19 66ApJ. . .143. .400S 46 6 45 93 8 relativistic particles,asinsupernovae.Stellarcollisions asanenergysourceingalactic radio galaxiesingeneralandbythequasi-stellar sourcesinparticular.Itisreason- nuclei havealsobeendiscussed recentlybyGold,Axford,andRay(1965),Spitzer (1965) account forthevastreleaseofenergyintheseobjects hasbeendiscussedbyWoltjer ably wellestablishedthatthesequasi-stellarsources radiateenergyataratebetween in contractinggalacticnucleimay,perhaps,account fortheenergyfluxradiatedby (1964), whosuggeststhattheshocksproducedby thecollisionsproducealargefluxof of thesystemisthereforesubstantialinterest. of theeffectswhichthesecollisionsproduceandtheir effectsonthecontinuingevolution 1965; Schmidt1965).Thepossibilitythatcolliding starsindensegalacticnucleimight 10 andergspersec,perhapsmaintainthis rateforsome10years(Burbidge time reachastageinwhichdirectstellarcollisions play adominantrole,aninvestigation apparently allstellaraggregationswithsufficientlylowangularmomentummustin inevitably toanincreasingnumberofcollisionsbetweenthestarsincluster.Since stellar density,itwouldseemthatthisprocessofacceleratingcontractionmustlead system hasenoughangularmomentumtoinhibititscontractionatarelativelylow tion oftheclusterandtoanaccelerationrateevaporation.Unlessstellar many fascinatingphysicalproblems.Itwaspointedoutseveraldecadesago(Ambart- These starswillleavethesystem,andthisprocessof“evaporation”leadstoacontrac- through randomcollisionsavelocityexceedingtheescapefromsystem. sumian 1938;Spitzer1940)thatinanisolatedclustersomestarswilloccasionallygain as yettothedegreeofcompactnessrequiredaccountforthesehighlyluminousobjects. those observedfromquasi-stellarradiosources,butitisuncertainwhetherexistingnucleihaveevolved flatter one,withamuchreducedluminosity.Theenergiesradiatedattheluminositypeakaresimilarto remain there,withrelativelylowrandomvelocities,andthesphericalsystemwillbetransformedintoa has adurationofabout1,000,000years.Forlatertimesthenewstarsformingindiskwilltendto yields apeakluminosity,whichisreachedatthisphase,aboutequalto10ergs/secforsystemwith will contractatanacceleratedrateuntilthetimeofrelaxationbecomessubstantiallylongerthan normal starsareproduced.Thesenewwillabsorbkineticenergyfromtheinnucleus,which falls towardthecenterofsystem.Ifanyangularmomentumispresent,gaswillformaflatdisk, liberation ofsubstantialamountsgas.Themasslosspercollision,f,insolarunits,isevaluatedona time requiredforastartolosethebulkofitsmassbycollisions.Theenergylostliberatedgas the diskwillexceedabout10atoms/cm,andthicknessbeafewhundredastronomicalunits. whose densitywillincreaseuntilgravitationalinstabilitysetsin.Fortypicalparameters,thein to about0.10forvequal4400km/sec.Thegasgivenoffincollisionscoolsrapidlybyradiationand fluid interactionssuchasshockswhichtransmitenergyandmomentumtransversetotherelativevelocity, 10 starsofsolarmass;thesystemradiusisabout0.05pc.Theluminositypeakforassumed very simplemodelconsideringonlytheenergyandmomentumofcollidingmasselements,neglecting system. Asthesystemcontracts,collisionsamongstarsbecomerelativelymorefrequent,leadingto Gravitational instabilitywillleadtotheformationofcondensationsindisk,andweassumethat considered inthispreliminaryanalysis.Thefirstphaseisthefamiliarcontractionproducedbyevapora- tion, themoreenergeticstarsescapingintoelongatedorbitsinwhichtheyareessentiallylostto V; meanvaluesof£rangefromabout0.04forarootsquarestellarvelocity,v1300km/secup s Sy © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Additional interestinthisinvestigationarisesfrom thesuspicionthatstellarcollisions The evolutionofanisolated,nearlysphericalaggregationorclusterstarsposes Successive evolutionaryphasesinthelifeofacompact,nearlysphericalgalacticnucleusorclusterare ON THEEVOLUTIONOFGALACTICNUCLEI Lyman Spitzer,Jr.,andWilliamC.Saslaw Princeton UniversityObservatory Received July2,1965 ABSTRACT 400 19 66ApJ. . .143. .400S -3 56 10 7 8 hot gasliberatedfromthecollidingstars.Itshouldbeemphasizedthatmuchof n isthenumberofstarsper unitvolume,tnthemeanstellarmass,andv root-mean- shall callthe“relaxationtime”withincluster, andwhichisgivenby cusses thelatephasesofevolutiongalacticnucleusandtotalluminosity a collisionistreatedinthethirdsection,whichtreatsrateofcooling,collapse stellar system,withparticularemphasisonthemoremassiveaggregationsatcenters where Nisthetotalnumberofstarsincluster, andTVsareferencetime,whichwe to delineatethecomplexphysicalprocessesoccurringincompactstellarsystems. analysis presentedhereistentativeandexploratory.Muchadditionalstudyrequired toward thegalacticcenter,andofstarcondensationfromthisgas.Thefinalsectiondis- pears tobeanaturalandreasonablyunavoidablestageintheevolutionofmassive, planation ofanyobservedphenomenon.Insteadweshallattempttoanalyzewhatap- us hasyetreachedsuchhighvaluesisconjectural. magnitude greaterthanthesevalues.Whetherthedensityinanyexistinggalacticnucle- stars pc.Toproducebystellarcollisionsthelargeradiativefluxesobservedfromquasi- star densitiesinthesevariouscompactsystemsaretherangeofatleast10to effects tobeexpectedwhentwostarscollide.Thegasejectedfromtheduring stage atthepresenttime. not infactbereachedmostgalacticnucleiuntillongafterstarshaveexhausted nearly sphericalsystemsingeneralandofgalacticnucleiparticular.Thisstagemay stellar radiosources,themaximumstardensitymustbeassumedseveralordersof mass of10Mo,whiletheupperlimitondiameteris100pc,Thecorresponding prominent Seyfertgalaxy,thelightfromapparentlystellarnucleuscorrespondstoa with