190 6Ap J 23. . 2 4 8C a hundred-thousandfold,andCanopusiscitedasanexampleof of thelight-ratio,p,is put equalto0.4andtheSun’sstellarmag- nosities intermsoftheSun’soutputlight takenasunity,and the parallax,tt,andstellarmagnitude,m, ofeverystar,viz., 0^762, andtheirstellarmagnitudesare—0.96and+0.40;i.e., exceeding thatoftheSunbyten-thousandfoldormore,possibly liancy. ‘Gill,Kapteyn,andNewcombhaveunderdifferingforms When theparallaxesareexpressedinseconds ofarc,thelumi- inversely asthesquareofdistance,wemay derivethefollowing for othercasesofasimilarcharacter,usingmethodslightly is morethan20,000timesasbrighttheSun.Ihavesought light, wemaysubstitutetheSuninplaceofaCentaurithiscom- when, inaccordancewiththebestdeterminations, thelogarithm that inthecelestialspacesintensityof radiant energyvaries allaxes ofthisstarandaCentauriarerespectivelyofoio the evidenceuponwhichthisdoctrineisbased,andextentto such astar.Itisthepurposeofpresentarticletoexamine announced theprobableexistenceofstarshavingaluminosity in theheavensofatleastafewstarsextraordinaryintrinsicbril- well-known relationthatmustexistbetween theluminosity,L, different fromthatofGill.Ifweassume,as iscommonlydone, parison, andfindfromtheexpressiongivenabovethatCanopus as theSun,andthereforepresumablyemitssamequantityof star, istherefore3.50 which thatevidenceisconfirmedorrefutedbyotherconsiderations. Canopus is3.50timesbrighterthanaCentauri.Thelightoftheone on StellarParallax,substantiallyasfollows.Themeasuredpar- But sinceaCentaurihasthesamemassandspectrum © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem As respectsCanopus,thecaseispresentedbyGill,Researches The prevailingopinionamongastronomersadmitsthepresence THE LUMINOSITYOFBRIGHTESTSTARS By GEORGEC.COMSTOCK 2m Lttp =C.(l) times asgreatthatoftheother. 248 190 6Ap J 23. . 2 4 8C in TableIII. resulting valueoftheluminosity,Z,,andthesevalueswillbefound puted foreverystarbrighterthanthesecondmagnitudewhich nitude istakenas—26.2,theconstantcinaboveexpression tion andthoseofcertainfaintercomparisonstars.Sincethettwhich i. e.,theyarethedifferencebetweenparallaxofstarinques- I havebeenabletofindareliableparallaxdetermination,the becomes sensiblyequaltoV2.WiththesevaluesIhavecom- with thesetheaverageangularpropermotionshownbystarsof the Sun’smotioninspaceandamountofthatperannum of thecomparisonstarsobtainedasfollows.Thedirection to theobservedrelativeparallaxassumedvaluesofparallaxes that thecomparisonstarsareinfinitelydistant,orwemustadd appears inEquation(1)isanabsoluteparallax,wemusteitherassume any givenmagnitude—e.g.,theseventh—andobtainfrom the givengroupofstars.Theresultsinvestigationthiskind made byKapteynforstarsto.theninthmagnitudeandextended sponding tothemagnitudesasgivenbyobserver. have beeninterpolatedparallaxesofthecomparisonstarscorre- sented inthesecondcolumnoffollowingshorttable,fromwhich by thepresentwriteroveranadditionaltwomagnitudes,arepre- combination theaveragelinearmotionanddistanceof (spectroscopically determined)beingknown,wemaycombine corrections arealwayslessthan0Î01,andsince thetrueparallax parison starsarerarelybrighterthantheseventh magnitude,these relative parallaxescannotbefarwrong,although inanyparticular case theymaybeeithertoogreatorsmall. Sincethecom- © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The observedparallaxesofthestarsareineverycaserelative; On theaverageinterpolatedcorrections totheobserved II.o 9.0 3-° 5-°< 7-° LUMINOSITY OFTHEBRIGHTESTSTARS Magnitude TABLE I Mean Parallax o.0046 0.0156 0.0320 0.0093 0.0063 Mean Luminosity 87 26 58 9 3 249 190 6Ap J 23. . 