192 9MNRAS. .90. . .17M infer asingledefinitefactconcerningtheinternaldensity-distribution. relativity. Einstein’sprincipleofrelativity takestheplaceof few terms:this,however,presentsnodifficulties. between observablesonly. Itfrequéntlyappears,whenthismode of from apieceofmathematico-physical reasoningmustbearelation necessary tointroduceintoourequationssymbols correspondingto Heisenberg’s “uncertaintyprinciple”takes itsplace.Itmaybe time, arecalculatedintermsofv. in asteroidcalculationsthenumberoftermstobetransformedisusually unobservables, butthephysicalcontentof assertionwhichresults sponding toquantitieswhichexperienceshows canneverbeobserved. avoids thisconcept,andtheresultingtheory offramesreferenceis observable ”;consequentlyadynamicsmust beconstructedwhich the transformation.Hansenaddsperturbationstomean not verygreat,andformostofthemitissufficienttoputv=w w +P—57.Inthisformulacontainsthedisturbingmassasfactor, Nov. 1929.Masses,,andTemperaturesofStars.17 observable ;quantummechanicsthereforediscards thisconcept,and The “velocityofasystemthroughtheaether ”provestobean“un- it isnotpossibletoinferfromtheobservedmasses,luminosities,and but theseperturbations,insteadofbeingcalculatedintermsthe (position, velocity)whenattachedtoanelectron provestobeanun- temperatures thattheinteriorsofstarsarenecessarilycomposed 2e sin(w—5T).Theremaybeasecondapproximationnecessaryfor and wecanthereforesubstituteinittheellipticvalueofv,namely, perfect gas;andthatitisnotpossibletodeducethevalueof anomaly :substantiallythesamethingisdoneinthistransformation, “ unobservable”velocitythroughtheaether. Thesimultaneouspair absorption coefficientforthestellarinterior.Instead,weareledto luminosities, andelectivetemperaturesofthestarsfromastandpoint This substitutionconstitutesthemainpartofnumericalwork,but Eddington inhiswell-knownwritings.Themainconclusionsarethat which isphilosophicallydifferentfromthatadoptedbyProfessor v Yale University: 2. Contemporaryphysicsmakesprogressbydiscardingideascorre- The analogytoHansen’sformmentionedin§1ismadeevidentby i. InthispaperIinvestigatetherelationsbetweenmasses, The Masses,Luminosities,andEffectiveTemperaturesoftheStars. 1929 May. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System v =w+E_0. By E.A.Milne. I. Introduction. w 2 192 9MNRAS. .90. . .17M 24 of theirangulardiameters. between massandluminosityforthemajority ofthestarsremainsfor been explainedtheoreticallywhytheefiective temperaturesofthestars have theirobservedorderofmagnitude.How exactisthecorrelation by appropriatechoiceofitsefiectivetemperature T-l,andithasnever Eddington allowsastarofgivenMtohave anarbitraryluminosity it hasneverreceivedtheoreticalexplanation ; theformuladerivedby these wasdiscoveredbyEddington,butitmust berememberedthat lawandtheobservedluminosity-colour law.Theformerof we havethetwosetsoffactswhicharesummed upintheobservedmass- theory ofhigh-temperatureionization.Thequantitativerelations— so fardeterminedmaybeliabletoconsiderableuncertainty. the observedtemperaturesofmaxima—amountinprincipletoafixation legitimate toassumeitforotherstars.Weshallfindthat(1),(2),and a linearsequenceconfirmsbothqualitativelyandquantitativelySaha’s of thephotosphericopacityfstar,thoughvalues structure ;aspecifictheoryoftheinteriorisnotrequired. because thestarsholdtogether;(2)followsfromtheirobservedspectra atmospheres. Thefactthattheobservedspectraofstarsfallinto equilibrium. Forallthesewehaveobservationalsupport:(1)follows (3) aresufficienttoyieldarelationbetweenL,M,Tjandtheinternal (from theobservedvariationofbrightnessoverdisc),butitseems that itsouterlayersaregaseous,(3)inradiative for manystarsgiveusafairknowledgeoftheefiectivetemperatures. we believethatthecolour-temperatureswhichhavebeendetermined (3) isinfactonlyknownfromdirectobservationthecaseofsun the electivetemperaturehasbeendeterminedforveryfewstars,*but the energyleavingeachsquarecentimetreofstar’ssurface.Actually stant, forthedefinitionofeffectivetemperatureTjisthat effective temperatureTj;LandTareequivalenttoadetermination pruned ofunnecessaryassumptions. they arenowshowntobetrueindependentlyoftheintroduction thought isfollowed,thatthesameresultsarededucedasbefore,but relations betweenthe“observables”ofastarwhichareindependent of itsradiusrbytherelationL=47rrcrT,wherecrisStefan’scon- of arbitraryassumptionsabouttheunobservableinterior. Yet thesamegeneralmarchofideasimpelsustoinquirewhatare we donotdoubtits“realexistence”inthesensecustomaryphysics. aether orthesimultaneous(position-velocity)pairofanelectron,for . Theinteriorofastarcanneverbedirectlyobserved.Itisnot concepts correspondingtounobservables.Theoryisenrichedbybeing of courseanunobservableinthesamewayasvelocitythrough 18 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, x x1 * Forthesun,bydirectobservation, andforafewotherstarsbymeasurement f SeeBakerianLecture(1929), Phil.Trans.Roy.Soc.,A.228,421. In additiontotheforegoing,whenweconsider thestarsasawhole We havealsoafairworkingknowledgeofthephysicsstellar We knowfurther(i)thatthestarisinmechanicalequilibrium,(2) The timeseemsripeforapplyingthismethodtothestructureof 3. TheobservablesofastarareitsmassM,luminosityL,and © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 192 9MNRAS. .90. . .17M sources andsinks.Itsluminosityisthendetermined.Theresultof to changehorsesinmid-stream.Wemustcarry ourcoolingproblem to thebitterend. the radiuswithoutaffectingconditions inside.| Thatwouldbe cooling problemfortheinnermostTT)of radius,andthenassume cooling problemboundaryconditionsareparamount, andtheyprove to besoinstellarproblems.Wemustnot treattheproblemasa notice isthattheproblemessentiallya cooling problem.Inany justsufficienttorestoretheitsstateatbeginningof ¡sources. Ifwecouldcalculatetherateofliberationsubatomic the energyflowingpastthispointcanflow throughtheouter^of star inequilibriumandcalculateitscooling. The importantpointto any otherquantitieswefinditnecessaryto introduce tobuildupthe the calculationwillbeaformulaconnecting LwiththemassMand to theamountradiatedbystarasawholespace.Wethen und wededucethattheenergyliberatedmustcomefromsubatomic particularly (formainsequencestars)byHertzsprung’srecentstudies* future investigation,buttheexistenceofafairdegreecorrelationis assume thattheactualstarisprovidedwithjustthisdistributionof the intervaldt.Thealgebraicsumofsourcesandsinksisequal cool down;othersmaygainenergyandwarmup.Wenowintroduce of thePleiades,isverywellestablished. whole radiatestospace.Intheprocesssomepartsloseenergyand information theyyieldastothestateofaffairsininteriorastar. a distributionofsources(andifnecessarysinks)energythroughthe to useanothermethod. energy pergramofmatterunderstellarconditions,simpleaddition summed upintheHertzsprung-E-usselldiagramandexhibitedmore mass asthegivenstar.Wethenletitcoolfreelyforashortperioddt. Nov. 1929.andEffectiveTemperaturesoftheStars.19 The differentpartsofthestarinterchangeenergy,andasa energy tobeableperformthecalculation.Wearethereforecompelled steady state.Unfortunately,inthepresentstateofphysics,weknow geological, andbiological—thatgravitationalcontractionisinsufficient, temperature, density,stateofionization,etc.—^holdingintheinterior of astar. too littleofthelawsgoverningrateliberationsubatomic through thewholemassofstarwouldgiveitsluminosityin conditions. Itisknownfromvariouslinesofevidence—astronomical, and thenattempttocalculatewhatenergywillbeliberatedunderthese a factwemusttakeintoaccount.Theluminosity-colourrelation, our inquirybyaskingwhatismeant“calculating”theluminosity of astar.Thereareinprincipletwowayscalculatingtheluminosity explained orpredictedinthelightofpresent-dayphysics;(b)what II. Wecanbuildupastarinmechanicalequilibriumofthesame I. Wemayreconstructasbestwecanthephysicalconditions— 4. Itiscustomarytospeakof“luminosity-formulæ,”andwebegin The generalproblemsare{a)whetheranyoftheserelationscanbe © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System f Foranillustrationtaken fromphysics,seeAppendix. * M.N.,89,674,1929. 192 9MNRAS. .90. . .17M 20 Prof.E.A.Milne,TheMasses,Luminosities,xc.i them withthepropertyofemittingindependentlyphysicalcon- For example,takeEddington’sparticularsolutionforthestarwith point-source ofenergythedistributionisknown, andonlythefirst ditions ;theymaybepicturedasradioactivesources.Nowbringup a point-sourceatitscentre.Inserttherequiredsources,{andendow M hasbeencalculatedbytreatingtheproblemasacoolingproblem. greater thanacertainminimum.fThisiseasilyprovedasfollows. in theformthatastarofgivenluminosityLcanhaveanymassM by adjustmentofitsefiectivetemperature.Iprefertoputtheresult from outsideanarbitrarymassM'andscatteritgentlygramby according towhichastarofgivenmassMcanhaveanyluminosity siderations gohasbeenfrequentlypointedoutbySirJamesJeans.* proves thisisnotso;TjfixedwhenwearegivensimplyL certain circumstances. mass together.Thusstartingwith(L,M) we canalwaysconstruct radiation-pressure (derivedfromL)insufficient toblowupMwill a newconfigurationofequilibrium—thereis nothing elseforittodo. efiective temperature^ofthesphericalaggregatematter.Analysis- that thequestionwouldbeansweredbyourbeingabletofind physical stateoftheaggregatewheninequilibrium?Wemightexpect Take anymodelofastarforwhichtheluminosityLcertainmass themselves throughthenewstar?In case ofthestarwith certainly beinsufficienttoblowupM+M', and gravitywillbindtho cannot generateenergy.ThenewmassM+M'mustsettledowninto of thenewsystemandhowdooldsources ofenergydistribute over thesurfaceofstar,massM'being“deadmatter”that This isborneoutbyProfessorEddington’sfinalluminosity-formula,, and M—theproblemstillhasadegreeofflexibility. is thatwecannot.AllcandotoobtainarelationbetweenL, to explainanythingsofundamentalasthemass-luminositylaw.This tecture ofthestar.Wecannothopebyananalysisacoolingproblem formulæ :howisit,sinceweknownothingofsubatomicenergy,that The weightoftheaddedmasswillcompress thepre-existingstar aggregate MgeneratesacertainenergyLpersecond—whatisthe states wemustregardLandMasindependentvariables;acertain (L, M+M');theonlyquestionsare:Whatis theefiectivetemperature discussion ofequilibriumstates.Whenwearediscussing must dependondeep-seatedprocesseswhichareentirelybeyondany M andothercharacteristicsofastar.Thereareaninfinitenumber The analysisremovesadifficultysometimesfeltrespectingluminosity- of stellarluminosity,thoughitisnotalwaysanalysedinthisway. r at itsownrate.AllwecandoistolinkupLandMwiththearchi- of waysbuildingupastarinequilibrium,andeachthecools we canmakeacalculationoftheluminositystar?Theanswer t Asufficientlysmallmass mightbeblowntoinfinitybyagivenluminosityunder J Iassumenosinksarerequired. TheydonotoccurinEddington’smodels. * E.g.AstronomyandCosmogony(1928),p.83. That LandMareindependentvariablesasfarequilibriumcon- 5. MethodII.isinfactthemethodalwaysfollowedcalculations © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 192 9MNRAS. .90. . .17M , ofEddington’smodel“constant.”fThenewstarwillcourse in thepoint-sourcemodel,diminutionofstrength ofthepoint-sourcewillresult singularity atthecentre. See §16below. smaller luminosityandequalmassbysimplydiminishing therateofsupply centration ofenergy-hberationtowardthecentreand equal tounityattheboundary. in acondensationofthestartowardscentre. 77 isdefinedin§18below. energy fromthesources.Initiallyeachelementof the starhasaccelerationzero, and thestarwillinitiallycoolatsamerateasbefore ;thismeansthatcooling will nowexceedtheenergy-supplyandstarslowly contract.Borexample, neutral thesurfaceconditionscanhaveacompletelydominanteffect. i —ß.See§20below,equation (51). We seefromthisexamplethatwhenthemechanicalequilibriumis coefficient hasthedesiredvaluegivenbyL=47rcGrM(i—ß)//^,Land M beinggivenarbitrarily(subjecttoapossibleupperlimitforL). small, thecooling(radiationtospace)willexceedsupplyofenergy mechanical equilibriumthephotosphericabsorptioncoefficientistoo the interior.Itcanmakethisadjustmentquiteeasilysinceitsmechani- and thestarwillalteritsradiusuntilphotosphericabsorption cal equilibriumisneutral.If,forexample,inanyconfigurationof of thestar.