgiven,treatmentofMdwarfs inpapersbyöpikand The lastcolumngivestheEinsteingravitational redshift.Itap- The meandensities,p,andthevaluesofsurfacegravity,g, Contr.Mt.W.Obs.,No.356,3,1928. 3 Scares,Pub.A.S.P.,28,281,1916. Since thetemperaturesforgiantsanddwarfsseemtobenearly THE EMPIRICALMASS-LUMINOSITYRELATION487 1938ApJ 88 . . 472K I 27 26 28 pears todiminishslightlyalongthemainsequence.Onlyfor looked thatthestandardradialvelocitysystemiscorrectedbyan 488 white dwarfsistheredshiftappreciable,althoughinstatistical studies dealingwiththeF,G,andKstarsitshouldnotbeover- 3210. 3475. 3169. 35 ■ 3483. 3264. rived herewillnotbegiven,sinceDrs.ChandrasekharandStröm- similar objects,inaforthcomingpaper. types should,therefore,showasmallnegativeKterm. sun wouldshowzeroradialvelocity.Thegiantsofthesespectral amount of0.64km/sec,theredshiftsun,inorderthat bit waspublishedbyvandenBos,whousedmeasuresuptodate. is arevisionoftablepublishedpreviously;forADS3135newor- have developed. gren intendtoundertakesuchadiscussionwiththemethodsthey the Hyadesshouldgiveareliabledeterminationofhydrogen As mentionedinthepreviouspaper,meanofsixobjects discovered recentlybythewriter,willbevaluable insettlingthe cant. Induetimethepair68Tauri(Tfp= +14, Tfoi=+0.9); content, butdeviationsfromthemeanarenotconsideredsignifi- and thebinariesinHyades,givenTable7.Thelattertable Figure icontainsalsoobjectsoflowerweight,takenfromTable5, tude. question ofpossiblevariationscomposition with absolutemagni- V b 26 27 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem A theoreticaldiscussionofthemasses,radii,andluminositiesde- The twowhitedwarfsofTable6willbediscussed,togetherwith The companiontoProcyonwasenteredinFigure 1asaprobable The valuesthemselvesareshowngraphicallyinFigures1and2. Ap.J.,86,176,Tables4and5,1937. Circ.UnionObs.,No.98,1937.^Pub.A.S.P., 49,341,1937. ADS 4- 59 4.52 4.48 3.66 3.02 5- 65 Ifpv G5 PS Fo F8 F6 F7 Sp. Afbol G. P.KUIPER TABLE 7 +0.69 + .4° + -03 -|- .06 + .07 -0.37 log L A logT t +0.118 + -054 + .039 + -032 + .046 — 0.028 +0. ii + .09 -0.13 — .06 — -03 — .06 log R +0.07 + -04 - .19 - .21 - -IQS -0.44 log Wt. ii 1938ApJ 88 . . 472K © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem + vO tJ- o + + + + vo vo o C5 cfi > j§ o & Dh W) M K I 1 J* a 1938ApJ 88 . . 472K 29 luminosity relationchanges;theexactslopewill depend,ofcourse, on thebolometriccorrectionsused,butthere isnodoubtthatthe luminosity forstarsofthesamemass.Formoremassivestars, ferences, particularlyamongthestarsoflowaccuracy,areprobably presented byeAurigae.Since,however,differentselectionfactors luminosity relationinthatregion. however, Chandrasekharhasfoundverystrikingdifferencesinmod- lium) content.Thisconclusionisbasedonanotherresult,byStröm- volving stillaparameterdifferentfromthemass.Strömgrenhas only onemass-luminosityrelationbutwithmanysuchrelations,in- Hyades andfHerculis,areseentobebrighterthantheaveragestar operate forthedifferentgroupsofstarsused,itisdoubtfulwhether cf. alsoParenago,loc.cit.,Gabovits,Pub.Tartu,30,No.1, 