19 93MNRAS.262. .350S 12 2 1 A modelfortheformation,évolutionandstructureofsolarcylinder Mon. Not.R.Astron.Soc.262,350-358(1993) Jesper Sommer-LarsenandVincenzoAntonuccio-Delogu decreasing metalabundance. function ofage.Itthusappears meaningfultomodelthe also showsanapproximatelymonotonie increaseofowith Accepted 1992October29.Received23;inoriginalformFebruary24 the heavyelementabundance ofdiscstarsisadecreasing function oftheagestars(e.g.Fuchs&Widen1987). Astrophysical Observatory,VíaleA.Doria6,1-95125Catania,Italy The starsareusuallyassumedtobecontinuallyheatedby abundance evolution ofthesolarcylinderwithin theframe- It isgenerallyappreciatedthattheverticalvelocitydisper- Clouds. NielsBohrInstitute,Blegdamsvej17,DK-1200Copenhagen0,Denmark stochastic processes,e.g.interactionwithGiantMolecular sion 0.5). the best-fittingmodel(7V=1),wefindthatageofsolarcylinderis on thebasisofalargesetobservationaldatamainlylate-typestars.Star Galactic globularclusters. about 8Gyragoatametalhcityof[Fe/H]=-0.6.Wefurthermorefindthatthestar discrete components,withthetransitionbetweenolddiscandthickoccurring considered. Infallofhalogasisassumedtohappenonacharacteristictime-scalet Schmidt type:2*°c2^,N=1-2.Modelswithathresholdforstarformationarealso disc darkmatterfractionPis0.02±0.07andthattheoldthickare ¿= 11.8±1.0Gyrandthat3.40.4Gyr.Wefindtheupperlimittolocal 0 H v 0 The infalltime-scalederivedinthepresentworkisreasonableagreementwith In conjunctionwiththeGalaxyformationtheoryofSommer-Larsen(SL), It issuggestedthatthethickdiscformedafteramergerbetweensatellitegalaxy The resultsobtainedarelargelyindependentofthestarformationprescription.For work ofstandardchemical-evolutionmodels(e.g.Tinsley model ofSommer-Larsen (1991b). tive predictionsbasedonthe cooling-flow Galaxyformation for discstars,todeterminethe agetofthesolarcylinder, local fractionPofdiscdarkmatter. characteristic time-scaletover whichitwasbuiltupandthe the solarcylinderwithalargesetofdataonmainlylate-type constrain modelsoftheformation,evolutionandstructure a maximum-likelihoodmethod.Inthisfollow-uppaper,we constructed onthebasisofarelativelysmallsetdatausing city dispersionofdiscstarsasafunctionabundancewas stars. Theaimistodetermine thetemporalevolutionofo 1980). 0 x w We alsowishtocomparethe results obtainedtoquantita- In Sommer-Larsen(1991a),amodeloftheverticalvelo- 19 93MNRAS.262. .350S -1 vertical heatingofthediscintermsstellarorbitaldiffusion, The observedrunofdensitywithz,p(z)forlate-typestarsis clusion. and inSection4theresults.In5weinvestigate and finallySection6constitutesthediscussioncon- we selectedasampleof403mainlyGandKdwarfsfromthe taken fromKuijken&Gilmore(1989),whoderiveditforK dwarfs. ThedataareshowninFig.1.o(z)istakenfromthe for the403starsisshowninFig.3,togetherwithdistri- the sampleiscr=22kms~ThecumulativeWdistribution same paper,withthedatashowninFig.2. 2 THEDATA bution expectedforanormaldistributionofvelocitieswith test showsthatthetwodistributionsdifferat95percent velocity dispersionof22kms.AKolmogorov-Smirnov Ghese (1969)catalogue.Theverticalvelocitydispersionof blom (1985).Thissampleconsistsof177mainlyGdwarfs in termsofa‘cold’thindisc,themainolddiscand‘hot’ with agescalculatedfromtheempiricalage-chromospheric late-type main-sequencestars,weusedthedataofSoder- thick disc. significance level.Thelocaldataaremuchbetterinterpreted blom, Duncan&Johnson(1991,theirequation3).Asthe emission relation.WeusedthelatestcalibrationofSoder- blom’s sample,weestimatethattherelativeerroronage