1977Apj. . .212. .347H the Astrophysical Journal, 212:347-359

1977ApJ. . .212. .347H max max max max max max max Tinsley’s (1975)“standardmodel”aswellotherrecent proposedmodelsofGalacticchemicalevolution,and they foundthatinadditiontothemeanage,Agives the ageofGalaxyattimesolarsystemformation them tothegasineitheragentleorviolentmanner.Agreatdifficultymakingsuchmodelsisthatthereexists no leave theinterstellargastobeincorporatedintostars,andasstarsproduceeitherretainthemorreturn The AstrophysicalJournal,212:347-359,1977March1 relationship T~2Amay beafeatureofourGalaxy,andthusanymodelwhich correctlydescribesit. true ofnearlyalltheobservationally allowedmodelsstudied,notjustofthose whichincludeassumptions to comparethemodelagessoobtainedwithmodel-independent A.Forthelimitoflong-livedchronometer determined byAforestimatingthetotalageof Galaxy. radionuclide anditsdecayproduct.ThequantityA is notafunctionofthemodel.Inmodelstheyexplored evolution models,themeanageofelements,measured backwardfromthetimeofsolarsystemformation, which requireittobetrue. Since thesemodelshaveallbeencreatedtodescribeour Galaxy,thissuggeststhatthe pairs itisfoundthattheage oftheGalaxy,T,attimesolarsystemformed is2A.Thisfoundtobe to afactorof2. can bewellapproximatedbyaquantityA,whichis afunctiononlyoffewmeasurableorotherwiseknown a largeclassofreasonablemodels. discussed inthispaper.ThesedetailedGalacticevolutionmodelsfollowtheabundancesofelementsasthey these productiontermsarejustthehistoryoneistryingtodetermine.Onecantryconstructaself-consistent tries, byradioactivedating,todatetheGalaxy,ordeterminehistoryofGalacticnucleosynthesis,since the SW’s classofevolutionmodelsmaybeviolatedinour Galaxy,andhasreexaminedtheuséofmeanageas quantities belongingtoalong-livedradionuclidechronometer pair,apairconsistingoftworadionuclidesor good physicaltheoryofstarformation.Itwouldthereforebedesirabletoableelicitfromthenucleochron- model forthephysicalandchemicalevolutionofGalaxywhich,amongotherconstraints,produces the relative abundanceswillbeaffectednotonlybydecaybutalsothetime-dependentratesofproduction.Infact, nuclides. Fortheseobjectsnucleosynthesishasceased,sotherearenosources(exceptperhapsfromthedecay of observed abundancesoftheelements,includingradionuclides.Severalsuchattemptsbyvariousworkers are other radionuclides),andoneneedonlykeeptrackofthedecay.Anadditionalcomplexityisaddedwhen ologies someinformationaboutthehistoryofstarformationandnucleosynthesisinourGalaxywhichistrue for straightforward, withtheprimarydifficultiesinvolvingaccuratemeasurementofabundancesradio- © 1977.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. 186187 The purposeofthispaperistocalculatetheage the Galaxyusingseveralnucleochronometerpairs,for Tinsley (1975)hasshownthatcertainobservationaldata suggestthatsomeoftheassumptionswhichdefined Our analysestakenoteofthe recentlymeasuredneutron-capturecrosssectionsof Osand(Browne Schramm andWasserburg(1970,hereaftercalledSW) proposedthatforalargeclassofGalacticchemical The datingofmeteorites,lunarrocks,andterrestrialrocksbytheuseradioactiveclockshasbecomerelatively © American Astronomical Society • Provided by theNASA Astrophysics Data System 187 9 187187 max 186187 9max reduced byanaccuratemeasurementofthehalf-lifeReandmoreknowledgeoneffect yield anagefortheGalaxyof7to20x10years.However,thisrangecouldbesignificantly to theeffectivenetrateofnucleosynthesis,isconstantformostphysicallyplausiblerecent for whichimportantdatahaverecentlybecomeavailable.Itisfoundthatalthoughtherateof chemical evolutionoftheGalaxy.SpecialattentionispaidtoRe/Oschronometer, and