§Thestarthereforeadjustsitsradiusuntilabove the pressureandtemperatureatphotosphere,soonradius pressure tototalthroughthestar,isaconstantdepending expression isequaltoL,therateofliberationenergythroughout simply onthemassandmolecularweight.Nowwilldepend temperature. ByEddington’scalculations,thestarcoolsatrate can existinmechanicalequilibriumwithanyradiusandsoefiective constant fromcentretophotosphere.Thensince77isequalunity mechanical equilibriumisofthetypeknownas“neutral”;star at thephotosphereconstantvalueofktjisequalto/c into theequilibriumofastar.Letussupposethatktjisrigorously in theouterlayersanddiffuseenergy-sourcesthroughadded of Katthephotosphere.ThepeculiaritythismodelJisthat in generalnotsatisfythisrelation;convectioncurrentsmayoriginate convection currents)isattained. point-source models.* example, wecanscatter“dead”matternottooopaqueoverthesurface Nov. 1929.andEffectiveTemperaturesoftheStars.21 as tothedirectconnectionofmasswithluminositybyconsidering constructed bytheaboveprocess.Itisclearthatwecaninfernothing of thesetwoquestionspresentsitself.ProfessorEddington’ssolution “ dead”materialuntilanewstateofradiativeequilibrium(without other solutionsofthesameluminosityandarbitrarilygreatermass, l5 of thisproblemisonlyonespecialsolution;thereareaninfinity § Fromanyphysicalformula fork,kisreadilycalculatedintermsofTand f Weshallseethatthispeculiarityissharedbyall modelsnotpossessinga * Similarly,givenastar(L,M)wecanalwaysconstruct another(L—L',M)of f /cisthemass-absorptioncoefficient;77aparameter measuringthecon- x If kisindependentoftemperatureandpressure, ande,therateof Consideration ofthemodel“^77=constant”givesusfurtherinsight 6. Thisprocessisapplicabletoanymodelwithafiniteradius.For © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System ß)Ik ergspersecond,wherei—ß,theratioofradiation- 1 192 9MNRAS. .90. . .17M 2 12-2 4 -2 infinite diffusion.TherelationbetweenL,M,andkforthismodel 22 Prof.E.A.Milne,TheMasses,Luminosities,xc.i is aparticularcaseofEddington’sfour-termrelationbetweenL,M, Eddington’s solution,butweseethatinordertosatisfytheboundary radius risgivenbyL=i7rr•67rcrP.Ifthecondition liberation ofenergypergram,isaconstant,theequilibriummore We arethendrivenbackonthesolutionwithP=o,namely,Professor thus contradictingoursuppositionthatthestarhadafiniteradius. P->P(P =j=°)isimposed,itfoundthatp=oonlyatinfinity, internal pressuresoftheorderiodynescm..Butwhen is againneutralandtheconfigurationverysensitivetoboundary conditions theradiusmustnowbeinfinite.Thestateisoneof a devastatingefiect.ThevalueofPis£•¿aT-,,whilsttheboundary equilibrium isneutralasmallchangeinboundaryconditionsmayhave boundary. ProfessorEddingtonhasarguedthatthissmallconstant, has afiniteradius(p=oforr),theconditionsofradiative taken totendzeroattheboundaryp=o*Actuallyifastar to Eddington’ssolution,couldrestinequilibriumwithanyradius. of theorderidynecm.,canbeneglectedincomparisonwith equilibrium requirethatPshalltendtoasmallconstantatthe conditions. InEddington’ssolutionthetotalpressurePhasbeen delicate still.Atfirstsightitmightappearthatthisstar,according k, andrj,inwhichrjhasbeenputequaltounity. This istruewithEddington’sboundaryconditions,buttheequilibrium hold, andtheequilibriumrequiresre-investigation.Thisiscarried out below.Weshallfindthatthephotosphericabsorptioncoefficient . Itdependsonthegeneral architecture ofthestar. y a certainlimit;(ß)thattheboundaryconditionsmustbetakeninto k isjustassignificantinthegeneralcasemodel“ktj= 0 stant ;thestarcould rest inequilibriumwithanyradius.fBut the density-distribution, weareenabledtoobtain ourluminosity-relation the troublewiththismethodisthatwecannot derivealuminosity- being aparametermeasuringthedistribution ofenergy-sources,but Professor Eddingtonhasingeneralinvestigated starsofgivenr],rj account incasesofneutralequilibrium. 0 0 explicitly. density-distribution, orratheracertainfunction whichfixesthe relation inwhichallthetermsarecalculable. Bybeginningwiththe on thecompletemarchofdensity-distribution fromcentreto and acertaincoefficientC.ThisCisnumberdepending to afour-termrelationconnectingL,M,thephotosphericopacity^ (a) thatastarofgivenluminositycaningeneralhaveanymassabove 0 we findthatusingonlyassumptions(i),(2),and(3)aboveareled 0 constant.” ± If, ontheotherhand,“kt]”isnotconstant,solutiondoes This digressionservestobringouttwopointsofimportance: The equilibriumforagivenfwouldbeneutral ifwereacon- 7. Whenweproceedtocalculatethecoolingofanymodelwhatever f Apartfrommoredelicate considerationsofthetypementionedin§6. * pdenotesthedensity. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 192 9MNRAS. .90. . .17M L, M,andaconstant adefiningthephotosphericopacityfrompurephysics. two equationsandunknowns,namely, C anda.Solvingfor The detailswillbeclearlater. from theSunandCapella.Wethentabulate theluminosities,masses, single undeterminedconstanta,aphysical constant dependingonly logous family.ThevaluesofKwillbedifierent, buttheyinvolve, value ofCisthesameforeachstar,theybelongtohomo- and efiectivetemperaturesofallothermembers ofthefamily. two members,Sun*andCapella*,areindistinguishable observationally and awehavethedatatoconstructourhomologous familyofwhich on thepropertiesofstellarmaterialinphotospheres. Wehavethus according tothetheoryofphotospheric absorptioncoefficient,a the samemass,luminosity,andefiectivetemperatureasSun family ofstars,i.e.tabulatetheirmasses,luminosities,andtempera- tures, andthencomparetheactualstarswiththisfamily.Thefamily basis ofcomparisonitisinteresttoconstructacompletehomologous I choosetoconstructisthatfamilywhichcontainstwomembersof type ofdensity-distributionfromstartostar.Inorderhavea piece ofinformationabout,equivalenttothemeangradient(f> our four-termrelationbetweenL,M,/q,andCforeachstar.The Capella, respectively.CallthesetwomembersSun*andCapella*. trustworthy. Whatwearereallyinterestedinisthevariation photospheric opacity/qisnotaccuratelydetermined,sothatthe These twohypotheticalstarsprovideuswithequations,namely, values ofCdeducedbytheforegoingprocedurewouldnotbevery from thecentretophotosphere.Whencf)isconstant,itsvalue(from deviation ofthedensity-distributionfromthatpolytropew=3. its definition)isunity,andthedensity-distribution—1thatof number associatedwith.ThevalueofCis,however,adefinite varying massandradius.Anyparticularactualstarwillbeamember We couldinthiswaydeterminethecoefficientCfromobservationfor Eddington’s model,theEmdenpolytropew=3.Sincevalue which thestarbelongs.FromitsobservedLandMaknowledge every star. of Cfixesthedepartureforma“homologousfamily,”definite © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 192 9MNRAS. .90. . .17M 1 envisaged asacommonratio ofmeandensitytocentraldensity. perfect gas,stilllessdoweneedtoknowtheir meanmolecularweight; posed ofperfectgas,andthattheirouterlayers areinradiativeequili- liquid intheirinteriors,assupposedbySirJames Jeans.Clearlythe brium. Wehavenotassumedthattheirinteriors arecomposedof tion intheirinteriors.Forallouranalysis is concernedtheymaybe we havemadenoassumptionaboutthedistribution ofenergy-genera- observed relationsbetweenthemasses,luminosities, andtemperatures stars, areinmechanicalequilibrium,thattheir outerlayersarecom- recall, however,thattheonlyhypotheseswe havemadearethatthe mass, luminosity,andeffectivetemperature;thepreviouslynoted in factformasinglehomologousfamily,withcommondensity- stars shouldturnouttohaveacommondensity-distribution.We It isnopartofthepurposepresentpapertodiscusswhy evolutionary implicationsareconsideredinapaperwhichistofollow. nosities nowimpliesthatthevaluesofCwouldhavecomeoutapproxi- agreement betweenourtableofluminositiesandtheobservedlumi- mately thesameforallstars. using ourfour-termrelation,foreachobservedstarfromits data, orsomethingclosetoit,wecouldhavecomputedthevalueofC, distribution^ have reversedtheprocedure.Takingopacitygivenbyspectroscopic is abouttwicethatwhichIhaveindependentlydeducedforSunand about thefunction,stars(giantsandmainsequencestars)do would havebeenanentirelyartificialconstruct;themembersofour actual stars.Ioriginallyconstructedthetablemerelyasanexperi- agreement is,however,profoundlysignificant.Itisclearthatwecould opacity differentfromthatofrealstellarmaterial.Thefactthe of entirelydifferentordersmagnitude,ourspecialhomologousfamily opacities necessarytoattributeSun*andCapella*hadcomeout opacities ofthephotospheresSunandCapellaphotospheric Capella fromspectroscopicconsiderations4Ifthe homologous familywouldhavehadtobeendowedwithaphotospheric the photosphericopacityistakentohaveacertainvalue.Thevalue fairly accuratelypredictedtheluminositiesofallknownstarsin same valueforalltheobservedstarsprovidedcoefficientadefining as theluminosityofactualstar.Inotherwords,stars of thephotosphericopacitynecessarytoattributeSun*andCapella* are observationallyindistinguishablefromthemembersofacertain of thecorrespondingstarinourhypotheticalfamilyiscloselysame 24 Prof.E.A.Milne,TheMasses,Luminosities,xc.I, heavens ;jthatistosay,forastarofgivenMandT-^theluminosity ment, asastandardofcomparison.Itthenappearedthatthetable homologous family. § Notofcourseacommon density.Acommondensity-distributionisbest t Withtheexceptionofwhitedwarfs.See§28. j BakerianLecture,Phil. Trans. Boy.Soc.192Q. f This meansthatasfarthecoefficientCdisclosesinformation io. Thisisthemainpositiveresultofpresentpaper.Its This isequivalenttothediscoverythatcoefficientChas © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 192 9MNRAS. .90. . .17M 1 ton’s “luminosity”formula oritsmorepreciseform(32);either steady state.Thisappears tomebethetruesignificanceofEdding- about thedistribution ofsourcesnecessarytomaintainthestarin a thecentralvalueof1—ß,withadetermined meanvalueof where 1—ßisdeterminedintermsofM,athree-termrelation, with thearchitectureofstars,fromobservation. paper permitsanactualdeterminationofacharacteristicconnected a homologousfamily.Themainpointisthatthemethodofpresent difficulty, thoughitmightthenbefoundthatthestarsdidnotform coefficient. Anyothertheorycouldbeintroducedwithouttheslightest absorption coefficient.Butthegeneralprinciplesofpaperare certain pointdependonanassumedformulaforthephotospheric quite independentofanyspecialtheorythephotosphericabsorption Eussell diagramandotherstellarcharacteristics. luminosity lawgoessomewaytowardsaccountingfortheHertzsprung- interiors butdeducedfromobservation.Butasucceedingpaperwill show thatthiscommonfeaturecombinedwiththeempiricalmass- common featureofthestarsnotdependingonhypothesesabouttheir law northeHertzsprung-Eusselldiagram.Wesimplydiscovera first instance,explainsnothing—neithertheobservedmass-luminosity —the distributionofsourcesandsinksnecessaryfortheobserved scope ofthepresentinvestigation. structure tobeasteady-stateone.Theseareproblemsbeyondthe reaction ofphysicalconditionsonthedistributionenergy-liberation up acommondensity-distribution;oritmaybethatthereissome composed ofperfectgas.Itmaywellbethatsomestabilitycriterion, homologous family,cannotbetakentoimplythattheinteriorsare of .thetypediscussedbyJeans,comesintocompelstarstake Nov. 1929.andEffectiveTemperaturesoftheStars.25 of thestars,nowpartlydescribedinformthattheybelongtoa 12. ProfessorEddington’sluminosityformulainitswell-knownform It shouldalsobepointedoutthattheresultsofpaperaftera 11. Itshouldbepointedoutthatthepresentinvestigation,inthe. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System _47tcGM(i-ß) L 192 9MNRAS. .90. . .17M be theequationofthiscurve.Thendensity functionz(x)isthat from centretoboundary,andiftheyareplotted againstoneanother solution of any actualconfigurationofequilibriumyand 2willvarysmoothly a smoothcurvewillresult.Let star ;wehavetwounknowns,yand2, onlyoneequation.In r fromthecentre;pdensityatthispointM(r)massenclosed pressure +radiation-pressure)atapointinnon-rotatingstardistant the computedinterioropacity. 26 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, by asphereofradiusr;Gtheconstantgravitation.Then determines thesource-distributionfromobservedLandM We removethecentralconstantsbymakingsubstitution so thaty—iandz=1atro.Then Let P,pbethecentralvaluesofandp.Put Introducing (2)in(1)anddifferentiatingwehave Then (5)becomes equation ofmechanicalequilibriumis where c 13. MechanicalEquilibrium.—LetPbethetotalpressure(gas- This equationisnotsufficienttofixthedensity-distribution inthe We proceedtogivethemathematicalanalysis. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System II. GeneralTheoryofStellarEquilibrium. 2 2 47rGp rdr\zdr; c 2 = r dr\pdr)^P 2 i d/rdP\ P ~Pc^ n x dx\zdx) • (9) 2Z 2 2 x dx\zdx^' M(r) =i^7rrpdr.(2) 2 i dfxdy ÆP _GM(r) dr r 2/ =/(2)•(8) \477-Gp c Jo x (4) (5) (3) (7) (6) Nov. 1929. and Effective Temperatures of the Stars. 27 which satisfies dz z — i, — = 0 at x = o. dx