t =4.6Gyr,wehavebinnedallstarswithemissionlevels lower thantheonecorrespondingtot=tintobinof ages arenotwellcalibratedgreaterthanthesolarage of astaris40percent(thiscorrespondswellwithSoder- stars olderthantheSun.FromfourdoubleinSoder- w w 0 e Figure 1.Theverticaldensitydistribution oflate-typestars.The observations ofKdwarfsbyKuijken &Gilmore(1989)arerepre- sented bypoints, the modelbysolidline. In Section2wedescribethedata,in3models To obtainthelocalverticalkinematicsoflate-typestars, To obtaintheagedistributionandcr-agerelationfor w © Royal Astronomical Society • Provided by the NASA Astrophysics Data System z [pc] -2 _ 1 blom’s ownestimate).Theresultsandthedataaregivenin Table 1anddisplayedinFigs45. disc fromSoderblom’sdata(ithasbeenassumedthat estimate thestarformationrate(SFR)overlifetimeof with theresultsobtainedinSection4).Thearedis- played inFig.6. ¿= 11.8Gyrand2*38Mpcatpresent-consistent The observationsbyKuijken&Gilmore(1989)arerepresented Figure 2.Theverticalvelocitydispersion,o,asafunctionofz. points, themodelbysolidline. 0 w Figure 3.Thecumulative|W|-distribution for403late-typestars from theGliese(1969)catalogue isshownbythedottedhistogram, the modelbysolidlineand anormaldistributionwiththe sample dispersion of22kmsbythedashedline. We haveusedadiscmodeloftheBahcall(1984)typeto A modelforthesolarcylinder351 19 93MNRAS.262. .350S formation rate and chemicalevolutionarealso presented. to theonepioneeredbyBahcall (1984).Modelsforthestar In thissection,self-consistent modelsofthelocalvertical 352 J.Sommer-LarsenandV.Antonuccio-Delogu disc structureareconstructed. Theapproachisverysimilar 3 THEMODELS type) andisshowninFig.8.Theobservationalerroronthe is takenfromSommer-Larsen(1991a)(thoughwithanew it isofnoimportancethatthesestarsarenotnecessarilylate- line andthemodelconvolvedwitherrorfunctioncharacteristic data for1300discFstars(notethatthefollowinganalysis (1987) (againusingequation20).Thisrelationisbasedon empirical relationbetween[Fe/H]and[O/H]-seeequation of thedatabydashedline. Figure 4.Thelocalagedistributionoflate-typestars.Datafrom [O/H] forthesestarsis0.07dex. Stecklum (1992),thissampleisessentiallyunbiased. [O/H] ofthesestarsis0.11dex.AsshownbyMeusinger& Soderblom (1985)arerepresentedbypoints,themodelsolid 20) andisshowninFig.7.Theobservationalerroronthe In total,dataforabout3000starsareusedinthisanalysis. Finally, thea(z)-[0/H]relationistakenfromStrömgren The localoxygenabundancedistributionof122Gdwarfs w Table 1.Theverticalvelocitydispersion,cr^,of Soderblom’s stars. age >4.6Gyr7224.0±2.0 3.0 Gyrow Gyr [km/s] 2/ v+4 G where ipisthegravitationalpotential, pthetotaldensityand For anaxisymmetricdisc,Poisson’sequationgives curve atR=8 kpcisapproximatelyflat,the firsttwoterms (R, ,z)theusualcylindrical coordinates.Astherotation _ 1dx¡)dip ^ Figure 6.SFR-agerelationforlate-typestars.Fordetails,see caption toFig.4. 3.1 Thediscmodels Figure 5.Thelocala^-agedistributionoflate-typestars.For *'-8Fsra?-’' * details, seecaptiontoFig.4. 