enableanagedeterminationtobemadewhichmightseverelyrestrictcosmologicalmodels. evolution modelsexamined.Thisconstantnetrateimpliesthatforthesetheageof star formationdoesvarywithtimeinthedifferentmodels,quantity0exp(v),whichisrelated Subject headings:nucleosynthesis—stars:abundancesevolution ments whichcouldbecarriedoutinthenextfewyearscanreducetheseuncertaintiestremendously mean agecanbeestimatedbytheparameterA,inwhichcaseofGalaxyis2A+ of stellartemperaturesontheOs/Osneutron-capturecrosssectionratio.Infact,experi- i©, wheretistheageofsolarsystem(4.6x10yr).ThepresentuncertaintiesinA Galaxy atthetimesolarsystemformedistwicemeanageofstableelements.This Q COMMENTS ONGALACTICEVOLUTIONANDNUCLEOCOSMOCHRONOLOGY Long-lived nucleochronologiesarecalculatedforseveralrecentlyproposedmodelsofthe Kem L.HainebachandDavidN.Schramm Enrico FermiInstitute,UniversityofChicago Received 1976June9;revisedAugust23 I. INTRODUCTION:DATINGTHEGALAXY ABSTRACT 347 1977ApJ. . .212. .347H 187 187 where misthemassofgasindisk,0ratebeingconvertedfromintostars,andR the where isthetotalmassindisk,orsomeunitofareaand/isratefallinginto the where Zisthemassfractionofmetalsinstars,andthatgas;Xaradionuclide gas anddustinthediskisallowed: first excitedstate,likelytobepopulatedinastellar^-processenvironment(Woosleyetal1976),whichallowone where m=Xmisthemass,ingas,ofaradionuclide ofinterest,andAisitsdecayrate. fraction oftheirmasswhichstarsreturntothegas(bysupernovae,etc.);and immediately aftertheyform),are: heavy nuclides.AnothermodificationallowsmatteraccretedbytheGalaxytobemetal-rich: Talbot (1973),thepossibilitythataveragemetallicityofnewlyformedstarsmaybegreaterthan the chronometersinmind,letuslookatGalacticchemicalevolutionmodels. cosmochronometers arethosesynthesizedbyther-process{rapidneutron-captureprocess).Thechronom- use aschronometers.Atthepresenttimeonlyradionuclideswhichhavebeenfullydevelopedfor unit ofmassretainedbystars.Usingequation(4),onecan rewriteequation(5)intermsofthemetallicityZ: where m=Zmisthemassofmetalsingas,andy isthemassofmetalsproducedandejectedbystarsper where /referstomatterfallingintothedisk. eters workparticularlywellsincetheyarepresumedtobesynthesizedalongwiththebulkofheavyelements, and willconcentrateonhowthetwotopicsinteract. bach andSchramm(1976).ThepresentpaperwillfullydescribetheGalacticevolutionnucleosynthesismodels to useReandOsasacosmochronometer(Clayton1964).TheRe/Oschronometernowtendsleantoward of theradionuclideingas, equation(6)becomes: disk; of interestinstars,Xisthatthegas;andFenhancementfactorassumedtobesamefor all and sincetheirabundanceratiosatthetimeofproductionarenottotallyunknown.Withr-processnature greater agesfortheGalaxythanpreviouslythought,allmodelsexamined.ThishasbeendiscussedinHaine- 348 1975; BrowneandBerman1975,1976)therecentlyrecalculatedneutron-capturecrosssectionofOs g s x zg z s Tinsley’s (1975)equationsofGalacticevolutionwillbeusedwithsmallmodifications.Inparticular,following The equationsofGalacticevolution,assuminginstantaneousrecycling(i.e.,starswhichejectmatterdosoalmost Writing m=AmN,where A=atomicweight,mmassofahydrogenatom, andN=numberofatoms Schramm (1974)hasreviewednucleocosmochronologyandlookedatradioactivespecieswhichmightbeof One alsohasforradionuclides: xa H © American Astronomical Society • Provided by theNASA Astrophysics Data System v J dt dN f ,_iOjiA) (zZJ dt m g = -\N+y x II. GALACTICCHEMICALEVOLUTION:FORMALISM ^ =—Z0(1-R)+yzM-R)Zf,(5) Sr = -Am*-X0(1—R)+^0(1f, Sf HAINEBACH ANDSCHRAMM Z =FZ,X, Z =FZ,,(1) f S ¿(1 --R) Am H dmtot _~ J 9 dt + Vz Mg 0(1 -*) 0(1 —R)Xf sf m XntgX\' g +z <'->;t f Vol. 212 (5a) (3) (7) (6) (4) (2) 1977ApJ. . .212. .347H 8 max No. 2,1977 and outofthegasforreasonsotherthannucleosynthesis: where N,t/i,anda>are,ingeneral,functionsoftime.FollowingTinsley,wedefine Defining theproductionrateP=y(}—R)¡Am,andaparametero>,whichdescribesmovementofmetalsin and thetimesolidparticlesbegantocondense.Short-livedchronometersdetermineA~1-2x10yr(Schramm and writethesolutionofequation(9): manner. Itwillbeshown,however, thatageneralrelationbetweenmeanageandtotal agedoesseemtoexistfor Thus eveninmoregeneral models thanthoseusedbySW,Aisadirectestimate ofthemeanageele- and usingtheassumptionsofequations(1)(2)thatX¡X—Z/Z,etc.,equation(7)becomes where themeantimeofstarformation,isnotthemeanageofstableelements intheinterstellargas;ratherT—iis(cf.deriva- v © American Astronomical Society • Provided by theNASA Astrophysics Data System Amax =J+— y [A +v(i>1MHn NUCLEOCOSMOCHRONOLOGY N(t) =í>-i*'Pip(t'). e ^ fz dt dN f max t v <^t> =J*tipdtjifidt. max r -ÄA, T -UKA. (18) = —XN+Pip-wN, w Ai _Ayex+ [^Ir- Uop»(owtJ m Znig XV g JJO-ITi-»l 1 A, Jo 1 Fi/Py fjo exp[Ají+v(t)]if>dt\ ■ InR(Ü), (16) (13) (12) (10) (17) (15) (14) ai) 349 (9) (8) 1977ApJ. . .212. .347H 187 40 v v 187 187186 187 232235238 40 235238 187 187187 186 87 87 187 times. Furthermore,long-livedchronometersareunaffectedbysmall-scalestructureinthenucleosynthesisrate. because iftoomanyhalf-liveshavepassedsincesynthesisofthenuclidebegan,informationislostaboutearly accretion. Inmorecomplexmodels,nolongerhasthissimplephysicalmeaning;however,itremainsauseful the rateatwhichmetallicityofgasincreases.Thesupernovaisproportionaltostarformation the Redecay,an¿-process-onlynuclidewhichliesonaflatandconstantportionofoNcurve.However, lived radionuclidechronometer.ThenuclideKmightserve,butitshalf-lifeisonly1.28billionyears;decay can donucleocosmochronologybyinsertingandv(t)intoequation(14)solvingforT,theageofGalaxy numerically, givensomeassumptionsregardingstarformation,oratleastassumptionaboutthehistoryof mately constantinmanymodels. quantity. Forwantofabetterterm,weshallcall^etheeffectivenucleosynthesisrateandshowthatitisapproxi- 0, whilethefactoreadjustsforremovalofmetalsfromgasbystarformation,andadditionGalactic chronometer isduetotheworkofBrowneandBerman(1976)hasbeenbrieflydiscussedbyus(Hainebach necessary forthe¿-processabundancecalculation,hadnotbeenmeasured.TheestablishmentofReasauseful the half-lifeofReisuncertain(30-60billionyears),andneutron-capturecrosssectionsOsOs, certain. Clayton,however,notedthatRewilldoverywellsinceitsdaughterproductOsis,otherthan for tell ussomethingaboutthehistoryofnucleosynthesisinGalaxy(cf.SchrammandTinsley1974). each oftheactinideelementsTh,U,andPu.AswasnotedbyFowlerHoyle(1960), U nucleosynthesis events(supernovae)thatproducethebulkofheavyelements.Thisprocessisthought to product, Ar,isvolatile;furthermorethenucleosynthesisofKnotwellknown.Along-livednuclideneeded, models forGalacticevolution.However,beforedoingthisletusfirstdiscussthechronometers. at thetimesolarnebulawasisolatedfromGalacticnucleosynthesis.Weshalldothisforseveralpublished i/* andm,ifnecessary,Z/Z,Z//Z,/.Ifthesefunctionsareavailableinamodel(orbyassumption),one 350 now noworsethanthoseofthewell-establishedu/Uchronometer.TheRe/Oschronometeralreadyprovides chronology couldproceeddespitethisdifficultyifthe¿-processportionofabundancebecalculated are everdeveloped,mightnotbeinagreementwithr-processchronometers.However,theirdevelopment different fromthehistoryofsynthesisbulkheavyelements,and¿-processchronometers,ifthey synthesize partorallofthemostneutron-richstableisobareach^4-chainaboveironpeak,including