adiameterofatmost2pc,andmassestimated10Mo.InNGC4151, rather scant.AtthecenterofM31isanucleuswhoseluminositycorrespondsto10Lo, square velocityinthreedimensions; Gisthegravitationalconstant. workers—see therecentsurveybyMichie(1964).Therealphenomenonisremarkably all ofthesamemassisgivenby tional potentialbyasquarewell.Onthisbasisthe rate ofescapefromaclusterwithstars the rateofstellarcollisionsinadenseclusterandanalyzes,onverysimplemodel, the cluster,andeffectsproducedbyaninitialangularmomentum.SectionIItreats of galaxies.Thefirstsectionreviewstherateevaporation,resultingevolution and whosediameterdoesnotexceedabout5pc.InM32thereisanevensmallernucleus, collisions. time, thereisnoquestionbutthatlargeamountsofenergywouldbereleasedinstellar and vandenBergh(1965).Ifdensegalacticnucleiareassumedtoexistatthepresent complex. OnesimpleapproximationisthatofSpitzerandHärm(1958),whoconsider the veryluminousradiosourcesmaybegalacticnucleiwhicharegoingthroughthis their nuclearfuelandgrowndim.Ontheotherhand,itispossiblethatatleastsomeof the exactequationfordiffusioninvelocityspacebut whoapproximatetheactualgravita- 8 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Our approachhereisnottoproposethatcollidingstarsingalacticnucleiareanex- The observationalevidencefortheexistenceofsuchdensenuclei,however,isstill The rateofevaporationfromasphericalstellaraggregationhasbeenstudiedbymany The followingsectionsdiscussvariousphasesintheevolutionofanearlyspherical I. STAREVAPORATIONANDEARLYEVOLUTION EVOLUTION OFGALACTICNUCLEI401 r= mi/2h. (2) D R27 3TrGmnIn(A/2)’ N dt887V 1 dN g (i) 19 66ApJ. . .143. .400S 7/2 2 processes whichhavebeen consideredinthisconnection,andshowntobenegligible, are are presentwhichreduce thisenergy,thenthecontractionwillbemore rapid.Two tidal interactionsbetween starsandinteractionsbetweenthe interstellar densed regionsofthesphericalaggregation. likely. Inanycase,theeffectonmostofstars makingupthesystemismuch gives ineithercasetherateatwhichstarsleave bulkofthesystem.Thenequation force oftheGalaxywilldisruptmoreelongated orbitsandleadtoactualescapeof for escapetoinfinity,thestarwillspendanincreasingfractionofitstimefarfrom stars, aspointedoutbyvonHoerner(1957),butfor agalacticnucleusthisresultisless tended orbits,farfromthecenterofsystem,and weshallassumethatequation(1) same whetherthemoreenergeticstarsescapeentirely orwhethertheycollectinex- stellar velocitieschangebydiffusioninvelocity space. Forglobularclustersthetidal center ofthesystem,andrateatwhichitsenergychangesbecomessmaller out byHénon(1960),asthetotalenergyofastarapproacheszero,valuerequired present exploratorydiscussionequations(1),(2),and(6)areprobablysufficientlyac- smaller; henceinanisolatedclusternostarsever entirely escape,ifweassumethatthe out fromthesystembutwhichcontinuetopassthroughcentralregions.Aspointed where thestardensityishigh,andtoaccumulateinelongatedorbitswhichextendfar In anactualsphericalsystemstarswilltendtogainenergybyencountersnearthecenter, curate. (6) willgivecorrectlythetimedependenceof numberofstarsinthecentrallycon- concentration tothecenterofsystem,variation7äwithIn^Y),etc.For b; 1960),whohasconsideredanumberofrefinementsinthetheory,includingdensity where T^oisthevalueofTßatinitialtime. will varyaboutasN,decreasingrapidlythepopulationofsystemdiminishes. remaining systemwillremainconstant,asthetotalgravitationalandpotential radius. Wethentake With thisvariationassumed,equation(1)canbeintegratedandwefind energies. ThesystemradiusRwillaccordinglyvaryasN,andtherelaxationtime,Tr, increments, anescapingstarwillhavenearlyzeroenergy.Hencethetotalenergyof where R'isthevalueofRexpressedinparsecs. and fromthevirialtheorem 402 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 n intherangefrom0to4(Spitzer1958).Ifequations(3)and(4)aresubstitutedinto expression ofequation(4)givesareasonablycloseapproximationforpolytropeswith where 0isthemeanpotentialenergyperunitmass;constantJinright-hand equation (2),weobtain © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Equation (6)isbasedonaconstanttotalenergyof thecluster.Ifdissipativeprocesses Another effectignoredinthissimplifiedmodelistheanisotropicvelocitydistribution. More preciseversionsofequations(1),(2),and(6)havebeengivenbyKing(1958a, To relatethismodeltoactualstellarsystems,wedefineRastheroot-mean-square Since gravitationalencountersamongthestarschangestellarvelocitiesbysmall 5 = 8.3xio7—r^j7TT7 7 2/7 No VTro)’ 2 N /0.040A fl* =—h 1/2 n =——.O) (w/mo) (logY—0.3) 3, 47^R 3N 7\7-l/2 P'3/2 GNm 1 2R years, (6) (5) (4) 19 66ApJ. . .143. .400S 8 is nofirmphysicalbasis for theseinterestingandpossiblyimportantmechanisms, we shall assumethatequation (6)givesthecorrectrateofevolutionduring earlylifeof the system. ker (1964)havesuggestedthatco-operativeinteractions betweenstarsmightmuch tions betweenstarsandgashasbeenproposedby Spiegel(1964).Sinceatpresentthere enhance therateofevaporation,andasimilarmechanism involvingenhancedinterac- Hence thisprocessalsomaybeneglected. here (seeTable1)itseemsmostunlikelythatany subunitscanbemoretightlybound. more tightlyboundthanthemainsystem,i.e.,unless thestellarvelocitydispersionin the contractionofclustermorerapidlythanis giveninequation(6).MillerandPar- remain thesame.Evidentlythisprocesscannotplay amajorroleunlessthesubunitsare potential energyfromthesubunitstomainsystem. Thetotalkineticenergymust clusters withinthemainsystem,amechanismsuggestedbyGoldetal.