0 9A i! ID 19 93MNRAS.262. .350S = zandis 2 Figure 7.Thelocaloxygenabundancedistribution.Datafrom ponent type].Hence model oftheBahcall,Schmidt&Soneira(1982)four-com- of theleft-handsideequation(1)approximatelycancelout the solidlineandmodelconvolvedwitherrorfunction where pisthedensityofdarkhalomaterial.Following characteristic ofthedatabydashedline. Sommer-Larsen (1991a)arerepresentedbypoints,themodel Bahcall, pisexpressedas [we findthistoholdagoodfirstapproximationfor where Pdisc,oPdiscUo)*assumedthatpis Phalo ^Pdisc,0>(3) Pdisc X!P•(4) vertical velocitydispersiona,sothatthediscdensityis dark haloisquiteround-seeSection4. approximately constantfor|z|<3CR.Thisrequiresthatthe |z| <3kpcatÆ=RbasedonareahsticGalaxypotential be expressedas The z-componentfirstmomentoftheVlasovequationforan expressed as found thelast term (the‘tilting’term)tobe relatively unim- subpopulations accordingtocharacteristicvaluesofthe (e.g. Binney&Tremaine1987). Sommer-Larsen(1991a) axisymmetric systemcan,fortheindividualsubcomponents, halo dz halo halo w 0 Again followingBahcall,thediscmaterialisdividedinto 4jt G{pfase“bPhalo)? i © Royal Astronomical Society • Provided by the NASA Astrophysics Data System í: J =ex l 2 J where p^p(z=Zq).Fromequations(2),(3)and(7), following. Figure 8.Thelocalo-[0/H]relation.DatafromStrömgren (1987) arerepresentedbypoints,themodelsolidhistogram portant incomparisontothefirsttwo,soweneglectit the databydashedline. and themodelconvolvedwitherrorfunctioncharacteristicof Without lossofgenerality,wecanassume^(R,z=0) 0 By integration,itthereforefollowsthat ents equation(5)simplifiesto p^z) canthenbeobtainedfromequation(7). interval (/-1,j)¿/10].Theo , /=1,10aretreatedasfree determined numericallyfromequation(8).Thedensities and, fromthesymmetryofdisc,dip/dziz=0)0. Pdisc(z) 2/o'=PÓP 0 ]n(p)_1dip Sommer-Larsen 1991a),sofortheindividualsubcompon- Bahcall (1984),andstarswith M>4.2(approximatelyG contributions totheobserved localdensityaretakenfrom 4jtG dz parameters to be constrainedwhencomparing modelsto dispersions o,/=1,10[bin / correspondingtoagesinthe and later)plusthewhitedwarfs andthegiantsarefurther subdivided into10agebins withcharacteristicvelocity w 0 0 w v w 2 l2 1 dip Observationally, a'(z)-constant(Bahcall1984; A discmodelisobtainedinthefollowingway.Therelative Given pi)andspecifyingavaluefor6,ip{z)canbe dz {o)* w w -2 piexp A modelforthesolarcylinder353 2 (o‘) w y>-ipo ^Pdisc,0* (8) (6) (7) 19 93MNRAS.262. .350S JK 7 1 /y S J 7 f'(ow)~r <7w2 The chemicalevolutionofthestar-formingGalacticdiscis This isthencomparedtothelocallyobservedWdistribu- d2* _2j, d2 determined bythefollowingequations: tion. tion as normal. Hencewemodelthelocalverticalvelocitydistribu- velocity distributionofcomponentjisapproximately Bahcall (1984),itfollowsfroma(z)-constantthatthe and againcomparedtotheobservations.Finally,asshownby 3.2 Modelsofstarformationandchemicalevolution P(W)dW= —dW. This isthencomparedtotheobservationsofKuijken& as Gilmore (1989).Similarly,aasfunctionofziscalculated p(z) =Ep(z)oexp lated as brown or‘black’dwarfs. would bethecaseifdiscdarkflatterwas,forexample, which accountsforthepossibilitythatdisccontains Pdisc,0 =(l+^)písc,0(H) dark matterisdistributedlikethelate-typestars,which unobserved darkmatter.Itisassumedthatanysuchdisc next subsection)andthefarenormalizedsuchthat where the2areobtainedfromstarformationlaw(see pi°cfj(j)X, (10) duce, followingBahcall,afactorPsuchthat contributions ofobservedmatterinthediscmodels. component j-seeTable2,whichspecifiesthedifferent right-hand sideofequation(10)isthelocalmassfraction ^ P(z)E/o'exp[-(V’-^o)/(<4)] are introduced.Hencethelocaldensityofbinjis quantities observations. FollowingSommer-Larsen(1991a),the f w w 354 /.Sommer-LarsenandV.Antonuccio-Delogu ? aw 0 dt dt t* The densityoflate-typestarsasafunctionziscalcu- To completethespecificationofdiscmodels,weintro- 2 J i 2V(z) (<4)EpictV)exp[-)/(o-v] 0 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2J Z [po/(f2noi)]exp{-W/[2[o]} w exp 2 Pa z i (14) 0 2 \¡r-_xpQ {¿w) &Z (16) (15) (13) (12) (9) Larsen (1991a). 