We note that/(i) = 1 by definition. 14. Homologous Families.—To every arbitrary function f(z) for which/(i) = i there corresponds a definite density-distribution z(x), the solution of (9). All stars having this density-distribution we speak of as the “ homologous family defined by/.” The classical researches of Lane, Jeans, and others have investigated the properties of homo- logous series of stars, i.e. a series of configurations of equilibrium of constant mass. A “ homologous family ” is a generalisation of this concept, as it includes stars of all masses and all radii having the same density-distribution. The function z(x) may at first increase or decrease with x. For functions/corresponding to physically existing stars z(x) will ultimately decrease to zero for some finite value of x. Let aq be this value. Then the radius r1 of the star is given by

Ti = (10)

We see that given/, i.e. given the homologous family, we still have two constants at our disposal, namely, Pc and pc. Thus there is a double infinity of stars of a given homologous family.

Let us now find the mass M of a given member (Pc, pc) of a given homologous family. We have

or, using the differential equation (9), x 3 2 2 P c ~x df(z)~ M - (II) 4.TrQ3pf\-Z dx _ where the suffix # = aq denotes that the expression is to be evaluated at a? = aq. Thus, for members of a given homologous family,

2 MocP//pc .... (12)

Given the mass, one constant remains still at our disposal. By (10) this may be taken to be the radius. As an example of a property of a homologous family we will find the ratio of mean density p to central density pc. We have

2 P = M = jTTjip/p^dr = ^¡zx dx 1 i df(z) (13) 3 z dx - Pc î^h Pc V x=xx

Tims plpc is the same for all members of a homologous family. Again, from (10) and (12) 2 PcxM V .... (14)