(1956).Itisclear equation (8).Weconcludethatthisdissipativemechanismisalsonegligibleincom- the subunitsexceedsthatinmainsystem.Inview ofthelargevaluesvinterest nucleus mightbethedisruptionofsmallerstellarsystems,suchasbinariesandsub- parison withevaporationofstars. that thismechanismdoesnotchangethetotalenergybutredistributesnegative approximation werenotvalid,aswouldbethecaseifgassufficientlyhottoex- tend throughoutthesystem,twouldbeappreciablyincreasedabovevaluegivenin where Mandarethetotalmassesofstarsgas,respectively,insystem. Since thegaswillcool,contract,andcondenseintonewstarswhileMisstillmuchless gas. Theenergygainedbythegas,asstarpassesthrough,isreadilycomputedif impulsive approximationisagainadopted,andweobtainapproximately than thestellarmass,M(see§III),twillbemuchgreaterTr.Ifimpulsive important. longer thantheevaporationtime88Tr.Sinceimpulsiveapproximationprobably tional interactionsbetweenstarsandgas,theenergydissipatedbeingradiatedby overestimates theenergydissipation,weconcludethattidaleffectsarelikelytobeun- value computedfromthesolarmodelgivenbySchwarzschild(1958)with50percentof in theneighborhoodof10,tidaldissipationtime,tisaboutanordermagnitude the originalhydrogenremaining.Comparisonofequations(1)and(7)showsthatforN grazing encounters;moreoverthestellarradiusofgyrationwassetequalto0.31ro, where Uisthetotalenergyand7#relaxationtimedefinedinequation(2)above. equation (7)anintegrationwascarriedoutoverallimpactparametersdowntojust escape velocityfromthesurfaceofeachstar.Inevaluatingnumericalconstantin Equation (7)isvalidonlyiftherelativevelocity,F,substantiallylargerthan for theimpulsiveapproximationcanbeapplieddirectlytopresentsituation.This s obtain radiated awaybeforethenextencounteroccurs.Withstraightforwardcomputationswe procedure assutnesthatthepulsationenergygainedbyastarincloseencounterwillbe by atavelocityF,hasbeenconsideredpreviouslySpitzer(1958),whoseequations medium. Tidaldissipationinastellarsystem,whenperturbedbylargemassmoving g sg g No. 2,1966EVOLUTIONOFGALACTICNUCLEI403 8g h It iscertainlypossiblethatothermechanisms,not consideredhere,mayaccelerate Another mechanismthatmightacceleratethecontractionofaclusterorgalactic © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Similarly, wemaydefinetasthecorrespondingtimefordecreaseofUbygravita- g ‘■'•u m Ma Ms R » (8) 19 66ApJ. . .143. .400S 1/2 2 23 2 cross-section, cr,foracollision betweentwostarsofidenticalradius,r,is given by If tidaldeformationisignored,andacollisiondefined asgrazingcontactorcloser,the where wehavereplacedtherelativevelocity,F,by 2fl,wherevisagaintheroot-mean- square stellarvelocity,andhavesubstitutedfrom equations(3)and(4)fornv. must awaitamoreprecisetheoryofstellardynamics. suggestion seemsunlikely;starsthathaverelativelylittleangularmomentum,and if theescapingstarscarryoffadisproportionatelylargeshareofangularmomentum in thesystem.Inviewofstrongcentralconcentrationmostsphericalsystems,this a verysubstantialdecreaseofRispossibleifesmall. flattened disk.AfactorofJfromequation(4)hasbeenignoredin(11).Fora than starswithhigherangularmomentum.However, detailedinformationonthispoint whose orbitsthereforepassclosetothecenter,mightbeexpectedescapemorerapidly and weobtainfinally proximate forcebalanceequation lip ticityofthesystemwillbeordere.Ifwesubstitutefromequation(4)forvqand Thus if€is0.1or0.01,Rd/Rowillequal5X10~2.5,respectively.Evidently determine ved,themeanrotationalvelocityofdisk,fromcorrespondingap- cluster orgalacticnucleuscontractingbyevaporationwehaveseenthatRvariesasA, In theseequationsNdisthetotalnumberofstarsinsystemwhenitbecomesa we obtainfromequation(9) where vistheinitialvalueofroot-mean-squarestellarvelocity.Evidentlyeequals mean-square value.Weshallemployaverysimpleorder-of-magnitudeargumentto the initialratioofrotationalvelocity,ve,tov.Itmaybenotedthatel- s determine theratiooffinaldiskradius,R^toinitialRq. s stellar systemwillbecomeaflatteneddisk,whoseradiusRdweagaindefineastheroot- rapidly thanthegravitationalforce.Ultimatelythesetwowillnearlybalance,and increases duringthecontraction,andresultantcentrifugalforcewillincreasemore have thesameangularmomentumperunitmassasdoesentiresystem;henceve initial angularmomentum,MJ,whereJisthemomentumperunitmass. Evidently JisaboutequaltoRvq,wherevethemeanrotationalvelocityofsystem. We assumethatasthesystemcontracts,Jremainsconstant;i.e.,escapingstars 404 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 s s0 s © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The meantimebetweencollisionsforanaverage star isgivenby Gold etal.(1965)havepointedoutthatthislimitationofthecontractionwillbeeased A dimensionlessparameter,e,isdefinedas We considernextthelimitationofcontractiontobeexpectedifsystemhasan II. STELLARCOLLISIONSANDLIBERATIONOEGAS c1,2 (U) ny0 2 ’Hi) “2(5-»)T' ! {V\3Gm^ 217/ 72 9.7 X10R' rVrN y =.