1938 (seen.25). el; whichmustbelargelyresponsibleforthescatterinmass- identified thisparameterwiththehydrogen(orandhe- spurious. Theconclusion,therefore,isthatwearenotdealingwith of thesamemass;thesedifferencesareconsideredreal.Otherdif- no interpolation-curvewasthereforedrawn.Somestars,suchasthe luminosity diagram,wemayconcludethatProcyonBisprobably Procyon Bfromtheempiricalmass-luminosityrelationmaythere- changes inmodelcannotbeheldresponsibleforthedifferences gren andChandrasekhar,thatthestarsofmasssmallerthanfiveto this meanempiricalrelationhasanydefinitephysicalsignificance; relation whichisusefulinthediscussionofsuchproblemsasthat not anormalreddwarf. Until otherlargedeviationsarefoundinthisregionofthemass- white dwarfforthefollowingreason.Observationsshowthat fore becomputedundertheassumptionthatstarisanormal change isreal.Theexplanationmaybesimply thatthestarsKrüger ten timesthesunarebuiltessentiallyonsamemodel,sothat 60 andoEridaniChaveasmallhydrogencontent, aswouldfollow dwarf. Thedeviationisinthatcase2mag.,orover0.4logm. close forthelowerpartofmainsequence.Thedeviation correlation betweenabsolutevisualmagnitudeandspectraltypeis 490 G.P.KUIPER 2 29 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem ThechangefoundnowisbetterdeterminedthaninFig.1 ofAp.J.,86,194,1937; For thestarswithmassessmallerthan0.6O slopeofthemass- The dotsinFigure1defineanaverageempiricalmass-luminosity 1938ApJ 88 . . 472K © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem + I + + + + + to to H ci to H q q ci O Ö S Ó I .3 ^3 S 'bü 1 0 1 1938ApJ 88 . . 472K 3 lation inFigure1,towardthefainterside.Itisofgreatinterestthat before thisconclusioncanbeaccepted.Ifitholds,mayexpect- from anextrapolationofStrömgren’stables.Amoredetailedtheo- 492 G.P.KUIPER retical investigationoftheselow-temperaturestarswillbeneeded ed thatstarswillbefoundwhichdeviatestronglyfromthemeanre- tor AB,hasprobablyafairlyhighhydrogencontent.Furthermore, theoretical relations.CastorC,connectedwiththeA-typestarCas- the whitedwarfoEridaniB(also)hasaverysmallhydrogencon- tent, whichfollowsfromthedatainTable6andChandrasekhar’s of 7]Cassiopeiaedeviateinthesamesensefrommeanrelation what brightforitsmass.Itisalsonoteworthythatbothcomponents . Itwould,therefore,notbesurprisingtofindBCsome- the brightercomponentof¡j,Herculis(A)lieswellabovemain sequence andmaythereforebeexpectedtorelativelypoorin promise tobeoffutureinterestinthestudyempiricalmass- luminosity relation.Theyareofspecialimportanceinparallaxand to whichdiscussionisprofitable. the directioninwhichfutureresearchmaybecontinued.Theaccu- Figure i. double-star observations.Spectraltypesmarkedwithanasterisk racy oftheobservations(parallaxes,temperatures)willsetlimits which willprovideasignificantextensionofthemass-luminosityre- lation beyondKrüger60B.Thevalueofthespectralparallaxthis 2 are myowndeterminations.OfparticularimportanceisWolf424, only one-halfofone-tenthwillstillgivequite uncertainresultsfor means thatevenestimatesofspectraltypewithaprobableerror star hasbeenunusuallyuncertainbecauseofthegreatsteepness to achangeof1.7inMorfactor2.2thespectralparallax.That the mainsequenceinthisregion.AchangeofM8toM7corresponds trary. Iftheevidenceavailableindicatedthat theparallaxis(or types laterthandM6.