of then, anditisworthwhile to reexaminethematter.OurbasicsourceisMorgan andLovering’s(1967)simul- Os)(c forcosmoradiogenicosmium).R(187)=[1 + Os/Re],andasexplainedinHainebach tween Clayton’s(1964)paperproposingtheRe/Oschronometer andthepresent,manynewdatahaveaccumulated. shall carefullyreexamine.Suchareexaminationisnecessary anddeservedbecauseduringthetimeelapsedbe- a lowerlimittotheageofGalaxy,sooneisjustifiedinusingitdespiteitsremaininguncertainties,which we and Schramm1976).AlthoughtheuncertaintiesconnectedwithRe/Oschronometerremainlarge,they are of thenucleosyntheticsupernovae(cf.Schramm1976).Thushistory¿-processnucleosynthesismaybequite occur inHe-burningshellsredgiantstars(Ulrich1973;Iben1975),whichareprobablynottheprogenitors where tistheageofsolar system,andor(186)a(187)aretheneutron-capture crosssectionsofOsand Unfortunately Srliesonasteepportionofthe¿-processoNcurve,makingitsabundanceratherun- and subtractedout.ThefirstobviouscandidateisRbwhichdecaystoSrwithahalf-lifeof4.7billionyears. are suitablelong-livedchronometers. Schramm (1976)canberewrittenintermsofpresent-day laboratory-measuredquantitiesas: Os. gs c cr+A Q 187 When adetailedmodelofGalacticchemicalevolutioniscalculated,onejustintegratesthedifferentialequations It caneasilybeshownthatinTinsley’smodels,withnoinfall,constantyieldy,ando>, Some heavier-than-ironnuclidesarealsosynthesizedbythe¿-process(slowneutroncaptureprocess)thoughtto One thereforeturnstolong-livedr-processnuclideswhichareusuallythoughtbemadeinthesameexplosive The nuclideswhichformthebulkofheavyelements(carbonthroughironpeak)containnosuitablelong- Some nuclidesaresynthesizedbybothther-and¿-processes.Clayton(1964)showedthatr-processcosmo- Using theformalismofequations(12)and(13),onecan setthingsupforthechronometerpairRe/(e+ Clayton (1964)gaveavalue of11.3for[Os/Re].However,agoodmanydata have becomeavailablesince z now © American Astronomical Society • Provided by theNASA Astrophysics Data System 187 187 Os] c ReJr+A 187 < OsRe f0s Os 187187 HAINEBACH ANDSCHRAMM III. NUCLEOCOSMOCHRONOMETERS a) TheRe/OsChronometer -¿f [exp(Ai)-1] me i) TheOs/ReRatio 187 e (Re/Re) exp(Ai) 187G *Tt' / vdZ 186 <(o39±oo3) 2 II 12131415 hi) TheOsNeutron-CaptureCrossSections [^n]„™.."'- <» NUCLEOCOSMOCHRONOLOGY i i)ti11inh111 187 ii) TheDecayConstantö»/Re 12.7 ±2.0bynumber. S (^/Re)as m (20) 351 1977ApJ. . .212. .347H 187 187 232238235 products ofnuclearweaponstests(Eccles1970). the neutron-richsideofvalleybetastability.Someinformation,however,isbeingobtainedbystudying the there mustbeseveralaccurateandconsistentindependentdeterminationsofthehalf-lifeRe.(SeeAddendum.) the Galaxy. progenitors whichdecaytothembyachainofa-andperhapsß-decays.Thepairsusuallyusedfordating are duction ratiosarereasonablywellknownsincedeterminingthemismainlyamatterofcountingß-stabler-process Part ofthismaydisappearwhenaplausibleastrophysicalsiteforther-processisdetermined(cf.Schramm1973). uncertainties duetothepossibleeffectsofdelayedfission(Wene1975). are usefullong-livedr-process-onlychronometers.Theirhalf-liveswellknown,andthenucleosynthesispro- make Re/Osthemostexactandusefuloflong-livedcosmochronometers.Beforethiscanhappen,however, Neutron-Capture CrossSections: certainty intheRehalf-life(cf.