(15) 3 8 7 8 6 6 3.1X10 2.7X10 3.4X10 3.2X10 4.6X10 2.8X10 TABLE 1 2Gm 2R 14700 1470 0.1 147 6 9 9 9 11 8 9.7 X10 8.6 XIO 5.2 XIO 1.08X10 1.09X10 1.46X10 6GM' V2>(i8) 4700 470 47 10 11 12 10 13 9 2.8X10 2.7 X10 3.1X10 3.4X10 3.5X10 4.6X10 1470 147 14.7 10 years 13 12 15 12 16 11 8.5 XIO 8.6 XIO 1.09 X10 1.08 XIO 1.09 XIO 1.46X10 470 47 100 4.7 405 (16) 19 66ApJ. . .143. .400S 2 escape fromoneofthetwo identicalstars.Thisassumptionshouldberealistic ifMi(A) modify thisresult. escape ifitsthermalenergy perunitmassexceedsfl/2,whereisthe velocityof course, bythetime-dependent gravitationalfield.Weassumethatanymass elementwill the gravitationalfieldsofmovingstars.This problem isverymuchcomplicated,of whether thecollidingmasselementsareliberated, i.e.,towhatextenttheyescapefrom very unlikelythatshockwavesandhydrodynamic disturbancesperpendicularto7can relatively minorone.Amoresubstantialapproximation isinvolvedinthecriterionasto same directionwiththeuniformvelocity. This assumptionwouldseemtobea when onlytheouterlayersoftwostarsareinvolved inagrazingencounter.Itseems from Figure1,thedifferentmasselementsineach starareassumedtobemovinginthe kinetic energyintoheatwhenthecentralregions of thestarsdonotinterpenetrate,and gas liberated.Forexample,thepresentmodelpredicts arelativelysmallconversionof substantial importance,itdoesnotseemlikelytobeadominantfactorintheamountof local densitygradient.Whilethistransversecouplingbetweenmasselementsmayhave energy andmomentumatanglesintermediatebetweentherelativevelocity7 in theinelasticcollisionbetweenthem.Inanactualcollision,shockwaveswillcarry that theresultingtimehistoryofthesetwomassesdependsonlyonenergyliberated œ model isthatthetwomassesnii{A)andMi(B)collideinelasticallywitheachother, integrated overastraightlineextendingthroughstarA.Thecentralassumptionofthe presumably beliberatedbythecollision. Ax andAy, mass ofstarAwithinthetubeisdenotedbyw(^4),whilemAB)denotes square cross-sectionandwithanaxisparallelto7willpassthroughthetwostars; parallel totherelativevelocity7),and(b)fromside(perpendicular7).Atubeof illustrated inFigure1,wherethetwostarsareshown(a)asseenheadon(inadirection not allthemassinvolvedwillbeliberatedifinequality(18)isonlyjustfulfilled,but exceeds 1520km/sec.Sincesomeofthegaswillbeexpelledwithalargepositiveenergy, the sametubeinstarB.Evidently,ifcross-sectionalareaofhasdimensions this inequalityissatisfiedbyasubstantialmargin,mostofthegasintwostarswill tremely simplifiedmodel,basedonconservationofenergyandmomentum.Themodelis approach betweenthecentersoftwostars.Forthispurposeweshallconsideranex- some indicationofhowtheamountmassliberateddependsonç,distanceclosest For twostarsofsolartype,liberationallthegasbecomesenergeticallypossibleifV 496 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 t © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem A numberofotherassumptionsarealsoincluded in thepresentmodel.Asisevident For non-centralcollisionsconditionsaremorecomplicated.Itisdesirabletoobtain mi{Ä) =AxAy/p¿fe,(i9) 19 66ApJ. . .143. .400S 1 with n=3whenVexceeds 1.7fl. the gasinbothstars.This resultmaybecomparedwiththeexact in inequality elements, criteria1and2areinagreement,V mustexceed2vforliberationofall reasonable. substantially exceedflforgasliberationtooccur, theseadditionalapproximationsseem other isignored,andVevaluatedwhenthestars arefarapart.Similarlyweshallset In thesetwocriteriatheincreaseofrelativevelocity asthetwostarsapproacheach corresponding distanceintheabsenceofanyforces betweenthetwostars.SinceVmust q, theactualminimumdistancebetweenstellar centersduringacollision,toÿ,the (17), whichindicatesthat completeliberationisenergeticallypossiblefor apolytrope while fortheliberationofmasselementnti{B) flow ofheatacrosstheinterfacebetweentwoelements.Theconditionthatmass element Mi(A)beliberatednowbecomes of gravitythetwomasselementsismotionless.Thisassumptionequivalenttono yields their kineticenergiesbeforeimpact,measuredinthereferenceframewhichcenter if not,thennoneis. If inequality(21)issatisfied,weassumethatallthegasintwoelementsliberated; proportion totheirmasses.TheconditionthatU/[nii(A)+mi(B)\exceedv^/2then We nowconstructtwoalternatecriteriaforliberationofthemasselements,asfollows: inelastic impact,sothatthevelocityofseparationvanishes.Itisreadilyshown ference betweenthekineticenergiesbeforeencounter,whentwomasselements œ are approachingeachotherwitharelativevelocityF,andaftertheencounter,whenthey 0 are movingtogetheratthevelocityoftheircentergravity;weassumeacompletely œ and Mi(B).Thetotalthermalenergyreleased,whichwedenoteby£/,issimplythedif- great aseitherstarseparately. zero energywillbeliberatedifitislocatedatthecenterofgravitytwoseparating to vj,sincethefluidmassproducedbytwocombiningstarshasatwiceas mi(B) forallmasselements,thethermalenergyrequiredliberationshouldbeequal stars. Ontheotherhand,foradirectcentralcollisionwith<7=0,whenMi{A)equals No. 2,1966EVOLUTIONOFGALACTICNUCLEI407 exceed vJ/2certainlyunderestimatestheamountofmassliberated,sinceevengaswith of gravitythetwostars.Forthissituation,requirementthatinternalenergy much exceedsinwhichcasethecenterofmasstwoelementswillbenearly two starscollide,thecollidingmasselementswillbeinitiallymotionlessnearcenter at restwithrespecttostarA.WhenMi(A)equalsntiiß)andonlytheouterlayersof It maybenotedthatforadirectcentralcollision Mi(A)equalsMi(B)forallmass © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 2. TheenergyUisdividedbetweenthetwomasselementsinsameproportionas A finalassumptionisconcernedwiththedivisionofthermalenergybetweenmi(A) 1. TheenergyUisassumeddividedbetweenthemasselementsMi(A)andmi(B)in > > [mi{A)m{B)}^ i mAA)+mAB) V‘ mi(A)+MAB) V’ Mi(A)+mdB) ^V* m AA)^fleo mAB) ^fleo P (20) (22) (23) 19 66ApJ. . .143. .400S values ofpwereweighted bytheirgeometricalcross-section,proportional topdp.Itis evident fromthisfigurethat forbothcriteriathemassloss,f,decreasesvery sharplyas v^/V increasesfrom0to 0.1,andthendecreasesmoreslowlywithafurther increasein p—the distanceofclosestapproachintheabsence ofmutualattraction—anddifferent criterion 2onlythelightermasselementwillbeliberated. Ontheotherhand,forveloci- criterion 2givesthehigherliberationrate. computation thedistanceofclosestapproachwas setequaltotheimpactparameter, is