° may be)inexcessof0^050,thepairwasincluded, providedthepro- jected separationbetweenthecomponentsdid notexceedabout v © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem These remarksareallofaprovisionalnature,buttheyindicate In conclusion,inTable8thevisualbinariesarecollectedwhich The limitssetinthecompilationofTable8are necessarilyarbi- 3° Ap.J.,87,593,Fig.I,1938. 1938ApJ 88 . . 472K method ofobservation;further,massratiosmaybedeterminedfor of periodsoverfivehundredyears).Forstarsverylargeparallax, sixty timestheparallax(thislimitwillalreadyincludemanypairs possible; valuesdependingonestimatesareindicatedbyacolon. particular interestareADS9655and16644.AHtheseobjectsde- such pairsfromrelativelyshortarcs).Boundary-linecasesareen- accuracy maybeobtainedforsuchpairs(onaccountofthelarger the limitwastakensomewhatwider,sinceamuchgreaterrelative observations couldbecombinedwiththespectrographicobserva- it wouldbeamostinterestingadditiontoTable8.Otherstarsof variable motion;ifthepaircanberesolvedatmaximumseparation, much smallerthanthecomputeddynamicalparallaxes. tered inparentheses. angular scaleoftheorbitandapplicabilityphotographic serve aprominentplaceinparallaxanddouble-starobservations. ably optical,thespectralparallaxesderivedbyauthorbeingvery Of themagnitudedifferences,Aw,however,manyvaluesdependon inthespectrum. problem ofthemass-luminosityrelation.Thesedatafallintotwo estimates. Twosystems,ADS1322and9476,wereexcludedasprob- tions. Theywillbefewinnumberbutofgreatimportanceindividu- variables (fortheseobjectsthemassesandradii maybefoundand groups: (1)double-linespectroscopicbinaries that arealsoeclipsing to haveparallaxesinexcessofo'(o5o. tio maybemeasuredandthemagnitudedifference foundfrom the luminositiescomputedbymeansof stellartemperature all thedataonspectroscopicbinarieswhichhaveabearing ally (luminousstars). scale) ;(2)double-linespectroscopicbinariesfor whichthemassra- 312 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The magnitudes,mB,arereducedtotheIPvsystemwherever Not mentionedinthetableisRoss614,foundbyReuyltohavea Finally, Ross492andapairannouncedbyLuyten,mayappear Other objectsnotmentionedarethoseforwhichinterferometer ^ Pub.A.S.P.,47,282,1935.vHarvardBull.,No. 906,1937. A i. Afteradiscussionofthevisualbinariesitisnaturaltocollect THE EMPIRICALMASS-LUMINOSITYRELATION493 SOME SELECTEDSPECTROSCOPICBINARIES 494 G. P. KUIPER

TABLE 8*