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 192 9MNRAS. .90. . .17M and BakerianLecture,1929, Phil.Trans.Roy.Soc. total pressure. increased toinfinity.Wesavealgebrabytreating thelayersoutside becoming nearlyconstantfor9verysmall and decreasing,suddenly the photosphereseparately. photosphere. Thevalueispracticallyattained atthephotosphere. matical device,anditcouldbeavoidedby choosing a(ftwhich,after photosphere. Thisintroductionofarestriction onisonlyamathe- We shalllaterhavetoexaminetheefiect of thelayersoutside longer representstheactualdensity-distributionoutsidephoto- sphere, but(f)maybechosensothatitdoesrepresent itfromcentreto (f)(9) tendstoadefinitefinitelimitas0o,andweshallhenceforward With thisrestrictiononthesolutionofdifierentialequationno consider ourhomologousfamiliestobedefinedbythefunction(f)(9). duce aparametricrepresentationfor/(z),namely, The boundaryvalueof9iszero.Weshallimposetherestrictionthat small ;hence/(z)isverysmall.Toavoidawkwardlimitswenowintro- further inwards.Forareasonwhichwillappearlaterweareanxious to calculatethevalueofi—/3atphotosphere,whichwewillcall photospheric level;thereafteri—ßchangesveryslowlyaswego value unityJtoaveryslowlychangingwhichitreachesatthe where aistheradiation-constant,Egas-constant. i —jSj.Atthisleveltheleft-handsideof(17)isfinitewhilstzvery enter thestarfromoutside,1—ßchangesrapidlyitsboundary T wehave and radiation-pressure tototalpressureatthispoint.(Wedonotassume mean molecularweightatthatpoint.Leti—/3betheratioof in theouterlayers. assume theouterlayersaregaseous,inaccordancewithobservation. We proceedtocalculatetheratioofradiation-pressuretotalpressure the starmayhaveanyequationofstate,gaseousorliquid.Letusnow 28 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, ï —ßisconstant.)Then * Jeans,M.N.,85,199,1925. Î Gas-pressureiszeroatthe boundary,sotheradiation-pressureisequalto f Woltjer,B.A.N.,2,171(No.66),1924.SeealsoMilne, M.N.,85,784,1925; It isknownfromtheresearchesofJeans*andWoltjerfthataswe Let Tbethetemperatureatanypointinouterlayers,¡jl 15. Sofarwehaveonlyusedtheequationofmechanicalequilibrium; © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 11)3T3 i4 -ß=°vw iaß* VPPc*z (i -=&T*....) S f(z) =¿(0)0*....(19) ßp =(R/^)pT....(16) s z =6(18) Eliminating (!?) 192 9MNRAS. .90. . .17M 2 34 relation, alsoduetoEddington, equation ofmechanicalequilibrium(1)and using(15)wehavethe where kisthemass-absorptioncoefficient. Combining thiswiththe 34 net fluxperunitarea,L(r)/47rr,isgivenbyEddington’s equation Let L(r)bethenetfluxofenergyacrossasphere ofradiusr.Thenthe in thecaseofSun,thatouterlayersare inradiativeequilibrium. distribution, andnotmerelyonthelocalvalueofWewrite hence sodoes1—ßThusdependsonthecompletedensity- depends onthecompletemarchoffromcentretophotosphere,and Since 8(x)isthesolutionofadifferentialequationcontaining-model,wemaylet Neglecting theexceedinglysmalldifferencebetweentruemass d ->ointhefactoronright-handsideof(21),obtaining Nov. 1929.andEffectiveTemperaturesoftheStars. the massequation(11)becomes 16. Letusnowintroducetheassumption,supportedbyobservation We nownotetheremarkablefactthatPandpareinvolvedin c The differentialequation(9)nowbecomes © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 32 2 i 7mGM^ M = 4 x A Vi' 4 L(r) = 2 47rr 2 L(r) M = 1 3 4ttGp C = 3 P 47rcGrM(r) d[(i—/?)P] c ttGVo' 3 K 4:2 3 4Po x(d9/dx) tJ 1 4 k _c_ d(JaT) P -x- 12 3 ^ -0 % dr dP Jd6_^ xf(dd/dx)f \dx; X = i 2 2 2 (7) (26) ■*(23) (5) (4> (22) (21) (20) 29 192 9MNRAS. .90. . .17M limit ofL(r)as0tendstopracticallyzero.LetthisbeL,the We wishtoascertainthefluxofenergyatphotosphere,i.e. luminosity ofthestar.Then 30 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, photosphere itself(whereT=Tj),maybecalculatedbyanalysisof the equilibriumoflayersoutsidephotosphere,forgiveng a functionofT.*Ifprovedtobeindependent{e.g.ifitwere (gravity) andTi.e.forgivenL,M,T^Itwillingeneralbe coefficient inthephotosphericlayers);thenvalueofkat form afour-termrelationbetweenL,M,Cwhichmustbesatisfied layers inradiativeequilibrium. f isgiven,itmustexpandorcontract(whichcandofreelysinceits L forourmodelofgiven<£.Equations(24)and(28)incombination a singlephysicalconstant),wecouldintroduceitin(28),calculate brium, onourmodel,andveryclosediscussionofboundaryconditions namely, thattheexpressiononright-handsideof(25)hasacertain and Mwecoulddetermineasinglefactaboutthedensity-distribution, photosphere isjustsufficienttomakekcomerightaccording(28). radius. Butif/CidependsonTtheradiusofstarisfixed.If minate. Wehaveseenthatowingtoourhavingbeenable,byamathe- numerical value.Buttheradiusofstarwouldbequiteindeter- for anystar. obtain 1-ßjasafunctionofMandConly,weshouldhavestillthe matical coincidence,toeliminatePandpsimultaneously,so and sofrom(24)computeC.Thisisthesingle pieceofinformation can computekfromanyphysicaltheory, derive 1—from(28), mechanical equilibriumisneutral)untilthetemperatureT^at would benecessarytoalterourmodelsoasgiveadeterminate we canderiveaboutthedensity-structureon thehypotheseswehave Our problemistheinverseproblem.FromobservedL,M,andTwe actual valueofpatourdisposal.Thestarwouldbeinneutraleqydli- account ofbelowwhenwe cometocalculatek de—mechanical equilibriumofthestaras awhole,gaseousouter x 1? be ameasureofthedeparturedensity-distribution fromthat departure ofCfromthevaluecharacteristic thepolytropen=3will has thesamevalueasinEddington’squartic equation.Thusthe tion (20)reducestoEmden’sequationforthe polytrope n=3.Cthen of Eddington’smodel.Wecanseeinageneral waythatthevalue i -/?!,andsofrom(24)determineC.ThusforastarofgivenL 1 1} c ± x c v ma * Aswellasofg.Theeffect ofthepressureinphotosphereisdulytaken Let ussupposewehavesomephysicalformulafork(theabsorption Since i—ßiisfixedby(24),wehavenowfoundtheluminosity 17. When<£(0)=constant1,thefundamental difierentialequa- © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 4 ^ P4<¿(0)0] c 4 77CI _ 47tcGtMd[(i-/3)P(fl)fl] _ 4^'^(~~ßi)(28) limc 192 9MNRAS. .90. . .17M 2 34 formula,” integrate(27)fromphotosphereto centre. Wefind presupposes, however,anaccurateknowledge ofkateachpoint. liberation throughthestarispossiblefromobservation. Themethod difierentiation. Inthiswayanestimateofthe distributionofenergy- the rateofliberationenergyatanypoint, e, couldnowbefoundby from observation),wecaninsertforitin(27).SinceL(r)=/^.irr^dr, This canbewrittenapproximatelyas given by(27).Solving(30)for1—ß((f)(9)beingnowconsideredknown the valueof(f)couldbeestimated,andsocoefficientMon (f)(9) forthestarunderconsiderationcouldnowbeconsideredknown, right-hand sideof(31)couldbeestimated.Hence1—ß estimated. Since, bythelinearapproximationfor(f)orotherwise,marchof substituting forP/pfrom(22)wehave Now atthecentre(9=1),(f)(9)1.Applying(30)toand radiative equilibriumthroughout.Ifitisgaseousthroughout,(17) out aboutthestarifweassumeittobegaseousthroughoutandin holds throughoutandwehave 1 information aboutthegeneralmarchof(f)betweencentreandboundary. c In thepresentpaper,however,Imakenoattempttodetermine c equal tothatdeterminedbyobservation.Inthiswaywecouldacquire of AadjustedsothatthevalueCcalculatedfrom(25)comesout deduced forCfromobservationisequivalenttothedeterminationof march oí(f)(9)fromtheobservedC. is notconstant,thenextsimplestformitcantakelinear the meangradientof(f)(9)betweencentreandphotosphere.If We canimaginethissolvedforvariousvaluesofA,andthenthevalue Nov. 1929.andEffectiveTemperaturesoftheStars. Inserting thisin(20)weshouldhave To bringthismethodintorelationwithEddington’s “luminosity- If thestarisinradiativeequilibriumthroughout,flowofenergy 18. Wewillnowdigresstoinquirehowmuchmorewecanfind © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 32 ia+9(^9, i -ft/Ky7raGM 4 3£[4{~M]-' ■• ft W^cf)^(d9ldx)^ i -ß/nyP/, ¿(0) =i_A(i-0)..(29) 3 - 7T^(Ö)] p c Jo (31) (30) 31 192 9MNRAS. .90. . .17M 2 interior. Givenanyequationofstatefortheinterior, andanytheoryofthe a density-distributionconsistent withtheobservedL,M,andT absorption coefficientkintheinterior,wecanalways ascertainthedistribution of sourcesintheinteriorfor a0compatiblewiththeboundaryconditions,i.e. for be independentofr.Thenwehavethe equation ofhydrostatic r inthephotosphericlayers.LetFbenetfluxofenergyperunit and theequationofradiativeequilibriumf Layers.—Let pbethegas-pressure,p'radiation-pressureatapoint v area, gtheaccelerationduetogravity;Fand gmayherebetakento equilibrium correction to(28)duetheeffectoflayersoutsidephotosphere. to showhowcalculate/q.Incidentallyweshallderiveaslight k isknown.* continue ourinvestigationwhereweleftitattheendof§16.Wehave L, M,andkprovidedefinitebutscantyinformationabout<£,this adjustable parameter).Theimportantthingistheprincipleinvolved; for anactualstar,grantedaknowledgeoftheabsorptioncoefficient. equilibrium providessomebutstillscantierinformationaboutr¡,if combined withtheassumptionofagaseousinteriorinradiative alone, anditdependsfurtheronassumingsomekindofformulaforcf) tration ofenergy-liberationtowardsthecentrefromobserveddata formula,” oritsanalogue(32);itdeterminesrjfromanobservedL on muchmoredrastichypothesesthandoesthedeterminationofC (such asthelinearone,oranysimpleexpressioncontainingan and Mif/cisknownfrompurephysics,i.e.itdeterminestheconcen- me tobetherealsignificanceofProfessorEddington’s“luminosity interior, kcanbeestimated,andso(32)evaluatesrj.Thisseemsto a hard-and-fastvalue,hasvalueestimatedfromobservation. 32 Prof.E.A.Milne,TheMasses,Luminosities, equation, namely(31),inwhichthecoefficientofM,insteadhaving meaning-up kyjisspecifiedandthat1—ßgivenbyacertainquartic Following Eddingtonbywriting ± affords anestimateofFormula(32)isanalogoustoEddington’s We haveseenhow1—ßniaybeestimatedfrom(31).Hence(32) “luminosity-formula” (§12),withthedifferencethatwayof we have c c f Thisequationisknown to holdverycloselyrightuptheboundary. * Themethodof§18applieswhateverequation stateisassumedforthe 20. TheMechanicalandRadiativeEquilibriumofthePhotospheric It shouldbeobservedthatthemethodindicatedin§§17,18depends 19. Wenowturnbackfromspeculationsabouttheinteriorand If wehaveaphysicaltheoryoftheabsorptioncoefficientkin © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System dp' +dp=-gpdr....(33) L(r)/M(r) -77L/M, =K7,dF dp’ =-/cpF/c....(34) 7rcGM'Pjo 4 XC. I, (32) 192 9MNRAS. .90. . .17M 116 x \p'llp' isequaltoT^/Ti,whichvariesfrom2/unitybetween 1 theory ofk.Theanalysis couldbemodifiedtosuitanyothertheory. f4 to varywiththedepth. Writing the boundaryandphotosphere.Weshallneglectthisexceedingly small variationandreplaceitbyunity.Wethenhave or Denote thecoefficientofu~onright-handsidebyA.Theterm where Then (39)transformsinto Hence by(35) be theaveragenumberofelectronsperion;fweassumeitnottovary with depthinthelayersnowunderdiscussion.Then where aisphysicalconstantandPtheelectron-pressure.Letx photo-electric effect.Then* Using p=|«Twehave Nov. 1929.andEffectiveTemperaturesoftheStars. Hence 6 * Thecalculationofthevaluekatphotosphere dependsonaparticular f Ihavenotsucceededin solving theequationwhenx,ionization,istaken To solvethismakethesubstitution Now assumetheopacityofphotosphericlayersisdueto © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 18 +P+a = Pi! ^duijegi+xpf1 yïi™ dfi76ÎVu y A U=X7ï"(^U]^(u-|-^2) 1 +98 dp'Fax(%a)lp dp egix i +p~r~,H—~~~ dff P' K =irá/)^ r Pi =w dp16u p =u ,du 17A V ,dp og_ = 2 K = A —u]^u 9s dp' /cF ax {ia\l Pi P I x aP„ ^9/2 * udu '17/16 x 1/16 V 9/8 3 (44) (43) (42) (40 (4°) (39) (38) (37) (36) (35) 33 192 9MNRAS. .90. . .17M f 4 involving ip.Sinceip is nearlyunitythiscoefficientunity. Inserting thisin(28)weget get byusingourcruderformula(28).The value ofby(36)and conditions aretakenintoaccount.Letus compare itwithwhatwe or constant aandC(involvedinßby(24))when thepreciseboundary (37), isgivenby 34 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, Now p' increaseswithdepth,urapidlyandtendsto Inserting thisin(47)andsolvingforLwehave we findonintegration Hence by(42) Hence by(46) photospheric levelp=p/wehave reached thevalue(Weevaluateif;below.)Applying(40)to photospheric levelT=Tj.Letussupposeatthisdepthithas But by(44) or 1 since u=owhenp'—^«T.Thisequationshowsthatas 0 The onlydifferencebetween (50)and(50')isinthecoefficient Equation (50)isthefour-termrelationbetweenL,M,opacity The parameteruwillnotquitehavereachedthevalueat log © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 2 477-cGMi+æ(1-^x)0 = A- 21 +A 1 — (i -ßj)^ \aa xßii+ßiixfiift”■!) 2 i -\-x'JaT/’Tji+ÆT^-ßi i+.¿(-^) L =Tt ax p_jSj U-± ~i~U<^ x 16/17 2 ßi ^ 477CGMi+ÆT^ I -ßi>A16(1-^^ F “77‘"IT~~L~' g _GM47rr47tGM 1 1 ßl A ——O. 1 \aa xßi [^og{r-~) +ulog 2 1 u -ßl ßx pi ='i'iPi aL x^a = i/r«!. -77. =O (So') (49) (Si) (48) (46) (So) (47) (45) Nov. 1929. and Effective Temperatures of the Stars. 35