insufficienttoliberatebothelements,butcan thelighterone;inthissituation ties onlyslightlygreaterthanthethermalenergy releasedinthelightermasselement results obtainedwithcriterion2.Forthehighvelocities criterion1leadstoagreater sufficient toliberatebothelements,andoncriterion 1thisprocesscanoccur,whileon amount ofmassliberation.Thisistobeexpected, since thelightermasselementexperi- between thestellarcenters.Thesecurveswerecomputedwithcriterion2,assumingthatnoheatexchange ences thegreatestincreaseinthermalenergy; for thehighvelocitiesenergyis computed withcriterion1fordifferentvaluesofv/V,whileFigure3givessimilar takes placebetweenthecollidingmasselements. between thestellarcenters.Thesecurveswerecomputedwithcriterion1,assumingfullheatexchange between thecollidingmasselements. in solarunits,asafunctionofq,thedistanceclosestapproach;Figure2givesdata the twocenters,whereMi(A)isnearlyequaltoMi(B)becauseofsymmetry.Results given bySchwarzschild(1958)with50percentoftheoriginalhydrogenremaining.In stars withavarietyofvaluesforVandç.ValuesMiwerecomputedfromequation applied numerically,withanIBM7094computer,tocollisionsbetweentwosolar-type the computationsareshowninFigures2-4.2and3givetotalmassliberated, the computationsmostofgasliberatedcomesfromregionequidistantbetween 408 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 œ (19) forAx=Ay10~Vo.Thedensityfunction,p,wastakenfromthesolarmodel © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Figure 4givestheaveragemassloss,f,insolar mass units,foreachv/V.Inthis Fig. 3.—Massofgasliberatedinastellarcollision,asfunctionq,thedistanceclosestapproach Fig. 2.—Massofgasliberatedinastellarcollision,asfunctiong,thedistanceclosestapproach To computetheamountofgasliberatedinnon-centralcollisions,criteria1and2were œ 19 66ApJ. . .143. .400S important. designed toindicateina general waytheconditionsunderwhichsucheffects mightbe velocity, F,ofthetwostarsissufficientlygreat, the temperatureofcollidingmass included inthemachinecomputations,butnumericalimportanceofthiseffectwas evolutionary processconsidered here,andaveryapproximateanalysishas beenused, lision, andtheeffectthesereactionshaveon liberationofgas.Whentherelative factor partiallyresponsibleforthisresultisthat Vmuchexceedinganygasthat for thedashedlinesaregivenonright-handscale,whichisamplifiedtentimes. proach, asafunctionoftheratioi^,escapevelocityfromsurfaceonetwoidenticalstars, preted asthemassliberatedinunitsofstellarmass. for starsalongthemainsequence.Inthismoregeneralcase,however,fmustbeinter- mass losscomputedfromcriterion2is0.044wocorrespondingto2.2percentofthetwo energy ofthegascanbeappreciablyincreasedduring thebriefdurationofcollision. elements willberaisedhighenoughtoproducerapid nuclearburning,andthethermal has itsmomentumchangedmuchbycollisionswill tendtobeliberated.ForFlessthan material remaininginthestarsisrelativelyunchanged bythecollision.Onephysical change. Themomentumcontributionfromeachelementthatwasnotliberated is notliberatedfromstar^4,willleadtoaretardationofAbecauseitsmomentum stellar masses. equal to1850km/sec,andaroot-mean-squarestellarvelocity,vof1300the This enhancementofnuclearreactionsdoesnotappear toplayadominantroleinthe zJqq asomewhatdifferentresultwouldpresumably be obtained,andcollisionswouldtend estimated approximatelyfromtheavailabledata.Therelativechangeinkineticenergy energy releasedwiththeliberatedgas.Hencekineticperunitmassof of eachstarisfoundtobeatmost0.5percent,lessbyanordermagnitudethanthe to F,therelativevelocityoftwostars.Theleft-handscalegivesvaluesfforsolidlines; any twoidenticalmain-sequencestars,sincethedensitydistributionisroughlysimilar to bethemorerealistic.ForvjVequalcorrespondingarelativevelocity,V vjV. Ingeneralweshalltakethevalueoffcomputedwithcriterion2,sincethisseems No. 2,1966EVOLUTIONOFGALACTICNUCLEI409 to slowdownthestarswithoutmuchliberationof gas. s © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We concludethissectionwithadiscussionofnuclear reactionsduringastellarcol- It maybenotedthatFigures2,3,and4areapproximatelyvalidforcollisionsbetween Fig. 4.—Averagemassofgasliberatedinstellarcollisions,averagedoveralldistancesclosestap- The materialinstarAwhichencounterslessmassiveelementsfrombut 19 66ApJ. . .143. .400S 0.005. Themorerecent determination ofthereactionratebyCaughlan andFowler with 50percenthydrogen bymassandwitharatioofcarbon-nitrogento hydrogenof generation ratewasthatgivenbySchwarzschild (1958, eq.[10.12])forthecarboncycle, which istakentobeL/V,whereLthelengthof amasselement.Thenuclearenergy reactions maybeexpectedtodoublethethermal energyintheeffectivecollisiontime, average initialdensityofeachelementbythesubstantially greatermaximumdensity. tion neglectsthecompressionofstellarmaterial bythecollision,butreplaces head-on collisionatdifferentvaluesofV;hereris the distanceofmasselementfrom subsequent totheshock,arepossibleinprinciple. equal tothemaximuminitialdensityintwoidentical masselements.Thisassump- over theinitialdensity;furtherincreasesindensitybyhydrodynamiccompression, shock waveproducedbythecompressiondensitycanincreaseatmostafactor4 the axispassingthroughcentersoftwostars. Inthiscomputationphasbeenset (1962) yieldsavalueequal tothatgivenbySchwarzschildatatemperature somewhat function ofrthedistancefromlinejoiningtwocenters.ForeachvalueF,relativevelocity collision doublestheinternalenergy. of thetwostars,asteriskdenotesconditionsforwhichreleasenuclearenergyduring ^ Fig.5.