Binaries and Globular Cluster Dynamics

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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aBINARIESANDGLOBULARCLUSTERDYNAMICSFredericA.Rasio,JohnM.Fregeau,&KritenJ.JoshiDeptofPhysics,MIT,Cambridge,MA02139,USAAbstractWesummarizetheresultsofrecenttheoreticalworkonthedynami-calevolutionofglobularclusterscontainingprimordialbinaries.Evenaverysmallinitialbinaryfraction(e.g.,10%)canplayakeyroleinsupportingaclusteragainstgravothermalcollapseformanyrelaxationtimes.Inelasticencountersbetweenbinariesandsinglestarsorotherbinariesprovideaverysigni cantenergysourceforthecluster.ThesedynamicalinteractionsalsoleadtotheproductionoflargenumbersofexoticsystemssuchasultracompactX-raybinaries,recycledradiopul-sars,doubledegeneratesystems,andbluestragglers.OurworkisbasedonanewparallelsupercomputercodeimplementingH´enon’sMonteCarlomethodforsimulatingthedynamicalevolutionofdensestellarsystemsintheFokker-Planckapproximation.Thisnewcodeallowsustocalculateveryaccuratelytheevolutionofaclustercontainingare-alisticnumberofstars(N～105 106)intypicallyafewhourstoafewdaysofcomputingtime.Thediscrete,star-by-starrepresentationoftheclusterinthesimulationmakesitpossibletotreatnaturallyanumberofimportantprocesses,includingsingleandbinarystarevolu-tion,alldynamicalinteractionsofsinglestarsandbinaries,andtidalinteractionswiththeGalaxy.

1.INTRODUCTION

Thedynamicalevolutionofdensestarclustersisaproblemoffun-damentalimportanceintheoreticalastrophysics,butmanyaspectsoftheproblemhaveremainedunresolvedinspiteofyearsofnumericalworkandimprovedobservationaldata(seeMeylan&Heggie1997forarecentreview).Therealizationoverthelast10yearsthatprimordialbinariesarepresentinglobularclustersindynamicallysigni cantnum-bershascompletelychangedourtheoreticalperspectiveonthesesystems(see,e.g.,Gaoetal.1991;Hutetal.1992;Sigurdsson&Phinney1995).Mostimportantly,dynamicalinteractionsbetweenhardprimordialbina-riesandothersinglestarsorbinariesarenowthoughttobetheprimarymechanismforsupportingaglobularclusteragainstgravothermalcon-tractionandavoidingcorecollapse.Inaddition,exchangeinteractions

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We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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betweenprimordialbinariesandcompactobjectscanexplainverynatu-rallytheformationoflargenumbersofX-raybinariesandrecycledpul-sarsinglobularclustercores(see,e.g.,Camiloetal.2000).Dynamicalinteractionsinvolvingprimordialbinariescanalsoresultindramaticallyincreasedcollisionratesinglobularclusters.Thisisbecausetheinterac-tionsareoftenresonant,withallthestarsinvolvedremainingtogetherinasmallvolumeforalongtime(～102 103orbitaltimes).Forexam-ple,inthecaseofaninteractionbetweentwotypicalhardbinarieswithsemi-majoraxes～1AUcontaining～1M⊙main-sequencestars,thee ectivecrosssectionforadirectcollisionbetweenanytwoofthefourstarsinvolvedisessentiallyequaltotheentiregeometriccrosssectionofthebinaries(Bacon,Sigurdsson&Davies1996;Cheung,PortegiesZwart,&Rasio2001).Thisimpliesacollisionrate～100timeslargerthanforsinglestars.Directobservationalevidenceforstellarcollisionsandmergersinglobularclusterscomesfromthedetectionoflargenum-bersofbluestragglersconcentratedinthedenseclustercores(see,e.g.,Bailyn1995).