iooo Parallax Source Star R.A. Decl. VAB Am Spec. 7r(d) Ir. dy. sp. ADS 48 0^00 Ifl4 +45° 16' 8.1 o. i *K6, K6 89+ 5 120 930 (ADS 61).... o 01.0 +57 53 6.0 0.8 G 106. I4±8 53 930, 91 PASP 47,282, o 07.7 +68 47 12.2: 0.3 *Ms3 56±7 ADS 433 o 26.0 +66 42 10.2 1.9 M3 103 ±7 ADS 490.... 0 30.1 - 4 09 5-2 0.6 F7 7- 52±8 (Si) 930 -69°44 1 il .6 — 69 21 7.4: 0.6 05 128. 52 98 -30052Q AB i 30.4 -30 25 7.2 0.2 05 4.6 36±9 56 91 AB, C 4.9 59 93 ADS 1733 - • 2 il. i — 18 42 7-8 i .0 *K3 74 (42) 930 ADS 1865.. 2 22.5 + 3 59 8.6: 0.2 *K6 25- 61 98 a For... 3 07. -29 23 4.2 3-5 F5 72±5 55 93 +68°278 BC, 3 38. +68 21 10.8 0.0 *M2 + 54± 9 (ADS 2995). 4 00. +37 49 7.1 1-3 *K2 30± 6 'SS' 930 68 Tau.. 4 IQ- +17 42 4.4 4.0 A2 Hyad ADS 3248.. 4 23. +15 56 6.6 i .0 F8 40. Hyad ’28' ■98’ ADS 3321 CD 4 30.2 +16 19 il.2: 2.4 * K6 Hyad HR 1504 4 38.6 -59 08 6.4 0.2 Go 50 93 104 Tau. ... 5 01.5 +18 31 5.1 0.0 Gi 2 ± ? 58±8 46 a Aur Hh. 5 10-0 +45 44 10. i 3-5 *Mi + 73+3 Intf. =63 (ADS 3900)... 5 I4-1 — 3 il 7.9- 2.8 * K4 66± 5 no 58 930 -51 1540 5 31.2 — 51 08 9.3: 0.2 Ko no 93 BDS 3112 AB 5 56.6 -31 03 7.6 0.4 *K5 + < i 53+11 79 (78) 93 AB, C . . o. i 2 51 93 HR 2162.... 6 02.2 -48 27 6-5 0. g5 2 52 93 4 —4603046 7 14-6 -46 49 7.6: 1. i Ko 73 + 9 14 93 ADS 6487 7 52.9 - o 33 8.3: 5.0 81 (69) 485 HR 3430 8 34-8 — 22 19 5-1 1.8 G6 62+6 36 76 93 ADS 7067 8 46.0 +71 il 8.6 O. 2 * K6, K6 93 + 4 90 66 930 tUMa BC. 8 52.4 +48 26 10.2 0.0 Mi 35 + 66± 6 30 A, BC. .. 3-2 7.1 A5 91 63 930 10 UMa. 8 54-2 +42 il 4.0 2.0 F 2 72 + 6 66 (02 UMa) 9 01.6 +67 .32 4.9 3-3 f 53 + 6 40 50 91 (ADS 7251)- 9 07.7 +53 07 7.2 0.0 *K6,4 K6 i62±3 (240) (152) 930 ADS 7284.. 9 12 +29 00 7.2 0. *K3 34- 60+8 57 48 930, 91 0 ^ Arg... 9 26.8 —40 02 3- 1. 2 Fo 35- 70+9 61 55 930, 91 7 ri LMi.. 9 29.7 +36 16 5-4 7-5 Ko 95+5 91 —6o°29ii... 11 20.3 —61 06 7.4: i. 2 Ks 234- 67+13 104 91 Wolf 424... 12 28.4 + 9 34 il.9 0.0 *M7(+) (220) (320) t 78 UMa. 12 56.4 +56 54 5-0 2.4 Fo UMa(?) 24 BAN 305 42 Com.. 13 05 +18 04 4- 0.0 F4 57+7 43 930, 91 7 (ADS 8841). 13 il +17 34 6.4 2.7 k 68+7 79 930 ADS 8887.. 13 18.9 +29 45 8.9 0.4 *K63 67+11 46 930 î ADS 9031.. 13 44-5 +27 29 7.0 0.4 K6, K6 156. 6o± 9 66 91 ADS 9090.. 13 58.5 +46 49 9-3 O. I *Mi, Mi 79+ 6 , 95 (ADS 9446). 14 51-6 — 20 58 5- Ks,M2 172 + 4 (200) (174) 93 4 ADS 9544.. 15 08.8 — o 58 o G8 ! ± ? 60 ± 4 42 77 CrB.. 15 19.i +30 39 0.3 F9 42. 68±8 (57) 44 930 ADS 9716.. 15 32.5 +40 08 0-3 *k3 57- 46+9 47 54 § \p Ser... 15 39-0 + 2 50 6-3 Gs 46 ± 10 89 52 485 * Spectral types marked by an asterisk (*) are my own determinations. The last column gives the ref- erences for the dynamical parallaxes; 930=^./., No. 930; 91, 93, 95, and 98 are the numbers of the Union Observatory Circulars; 451 and 485 are the numbers of the Lick Observatory Bulletins. t Two good spectra available for this star give M7+ and M7, respectively. The star is, therefore, defi- nitely somewhat earlier than Wolf 359. îThe spectral parallax is about 0T06. It seems unlikely that the parallax should be much less than 0T05 or 0T04, as the dynamical parallax would indicate. The value of the latter was confirmed by the in- clusion of recent measures. Hence, the must be high (in which case the binary should soon close in rapidly), or the mass of the binary is unusually small. Additional trigonometric determinations will be valuable v § Derived from Pitman’s orbit, Pub. A.S.P., 46,196, 1934.