2 4 8C sible errorsofunknownamount.Wemayconjecture thatthelumi- parallaxes areclearlyimpossibleandshowonly thepresenceofsen- nitude, theirrelativeparallaxesareapproximately zeroandindistin- light astheSun,and,introducingintoEquation(1)valueL= nosities ofthesefourstarsaregreaterthantheaverage, buthowmuch guishable fromerrorsofobservation.Table III thereforecontains are suchverybrightstars,apparentlyfainterthanthesecondmag- we findatoncefourinadmissiblecases.The negativeabsolute all theobtainabledataforourquest,andin considering thesedata the comparisonstarscommonlyemployed,andthereforeifthere sponding toZ=ioooarenotgreaterthanthemeanparallaxesof does, containcasesinwhichtheadoptedvaluehasbeenmade tion fromrelativetoabsoluteparallaxes.Itmay,andprobably table. a starcorrespondingtothemagnitudes,m,showninfollowing definition ofsuchstarsthattheyshallemit1,000timesasmuch that thereexiststarsofextraordinarybrilliancy,Iadoptasthe several hundredthstoosmall. star. Asaresult,TableIIIcontainsnocaseinwhichtheadopted parallax hasbeenmadesomuchasofoitoogreatinthetransi- responding errorofdefectintheresultingparallaxprincipal and probablywillthusexceeditinsomecases,producingacor- the tabularvaluebyseveral,ormany,hundredthsofasecond, other hand,thetrueparallaxofcomparison*starmayexceed allax oftheprincipalstartoogreatbysomuchasofoi.On that innocasewillthecorrectionmakeresultingabsolutepar- of thecomparisonstarcannotbelessthanof00,weareassured 1,000, findthefollowingvaluesofabsoluteparallaxsuch 2SQ © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem To rendermoreprecisetheconditionsofoursearchforevidence Below thesecondmagnitudetheseabsoluteparallaxescorre- GEORGE C.COMSTOCK TABLE- II 190 6Ap J 23. . 2 4 8C means ofEquation(1).Withoutnecessarily attributingahigh a Tauri greater isquiteunknown,andthestarsmustbedroppedfromcon- sideration asunavailableforthepresentpurpose. Inhere isshownthemeanluminosityofstars ofthethird,fifth, great bythethreeabnormalcases.In last columnofTable parallaxes havebeenleftdecidedlytoosmall, andhaveproduced than themean.Thisdistributionofvalues,improbableinitself, less thanthemeanvalue,whilethreehavevaluesmuchgreater ually 3820timesasbrighttheSun. the averagetwenty-fivestarsunderconsiderationareindivid- mean ofthenumbersinluminositycolumn,andfindthaton To testthecharacterofevidencethattheyafford,weform seventh magnitude,etc.,computedfromthe meanparallaxesby correspondingly greatvaluesoftheluminosity. result ofthetransitionfromrelativetoabsoluteparallaxes.Afew corresponds entirelytowhathasbeenaboveindicatedasaprobable added Arcturus,whoseluminositycomesjustwithinthelimit. nary brilliancyasdefinedabove,andpossiblytotheseshouldbe a Crucis Sirius 7 Geminorum ß Tauri.: a Orionis Rigel a Persei a Eridam Capellà a UrsaeMajoris Pollux P rocyon Regulus Canopus Castor © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem There remainthreestarstowhichthedataassignextraordi- b) Themeanvalueitself,3820,isfartoogreat, andismadetoo Two circumstancestendtoimpairconfidenceinthisvalue: a) Twenty-twoofthetwenty-fivestarshaveluminositiesmuch. Star LUMINOSITY OFTHEBRIGHTESTSTARS — i.o Mag. °- 5 °- 5 0. 2 0.7 2. o 1.9 2. o i. 2 I .o 1.8 1-5 1.9 1. 2 1.8 — O.OI2 Parallax 0.088 o. 117 0.076 0.050 0.008 0.067 0.008 O.O57 O.O32 O.064 O.342 O.O28 0.031 O.O58 °-37<5 TABLE III 54950 13800 Lum. .263 T 490 355 288 151 73 33 34 43 87 60 66 6 2 ß Centauri e UrsaeMajoris. ß Crucis Antares Arcturus Spica a Cygni Altair a Centauri 7] UrsaeMajoris.. a Gruis. a PiscisAustrinus. Vega Mean Star Mag. 0. 4 0. 4 0-3 i. 10 1.4 1.9 i. 6 1. i 1. 2 i. 2 1.9 1-5 1.4 1.8 — o.003 — 0.042 — 0.012 Parallax ofo962 0.137 0.033 0.023 o. 240 0.090 0.030 0.758 0.054 0.088 22400 o. 00430 1 Lum. 6 3820 25 996 45 525 120 160 21 2 i 190 6Ap J 23. . 