2.MONTECARLOSIMULATIONSOF

CLUSTERDYNAMICS

The rstMonteCarlomethodsforcalculatingthedynamicalevolu-tionofstarclustersintheFokker-Planckapproximationweredevelopedmorethan30yearsago.Theywere rstusedtostudythedevelopmentofthegravothermalinstability(H´enon1971a,b;Spitzer&Hart1971a,b).MorerecentimplementationshaveestablishedtheMonteCarlomethodasanimportantalternativetodirectN-bodyintegrations(seeSpitzer1987foranoverview).ThemainmotivationforourrecentworkatMITwasourrealizationafewyearsagothatthelatestgenerationofparallelsupercomputersnowmakeitpossibletoperformMonteCarlosimula-tionsforanumberofobjectsequaltotheactualnumberofstarsinaglobularcluster(incontrast,earlierworkwaslimitedtousingasmallnumberofrepresentative“superstars,”andwasoftenplaguedbyhighlevelsofnumericalnoise).Therefore,theMonteCarlomethodallowsustodorightnowwhatremainsanelusivegoalforN-bodysimula-tions(see,e.g.,Aarseth1999):performrealistic,http://doc.guandang.netingthecorrectnumberofstarsinadynamicalsimulationensuresthattherelativeratesofdif-ferentdynamicalprocesses(whichallscaledi erentlywiththenumberofstars)arecorrect.Thisisparticularlycrucialifmanydi erentdy-namicalprocessesaretobeincorporated,asmustbedoneinrealisticsimulations.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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OurimplementationoftheMonteCarlomethodisdescribedindetailinthepapersbyJoshi,Rasio,&PortegiesZwart(2000),Joshi,Nave,&Rasio(2001),andJoshi,PortegiesZwart,&Rasio(2001).Weadopttheusualassumptionsofsphericalsymmetry(witha2Dphasespacedistri-butionfunctionf(E,J),i.e.,wedonotassumeisotropy)andstandardtwo-bodyrelaxationintheweakscatteringlimit(Fokker-Planckapprox-imation).Initssimplestversion,ourcodecomputesthedynamicalevo-lutionofaself-gravitatingsphericalclusterofNpointmasseswhoseorbitsintheclusterarespeci edbyanenergyEandangularmomen-tumJ,withperturbations Eand Jevaluatedonatimestepthatisafractionofthelocaltwo-bodyrelaxationtime.Theclusterisassumedtoremainalwaysveryclosetodynamicalequilibrium(i.e.,therelaxationtimemustremainmuchlongerthanthedynamicaltime).Ourmainim-provementsoverH´enon’soriginalmethodaretheparallelizationofthebasicalgorithmandthedevelopmentofamoresophisticatedmethodfordeterminingthetimestepsandforcomputingthetwo-bodyrelax-ationfromrepresentativeencountersbetweenneighboringstars.Ournewmethodallowsthetimestepstobemademuchsmallerinordertoresolvethedynamicsintheclustercoremoreaccurately.

Wehaveperformedalargenumberoftestcalculationsandcompar-isonswithdirectN-bodyintegrations,aswellasdirectintegrationsoftheFokker-Planckequationinphasespace,toestablishtheaccuracyofourbasictreatmentoftwo-bodyrelaxation(Joshietal.2000).Fig.1showstheresultsfromatypicalcomparisonbetweenMonteCarloandN-bodysimulations.Fig.2showstheresultsobtainedwithourcodeforHeggie’sCollaborativeExperimentrun(Heggieetal.1998).

3.SUMMARYOFRECENTRESULTS

OurrecentworkhasfocusedontheadditionofmorerealisticstellarandbinaryprocessestothebasicMonteCarlocode,aswellasasimplebutaccurateimplementationofastatictidalboundaryintheGalac-tic eld(Joshietal.2001a).Asa rstapplication,wehavestudiedthedependenceoninitialconditionsofglobularclusterlifetimesintheGalacticenvironment.AsinpreviousFokker-Planckstudies(Cherno &Weinberg1990;Takahashi&PortegiesZwart1998),weincludethee ectsofapower-lawinitialmassfunction(IMF),masslossthroughatidalboundary,andsinglestarevolution,andweconsiderinitialKingmodelswithvaryingcentralconcentrations.We ndthatthedisrup-tionandcore-collapsetimesofourmodelsaresigni cantlylongerthanthoseobtainedwithprevious1D(isotropic)Fokker-Planckcalculations,butagreewellwithmorerecentresultsfromdirectN-bodysimulations