This discrepancy simply represents the imperfection of our fit of the

layers outside T = Tx on to the layers inside T^ It is not certain that our approximations are trustworthy to the extent of retaining iff, but it is useful to find the size of the correction. 21. Determination of ifj and the i/j-Factor.—From (44) we have

Tk(u2 ~ ui) == I-

Substitute this value of u2 in (45) and then apply (45) to the photo- spheric level. At this level p' = p^ = 2p0' and u = xfju^ Hence

- 16/17 log2 = - i/r) + (16/17+ Mjjlog 2^+ 16/17 16/17 + ujj

Now express % in terms of ß± by means of (46). After some reduction we find 1 16 -Kl-ft) 17 , iKi-A) 17 ß. __ '7 ßi (52) i ip ~ . , l6 I ß-, «/r1 + I + T I? ßi

This equation determines ip in terms oí ßv It must be solved by trial. The following short table is calculated from (52).

Table I. Mi+ßiCW1-!)]. 0*0 0*878 0*725 0*1 0-888 0*754 0*2 0*900 0*786 0*3 0*913 0*819 0-4 0*928 °48S3 °'5 0*944 0-888 0-7 0*980 0-956 1*0 1*000 1*000

22. Determination of the Ionization Factor.—Before (50) or (50') becomes an explicit formula we have to show how to calculate the factor (1 + x)/x the photospheric level. This depends on an estimate of the significant ionization-potentials. We shall suppose for definite- ness that the material has a first stage ionization-potential of Xi = 6*08 volts and a second-stage ionization-potential of Xz = 11*82 volts. These are the values for calcium, but of course this does not mean we are assuming the photospheric layers to be entirely composed of calcium. But it is particularly desirable to avoid the appearance of choosing I.P.’s to suit the later results. All we want are some ionization- potentials that determine the abundance of free electrons. First-stage ionization will go on with increasing temperature until it is practically complete. Thereafter second-stage ionization will set

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 192 9MNRAS. .90. . .17M an(are decreases withincreasingtemperature,Æ-*2. it willbeobservedthat(54)and(55)both reduce toÆ=1.As ionization correspondstodverysmall, large,forwhichvalues 36 Prof.E.A.Milne,TheMasses,Luminosities,xc.i. As before,xisafunctionofTandßonly. Completion offirst-stage where is x. Since pi=isafunctionofT!andßonly,henceso where say. Also whence whence x2 say, inawell-knownnotation.* x1 Eliminating andwehavefrom(53) and in. Thetwoionization-potentialsXiiX2sufficientlyfarapart ± for ustobeablecalculatethetwostagesseparately. T =Tj, 1 For thesecondstagewehave,puttingx=+ÿ, The numericalvaluesofthe“weight-factors ” forcalciumare Now, For thefirststagewehaveforionizationatphotosphere © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System * R.H.Fowler,Statistical Mechanics,p.370,equation(1072). v8 1 2(277)S i -ÿh^ -ÿ— (Pe)!=r-T^e-x./«.K. 2 2 % ^ (PJi _¿ Æ (Pe)l 1 = ^! +*Pi-ßl T_ 2b ~P 2(27Tm)*lkl i —x d ^ = 2 ÍC = i +^’ 2(27Tt)Í¿S ” iix./M.=K i-^K, 1 — = s1 ßi pi_ ßl-^-2 i-ßK ' (1 + 1 h q 0 ßi Pi_ ft K’ Pi' i -\-d- 2 0*664. 2 1 _A_ -0i* (SS') (S3') (54') (SS) (54) (S3) 192 9MNRAS. .90. . .17M : 2 !. 3000.3500. ; *0236*1126 :■ *344-868 ; *331-86o i -319-849 : *3078*838 ‘ *2950-828 : *2824‘816 : -2712-803 : *2590-788 ’ *2472-774 ' *2228-739 : *2100-718 : *1976*694 : -1845-667 :■ *1710*640 i -1571-605 ‘ *1419-567 : *1341*544 :■ *1200*520 :■ -1173-493 > -1082-461 : -0983-427 ;■ -0875*388 *0755’342 ; -0687’SIS ; -0613-2835 ; -0530-2470 : -0473*2i8 ) -0409‘1929 :■ -0334'1588 ■ -2348-759 ues ofIonizationx(NumberFreeElectronsperAtom)atthePhotosphere,forStarsgiven 1 x having MeanFirst-stageandSecond-stageIonization-PotentialsequaltothoseofCalcium. —ßi,andEffectiveTemperatureTheCalculationshavebeencarriedthroughforaOas for given1—ßxandTj. Nov. 1929.andEffectiveTemperaturesoftheStars.37 The followingtablegivesthevaluesofxcalculatedfromtheseformulae o © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System •923 *996 •968 -998 •960 -998 •909 -996 *957 *997 •685 -976 •983 -999 •982 -999 •977 ‘999 •975 '999 ■942 -997 •875 *994 •834 -991 *757 *985 •727 *981 •631 -970 •952 -997 *935 *997 •900 *995 •889 -995 •984 1*00 •980 -999 •973 -998 •971 -998 •803 -989 •532 *95° •965 -998 •857 *993 *457 -930 •340 *873 •588 -963 4000. 5000. i 1*009 1*145 1*022 1*292 I*Ol8 1*20 1*007 1*122 1*001 1*027 1*001 1*020 1*031 1*370 1*028 1*340 i*oi6 1*231 1*012 i*i88 1*002 1*041 1*002 1*034 1*014 1*210 1*005 1*087 1*036 1*408 1*039 *428 1-033 1-390 1*026 1*332 1*023 1*312 1*010 1*167 i*oo6 i*iii i*oo6 1*099 1*003 1*063 1*020 1*272 1*001 1*012 i-ooo i*oo8 1*000 1*004 1*004 1-075 1*003 1*051 i-ooi i*oi6 1*043 1-446 6000. 7000. o 1 1*612 1*882 1*152 1*475 1-191 1-545 1*778 1*950 1*738 1*940 i*66o 1*911 1*536 i*86o 1*259 1-640 1*226 1*598 1*715 1*932 1*689 1*923 1*822 1*960 1-792 1*953 i*444 1*807 1*364 1*625 1*069 1-277 1*035 1-857 1-969 1*846 1*967 1*833 1*964 1*809 1*956 1*760 1*945 1*625 1-899 1*510 1*845 1*322 1*703 1*098 1*358 1*867 1*972 1*477 1-830 1*406 1*781 1*126 1*423 1-875 '974 8000. 9000. Table II. I0 I 10,000. 12,000.14,000.16,000.18,000.20,000. 1*967 1*996 1*770 I*960 i*8i6 1*970 1*986 1*998 1*983 1*998 I-870 I*980 1-847 1-975 1*990 1*999 1*982 1*997 1*980 i*997 1-977 1-997 1*960 1*995 I*580 I*905 I*4l6 I-830 1*989 1*999 1*988 1*998 1*946 1*991 1*920 1*988 1*899 i*985 I-670 I*936 1*992 1*999 1*991 1*999 1*985 1*998 1-973 1-996 1*992 1*999 1*956 i*994 *95 ï-993 1*730 1*951 1*993 1*999 1*932 1*990 1*993 *9 OO 00 1 1*990 1*996 1*999 2*0 1*999 '9 1*992 1*997 1*999 2*0 1*999 2*0 1*995 1-998 1*994 1*998 1*952 1*982 1*999 2*0 1*976 1*990 1*999 2*0 1*999 2*o 1*999 2*o 1*999 2*o 1*999 2*0 1*999 1-999 1*998 1*999 1*997 1*999 1*996 1*999 1*985 1*994 1*999 2*0 1*999 2*o 1*999 2*0 1*988 1*995 1*999 2*o 1*999 1-999 1*999 2*o 1*998 1*999 1*998 1*999 1*999 2*0 1*998 1*999 2*0 2*0 2*0 1*999 1-999 1*999 1-999 1*992 1*995 2*0 2*0 2*0 2*0 2*0 2*0 2*0 2*0 1*999 2*0 1*996 1*998 2*0 2*0 1*999 2*o 2*0 2*0 i*997 1*998 2*0 1*998 1*999 2*0 2*0 2*0 2*0 2*0 2*0 192 9MNRAS. .90. . .17M 12 12 Ii 120 ,lI 1202, , *480 -382-892-987i-oo1-0541-5041-8981-9801-995‘9'° •460 *369-885-986i-oo1-0501-4841-8901-9781-994i*999'9°‘ •440 -357*877-985i-oo1-0461-4651-8831-9761-994i’999i*9992-0 •54O -422-912*9901-00I-0Ó81*5621-9181-9831-9982-0 •580 -450-923-992i-oo1-077*593°’87i*9992-02-o •500 -395-899-988i-oo1-0581-5231-9061-9811-995’9‘° I —/Sj.3000.3500.4000.5000.6000.7000.8000.9000.10,000.12,000.14,000.16,000.18,000. •660 -513-943-994i-ooio51-673'9°‘* •620 -480-932-993I-oo1-0921-6351*9401-9901*9992-020 •700 -550-952*995I-ooI-I221-7101-9601-9942-02*0 38 x isgivenbytheprecedingtableasafunctionoíi—ßandTl.When accurate workisrequired,(50)tobepreferred(50').Theco- where (f)(9)isafunctionof9equalto1when=andtending efficient Cisgivenby finite limitas0->o,andisafunctionofxequalto1atæ=o equal tooatæ=aqgivenbythedifierentialequation takes thevalueofEddington’sconstant; (24) takestheformof with theconditiond9fdx=oatæo. mention ofTinthefour-term relationascomparedwith(L,M,kC). differential equationforthepolytropen= 3,andthecoefficientC L, M,TandthecoefficientC.f the photosphere;and(50')isalwaysequivalent (whateveristheform Eddington’s “quarticequation”forß,except thatßandreferto ±1 of (f>)to x v 1 ± © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System oooooo ooooooo f Wehavereplaced/butonlythesinglecoefficientCdependentonit.Wecon- use itasastandardofcomparison.Actuallyweneednotthewhole Hence equations(50')and(24)appliedtothese twostarsbecome Capella*, sincetheyhavethesamespectrum. Thephotospheric into thefactor(1-fiæ)/îc,thiswouldbeverycomplicated;furtherwe molecular weight^mayalsobetakenthe sameforthetwostars. Further, theionizationissameinatmospheres ofSun*and struct thisfamilybychoosingthewhich containstwostars struct onedefinitehomologousfamily{i.e.choosesome)and have noreallyaccurateknowledgeofa.Insteadweproposetocon- depart fromunity.Owingtotheintricatewayinwhichßenters introduction, wecaninprinciple,forastarofobservedL,M,andT being providedwithenergyL,theelementswillcontractorexpand, given aswell,theradius,andsoefiectivetemperature,mustadjust formulae. Forgiven<^,thestarisinneutralequilibriummechanically, that thevalue1—ßa>tphotosphereisnotwidelydifierentfrom generalisations ofEddington’sequations,reducingtothemwhenhis Capella. Sun* andCapella*indistinguishableobservationally fromSunand we couldcomputeC,andsoformanestimateoftheway($) isthat(j>thesameforactualstarsSunandCapella. (2) and(3)thattheirouterlayersaregaseousinradiativeequili- evidence asthevalueofCprovidestoformfunction and C'cameoutpracticallythesamenumber.Inotherwords,such ments showthatifwehadreversedourprocedureandusedthespectro- belong tothesamehomologousfamily.fButnumericalagree- say thattheagreementispracticallycomplete.Whatdoesthisagree- and Capella. 42 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, observed (L,M,T-l)and(I/,M',T/)weshouldhavefoundthatC scopic determinationofatocomputethevaluesCandC'from ment signify?WehaveurgednoreasonwhySunandCapellashould consequence about4timesasbigforSun*andCapella*Sun about twicethoseofSunandCapella;thelevelsatwhich Capella* abouthalfthoseforSunandCapella;theconstantaisin methods pickoutthephotosphereareatpressuresforSun*and 4 Thoughtheyareobviouslylikelytodosoongeneral groundsofsimilarity. To calculatetheluminosityofamemberourhomologousfamily This agreementoftheopacitiesshowsusthatSun*andCapella* I washtoemphasizethatthisisanobservationalresult;whether Considering theuncertaintiesinspectroscopicmethodwemay The comparisonisstriking.Sun*andCapella*haveopacities © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 59; 10 L® M®2*V1-&®/ftVTj®/• • ^ il =Mi+^/-ßiYßißiTy x -ßi©=0-06804,Ti©S74° 12 -ßi=o-ogoaß^M/M®).• (58) Nov. 192g. and Effective Temperatures of the Stars. 43