§III)andcalculationof/. However, partofitstemsfromourinabilitytocreate,andmeasure,thenuclearpropertiesnucleifarout on events contributingtothesolarsystemanditsformation,ratherthanoverallnucleosynthesispatterns in burg 1976),willnotbediscussedindetailheresincetheinformationtheyproviderelatestodetailsoffinal Relative Abundances: Th/U and/.DataforthesechronometersaretakenfromthosecompiledinTables12 of 352 ratio Os/Re(cf.Fig.1).Someofthisuncertaintymayslowlyrecedeasmoremeteoritesareanalyzed,and the ratios oftheseelements.Onemeasuresthemincarbonaceouschondriticmeteorites,believedtoberelatively Schramm’s (1974)review.TheyarelistedinourTable2,whichalsoincludesextensionsoftheproductionratio statistics improve. unfractionated. Yetsomechemicalfractionationisinevitable,asseenforexampleinthespreadofvalues the Decay Constants: 2321023582389 24412926 186 187 187 1871 / =0.82(range0.7to1.3) The threenuclidesTh(t=1.40x10yr),U(t7.04and4.47r) The majoruncertaintyintheU/UandTh/Uagescomesfromr-processproductionratios. The short-livedchronometersPuandI,thenewlyestablishedA1(cf.Lee,Papanasstasiou,Wasser- As mentionedbyHainebachandSchramm(1976),thelargestuncertaintyinRe/Osageisduetoun- a(186)/a(187) =/x(0.39±0.03) Os/Os =0.0159 Re/Re =0.625 A(Re +Os)=0 Part oftheuncertaintyinTh/UandRe/Osagescomesfrompresent-dayabundance A(Re) =A71.6±0.5xIQ-^yr" [Os/Re] =12.7±2.0 1/2 0 18 now © American Astronomical Society • Provided by theNASA Astrophysics Data System r-Process ProductionRatios: Relative Abundances: Decay Constants: 1 -10 -10 R235IR238 [=1.5±0.5 P232IP23B [=1.9(+0.2,-0.3) (232238 2.4(+0.35,-0.15) (235/238 o3 A =4.95±0.05x10-yr" A8 =1.551±0.014x10yr A35 =9.85±0.09xlOyr“ Xh/U)r+A = UU)r+A =13±026 232 23 2 L= 1.5to1.7 L= 0.89to1.2 => 1.9(+0.2,-0.4) ^ 1.5(+0.5,-0.61) HAINEBACH ANDSCHRAMM Data fortheRe/OsChronometer tí) TheActinideChronometers c) DiscussionofUncertainties Browne andBerman1976 Woosley etal.1976 Nier 1937,referencedbyLedereretal.1967 Schramm 1974;alsoseetext Our analysis(seetext)ofdataMorganandLovering1967Anders1976 Holden andWalker1972 Actinide Chronometers Wene 1975J Schramm 19741 Wene 1975J Schramm 19741 U.S. AtomicEnergyCommissionReport Adopted combinedrange Jaffey etal.1971;rangeincludesLederer1967 See AppendixinSchrammandWasserburg1970 Holden andWalker1972;rangeincludesLedereretal.1967 Jaffey etal.1971;rangeincludesLederer1967 Adopted combinedrange TABLE 2 TABLE 1 Vol. 212 1977ApJ. . .212. .347H 235238232187187 v max max max v dwarfs (cf.vandenBergh1962;Schmidt1963).Galacticmodelswhichbeginwithaprimordial(i.e.,bigbang)low metal composition,ahighstarformationrate,andthecurrentlyobservedspectrumofmassesnewlyformed has beencalculatedforthreelong-livedchronometerpairs:U/U,Th/andRe/Os.Uncertainties Thuan 1974).TheageoftheGalaxy,T,attimesolarnebulawasisolatedfromGalacticnucleosynthesis No. 2,1977 A varietyofsolutionstothisproblemhavebeenproposed,andaredescribedbelow. bution ofGdwarfmetallicitiesinthesolarneighborhoodpeaksatahighervaluethansuchmodelswouldpredict. stars, producemanylow-metallicity,low-massstars(e.g.,Gdwarfs)atearlytimes.However,theobserveddistri- nucleosynthesis givearangeofallowedagesforeachchronometerpair. in radioactivedecayrates,presentabundanceratios,and(exceptforRe/Os)productionratiosatthetimeof as the“exponentialmodel,”itshouldnotbeconfusedwithslightlydifferentexponentialmodelofFowler star formationmodel(TalbotandArnett1973;Talbot1974),(4)thecombinedhalo-disk(Ostriker the standard,whichassumestimedependencesareexponential(Tinsley1975),(2)initialspikemodelsofTinsley years. where mQ=(0).Byequation(10),vœt.Thusawil(—R),andtheeffectivenucleosynthesisrate is hood. rate i/jsmooth,inordertoimitateanearlyburstofmassive starformation.Thus0eremainsconstant.