—Gastemperatureofcollidingelementsafteracentralcollisionbetweentwostars,as symbols havetheirusualmeaning.Thedensity,p,inequation(24)mustbesetequalto where pisthemeanmolecularweight,wmassofunitatomicandother Mi(B% wethenhave y the gasdensityaftercollision,whichmaysubstantiallyexceedinitialdensity.In sumption thatallthekineticenergyofcollidingmasselementsgoesintothermal energy. Ifweconsideronlythecaseofequalmassesintwoelements[mí{A)to 410 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Shown byasterisksinFigure5arethelimitsof centralregioninwhichnuclear Figure 5showsthetemperaturecomputedfromequation (24)asafunctionofrfor The increaseintemperatureproducedbythecollisioncanbecomputedonas- lF2 =|M]4 p+ar(24) 2hmq 19 66ApJ. . .143. .400S 1/2 8 fact someofthegaswillbe expelledatsubstantialvelocities,withtherest not escaping. basis ofthepresentmodel. Themodelissomewhatunrealisticinthisrespect, sincein which justsatisfythecriterion forliberation,theenergyatinfinityshouldbe zeroonthe gas maywellbeejectedentirelyfromthestellarsystem. However,fortheevolutionary would bedesirable. velocity muchexceeds5000km/sec.Evidentlymore refinedcalculationsoftheseeffects high velocitiestheliberationofgasfromstarsmaybeaidedbyanuclearexplosion, generation istoplayanimportantroleinthedynamicsofcollision.Atthesevery model discussedin§IVtherewillberelatively fewcollisionsinwhichtherelative unusual conditionsremainstobeexplored.Ifnuclear explosionscanoccur,theliberated hydrogen-burning cycleinvolvingthelightelements canproceedrapidlyunderthese required toestablishthisconclusion.Theproton-proton reactionisnotsufficientlytem- perature-sensitive toyieldanuclearexplosionunder theseconditions,andwhethera although amoredetailedstudyofthespecificnuclearreactionsoccurringwouldbe tions, relativevelocitiesgreaterthanabout5000km/secareneededifnuclear-energy for theconditionsinTable2. less. TherelativelyslowvariationofthenuclearenergyreleasedwithVresultsfrom dominance oftheradiationterminequation(24),whichleadstoavariationTasF metry inFigure1;forothermasselementsthereleaseofnuclearenergywillberelatively collision time,L/F,tothetotalthermalenergyreleasedbycollision.Theseratios are takenfromtheparticularmasselementnexttomid-pointoflinesym- in Table2,whichgivestheratioofnuclearenergyreleasedduringanassumedeffective of thecarbon,nitrogen,andoxygenisotopesfromassumedvalues.Obviously long lifefortheß-decayprocessesincarboncyclewillchangerelativecomposition present assumptionsgiveonlyacrudeandpreliminaryrepresentationoftheactualsitua- However, thetreatmentisveryapproximateinanycase,sincefortheseshorttimes tion. by some2ordersofmagnitudethantheearliervalue.EvidentlyuseCaughlan- greater thanIO°K;extrapolationtoatemperatureof2.5X10Kgivesratehigher Fowler reactionratewouldlowersomewhatthepositionofasterisksinFigure5. No. 2,1966EVOLUTIONOFGALACTICNUCLEI411 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem As thecollidingstarsseparate,gaswillstreamout into space.Forthosemasselements It isevidentfromTable2that,onthebasisofpresentveryapproximatecalcula- The dependenceofthenuclear-energygenerationrateonrelativevelocityisshown 30000. 26000. 22000. 18000. 14000, 10000, 6000, III. COOLINGANDCOLLAPSEOETHELIBERATEDGAS Velocity V Relative (km/sec) Nuclear EnergyReleasedduringCollision Ratio ofNuclearEnergyReleasedtoThermal 170 133 100 Energy, forDistanceofClosestApproach 43 21 71 6.8 TABLE 2 of StellarCenters,Equalto: 0.4 r/ 13 10 4.5 0.3 7.2 2.6 1.2 3 3 4 4.0X10-2 5.9X10-2 2.6X10-2 2.7X10- 7.7X10- 5.6X10- 1.5X10-2 0.8 r O 19 66ApJ. . .143. .400S 7 2 2 7 the coolingprocessaccelerates rapidly.Asthegascloudcoolsandshrinks, thetem- of thegasremainsconstant untilthedensityrisessufficientlytomakeradiation efficient; typical caseshowninFigure6,takenfromcurvescomputed bySaslaw(1965).Theradius unity, whichhasbeensetequalto0.54.Thelastterm inequation(25)givesthecooling liberated gas,withNequaltothetotalnumberofinterstellargasparticlesinsystem, the initialgasdensityat¿ =0isassumedtobezero.Oncethetemperature startstofall, larly byOvin,doesnotchangethetimescaleappreciably. dT/dt throughuseofthevirialtheorem,whichgives adiabatic heatingofthegasasitcontracts;changeinvolumeisdeterminedfrom and 3kTi/2theirmeanenergyinitiallyafterliberation.Thesecondtermresultsfromthe The firsttermontheright-handsiderepresentsheatinfluxassociatedwith and vistheroot-mean-squareparticlevelocityin the gas,whileXisaconstantoforder where RnandRaretheradiiofgalacticnucleus andofthegascloud,respectively, In anycase,however,someoftheescapingmasselementswillsatisfyliberation in thereferenceframeofstellarsystemescapingmaterialhasatotalpositive criterion byawidemargin,ifissubstantiallylessthanV.Weshallassumeherethat gas cloudwithinagalacticnucleuswasfollowedinnumericalcomputation.Thebasic by free-freetransitions.Considerationoftheradiation emittedbyimpurities,particu- nitrogen, andoxygengroupwillincreasetheradiationratesomewhat. become interstellartheenergywillfinallybeconvertedbackintorandomparticleenergy, stars ismotionless,togetherwithanequalenergyresultingfromthemotionofcenter mass ofJ(F/2)inthereferenceframewherecentergravitytwocolliding 412 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 rises aboveafewprotonspercubiccentimeter,becauseofgasliberationduringcollisions, 4 X10°K,correspondingtoanrmsprotonvelocityof1000km/sec.Asthedensity within asphericalstellarsystemtheamountofgaspresentinequilibriummustberela- perature gaswillinfactcoolrapidly,asshownbySpitzer(1942),whopointedoutthat thermal balanceequationusedis the temperaturewillbegintofallrelativelyrapidly.Radiationbyionsofcarbon, rms