© American Astronomical Society • Provided by the NASA Astrophysics Data System THE EMPIRICAL MASS-LUMINOSITY RELATION 495

TABLE 8—Continued

iooo Parallax Source Star R.A. Decl. 'AB Aw Spec. tr. dy. sp

+45 2505... 171109032 +45° 5o' 9.4 0.4 *M3 + < i 143 ±6 US 36 Oph.. 17 09.2 — 26 27 0.0 Ki, K3 4 i79±5 52 174 93II ADS 11632. 18 41.7 +59 29 0.8 *m4, m4 16 282 ± 4 (240) (282) 930 T CrA.. 18 59-7 -37 12 0.0 F7, F7 119. 62 51 91 17 Lyr C 19 03.6 +32 21 0.4 *m5 123 + 6

Ross 165... IQ 41-7 +26 55 0.9 *m5+ n6± ii ADS 12889. 19 41.8 +33 22 0.2 *k3 48+ 5 57 44 930 If HR 7703.. 20 04.6 -36 21 5-4 k4 177 + 9 191 +4°45io... . 20 34.6 + 4 37 I *k5 57±4 87 Fur un 54. . . 20 56.2 +39 41 1.9 *M2 + 97 ± 8 87

(610 Cyg) 21 02.4 +38 15 0.7 *Ks, K6 26 299Í 3 (360) (276) 930 -S8 7893.. 21 39.4 — 58 08 0.1 Ma 6.8 49 i 5 54 95 (/* Cyg) 21 39.7 + 28 17 1.4 F6, F3 < i 46 ± 8 66 44 930 ** HR 8501. 22 ii.7 -54 06 5 Go 3 8o± 7 II Another case (n. Î, above) where the dynamic parallax is much too small. If Strand (B.A .N., 8, 206, 1937) has computed a provisional orbit for this star and found the dynamical parallax o'ii7. It is almost certain that this value is too large, probably as a result of the fact that the orbi- tal inclination found by Strand is too close to 90o. The trigonometric parallax, the Mount Wilson spectro- scopic parallax, and my spectral parallax (o'o5o) agree very well. Also Strand’s doubt whether this binary is physically connected with 17 Cygni seems unjustified. , , and parallax sug- gest that the connection is real. The nature of this case is apparently the opposite of cases Î and ||, above. ** The secondary was classified at Mount Wilson earlier than the primary.