3.2 4.2ageG (Lacey &Fall1985;seealsoSommer-Larsen1991b). density thresholdforstarformation. threshold]. Wealsoconsidermodelswithagassurface linear) starformationlawsareanalysedastwoextremecases Following Wyse&Silk,theN=1(linear)andN=2(non- Gas &Dust + WhiteDwarfsageG + GiantsageG d2¡ 2 deci- is, followinge.g.Wyse&Silk(1989),expressedas g fraction. [Kennicutt (1989)findsN—1.3aboveastarformation Sommer-Larsen 1991a).Inthispaper,thestarformationlaw diffusion onthechemicalevolution(seediscussionin tion time-scale,whichmaydependonthegassurfacedensity, Table 2.Thelocaldistributionofobservedmatter. Note thatthistransformation isdifferentfromthatusedin and forsimplicityweneglecttheeffectsofradialflows (Sommer-Larsen 1991a),where2and2*arethegas Total observeddensity=0.0958 Sommer-Larsen (1991a).Themodelswereused tocalculate stellar surfacedensitiesrespectivelyandisthestarforma- [G/H] =0.7[Fe/H],[Fe/H]>-1.7. (20) and [O/H] usingtherelation d(Z2) V v v 0 g g dt t x dí When comparingwithobservations,oxygenratherthan The infallrateisparametrizedas Following Pagel(1992),[Fe/H] hasbeenconvertedto dt +T(1-Z) age 6 age G age G age G age G age € -4fl 10 9 8 4 3 2 1 7 6 5 M pc.LMF:localmass [0.9,1.0]t /(^)Eajß [0.8,0.9]t /(^)S<7^ [0.7,0.8]t /(^)Sa*r [0.3,0.4]t /(^)S<7^ [0.2,0.3]to /(^)S<7^ [0.1,0.2]t /(^)S [0.0,0.1]t /(^)S<7^ [0.6,0.7]to /(°vv)^ [0.5,0.6]t /(^)2^ [0.4,0.5]¿o /(0yp)£<7^r 0 o o o o o o o a LMF aw 0.015 8 0.021 4 0.469 4 0.031 11 [km/s] (19) (18) (17) 19 93MNRAS.262. .350S j y 1 y ; fraction of25percent(Kuijken&Gilmore1989)andagas tion rateandyieldwerechosentogiveapresentdiscgas the relativesurfacedensityofstarsasafunctiontime.The which theinfallinggasiscoming.Inthisway,relative given valueoftheinfalltime-scalet(andN),starforma- calculations werecarriedoutinthefollowingway.Fora For theinfallinggas,avalueof[O/H]^-1.05wasused, folded withtheappropriateobservationalerrorfunctions, Larsen 1991a).Thelocalcr-[0/H]relationisalsocalcu- By weightingthe2'andSby/',localage[O/H] corresponding tothe[Fe/H]--1.5typicalofhalo,from abundance of[0/H]=+0.07(fromequation20thiscorre- to thedata. binned inthesamewayasdataandthenfinallycompared lated (thelocala-ageisalreadygivenbythechoiceof distributions oflate-typestarsarecalculated(cf.Sommer- sponds to[Fe/H](¿)=+0.10dex,asfoundintheHyades). surface densities2,y=1,10arecalculated.Foreachofthe The datasetwasfittedusingamaximum-likelihoodtech- flattened byafactorof2[thelimitinglinescorrespondtothe velocity ofv=220kms"atthesolarpositionimpliesthat we assumethateisconstrainedto6^0.15.InFig.9,the the parametere>0.15,probabilitythatlocaldark nique asdescribedindetailKuijken(1991).Forvaluesof combination withthediscshouldgiveatotalrotational resulting discsurfacedensityofthesolarcylinder2is matter fractionPisnegativemorethan95percent.Hence 4 RESULTS spherical andabovethelineg=0.5ifdarkhalois Z(e) shouldlieabovetheline^=1.0ifdarkhalois shown asafunctionofe.Theconstraintthatthedarkhaloin 10 bins,thequantities=2/Alog(Z)arealsocalculated. 