The form of the ionization factor arises from the fact that [(1 + #)/#]o = 2. To derive the absolute bolometric magnitude m we use the formula

w = 4-85 - 2-5 logjQ (L/L0) . . . (60)

Tables III. and IV. give the results. The procedure is to start with a value of 1 — ßlf and compute M from (58). Then derive L from (59), using the previously given table (Table II.) to give the ionization factor. Table III.

Values of i — ßx for given Mass M, for a certain Homologous Family. M. i-ßi. M. i-fr. M. x-ß,. 0*0 *00 2*2 *1889 7- 0 -4387

0*2 *0036 2*4 ‘2063 7’5 *4533 0*3 *0079 2*6 -2227 8- 0 *4668 o*4 -0137 2*8 *2381 9- 0 -4909 0*5 *0210 3*° *2529 10*0 *5119 o*6 -0289 3*2 *2669 11- O *5305 0-7 -0379 3-4 *2801' 12- 0 *5470 0-8 *0475 3- 13- 60 -2927 *5618 0- 3*8 9-3046 14- -0576 0 -5753 1- 4- o15- -0680 00 '3160 *5875 i*2 *0893 4'5 -3421 16- o -5987 i*4 -1106 5- i8-o -6185 0 *3655 1-6 -1314 5*5 *3865 20*0 -6357 1-8 -1515 6- 0 -4055 2*0 *1707 6-5 -4229

If accurate work is required the luminosities of Table IV. need small corrections for the value of the factor involving Table V. (p. 46), derived from Table I., gives the correction in magnitudes, the zero having been adjusted to make zero correction for the Sun, since this has been taken as standard. The corrections are to be added to the bolometric magnitudes of Table IV.