The first group gives fundamental data for the problem of stellar structure, and the second group is useful in a study of the scatter of the mass-luminosity relation33 and in giving data that may be used statistically in a determination of the mean position of the mass-luminosity relation (the mean absolute magnitude may be de- termined from the spectral types; the mean mass, by substitution of the mean value of sin3 i into the mean value of m • sin3 i). A review of the available data showed that for many close pairs contradictions seem to be present;34 it was clear that for many ob- jects new observations would be required in order to find the inter- pretation of such discrepancies. Particularly close pairs with very unequal components are likely to present apparent anomalies; the systems with nearly identical components, however, are more easily interpreted. In the present paper we shall discuss only a few objects of the latter class ; the discussion of additional spectroscopic binaries will be made later by Dr. Morgan and the present writer. 33 Cf. Pub. A .S.P., 47,144, 1935; the scatter found in this manner will be a minimum value, since the components of binaries seem, at least statistically, to have common properties (hydrogen content). Cf. Harvard Bull., No. 903, 1936. 34 E.g., Wyse, Bull. Lick Obs., No. 464, 1934.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1938ApJ 88 . . 472K 36 35 stars; x,y,zrefertoarotatingco-ordinatesystem. were plottedonlarge-scalediagramsfromwhich theratioof ys-plane throughthecenterofonecomponents. Theresults orbital plane,andx=yowiththecenterofgravity;r curves, £2=C,bothforthexy-planeanda planeparalleltothe are thedistancesofanypoint(x,y,z)fromcenterstwo circular, withradiusa;thexy-planeisassumedtocoincide which aconstantvalueisadopted;Gthegravitationalconstant; and Misthemassofeachcomponent.Theorbitsupposedtobe .Inthiscasetheequipotentialsurfacesaregivenby The symbolshavethefollowingmeaning:£2ispotentialfor of negligiblemass,inwhichtheperiodsrotationareequalto the formula very massivestarsmaybeexceptionstothisrule.Hencethe the modelconsideredwillhavetwoequalcomponentswithenvelopes the presentweshalllimitourselvestopairsofnearlyidenticalstars, the componentswill,ingeneral,betwoofthesesurfaces.Sincefor the determinationofequipotentialsurfaces;boundaries ellipsoidal inshape.Wehaveagoodtheoreticalevidencethatthe of interestherearebinarieswhichshowtwospectra. will bemadeinthismanner.Fortunately,theonlycaseswhichare Roche, orpoint-mass,modelshouldgiveacloseapproximationin stars possessstrongcentralconcentrations;onlycertainclassesof radii ariseswhenthepairisveryclose,sothatcomponentsare with theratioofsurfacebrightnessesdeterminedfromlight- relative strengthofthetwosuperimposedspectra;incombination necessary todeterminetheratiooftotalbrightnessesfrom of theradiicannotbefoundwithanygreatprecision.Itistherefore x2 curve theratioofradiiisfound.Allourdiscussions 496 6 With theaidofthisformulawriterhascomputed afamilyof © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 3 E.g.,Jeans,ProblemsofCosmogony,p.162,1919. 35 Chandrasekhar,op.cit.,chap.viii. 3. Asecondquestioninconnectionwiththedeterminationof 2. Itiswellknownthatinmanyclassesoflight-curvestheratio = —GM G. P.KUIPER 3 - + /*! ra 2 + ÿ ]■ (3) 1938ApJ 88 . . 472K 02 37 between thecomponents).Acertainamountofrounding-offwasap- nents willstillappearseparated. tioned above,maybeexpectedtohavealowintensity(apartfrom values of¿u/atherewillbeabridgebetweenthem,which,asmen- tity maybefoundfromthelatter,whichisobserved.Thequantity R ='s/aJbiC^whereasthelight-curvegivesorandh.We plied tothepointedsidesofstarsfacingeachother,sincetheo- lar value.(Wesselinkhadalreadypointedout thisagreementwith 0.935 ±•(ni.e.).Thishappenstoagreeprecisely withthetabu- reflection effects),sothatinthephotometricsolutioncompo- rounded offtothenearesthalfof0.01.Itshouldbenotedthatfor ratio maybeobtainedobservationally.Table9containstheresults, shall thereforetabulateR/aagainst¿u/a,sothattheformerquan- should below.Weareinterestedinknowingthemeanstellarradius, retically theintensityofthesepoints(apartfromreflectioneffects) 0.9323+ .0020(m.e.)intheuniformsolution,forwhich¿u/a=0.38, dalis. Fromanunusuallyaccuratelight-curvehefindsbja^= ¿u/a =0.43,thetwocomponentsjusttoucheachother;forlarger b/a tobeexpectedfromtheRochemodelisalsogiven,sincethis axes, a&!,andwerereadoffforgivenaja(ifisthedistance the Rochemodel.) this starisabout0.1,andwemayadopttheempirical valuebjcir= fri/