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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Figure1EvolutionoftheLagrangeradiiforanisolated,single-componentPlummermodel(frombottomtotop:radiicontaining0.35%,1%,3.5%,5%,7%,10%,14%,20%,30%,40%,50%,60%,70%,and80%percentofthetotalmassareshownasafunctionoftime,giveninunitsoftheinitialhalf-massrelaxationtime).TheresultsfromadirectN-bodyintegrationwithN=16,384(noisierlines)andfromaMonteCarlointegrationwithN=105stars(smootherlines)arecompared.TheMonteCarlosimulationwascompletedinlessthanadayonaCray/SGIOrigin2000parallelsupercomputer,whiletheN-bodyintegrationranforoveramonthonadedicatedGRAPE-4computer.TheagreementbetweentheN-bodyandMonteCarloresultsisexcellentovertheentirerangeofLagrangeradiiandtime.ThesmalldiscrepancyintheouterLagrangeradiiiscausedmainlybyadi erenttreatmentofescapingstarsinthetwomodels.IntheMonteCarlomodel,escapingstarsareremovedfromthesimulationandthereforenotincludedinthedeterminationoftheLagrangeradii,whereasintheN-bodymodelescapingstarsarenotremoved.NotealsothattheMonteCarlosimulationisterminatedatcorecollapse,whiletheN-bodysimulationcontinuesbeyondcorecollapse.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

BinariesandGlobularClusterDynamics

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Figure2CorecollapsetimetccandclustermassatcorecollapseMccascomputedbymanydi erentnumericalcodesinHeggie’sCollaborativeExperiment,andwithourMonteCarlocode.Thesolidrounddots,trianglesandsquaresarefromvarioussetsofdirectN-bodysimulations.OpensymbolsarefromFokker-Plancksimulationsandthestarsarefromanisotropicgasmodels.Ourdatapointisindicatedbytheplussymbol,correspondingtoacorecollapsetimetcc=12.86GyrandamassatcorecollapseMcc=4.73×104M⊙.TheinitialconditionisaKingmodelwithadimensionlesscentralpotentialW0=3,atidalradiusrt=30pc,andcontainingamassM=6×104M⊙.Theclustercontainssinglestarsonlywithapower-lawIMFofslope 2.35(Salpetermassfunction)between0.1M⊙and1.5M⊙.Allsimulationsaredoneassumingnostellarevolution.Heatingby“3-bodybinaries”isnotincludedinourMonteCarlosimulation(thisonlya ectstheevolutionbeyondcorecollapse).SeeHeggieetal.1998formoredetails.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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and2DFokker-Planckintegrations.Inagreementwithpreviousstud-ies,ourresultsshowthatthedirectmasslossduetostellarevolutioncausesmostclusterswithalowinitialcentralconcentrationtodisruptquicklyintheGalactictidal eld.Thedisruptionisparticularlyrapidforclusterswitharelatively atIMF.Onlyclustersbornwithhighcen-tralconcentrationsorwithverysteepIMFsarelikelytosurvivetothepresentandundergocorecollapse.

Inanotherrecentstudy,wehaveusedourMonteCarlocodetoex-aminethedevelopmentoftheSpitzer“massstrati cationinstability”insimpletwo-componentclusters(Watters,Joshi,&Rasio2000).Wehaveperformedalargenumberofdynamicalsimulationsforstarclusterscon-tainingtwostellarpopulationswithinpidualmassesm1andm2>m1,andtotalmassesM1andM2<M1.WeusebothKingandPlummermodelinitialconditionsandweperformsimulationsforawiderangeofinpidualandtotalmassratios,m2/m1andM2/M1,inordertodeter-minethepreciselocationofthestabilityboundaryinthis2Dparameterspace.AspredictedoriginallybySpitzer(1969)usingsimpleanalyticarguments,we ndthatunstablesystemsneverreachenergyequipar-tition,andaredriventorapidcorecollapsebytheheaviercomponent.Theseresultshaveimportantimplicationsforthedynamicalevolutionofanypopulationofprimordialblackholesorneutronstarsinglobularclusters.Inparticular,primordialblackholeswithm2/m1～10areexpectedtoundergoveryrapidcorecollapseindependentoftheback-groundcluster,andtobeejectedfromtheclusterthroughdynamicalinteractionsbetweensingleandbinaryblackholes(seePortegiesZwart&McMillan2000andreferencestherein).WehavealsousedMonteCarlosimulationsofsimpletwo-componentsystemstostudytheevapo-ration(orretention)oflow-massobjectsinglobularclusters,motivatedbythesurprisingrecentobservationsofplanetsandbrowndwarfsinseveralclusters(Fregeauetal.2001).