atomicvelocitiesareassumedtoequalthoseofthestars.However,high-tem- mixing throughoutthesystemifinterstellardensityisverylow.Iftherewereno either byrandomizationwithinashock,iftheinterstellardensityisappreciable,or rapidly. Instead,itseemsmorelikelythatthehotgaswillexpandnearlyadiabatically, energy asX-raysimmediatelyfollowingthecollision.However,opacityofhot energy andradiationenergy.Milford(1961)hasconsideredtheescapeofthis stars. Itmaybenotedthatthisresultisexactifthegashasapositiveenergyperunit energy perunitmassaboutequaltov/2,whereisthemean-squarevelocityof cooling, thegaswouldhavesamedensitydistributionasstars,sinceinitial converting itsthermalenergyintokineticofsystematicmotion.Afterthegashas gas isveryhigh,anditseemsunlikelythatmuchradiantenergycanleakout time resultingfromfree-freetransitionsisabout6X10/nyears,ifthetemperature tively smallbecauseofthelowgastemperature.Accordingtohisresultscooling of gravitythetwostars.Aswehaveseeninlastsection,ifVexceeded5000km/sec, the positiveenergyofescapinggasmightbemuchincreasedbyanuclearexplosion. g g g s p © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Equation (25)hasbeenintegratednumericallyon acomputer,andtheresultsfor To followthecoolingandresultingcollapseincloserdetail,evolutionofaspherical Immediately followingthecollisionthisenergywillbeinformofrandomkinetic Ul 1}( ¿I-rr —T)12TdV2mJ f dt NatSV3k g Rn Vs v _ ga (26) 19 66ApJ. . .143. .400S 5 is assumedtocoolbyfree-freecollisionsandcontractin hydrostaticequilibrium. Mass atcollapse(Mg) Number ofstarsinnucleus Cooling timeforstaticgas(years). . Collapse time(years) free-free emission)aregivenforahypotheticalstaticgascloudwhosevolumeremains interstellar gaswillcollapsewhenonlyarelativelysmallamountofhasbeenliberated from thestarsinnucleus. values ofthecoolingtime(definedasrequiredforgastocool10°Kby perature dropssorapidlythatthepressuregradientcannotsupportgasandfreefall Radius ofnucleus(pc) cloud isappreciablyacceleratedbythecontraction.Moreover,itevidentthat outside thecontractingcloudwasignored.Forcomparisonwithcollapsetime, total massoftheinterstellargasisgivenatcollapsetime.Throughoutthisseries these arethetimesatwhichcloudvelocitybecomessupersonic.Inaddition, sonic, andthecomputationsstopatthispoint. time asgasisliberatedfromthestars.ItevidentTable3thatcoolingof constant, andequaltothevolumeofnucleus,butwhosemassincreasessteadilywith occurs. Shownonthefigureispointatwhichvelocityofcollapsebecomessuper- computations fwasvariedinaccordancewiththevalueofv/v;gasliberatedbystars No. 2,1966EVOLUTIONOFGALACTICNUCLEI413 œs © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 6.—Massandradiusofacloudgaswithingalactic nucleusasafunctionoftime.Thecloud The timesrequiredforcollapsewithdifferentvaluesofNandR'aregiveninTable3; Times forCoolingandCollapseofGas 4 2 3 2.5X10 2.0X10 5.3X10 9 10 1 TABLE 3 7 1 7 3.3X10 3.1X10 1.7X10 9 10 10 3 3 2 4.1X10 3.2X10 2.3X10 10 10 1 6 3 5 2.5X10 2.0X10 5.5X10 10 10 10 414 LYMAN SPITZER, JR., AND WILLIAM C. SASLAW Vol. 143

As with the contracting stellar system, the presence of an initial angular momentum will terminate the collapse of the cloud, and a flattened disk of gas will form at the center of the galactic nucleus. If we define e as before in equation (9), the ratio of the disk radius, to the initial radius, R0, is given in equation (11). In this equation, N is the number of solar masses within a sphere of radius R. Since the stars do not participate in this collapse, Nd may be set equal to NRdz/Rz, and instead of equation (12) we now have

/4 J?, /2/MY (27) R \ NR J ‘

The amount of contraction of the gas is now much more seriously restricted by angular momentum than for contraction of the stellar system. We turn now to a discussion of the properties of the gaseous disk which may be ex- pected to form at the center of a dense stellar system, in which an appreciable amount of gas is being liberated by collisions. The radius of the disk will be determined by the angu- lar momentum, in accordance with equation (27). The thickness of the disk will be de- termined by the temperature and by the gravitational field transverse to the plane of the disk. If the density of gas is assumed no greater than the local stellar density, then as shown by Spitzer (1942) we have approximately

Zg V g (28) r^Ts1

where zg is the root-mean-square height of the gas above the disk, and R is again the root- mean-square radius of the stellar system; vg and vs are again root-mean-square velocities for the gas and the stars, respectively. Since vg is likely to be less than v8 by a factor of 10-2 to lO-3, the disk is relatively very thin. The density of gas in the disk will tend to increase until gravitational condensation can begin. As shown by Safranov (1960), gravitational instability in a disk rotating with an angular velocity Í2 will not occur unless the gas density pg satisfies the condition

1.5 2 Pa > ti . (29) ttG

2 If the density of stars is uniform throughout the disk, O will equal 47rGps/3, and condi- tion (29) becomes = Pg 2ps * (30)