(Seenote 2 2A . where |Am|=0.9M isthemassconvertedtoblackholesorotherremnants bythefirstgenerationof model discussedbelow.Specifically, These featuresareillustratedinFigure2,whichTisplottedasafunctionofAfortherangeuncertainty mathematical simplicity,Tinsley’sinitialnucleosynthesisspikeisomitted.Note,however,thatsucha is model takesover.Thisgivesaninitialspikein0e,which isthereafterconstant,muchlikethecombinedhalo-disk obey equation(24). one, andpointFisfor/=1.3.Thereadermayadjustthislowerlimitaccordingtohisownjudgmentas the of eachchronometerpair.ThepointsA,B,andCarethecorrespondingtomidpoints(orbestvalues) gg0 v 9jSplkeg0 One oftheobservationalconstraintsonmodelsGalacticevolutionispaucityold,low-metallicityG Truran andCameron’s(1971)promptinitialstarformation modelputsthespikeinafterwhichstandard Using theformalismdevelopedin§II,severalpublishedmodelsforGalacticevolutionhavebeenanalyzed:(1) These modelssolvetheGdwarfproblembyenriching gasinmetalswithaninitialburst,orspike,ofnucleo- By equation(8),o>isaconstant.equations(8)and(4), Tinsley’s (1975)initialspikemodelputsaintothe yield ywhilekeepingwconstantandthestarformation It isalsoseenthatinthestandardmodelthreelong-livedchronometersgiveaconcordantageforGalaxy, z © American Astronomical Society • Provided by theNASA Astrophysics Data System This modelhasmathematicalsimplicityandwidespreadusage.Althoughthisissometimesreferredto v + i/re =-Aw,S(0+“ m(Q)H(t),(25) g3plkel Rg IV. GALACTICEVOLUTIONMODELS:RESULTS NUCLEOCOSMOCHRONOLOGY meiat23 «A =j-zi?y^-r°°-’() max d) TheStandardModel b) InitialSpikeModels T ä2A(long-lived)(24) 353 1977ApJ. . .212. .347H max + 12 max +1 v 7max max+ + 12+ max235238232187187 max max requirement ofconstantgas masscoupledwiththepresentlyobservedmlm£ 0.1hastheGalaxyaccreting recent modelsdiscussedbelow. tion (26)wouldyieldtsuchthatT=1.7A,whilea massofblackholesequalto0.25m(0)wouldgiveT= then therotationcurveofourGalaxywouldbeverymuch differentfromthatobserved.Thusthismaynotbe unobservedly largemass-to-lightratioswouldresult.Nor couldtheyhavecollapsedintotheGalacticnucleus,for 0.9 Mgo#1.4x10M©leftby8-645stars.These couldnotbeobservedbythegravitationallenseffectof a completelyconsistentGalacticmodel.Theauthorsadmit thattheirexampleisanextremeone.If,forexample, Press andGunn(1973;1976);however,theycould notbeassociatedwithanystellarglobularclusters,since pared withtheremaininggasindisk,Tä2Aremains validforit. curves aretheallowedrangesforeachpair.Theconcordantintervalisshaded.pointsA,B,andCbestvaluesof the a consistentmodelcouldbeconstructedwithmassof blackholesequaltom(0)=1.6x10M©,thenequa- should notethatinthismodelthereisaresidueofblack holesarrangedinglobular-likeclustersandconstituting below, exhibitsasimilarformof^(cf.Fig.3c);butbecause thecontributionfromitsinitialburstissmallcom- mass intheinitialspikeislargecomparedwithremaininggasmass.Thecombinedhalo-diskmodel,discussed With e-™=0.05and\—R0.7,onefindsf0.16r,AxTt0.84/,therefore,ä1.2A . age foreachpair;PointDisthelowerextremeRe/OsifWoosley’scorrectionfactor/=0.82;Efor/1.0;F/ 1.3. Equation (24),Tä2A,isviolatedbecause|mp|>curm(0)/(l-R)inequation(26);i.e.,thelossof gas after (0)theinitialspike;andH(t)isHeavisidestepfunction=0(¿<0);1(/>0).Evaluatingt by formula (17),onefinds massive stars(5Sikeg v g0? ? max Tinsley’s (1975)modelwith metal-freeinfallandconstantgasmassinflowrate hasbeenexamined.The We haveseenthattheTruranandCameronpromptinitial starformationmodeldoesnothaveT~2A.One Fig. 2.