10 parametersa,7=1,10).Thelocalrelationsarethen x w w 0 function ofe.For details,seetext. Figure 9.Thediscsurfacedensity ofthesolarcylinder,2,asa c 0 H 0H 10 w disc © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2 2 been assumedthatthediscisradiallyexponentialwithscale- y= 1,10.WeallowthestarformationlawparameterNto length 4.0kpc].AscanbeseenfromFig.9,ifthedarkhalois appendix ofSommer-Larsen&Christensen1989)andithas zero coreradiuscase,a=jR+z/qH(see (arguments forthedarkhalobeingnearlysphericalaregiven tions, henceq>0.5.Inthefollowingweshalladopt6=0.10 take thetwolimitingvaluesN=1andN=2.Thelinearcase upper limittoPisobtained. in Sommer-Larsen&Christensen1989).Solutionsfor spherical then0.10^6^0.15.Ifq=0.5therearenosolu- case, weobtaint=1.8±0Gyr,^3.4±0.4 0.10^6^0.15 areverysimilarand,byassuming6=0.10,an (Af= 1)givesthebestfit[interestingly,Kennicutt(1989)finds Kuijken (1991).FurtherchecksusingMonteCarlogenerated pseudo-data indicatethatthestandarderrorsgivenare errors havebeencalculatedbythemethoddescribedin TV-1.3, i.e.avalueclosetothelinearcase].For conservative. Theinverseoftheproportionalityfactorin P<0.02±0.07 andtheOj,j=l,10giveninFig.10.Standard to beoftheorder9percent. contribution tothelocaldensityoflate-typedwarfsisfound unchanged bytheintroductionofthisthreshold. tions. Wechooseathresholdvalueof2=8Mpc'(Pagel equation (18),4,isfoundtobe5.5Gyr.Thethick-disc Figure 10.Thea^-agerelation, with thefitstomodelbasedon u surface densitythresholdforstarformationinthecalcula- H thsidashed (equation 23)linesrespectively. diffusion theory.TheLacey(1984) modelisrepresentedbythe solid line,theWielen(1977)models bythedotted(equation22)and 0 i =10.3Gyr,¿¡2.8GyrandjP^0.02±0.07,thatthe pendent ofthestarformationprescriptionused. 1992). Wefindthattheresultsobtainedarevirtually Gp 7=1,10isquitesimilartothatfoundinthelinearcase. t0 0 The 13freeparametersinthefitarehencet,Panda For thenon-linearstarformationlaw(AT=2),wefind Following Kennicutt(1989),wealsointroduceagas 0vy We thusfindthattheresultsobtainedarelargelyinde- A modelforthesolarcylinder355 19 93MNRAS.262. .350S 2 G=a 0=a13 1/2 that theolddiscandthickarediscretecomponents. the dataisverysatisfactory. plotted themodelpredictionsinFigs1-8.Overall,fitto that theageofhaloisapproximately/i=14±1Gyr,in greater thantheageofsolarcylinder.Itthereforefollows predicts thattheageofmetal-weakhaloisabout2.3Gyr discontinuity. We discussbelowpossibledynamicalimplicationsofthis very goodagreementwithrecentabsolutedatingofthe fitted threedifferentlawstothedata,namelysolution However, aspointedoutinthatpaper,thistheoretical metal-weak Galacticglobularclusters(seealsobelow). 8 Gyragoatametallicityof[Fe/H]=—0.6stronglyindicates 356 /.Sommer-LarsenandV.Antonuccio-Delogu given byLacey(1984)forscatteringfromGMC: character ofstellarorbits,andsoon.InFig.10,wehave between thetwodifferentestimatesofisfairlyreasonable. the rotationalstateofyoungproto-galaxy.Theinfall prediction ishighlysensitivetoasyetunknowndetailsabout cent largerthanthatestimatedinSommer-Larsen(1991b). for aconstantdiffusioncoefficient: The resultingtemporalevolutionofodepends,however, perturbations oftheforcefield(Barbaras&Woltjer1967) but theyalsoshowthat,afteraphaseofsteadyincrease preceding sectionssupportandstrengthenthisconclusion, Previous work(Wielen1977;Fuchs&Widen1987)has DISC full dataset. figure, thedifferencebetweenresultsisnegligible, time-scale estimatedinthatpaperwasbasedonastrongly tion oftheGiantMolecularCloudssystem,about on someunderlyingphysicalassumptionsabouttheevolu- (GMC) (Spitzer&Schwarzschild1951)andfromlocal local <7-[0/H]relationisshowninFig.11,togetherwiththe data usedinthefitting,weexcludedStrömgren simplified modeloftherotationalstate.Thusagreement can inducediffusioninphasespaceofapopulationstars. solar neighbourhoodincreaseswithage.Theresultsofthe shown thattheverticalvelocitydispersionofdwarfsin