[Table

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 192 9MNRAS. .90. . .17M 44 H < w ij © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System -o «1 s2 § o ^1 1,1 •-S >s Ö Ös g o i.S ! I £ ^ § . !< 00 I ?s 8 LO lo ov § §5 Pro/. E.A.Milne,TheMasses,Luminosities, 00 , .vo 0 o .00MCO - D ^”tf-Lori-lo -j? coVO 3 “8 % 2 o I o M Oco^oo O COMÓ O T*LO?"VO o ¿OM6ONob g coLOrtf-VO OOO00 H M o' CO 8 Ti- O Ci ^ COMOON00C^»VOLOcoCi ^ COMÓ 4 CO § p i_i M^H M g Ci0Í O M 00 VO VO M Ci LO o co M W l-lM CO M6‘ O COVO O LOrtf-(N) LO VO rtf- On M LO VO CO Mo Cl CO i-t M CO co LO CO o 1 on cb ? T LO CO Ci oo LO CO LO rj-r- M oONOn00 t~'» t"*oovo M óOnCOoo o Ci Ci ON Ci CO o O M CO ON On 00 lo 00 Ö On m LO CO M Ci LO 00 rtf- vo CO LOOnM M CO co ó rtf- M rtf- ON Ci Ôn ob ON 00 l>- M On 00 CO W On co VO On oo On ob lo 00 Lo 00 O CTv b\ ob Tj- 00o G\ dbvóv¿ VO LOrtf- ON ON Lo M rtf- o O CO 00 o O CiVOH CO LOrtf- o M co O M O CÍVO ^ Thrt- CO t''» lorj- Ci ON LOM o o rtf- LOvo O rtf-CO co 00 rtf- ri- rtf- o M ON00 CO VOM Ci VO rtf- Ci CO o 00 VOlo o 0\ On 00 lo oo C- VOvoLOrtf-rtf-rtf-cOCOCOCiCl o Ci O O OVCO ^ coo o CO rt- o Ci VO rtf* L~— L>*VO LOLOrtf-rtf-rtf-fOCOCi Ci rtf-00 CO VOrtf- Ci t— VOLO L>* VOLOrtf-CO i-i co On 00 vo Tt t^VO LOrtf-COCi M OVO M COVO í>* VOLO^coCi O Oí CO M On m VO LOLOrtf-rtf-rtf-cOCOCi v¿ lo o O v¿ LO Ci rtf- 00 VO LOTtf-rtf- CO co vo m LO CO ri- O 0Í 00 O CO00 00 VO LO VO o b o rtf- CO O LO rtf- rtf- M 00 LOON LO LO O LO o TÍ* Ov^OX> Tf MVO co On om Ci Ml-l CO 00Ci CO Ci O M o ON LO ON oo CO ON co 00 Ci OO ON rtf- Ci M NO CO oo r-~ cí On lo t'- 00 Cl rtf- o o t-. COOnVO 00 rtf- loOn f'" O r- co o ri- rj- o On o O 00 Ci CO LO M00 00 O LOCO rtf- rtf- C0 ^tf- co CO OnVO l-l LOrtf- co rtf’ o Ci vo ON oCirtf-vo Ci o O M o On vo o ON VO O r^ CO coCi ■sj- o CO 00 LO LO O On vo M O VO co oí LQ 0O COOí LO O CO rtf-VO O 00 CO ONrtf- Ci M CO VO VO CO Ci CO coCi Ci 00 CO ON co Ci 00 M M CO 00 CO 0Í 0Í VO'O 00 o M VO O MC— On oc— O VOM w vo f- oo CO Ci M VO oo 00 M VO CO rtf- O lo vo On tJ- VO vo rtf- CO t'- CO ON VO xc. i 00 LO % rtf- CO co oo Oí 00 00 00 oo Ci LO Ci 00 ON Tt- 0Í 192 9MNRAS. .90. . .17M Nov. 1929.andEffectiveTemperaturesoftheStars. 00 Ocq 00 vo o m cq OOO VO 00 o o Ov vo 10 Ti- CO 00 00 VO M O OV vo 00 00 í^. M r— M VO Tf o M O cq IO M to 00 M o ov cq00voto CO 'O vo CO cq co to00 cq vo 10 M o 00 to M 00 O vo Tí- O H í^- to M O VO Ti- TÍ* Tf* CO ovvo tí* to00 IO M cq O VO © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System co Ti- cq o m o 00 M Ö M 00 tí- cq o Ov ó 6 Ov to 00 cq Tf VO00 OOO + 00 00 O Ó Ö 00 tocq Ov VO Tf- Tj- ó 10 tovo TÍ- bob 10 tovo Ov to Ö b o 00 o cococo +TfTftotovó to CO tO TÍ- tí- m Ti- to í^* 00 cq TÍ* M b CO O cq CN o Tf O + ó to Tf VO Tf 00 + Ó co + cq Ó TÍ- + 111 00 cq Ö o + + 00 vocq co 10cq00tí-oví- O bob O covo co CO O cqTfVO00 OOOOO + + I cq t^-voooTftom o r-~ 00vo o cqTf CO 00 Ó O cq ó 0 TÍ- Ö I 0 cq 00 O cq 1 b 0 CO 10 1 1 6 b Tf* vo cq ó I I O TfCOco00oX-- Ov Ó bob CO 1000 Ov 10 b CO vo b co lo00 Tí- OvTi- bob co lo00 ö 6 co vo00 I Ti- to CO to M b I Ov too I M W M coto00 Tf vo00O b Tf to I tJ- Ov 00 lili co Ov co Ov ó Ov CM t-— ovcq CO M VOH Ó M OV M 0 to O M O WM 1 I VO M Ov I M cq o I M . cq I i Ov M ^ 00 m cq m cq Ov cq to o cq ó?c^ cq ^lr^Ov tí- 00hvo I Tf vo00 Tf toVO Ov cq Ov M o Tf cq O cq co vo vo O cq to Ov 0 O O cq I cq cq to Ov co 1 cq CO vo cq 7*" ^9 cq vo I I I vo 00 Ov cq lo 00vo cq ¿q M CO00 cq cq to OvVO CO 10O I cq Th Tf Tf to vor- OV MTf O Tf00cqvoo 000000 Tf toco cq 00 cq Tf 00 to 0 00 cq vo mcq cq to ov I cq 00 1 I to Ti- cq I co ov voto I Tt- Ov CO00ov O 00J>*M vo Ov CO co Tf cq coTf CO co lili CO co M O o to co CO coTi*Ti- CO Tt-to cq I Tf- co ^rt- co 00 cq m I CO co I CO co Tt- Ov I \ I CO 00 Tf CO Tf co 00 Ov CO Tf to lo CO co Tf Ti- CO Tt- Ti- ’d- I I co 00 00 to Ov10 M COov +) Tf Ov 10 10 00 Tf O to ov Ti- to 00 Ov + TÍ" Ti- Ti* TÍ*10 I cq co 00 m to I cq C— CO Tf CO r- co CO ov Tf Ov w 00 Tf I to CO I Ov to 10 CO O + Tf O Ov I to o cq I tO I 0 Ti- to to CO ■"tf- 1 cq Ti- to I I to ir> I rú o rO 'Sb <4-+ G> CÖ o > a oá 5 tí o 'o tí o •+> 02 0) V o- O & o > tí CÔ o u o cd o o tí bù 45 192 9MNRAS. .90. . .17M 46 same massMandeffectivetemperatureastheactualY-, with thoseofourhomologousfamily,wewillspeciallycomputethe From (58)wefind luminosities ofY-Puppis*andthetwocomponentsKrüger60*.By For thisstarwemaytakethephotosphericionizationtobeÆ=2-0. V-Puppis* Imeanamemberofourhomologousfamilywhichhasthe assumed ionization-potentials woulddecreaseboththecalculatedbrightnesses.] Formula (59)thengives and similarlyforKrüger60*. computations weremade,SirJamesJeanshasdrawn myattentiontoAitken’s ton’s distributionofmassbetweenthetwocomponents. [Sincetheabove found the(observed)absolutebolometricmagnitudeofV-Puppisto whence theuncorrectedmagnitudeis—4*52.Thei/r-correctionfor magnitude ofY-Puppis*is Krüger 6o*bfainterand Krüger 60*/brighter.Averysmallreductionin the determinations (L.O.B.,365) whichgivemasses0*25and0*20.Thesewouldmake be —4-75.ThevaluecalculatedbyEddington fromhis“mass- From theobservedradius(Y-Puppisisaneclipsingvariable)Eddington ing toEddingtontheobservedfdataare: luminosity ”lawwas—5-06. i —ßi=0-6290-24,whencetheabsolutebolometric f Actuallyonlythecombinedmassisknown,butI havetakenoverEdding- Faint componentM=o-i6m=+i2- For V-PuppisEddingtongives 25. Beforecomparingtheluminositiesofactualstarsasawhole For Krüger60*weconsiderthetwocomponents separately.Accord- © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System component M=o-27m+9- Prof. E.A.Milne,TheMasses,Luminosities,xc.i, 0*06804 o*7 0*1 0*3 0*2 1*0 o*4 0*5 o-o o i-ft. M =19-2,T19,000. 1 3 L/L© =5-61x10, Correction foryj-factor. i —ßi~0-629. m =—4*76. Table V. -0*32 —0*27 -0*15 — 0*10 — 0*06 — 0*01 — 0*19 + 0*03 Am. 0*00 82 | T,= 32J e 3100 192 9MNRAS. .90. . .17M m m m m duce SunandCapella, alsoreproducesstarsatsuchextremeparts of Krüger 60.Inother words, ourhomologousfamily,chosentorepro- certain homologousfamily.Theresultisastonishing. V-Puppis*and luminosities ofactualstarswiththe ofthemembersa luminosities ofactualstars.Wearesimply comparingtheobserved gave Y-Puppiso-3itoobright,Krüger60b i-25toofaint,and Krüger 60*havealmost exactlytheluminositiesofY-Puppisand that Krüger60*6is0^29brighterthan 606;andthatKrüger 60*/ iso-47brighterthanKrüger60/.Eddington’s computations Krüger 6ofi-23toofaint. Eddington’s computedvalueforKrüger60/was 13-55. correction isAm=+0*03,whencetheabsolutebolometricmagnitude whence theuncorrectedbolometricmagnitudeis11-82.Theip- of Krüger60*/is Formula (59)thengives whence from(54), From (S3), Eddington’s computedvalueforKrüger606was+11-07. whence theuncorrectedbolometricmagnitudeis9-50.Thein- correction isAm=+0-03,whencetheabsolutebolometricmagnitude Hence of Krüger60*6is Formula (59)thengives Nov. 1929.andEffectiveTemperaturesoftheStars. Hence whence from(54), From (53), I wishtomakeitquiteclearthatamnot attempting topredict We seethatY-Puppis*haspreciselytheluminosity ofV-Puppis; For Krüger60*/wefindfrom(58), For Krüger60*6wefindfrom(58) © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 3 -2 L/L® =1-636xio-, L/L® =1-386xio, x =1/26-4. i —ß=0-00228. 1 i —ßi=0-0064. m =+11-85. m =+9-53. x =1/15-80. —7- =137- I +æ I -f"^ d =698, 1 d-L =248, 2X 2X = 8*40. 47 192 9MNRAS. .90. . .17M have expectedittointerpolateluminositiesreproducingactualstars. the rangeasV-Puppis(highmassandluminosity)Krüger60(low 48 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, this range. mass andluminosity).InbetweenSun*Capella*weshouldrather the logarithmsoftheir massesinfig.i,withtheefiectivetemperature actual starsgiveninEddington’s invaluabletablesareplottedagainst But wehavenoreasontoexpectitreproduce luminositiesoutside © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 26. Wenowturntothe starsasawhole.Theluminositiesofthe 192 9MNRAS. .90. . .17M o o appropriate toitsmass, luminosity,anddensity-distribution.If its density-distribution. Thenforitsactualmass andluminosityitmust to alaterpaper,butoneremarkmaybemade here. pressure andsolowopacity.Discussionof these curvesispostponed pressure photospheresoflowtemperature,and sincebothlowtempera- find aconfigurationof equilibriuminwhichitssurfaceopacityisjust ture andhighpressureretardionization,they giverisetolowelectron- low massandeffectivetemperature;for suchstarshavehigh- consequence oftheverylowionizationin photospheresofstars these mass-linesisthesteepcurve-upon left-handside.Itisa plotted onthesamediagram.ThediagramamountstoaHertzsprung- homologous familystampeduponit.The most strikingfeatureof information thatcanbegleanedfromthepresentinvestigation,forI siderations maycomein,asJeanshassuggested,andimposethe the coefficientC,nowseentobeastellarconstant.Stabilitycon- have notdiscussedwhatcanbelearnedfromthenumericalvalueof Eussell diagramturnedrighttoleft,withthemass-linesofacertain common density-distribution. in themass-luminositydiagramwecanbynomeansinfergaseous against temperatureforconstantmass.Theobservedstarsare are plottedinadifferentwayfig.2.Hereluminosityis follow acommondensity-distribution.Whatthissomethingis matter forfutureresearch.Ihavenotofcourseextractedallthe nature oftheinteriors.Acommondensity-distributionaloneis view ofouranalysis,fromthedistributionactualluminosities indicated. any assumptionsastothephysicalstateofinterior.Clearly,in the samedifferentialequation,issufficienttogiveobservedlumino- sities ofthestarsforgivenmassandeffectivetemperature,without have acommondensity-distribution,i.e.aredescribablebyoneand a stellarconstant.Converselythesingleassumptionthatstars tures discloseanyinformationabouttheinternaldensity-distribution, reproduces anyobservedscatterduetotemperature.Whatis,however, is ofcourseconsiderable,andwecannotsaythatthehomologousfamily is difficulttodetectanysystematicdeviationsavethatthe3000 closely reproducestheobservedluminositiesthroughoutrange;it striking—and thisistheimportantconclusionofourcomparisonwith that informationisthesameforallstars.Cbearsappearanceof other words,asfartheirobservedluminosities,masses,andtempera- observation—is thattheactualstarsmimicahomologousfamily.In curve isalittlehighatlowmasses.Thescatteroftheobservations are appreciablydifferent.Itwillbeseenthatthehomologousfamily family areplottedtascurvesonthesamediagram,fortemperatures written againsteach.Theluminositiesofourspecialhomologous Nov. 1929.andEffectiveTemperaturesoftheStars.49 3000 and6000,withshortcurvesfor350020,000wherethese Let ussupposesomethingcompelsthestar to adoptaparticular We conclude,then,thatsomethinginnaturecompelsthestarsto 27. ThecontentsofTableIV.,correctedaccordingtoV., I WeplotthefiguresofTable IV.,correctedaccordingtoTableV. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 4 cr> j 50Prof.E.A.Milne,TheMasses,Luminosities,xc.i, ^ photospherebecomesmoretransparent,ultimatelyveryrapidly,and g} willdecreasestillfurther.Thismeansthatwithexpandingradiusthe r- I pressure,andasitexpandsitstemperaturefalls,theionization I massislow,ionizationwillbelowinitsphotosphereowingtohigh s