2 4 8C there existsintheheavensanystarwhosebrilliancy exceedsthat observed relativeparallaxesofthethreestars named,orthrough either intheadoptedparallaxesofcomparison stars,orinthe The assumptionthaterrorsofthismagnitude areinadmissible the limit1000,wefindfollowingresults: be droppedfromthelist. character fromtheircomputedluminositiesandreducethesebelow fairly wellwiththemean,198,ofallvaluessavethreethatare a combinationofbothsources,constitutesthe entireevidencethat allaxes ofthreesuspectedstarsinordertoremovetheabnormal otherwise suspicious.TheagreementisstillbetterifArcturus with themeanvalue,3820,foundfromTableIII,whileitaccords on thisaccounttheextrapolatedvalueofLismaderelativelytoo luminosity ofthefaintstarsmore.thanthatbrightones,and in agroupofstars,andonthisaccountthevaluesLTableIII account ofitsnon-linearcharacter,Equation(1)isnotstrictly graphical processIhaveextrapolatedfromthesedataavalueof the resultfoundabove,£=135,itremainsabsolutelyinconsistent error intheconcludedluminosityoffirst-magnitudestars,but great. Itisnotfeasible,atpresent,toevaluatetheresultanttotal tion oflightinitstransmissionthroughspaceaffectstheapparent with themeanmagnitudeofTableIII,andfind¿=135.On the luminosityformagnitude1.o,correspondingapproximately of luminositywithincreasingapparentbrilliancy.Througha therefore asfurnishingatleastaroughindicationoftheorder rest theparallaxesandluminositiesofTableIII.Weregardthem stantial valuewithoutoverturningthefoundationsuponwhich whatever reasonablemarginofuncertaintymaybeattributedto applicable tothemeanrelationbetweenparallaxandluminosity are probablysomewhattoosmall.Ontheotherhand,absorp- brilliancy ofthesestars,showing,astheyought,aprogressiveincrease degree ofaccuracytotheseresults,wecannotimpugntheirsub- 252 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem If weinquirewhaterrormustbeattributedtotheadoptedpar- ßCrucis 0^015 Canopus 0.52 Riçel 0.2 GEORGE C.COMSTOCK 190 6Ap J 23. . 2 4 8C 5,1 of theSunathousand-fold.Opposedtoitstandconsider- author theyareofpreponderantweight. ations abovedesignated{a)and(6),inthejudgmentof stars ofthesameluminosityasthatSun,1starwith100,000 a paperissuedasNo.noftheGroningenPublications,andthere writer incredible.Theyareconsequences,and,insofarasrelates luminosity thantheSun.Theseresultsappeartopresent times greaterluminositythanthatoftheSun,”38starswith10,000 statement: “Therewillbeinaspacewhichcontains2,000,000 embodies apartofhisconclusionsinthefollowingremarkable times greaterluminosity,1800starswith1,000lumi- to thebrighteststars,extrapolations,fromanarbitrarilyassumed nosity, etc.,togetherwithmorethan12,000,000starsofsmaller law ofthedistributionstarsin'space—ain-supportwhich to theSun,increasesrapidlyamaximumataveragedistance of starsthefourthorfifthmagnitude,andthereafterdiminishes ber ofstarsperunitvolume,isaminimumintheregionadjacent thus madeleadstotheconclusionthatstellardensity,num- tributed inaccordancewiththelawoferror.”Theassumption to admitseemssupposethatthequantitiesz....aredis- lack ofsupportforthisassumptionanditsimprobableconse- in hisequations,butwithnoconsiderablesuccess.Theresult to theassumedlawfromwhichtheyarederived: “Ineednot is inherentinhisfundamentalassumption,andviewoftheentire endeavors toremoveitbyasuitablechoiceofconstantcoefficients the improbabilityofsuchavariationstellardensity,Kapteyn slowly tothesupposedlimitsofstellarsystem.Recognizing attributing oneandthesameparallaxtostars of adeterminedmag- expect togetagoodstepbeyondwhatmay be reachedbysimply say thatIdonottakethesuppositionmade hereforanexpres- no evidenceisadducedbeyondthestatement,“thebestthingfurther nitude andpropermotion. sion ofthetruth.Iamonlyopinionthat by makingitwemay above anygreatermeasureofcredencethanKapteynhimselfassigns quences, itdoesnotseemnecessarytogivetheconclusionsquoted © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Kapteyn hasinvestigatedtheluminositiesoffixedstarsin 1 Pub.Groningen,No.8,p.21. LUMINOSITY OFTHEBRIGHTESTSTARS 253 190 6Ap J 23. . 2 4 8C of opinionthatthemeanluminosityfirst-magnitudestars cannot berepresentedbyanumberwithlessthanthreesignificant requires morethanthreefiguresforitsexpression.Heisfurther any adequateevidencethatthemaximumofstellarluminosity figures. 254 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Upon asurveyofthewholecaseauthorisunabletofind Washburn Observatory, Madison, Wis. GEORGE C.COMSTOCK