result fromthefulldatafitting.Ascanbeseen result offittingtheremainingdataintermspredicting (1300 starsoutofatotalabout3000)fromthefit.The 5 VERTICALHEATINGOFTHEGALACTIC and twoofthesolutionsgivenbyWielen(1977),namelyone strongly indicatingthatthemodelisoverconstrainedby The fitsareshowninFig.10. Anon-linearxfittothedata 10). ThescatteringofstarsfromGiantMolecularClouds o, afastheatingofthediscoccurredabout8Gyrago(Fig. ha0 shows thatthe best-fitting solutionisLacey’s (1984),witha and theotherforavelocity-dependent diffusioncoefficient: w{L+(21) w w (2+bwt)/- (23) <7=(ai +¿?¿), (22) w w 2 wW The discontinuousjumpinverticaldispersionabout For thelinearstarformationmodel(7V=1),wehave The GalaxyformationtheoryofSommer-Larsen(1991b) The infalltime-scalefoundinthisworkisabout50per To testthesensitivityofsolutionstoamount © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2 Ä 2 function, theaboveimplicitassumptiontoholdavery bution ofGandKdwarfsrepresentstheSFRsincesolar In thispaper,wehaveimplicitlyassumedthattheagedistri- tracks ofVandenBerg(1985)and&Bell cylinder startedtoform.Basedonthestellarevolutionary found inthispaper.Itispossiblethatsystematicdifferences work isinprogress.Freeman(1991)finds(basedon a disc tothickwhichoccurredabout8Gyrago.Such discontinuous jumpinverticalvelocitydispersionfromold good firstapproximation. (1985), wefind,foranyreasonablechoiceoftheinitialmass 6 DISCUSSIONANDCONCLUSION where theStrömgrendatafor1300starsareomitted,bydashed by thesolidhistogramandmodelfittedtoarestrictedsample, Figure 11.Thelocal(/^-[O/H]relation.modelisrepresented between themetallicitycalibrationscancausethisdifference. disc occursat[Fe/H]-0.4dexratherthanthe0.6 He alsofindsthatthetransitionbetweenolddiscandthick able interest,asthiswouldenabledirectobservationofthe line. that theolddisc andthethickdiscarediscrete components, the (age)-zrelation,([0/H])-z relationandthe[0/H]-age smaller sampleofstars)indicationssuchadiscontinuity. have usedonlythefirstsevendatapointsinfit).Wecan continuous infallofgasfromthe halooveraperiodofabout relation, whicharedisplayedin Figs12-14,respectively. relations, whichmightbeobservableinthefuture.Theseare conclude thatouranalysissupportsLacey’sheatinglaw, although theotherpossibilitiesarenotruledout. started formingabout12±1Gyr ago,thatitwasbuiltupby 23), oneobtainsx~0.968and0.457,respectively(we 3 Gyr,thatthelocaldarkmatter fractionisnegligibleand X~ 0.183;forthetwoothersolutions(equations22and Improvement ofthedatingGandKstarsisconsider- From themodel,wehavederivedthree‘theoretical’ The mainconclusionofthispaper isthatthesolarcylinder 19 93MNRAS.262. .350S 9 Figure 12.The(age)-zrelation.modelisrepresentedbythe solid line. Figure 13.The([0/H])-zrelation.modelisrepresentedbythe solid line. with thetransitionoccurringabout8Gyragoatametallicity of [Fe/H]=-0.6.Wespeculatethatthediscontinuitywas of 5percentthemass disc. probably oftheorderafew times10M,i.e.oftheorder caused byasatelliteaccretionevent(Freeman1990).A metal-weak halois14±1Gyr, inverygoodagreementwith Sommer-Larsen (1991b),we estimatethattheageof simple calculationshowsthat themassofsatellitewas age estimates of metal-weakGalacticglobular clusters 0 In conjunctionwiththeGalaxy formationthoeryof © Royal Astronomical Society • Provided by the NASA Astrophysics Data System the solidline. Figure 14.The[0/H]-agerelation.modelisrepresentedby by twoquitedifferentapproaches. 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