Absolute Bolomebnc Magnitude © FirstClassdetermination(binaries).0Second (binaries).0Cepheids. o o the starthereforecoolsatanincreasingrate.Whenrateofcooling is equaltoLthestatewillbecomesteady. It isnowclearthatthe indicate whydwarfstars arenotfoundattemperaturesbelow3000 ; rate exceedingthegenerationofenergy.The sharpup-turnsonthe mass-curves offig.2actasbarrierspreventing thestarfromtaking star cannotexpandtoofaratlowtemperatures withoutcoolingata a verylowefiectivetemperature. Itisnotablethatthissharpbend occurs fordwarfsatbetween 3000and4000.Thisseemstome to © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System x Eclipsingbinaries Fig. 2. 192 9MNRAS. .90. . .17M o spheric opacity.Inthis analysisnohypothesisismadeasto the spheric opacity),andC,acoefficientdepending onthedensity-distri- nan takeupanyradius.Butwhenitsluminosity isgivenitadjusts we haveseenhowequilibriumconsiderationscanneverofthemselves its radiusuntilthesurfaceopacityissuchthat itcoolsatjusttherate general methodistheparttowhichIattachimportance.Theresult the Russelldiagram—inotherwords,thattherewasasystematic is inneutralequilibriumasfaritsmechanicalgoes,and prediction. change ofdensity-distributionovertheRusselldiagram.Themethod, tinguishable observationafiy(whitedwarfsexcepted)fromasingle method ofthepaperandparticularresultsitarrivesat.The dwarf stars;inotherwords,giantMorNcanberedderthan distribution) thatgiantstarscantakelowertemperaturesthan bution fromthevalue of C.Wecan,however,computeCforanystar centre andboundary.Wecancomputeitfrom theformofdensity- cooling ;itisatelescopedversionofthestructure ofthestarbetween relation connectingL,M,T(oritsequivalent, theradiusorphoto- ways inhomologousfamilies.Amemberofagivenfamily in mechanicalequilibriumallpossibleways,wehaveclassifiedthese to setupamass-luminosity-temperaturerelation.Buildingstars determine anythingsofundamental,sinceagivenluminositycancorre- stars. ItmightthenbediscoveredthatCvariedsystematicallyover modified byanimprovedtheoryofphotosphericopacityandionization, aspects ofthedensitydistributionwhichare significant forthesurface bution. ThecoefficientC,asitwere,recapitulates initselfthose spond toanymassaboveacertainminimum.Norhavewesought however, isindependentofthespecialresultsfound. homologous familywithacommondensity-distribution—maybe dwarf M’s.Ibelievethefactsareingeneralaccordancewiththis distribution ;wecanonlyinfersomething about thedensity-distri- at whichenergyisbeingliberated.Thisprovides uswithafour-term or bybetterdatafortheluminosities,masses,andtemperaturesof show thattheefiectislessconspicuousforhighermasses;here the gasesaresotransparentatstar’sboundarythatenergy I haveobtainedbyapplyingit—namely,thatthestarsareindis- tion thatsomethingcompelsthestarstohaveacommondensity- reduced pressurefacilitatesionization,andsotheopacityisless sensitive todecreasingtemperature.Weinfer(alwaysontheassump- ionized sufficientlytogiveappreciableopacity;atlowertemperatures of knownL,M,andTj providedweknowhowtocalculatethephoto- leaks awaytoofreelythroughtheouterlayers.Thecurvesoffig.2 is justthetemperatureatwhichunderstellarpressuresgasesbecome contract andtheirsurfacetemperatureswouldincrease.3000orso lower thanthistheywouldcoolsoveryrapidlythatlose if theywereputinaconfigurationwhichthetemperaturewasmuch Nov. 1929.andEffectiveTemperaturesoftheStars.51 more heatthantheirenergysupplyprovided.Theywouldtherefore 1 We havenotsoughttosetupamass-luminosityrelation;indeed, 28. BywayofconclusionIwishtodistinguishbetweenthegeneral © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 192 9MNRAS. .90. . .17M concerning evolution. to haveanaccuratetheory ofphotosphericopacity,soastoinvestigatethecorrect accordance withtheHertzsprung-Russelldiagram. It wouldbehighlyinteresting march ofthecurvesfig. 2 andthecorrectnessofconjectureshereputforward corresponding tothepossiblebirthofastarasan F- orG-star.Thisisalsoin would giveadditionalintersectionswithahorizontal lineL=incertaincases, valuable criticismsofanearlierdraftthis paper, thoughheisinno features oftheHertzsprung-Busselldiagram thenresults,butIleave way responsibleforthecorrectnessorotherwise ofthepaperinitspresent the discussionofthesespeculationstoalater paper.| nature oftheinteriorstar;weassumeonlythatouterlayers born eitherasaB-starora.giantM-star,starofanentirely different structure.Aschemeofstellarevolution reproducingthemain a whitedwarfresults.Thusweareledtothespeculationthatlarge that thehorizontallineL=neverintersectscurveM two configurationsofequilibriuminsidethehomologousfamilyor tinguishable fromtheSunandCapella.Thisfamilyisfoundtoconsist 52 Prof.E.A.Milne,TheMasses,Luminosities,xc.i, aggregate ofmatterMgeneratinganarbitraryluminosityLwillbe and noconfigurationofequilibriumexistsinsidethehomologousfamily, none. Forlargemass,thetwoconfigurationscorrespondtoaB-star curve intwopointsornone.Thus,givenLandM,thereareeither based ontheenergy-generatingprocesseswouldberequiredtoeffect then thefour-termrelationreducestoathree-termbetween value ofthephotosphericopacity,butbyconstructinghomologous are gaseousandinradiativeequilibrium. or agiantM-star;itistemptingtosupposethatwheresosmall genuine mass-luminositylawfromobservationofthetypeL=L(M), family (Cconstant),whichcontainstwostarsobservationallyindis- then aglanceatfig.2showswhystellartemperatureshavetheir L, M,andT-^TheactualscatterofforgivenMvaryingT observation andnotdependentonanytheoryoftheinterior. common density-distribution.Thisisaninferencebasedsolelyon have ascertainedthisnotdirectly,owingtouncertaintyasthetrue observed values. a calculationofstellartemperatures.If,however,wetakeover stellar range,buttheintroductionofagenuinemass-luminositylaw as givenbythisrelationissmallfortemperaturesoftheobserved family isclosetothatobtainedfortheactualstarsfromspectroscopic material. Theinferenceisthattheactualstars(excludingwhite with theluminositiesofactualstars.Further,photospheric dwarfs) doinfactconstituteasinglehomologousfamily,with of starswhoseluminosities,forgivenmassandtemperature,coincide opacity wefinditnecessarytoattributeourconstructedhomologous data. Thustheconstructedfamilycouldbemadeofrealstellar lr 0 f Thecurvesoffig.2showsmallhumpsintheregion 6ooo-7ooo°.These A givenhorizontallineL=inthisdiagrammeetsanymass- If Cisinfactastellarconstant,asobservationappearstoindicate, The valuesofCfortheactualstarsseemtobeaboutsame.We 29. IshouldliketoexpressmythanksProfessor Eddingtonfor © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 192 9MNRAS. .90. . .17M copper inS',andsoS'wouldbelargerthan Sinradius.Thusby tribution ofopacitywe couldinfersomethingastothedistribution amount ofasbestosasS'butconcentrated in asphereatitscentre, tion oftemperatureisthesameintwocases. could infersomethingofthedistributionconductivity. ductivity) whichisimportant.IfSwere provided withthesame of arguingthatsincetheflowheat^ theradiusinfrom observation oftheradii,givendistribution oftheenergy-source,we the copperinSwouldstillbemuchcolder than theequalamountof boundary isthesameintwocases,therefore theinternaldistribu- copper tohaveaverylargecoefficientofexpansion.ThenS',being internal temperaturedistributionswillbetotallydifferent.Thesphere effect, S'beingoflargerradiusthanS.But wemustavoidtheerror sphere S;foritisjacketedbyabadconductor.Nowsupposethe hotter, willexpandmorethanS,andweshallhaveanobservable surface temperaturesofthetwosphereswillbeequal,andflow are ofequalradiusandtheheatingcoilsstrength, vided atitscentreasbefore.Letitcoolfreelyinspace.Ifthespheres S' coatedwithasbestoswillbemuchhotterinsidethanthecopper of heatatandjustinsidetheirboundarieswillbeequal.Butthe like asbestos.Letitssurfacebeblackenedandaheatingcoilpro- which wesupposereplacedbyalayerofsomebadlyconductingmaterial second sphereS',alsoofcoppersavefortheouteritsradius, provided withasmallelectricheatingcoilnearitscentre.Letitcool freely inspace,ultimatelytakingupasteadystate.Nowimagine distribution isimportant. it maybeofinteresttohaveaphysicalmodelinwhichtheopacity inside theboundary.Idonotputforwardthisasamodelofstar, through aspherewemustnotleaveofftheinvestigationatpoint illustrates thepointthatinsolvingproblemofflowheat because inthestarmechanicalequilibriumrequiresconsideration.But owes muchtothewritingsofSirJamesJeans. tion ofstateagasappliedtotheouterlayersonly.Thepaperalso note), Vogtpointsoutthatthefundamentalequationsconnecting luminosity withtheothercharacteristicsofastardependonequa- of H.Vogt.Inparticular,inA.N.,Bd.233,Nr.5569,p.13(foot- of thepresentpaperowetoProfessorEddington’swritings.Ishould Kussell, Dugan,andStewart’sAstronomy;tothevariouswritings also liketoexpressmy.indebtednesspp.895-902ofvol.ii. form. Itwillbeobvioustoanyreaderbowmuchtheinvestigations Nov. 1929.andEffectiveTemperaturesoftheStars.53 The stellarproblemis akindofconverse;ifweknewthedis- In thisexampleitisthedistributionofinternal opacity(orcon- Let usimagineasphereofcopper,S,blackenedatitssurfaceand The followingsimpleexampletakenfromelementaryphysics © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System The CoolingSphere. Appendix. 54 Prof. A. S. Eddington, XC. I,

of energy-liberation. It is shown in § 18 above how this can be carried out. Description op Figures.

Fig. i.—Curves of absolute bolometric magnitude against mass (log10 M) for constant , for a certain homologous family. The plotted points show the observed bolometric magnitudes of actual stars (taken from Eddington’s lists. Internal Constitution of the Stars, pp. 154, 155), plotted against the logarithms of their observed masses, with the observed effective temperature written against each. Fig. 2.—Curves of absolute bolometric magnitude against effective temperature (T) for constant mass M, for a certain homologous family. The plotted points show the observed bolometric magnitudes of actual stars plotted against their observed effective temperatures, with the mass written against each. (Data the same as in fig. 1.)

Internal Circulation in Rotating Stars.

By A. S. Eddington, F.B.S.

I. In 1924 H. von Zeipel * published an important theorem giving the relation € oc (1 — co2/27rGp) . . . . (1) between the rate of generation of energy e and the density p in a star assumed to be rotating as a rigid body with angular velocity m. It was pointed out by H. Vogt f and the writer J that this theorem gives no grounds for expecting that the relation (1) will be fulfilled in actual stars ; its non-fulfilment will lead to circulating currents in the interior. Since the conditions in a rotating star are not spherically symmetrical, the flow of radiation will in general tend to heat the polar and the equatorial regions unequally ; the unequal heating causes matter to flow in large-scale convection currents. Von Zeipel’s condition deter- mines a singular case in which there is no unequal heating, and the star is free from convection currents ; it can thus conform to the artificial restriction of rigid body rotation. My purpose in this paper is to estimate the magnitude of the circu- lating currents that can be maintained by this unsymmetrical flow of radiation. Some results of interest follow from the mere existence of a circulation even if it be very slow. It produces a mixing which can scarcely fail to counteract the excessively slow diffusion of heavy elements to the centre and light elements to the outside of a star ; thus we are led to believe that the interior of a star has nearly uniform chemical composition throughout. The mixing also tends to equalise the angular momentum at diflerent levels, and so to bring about an increase of angular velocity inwards—a condition also caused inde- pendently by radiative viscosity as shown by Jeans. But in other * Seeliger Festschrift, p. 144, 1924. f Astr. Nach., No. 5342, 1925. Î Observatory, 48, 73, 1925.

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