Muchofourcurrentworkconcernsthetreatmentofdynamicalinter-actionswithprimordialbinaries.Weareintheprocessofcompletinga rststudyofglobularclusterevolutionwithprimordialbinaries(Joshietal.2001b),basedonthesamesetofapproximatecrosssectionsandrecipesfordynamicalinteractionsusedintheFokker-PlancksimulationsofGaoetal.(1991).TypicalresultsareillustratedinFigs.3–6.Theheatingoftheclustercoregeneratedbyasmallpopulationofprimor-dialbinariescansupporttheclusteragainstcorecollapseforverylongtimes,althoughthedetailsoftheevolutiondependsensitivelybothonthenumberofstarsNandontheinitialbinaryfractionfb.ClusterswithsmallerNtendtoevolvefastertowardsdeepercorecollapse(com-pareFigs.3and4,andnotethatthemaindependenceonNthrough

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

BinariesandGlobularClusterDynamics

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Figure3ResultsofaMonteCarlosimulationfortheevolutionofanisolatedPlum-mermodelcontainingN=3×105equal-massstars,with10%ofthestarsinprimor-dialbinaries.Thebinariesareinitiallydistributeduniformlythroughoutthecluster,andwithauniformdistributioninthelogarithmofthebindingenergy(roughlybe-tweencontactandthehard-softboundary,i.e.,nosoftbinariesareincluded).Thesimulationincludesatreatmentofenergyproductionandbinarydestructionthroughbinary-singleandbinary-binaryinteractions.StellarevolutionandtidalinteractionswiththeGalaxyarenotincluded.Timeisgiveninunitsoftheinitialhalf-massrelaxationtimetrh.Theupperpanelshowstheevolutionofthetotalmass(ornum-ber)ofbinaries.Thelowerpanelshows,fromtoptobottominitially,thehalf-massradiusoftheentirecluster,thehalf-massradiusofthebinaries,andtheclustercoreradius.Thesequantitiesareinunitsofthevirialradiusofthecluster.Notethelong,quasi-equilibriumphaseof“binaryburning”lastinguntilt 60trh,followedbyabriefepisodeofcorecontractionandre-expansiontoanevenlongerquasi-equilibriumphasewithanevenlargercore.Byt～100trh,onlyabout15%oftheinitialpop-ulationofbinariesremainsinthecluster,butthisisenoughtosupporttheclusteragainstcorecollapseforanother～100trh.Formostglobularclusters,trh～109yr,andthisiswellbeyondaHubbletime.Theevolutionshownhereshouldbecontrastedtothatofanidenticalcluster,butcontainingsinglestarsonly(Fig.1),wherecorecollapseisreachedatt 15trh.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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Figure4SameasFig.3butforN=105equal-massstarswith10%binaries.Notethesomewhatfasteranddeeperinitialcontraction,butstillfollowedbyre-expansionintoalong-lived,quasi-equilibriumphaseofbinaryburning.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