—TheageoftheGalaxyattimesolarnebulaisolation{scaleonleft),andpresentlyright),asafunction © American Astronomical Society • Provided by theNASA Astrophysics Data System _1 f =[l-Am,,/«>ry^J•(26) vspike HAINEBACH ANDSCHRAMM c) Metal-poorInfallModel s x Vol. 212 1977ApJ. . .212. .347H max 235 29-1 max max max max v 9max max No. 2,1977 lived. ThemodelclearlydoesnotobeytherelationTx2A.Infact,allchronometersinthismodel,because of happens intheothermodelsexamined,toU/UchronometerforlargeT,whenUbecomesnolongerlong- pc (10yr)(Oort1966)andthecurrentmassdensityof75M©.Theeffectsuchmassivedilutionis to (a) thestandardmodelwithnoinfall,(b)metal-enhancedstarformationmodel,and(c)combinedhalo-diskmodel. cause allofthechronometersto“saturate,”i.e.,TversusAhaveaverysteepslope,similarwhat most ofitsmasssinceformation,indisagreementwiththecurrentlyobservedaccretionrate0.8-1.6M© the saturation,areuselessfordeterminingT,althoughA,asalways,ismeanageofchronometersin gas different hadaslightlyhutequallyplausiblevalueofAbeenchosen. Table 3givesTforseveralvaluesofAandtwochronometersinthismodel.Theconcordantintervalis7 < at anytime.Inthismodeljthemeanageisameasureofoveralldilutionrateratherthanradioactivedecay. the quantityi/je,effectivenucleosynthesisrate,isapproximatelyconstantinMESFmodel(cf.Fig.3è). It varies withtimebetween1.7and2.3,not1.0asinthestandardmodel.Itisneverthelesstrue,remarkable,that consider accretiontotheGalaxy. a physicallymotivatedtheoryofstarformation.Inthistheory,theformationratestilldependsongasdensity, T <49x10yr.ItisseenthatTinsley’svalueof=22.4yrforA3wouldbegreatly calculations presentedinHainebachandSchramm1976, wehaverecalculatedthechronologiesinMESF interstellar gasintoalow-temperaturehigh-densitycloud,perhapstriggeringstarformation.Thustheaverage gas anddecreasesthecriticalpressureatwhichthermalinstabilitysetsin,convertinghigh-temperaturelow-density model tobe3 O + Vol. 212 1977ApJ. . .212. .347H 187 vmax 187 10 9 3 max9 9 max No. 2,1977 really concernedaboutthehighRe/Osage,onemight try toinvokeenhanceddecayofRedueionizationin years, andF=11billionyears.Eventhisgivesaconcordant agebetween11and19billionyears.Ifonewere and lowerextremesfortheRe/Osage(pointsC,D,E, andFinFigs.2,4,5)wouldmoveuptovaluesof Already Dyson(1972)reportsayet-unpublishedmeasurement byR.W.P.DreverandJ.A.Payneofthehalf- and theconcordantintervaltheyproduceisbroad.However, theintervalmightbenarrowedinnextfewyears. present ageoftheGalaxyis21billionyearsforRe/Os,whileconcordantinterval7to19years.How- (the presentGalacticage)wouldbecome:C(bestvalue) =22billionyears,D16E14 in goodagreementwiththeindirectmeasurementofHirt etalIfthismeasurementisadopted,thenthebestvalue ever, wefeelitprematuretobetooconcernedwiththis discrepancy,sincetheuncertaintiesindataarereal compass theRe/OsbestvalueforageofGalaxy.Forexample,inMESFmodel the intervals areessentiallythesameinallsuchmodels,withuncertaintiescomingfromchronometer follows fromtheconstancyof0eresultthatT£2Aforlong-livedchronometers,andconcordant more than1-2billionyearsinhisGalacticevolutionmodels. stars (Clayton1969;Perrone1971);however,Talbot(1973) foundthatthiseffectwouldreducetheRe/Osagebyno life ofRe.Theyobservedtheß-decayelectronsinaproportional counterandfoundt=4.7±0.5x10yr, simply thatofthestandardmodel.However,MESFmodelinnowaycontainsphysicalassumptions the T whichcanbereadoffofthegraphs.Forexample,for theMESFmodel(Fig.4)valuesofT+4.6x10yr standard model.Perhapstheconstancyofeffectivenucleosynthesisrateisevenaphysicalpropertyour billion yearsfortheGalaxy. AmoreprecisedeterminationoftheageGalaxy wouldallowustopredictthe Galaxy, andthusofanymodelwhichcorrectlydescribesit.(Oritmaysimplybeacuriouscoincidence.)There of heliummassfractionY= 0.3to0.2,andformetallicityZ=10“.Thisisconsistent withourrangeof7to19 Universe. Iben(1974)calculates theagesofglobularclusterstobebetween10.8and 17.4billionyearsforarange A: C=9.5x10yr,D5.8E4.7 yr,andF=2.9x10correspondingvaluesof in A. 1/2 Y ofglobularclusters. It isclearfromFigures2,4,and5thatwhiletheconcordantintervalsaregenerouslylarge,theyfailtoen- Finally, letuscomparethe range ofagescalculatedforourGalaxywiththe ofotheroldobjectsinthe © American Astronomical Society • Provided by theNASA Astrophysics Data System Fig. 5.