BinariesandGlobularClusterDynamics

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Figure5SameasFig.3butforN=105equal-massstarswithonly1%binaries.Heretheinitialcorecontractionisfollowedbymorerapidgravothermaloscillations,butthetime-averagedcoresizeremainssimilartowhatisseeninFigs.3and4.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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Figure6Evolutionofthehalf-massradiustocoreradiusratioforthethreecasesillustratedinFigs.3–5.Forcomparison,mostobservedglobularclusterswithresolvedcoreshaverh/rc 2 10.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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trh∝N/logNhasbeenscaledout),andsodoclusterswithsmallerini-tialbinaryfractions(Fig.5).Aftertheinitialcorecollapse,gravother-maloscillationspoweredbyprimordialbinariescancontinueforverylongtimes～50 100trhevenwithaninitialbinaryfractionassmallas1%.Withaninitialbinaryfractionof10%,weobserveasingle,verymoderatephaseofcorecollapseduringwhichthecoreshrinksbyafac-tor～5 10inradiusandthenre-expandsrapidlyonatimescaleofafewcentralrelaxationtimes(thisoccursatt/trh 60inFig.3andatt/trh 20inFig.4).Theclusterthenentersasecondquasi-equilibriumphaseofprimordialbinaryburningwiththecoreradiusincreasingslowlyuntilwellbeyond～100trh.

Sincetrh～109yrformostclusters,ifthetypeofevolutionillus-tratedinFig.3appliedtoallglobularclusters,therewouldbeno“core-collapsed”clustersintheGalaxy(10–15%ofallGalacticglobu-larclustersareclassi edobservationallyas“corecollapsed”).However,thetimescaleonwhichrealclusterswillexhausttheirprimordialbi-narysupplyandundergo(deeper)corecollapsedependsonanumberoffactorsnotconsideredhere,includingtheorbitoftheclusterintheGalaxy(thesimulationsareforanisolatedcluster,butmasslossandtidalshockingcanacceleratetheevolutiondramatically),andthestellarIMF(theclustersshownherecontainallequal-massstarsandbinaries;amorerealisticmassspectrumwillalsoacceleratetheevolution).Inaddi-tion,someclustersmaybeformedwithmuchfewerprimordialbinariesthaneventhe1%consideredinFig.5.However,thesimplepicturethatemergesfromthesesimulationsmaywell,to rstapproximation,describethedynamicalstateofmostGalacticglobularclustersobservedtoday.Indeed,foraclusterinthestable“binaryburning”phase,theratioofhalf-massradiustocoreradiusrh/rc～2 10(Fig.6),whichispreciselytherangeofvaluesobservedforthe～80%ofglobularclustersthathaveawell-resolvedcoreandarewell- ttedbyKingmodels(see,e.g.,Djorgovski1993).Someoftheseclustersmayhavegoneinthepastthroughabriefepisodeof“moderatecorecollapse”(asshownaroundt 60trhinFig.3).Yet,theyshouldnotbecalled“core-collapsed”(norwouldtheybeclassi edassuchbyobservers).Unfortunatelysometheoristswillevencall“core-collapsed”clustersthathavejustreachedtheinitialphaseofbinaryburning(t 10 50trhinFig.3).Sincethecorehasjustbarelycontractedbyafactor～2 3bythetimeitreachesthisphase,itseemshardlyjusti edtospeakofa“collapsed”state.TheadditionofbinarystellarevolutionprocessestooursimulationswillallowustostudyindetailthedynamicalformationmechanismsformanyexoticobjectssuchasX-raybinariesandmillisecondradiopul-sars,whichhavebeendetectedinlargenumbersinglobularclusters.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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Forexample,exchangeinteractionsbetweenneutronstarsandprimor-dialbinariescanleadtocommon-envelopesystemsandtheformationofshort-periodneutron-star/white-dwarfbinariesthatcanbecomevisiblebothasultracompactX-raybinariesandbinarymillisecondpulsarswithlow-masscompanions(see,e.g.,Camiloetal.2000,onobservationsof20suchmillisecondradiopulsarsin47Tuc).Rasio,Pfahl,&Rappaport(2000)presentapreliminarystudyofthisformationscenario,basedonsimpli eddynamicalMonteCarlosimulations.Inadenseclustersuchas47Tuc,mostneutronstarsacquirebinarycompanionsthroughex-changeinteractionswithprimordialbinaries.Theresultingsystemshavesemimajoraxesintherange～0.1 1AUandneutronstarcompanionmasses～1 3M⊙.Formanyofthesesystemsitisfoundthat,whenthecompanionevolveso themainsequenceand llsitsRochelobe,thesubsequentmasstransferisdynamicallyunstable.Thisleadstoacommonenvelopephaseandtheformationofshort-periodneutron-star/white-dwarfbinaries.Forasigni cantfractionofthesebinaries,thedecayoftheorbitduetogravitationalradiationwillbefollowedbyaperiodofstablemasstransferdrivenbyacombinationofgravitationalradiationandtidalheatingofthecompanion.Thepropertiesofthere-sultingshort-periodbinariesmatchwellthoseofobservedbinarypulsarsin47Tuc(Fig.7).AsimilardynamicalscenarioinvolvingmassiveCOwhitedwarfsinplaceofneutronstarscouldexplaintherecentdetectionsofdoubledegeneratebinariescontainingHewhitedwarfsconcentratedinthecoreofadenseglobularcluster(Hansenetal.2001;seeEdmondsetal.1999andTayloretal.2001onobservationsofHewhitedwarfsinthecoreofNGC6397).