—Like2,butforthecombinedhalo-diskmodel NUCLEOCOSMOCHRONOLOGY O CO X 357 1977ApJ. . .212. .347H 187 1- max max v very neartheHubbleage,andwouldbeconsistentwithfindingsofGottetal.(1974).AcriticalUniverse in/could bereduced,thentheRe/Oschronometermightabletousedinsupportofopenmodelsfor have anagefx1IH,or12to14billionyears,stillgreaterthanourlowerboundfortheGalaxyof7years. 358 the nucleochronologiesindicateaconcordantpresentageofGalaxy7to19billionyears.Itisfoundinthese However, ifnewmeasurementsoftheRehalf-lifeconfirmthatDreverandPayne,uncertainty 3 kms“-Mpc(Kristian1976;SandageandTammann1976).Anopen,unboundUniversewouldhaveanage 9 max which isafunctionofthechronometersthemselvesandnothistorynucleosynthesis.Itsuggestedthat nucleosynthesis isjusttwicetheageparameterA,whichmeanofstableelementsingas,and 2A forlong-livedchronometers;i.e.,theageofGalaxyattimesolarnebulawasisolatedfromGalactic same modelsthattheeffectivenucleosynthesisrate,wasestimatedtobe z ~0.15 (10yr)fromaconstant ÿmodelwhichproducesthecurrentlyobservedmg/m^ ~0.1;A—wastaken m&x The ageoftheUniverseitselfisboundedbyHubbleage,which18to21billionyearsforH=50.3± In summary,letussaythatinmostoftherecent,observationallyallowedGalacticevolutionmodelsexamined, We thankB.L.Bermanfordetaileddiscussionsofhisosmiumcrosssectionresultsandtheirimplications. 0 Press), p.224. Explosions (symposiumatLasVegas1970),Conf.700101, Press). Virginia (1970),p.1269. USAEC Div.ofTechnicalInformation,Springfield, Salam andE.Wigner(London:CambridgeUniversity 207, L79. B. M.1974,Ap.J.,194,543. Schramm andW.D.Amett(Austin:UniversityofTexas Earth ScienceandMeteoritics,ed.J.GeissE.D. (Schenectady: GeneralElectricCo.). Goldberg (NewYork:Interscience),p.273. and Essling,A.M.1971,Phys.Rev.C,4,1889. Addendum.—The experimentalcheckingofleveldensities ofOsisotopes,mentionedin§Ilia,hasnowbeen Note addedinproof.—One needs tobecarefulinapplyingourstatementconcerning TversusAtomodels Table ofIsotopes(NewYork:Wiley). © American Astronomical Society • Provided by theNASA Astrophysics Data System HAINEBACH ANDSCHRAMM REFERENCES Lee, T.,Papanasstasiou,D.,andWasserburg,G.1975, Nier, A.D.1937,Phys.Rev.,52,885. .1976,Nukleonika,21,727. .1974,Ann.Rev.Astr.Ap.,12,383. Perrone, F.1971,Ph.D.thesis.RiceUniversity. Press, W.H.,andGunn,J.E.1973,Ap.J.,185,397. 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Schramm andW.D.Arnett(Austin:UniversityofTexas Vol. 212 1977ApJ. . .212. .347H max 9 max 97max max1118 max max9 finds thatsolutionsexistfor values ofAareexcluded. finds inthismodelthatmlm=exp[—w(r+4.6x10yr)]0.25.Furthermore,onesolutionsto the GalacticevolutionequationsexistinthismodelonlyforA0forA,T;and/< , Kem L.Hainebach:SpaceSciencesLaboratoryA6/2447,TheAerospaceCorporation,P.O.Box92957,Los which hasazeroat David N.Schramm:EnricoFermiInstitute,LASR,TheUniversityofChicago,933East56thStreet, A hasaminimumvalueofjustunder4x10yr(pointDinFig.2),andthismodel,constrainedsothat Amax ^maxjfvalues.Numericalsolutionofequation(Nl)indicatessolutionsalong a gtot gtot 0 0 0 Re >()j areoresser 9 If oneconstrainsœinthismodeltogivetheobservedniglm^~0.1atpresenttime(T+4.6x10yr), © American Astronomical Society • Provided by theNASA Astrophysics Data System max9 A =i/2(ln10-1)1.8xyr,T2A3.5yr.(N4) 0 max /(A, T)=T— NUCLEOCOSMOCHRONOLOGY max T >2(ln10)A-t. Q max = 2(ln10-l)At, e (N2) (Nl) (N3) 359