WearealsocurrentlyworkingonincorporatingintoourMonteCarlosimulationsamorerealistictreatmentoftidalinteractions,and,inpar-ticular,tidalshockingthroughtheGalacticdisk(basedonGnedin,Lee,&Ostriker1999).Tidalshockscanacceleratesigni cantlybothcorecol-lapseandtheevaporationofglobularclusters,reducingtheirlifetimesintheGalaxy(Gnedin&Ostriker1997).Futureworkmayincludeafullydynamicaltreatmentofallstrongbinary-singleandbinary-binaryinteractions(exploitingtheparallelismofthecodetoperformseparatenumerical3-or4-bodyintegrationsforalldynamicalinteractions)aswellasafullydynamicaltreatmentoftidalshocking(performingshort,N-bodyintegrationsforeachpassageoftheclusterthroughtheGalacticdiskornearthebulge).

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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Figure7ResultsofourinitialMonteCarlostudyofbinarymillisecondpulsarfor-mationinadenseglobularclustersuchas47Tuc.Eachsmalldotrepresentsabinarysysteminoursimulation,whilethe lledcirclesarethe10binarypulsarsin47Tucwithwellmeasuredorbits(theerrorbarsextendfromtheminimumcompanionmasstothe90%probabilitylevelforrandominclinations).Thereare3principalgroupsofsimulatedbinaries.Systemsinthediagonalbandontheleft(A)areneutronstar–whitedwarfbinariesthatdecayedviagravitationalradiationtoveryshortorbitalperiods(～mins),thenevolvedwithmasstransferbackuptolongerperiodsunderthein uenceofbothgravitationalradiationandtidalheating.ThelargegrouplabeledBontherightcontainsneutronstar–whitedwarfbinarieswhichhadinsu cienttimetodecaytoRoche-lobecontactviatheemissionofgravitationalradiation.Theneutronstarsinthisgrouparenotlikelytoberecycledsincetheymaynothaveaccretedmuchmassduringthecommonenvelopephase.Finally,thesystemslyinginthethindiagonalbandtowardlongerperiods(C)arethoseinwhichthemasstransferfromthegiantorsubgianttotheneutronstarintheinitialbinarywouldbestable.Thesehavenotbeenevolvedthroughthemasstransferphase;themassplottedissimplythatoftheHecoreofthedonorstarwhenmasstransfercommences.Therearemanymoresystemsinthiscategorythathavelongerperiodsbutlieo thegraph.SeeRasio,Pfahl,&Rappaport2000formoredetails.

We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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Acknowledgments

ThisworkwassupportedinpartbyNSFGrantAST-9618116andNASAATPGrantNAG5-8460.OurcomputationswereperformedontheCray/SGIOrigin2000supercomputeratBostonUniversityunderNCSAGrantAST970022N.References

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We summarize the results of recent theoretical work on the dynamical evolution of globular clusters containing primordial binaries. Even a very small initial binary fraction (e.g., 10%) can play a key role in supporting a cluster against gravothermal colla

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