If the compact stellar system has a density of 107 Mo pc“3, the gas density must rise -15 3 9 3 2 to about 10 gm cm“ or a particle density of 10 cm“ . If zg is 10“ pc, the number of particles per cm2 column through the disk will be 3 X 1025, about the same value as for the atmosphere of the Earth. If the composition were the same as in the Earth’s atmos- phere, the disk would be transparent in the visible but opaque in the infrared. If solid particles form with the same relative abundance as in interstellar space, the optical thickness would be about 104. If the temperature is between 103 and 104 degrees, opacity by excited molecules, atoms, and H“ ions would produce a very large optical depth. Evidently such a gaseous disk resembles more a disembodied planetary or stellar atmos- phere than it does the normal interstellar cloud. In the evolutionary states discussed in the next section, the gas density within the disk will be several orders of magnitude greater than the values discussed here. When pg exceeds 2ps and gravitational instability sets in, condensation is likely to occur very rapidly. It is readily shown that at the high densities under consideration the normal times required for star formation, such as the time of free fall, for example, are

© American Astronomical Society • Provided by the NASA Astrophysics Data System 19 66ApJ. . .143. .400S 8-8 ratio ofthecollisiontime,t,torelaxation Tr.Fromequations(2),(3),(4), dominant, thecorrespondingtimeis2/v,asmay be seendirectlyfromequation(6). required forRtovanishonthebasisofequations (32),(2),and(13).Ifevaporationis where tisdenotedasthe^evolutiontime”of stellarsystem.Inthelimitwhere collisions arethedominantprocess,valueof tv atanytimeistheadditional until thecontractionhasgoneratherfar,thesenewstarswillcometoequipartitionwith (13), and(14)weobtain liberation bycollisions,collapsetoadisk,formationofnewstarswhichthenmixwiththe old starsleavesthestellarmassunchanged,butdecreasestotalenergy.Duringthis in whichthegasliberatedfromstarswillleadtoproductionofnewstars.Since traction ofthegalacticnucleusisnowgivenby time evaporationwillcontinuetodecreasethetotalmassofsystem.Theratecon- disk willriseuntilcondition(30)issatisfied,atwhichtimethecondensationofgasbegins the oldstars,risingoutofdiskinwhichtheywereformed.Evidentlythiscyclegas and weassumethatnewstarsareformed.Astatisticallysteadystatewillbereached, Tr, whichisessentiallytherelaxationtime,smallcomparedtocollisiont, gas whichwillcollectinthediskdiscussedprecedingsection.Thedensitythis accelerating rate.Duringthelaterphasesofthisprocess,stellarcollisionswillliberate important. Asevaporationproceeds,wehaveseenthatthestellarsystemcontractsatan c increased. IfNis10,R'1pcandrequalsre,thenWabout4XTr density n(R)isuniform;ifthecondensationtowardcenterconsidered,Wwillbe where Nisagainthetotalnumberofstarsandwehavesubstitutedfromequation(3)for the diskafterliberationfromstars,apointdiscussedinfollowingsection. ture mayexceedthisvaluebecauseofenergyinfluxfromnewgascontinuallyreaching will be70°ifthesurfacetemperaturesofstarsare5000°K.Theactualgastempera- the dilutionfactorW—thefractionofskyfilledbystars—isgiven responding totheenergydensityofstellarradiation.Nearcenternucleus nucleus. e ev e sumptions wouldgivedifferentresultsforthelaterevolutionaryphasesofgalactic stars, ortobodiesthesizeofplanetssmaller,aggregationsmuchmoremassive the stardensity,n.Theintegralhasbeenevaluatedonassumptionthat sumption thatstarsareformedwithaboutthenormalmassdistribution.Differentas- physical processofgravitationalcondensationmayconceivablybequitedifferentfrom under theseconditions. less thanafewhundredyears.However,itisnotclearwhattypeofsystemwillform than normalstarsisanopenquestion.Weshallheremakethearbitraryifplausibleas- that whichtakesplaceinspiralarms.Whethersuchcondensationwillleadtoordinary c No. 2,1966EVOLUTIONOFGALACTICNUCLEI415 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The evolutionarystateofthecontractinggalacticnucleus willdependprimarilyonthe We considernowthevariousprocessesofevolutionthatoccuraftercollisionsbecome The temperatureofthegasinsuchadiskwillnotbelessthancor- At adensitysomuchgreaterthantheofnormalinterstellarclouds, IV. EFFECTOFCOLLISIONSONEVOLUTION 2 W =Trrj^n(R)dR-~{ÿ,OD R dt88rt74v’ Äc 1 22_ 3/2 4 Tr 321+7UJ ’ t _3\n(N/2)/flco\ c 02 r 3N/r\ (33) 19 66ApJ. . .143. .400S 418 8 2 1/2 2 8 less thanthestellarluminosity ofabout10ergs/secforstarssolar type;ifthe formed inthegaseousdisk.Inadditionsupernova explosionsmayproduceinteresting mass-luminosity ratiofor thestarsinnucleusismuchgreaterthanunity, asseems dynamical consequencesinthedisk.Theluminosity computedforstate1inTable4is for Vsoneartothetheoryin§IIbeginsbeinvalid. row ofTable4.Thevaluevinthethirdcolumnisobtainedfromequation(33)with extrapolated tohighvaluesofv/V.Thisextrapolationisevidentlyquiteuncertain,and released. Ifequations(4)and(16)aresubstituted intoequation(34),weobtain N setequalto10,andtheescapevelocity,620km/sec,valuefor end whentisaboutequaltoTr;thesystemparametersinthisstatearegivenfirst An amountJvperunitmassliberatedislostby thecoolingofgas,andweassume 2 v,andthevalueoffinfifthcolumnisobtainedfromlowercurveFigure4, Sun. Inthenextcolumnrmsrelativevelocity,V,betweentwostarsissetequalto with resultingcontractionofthecluster,alreadydiscussedin§I.Thisphasecomestoan that anequalamountislostasthegasfallsinto disk,radiatingthepotentialenergy pends essentiallyonvalone. will beproportionalto\/N.Forconvenienceallstarsareassumedhavethesolarmass. properties ofthestellarsystematsuccessiveevolutionarystates,eachcharacterizedby pendent onIV,thenumberofstarsinsystem,andisnotdirectlydependent where istheescapevelocityfromsurfaceofstars,assumedalltobeidentical. cuss thesuccessivephasesinevolutionofgalacticnucleusandshallcompute cluster radius,R.Sincefalsodependsonlyonv/v,thesolutionofequation(32)de- differ muchfromthoseinTable4.Thevalueoftheevolutiontime,4v,andtTr also, willvaryasN,whilethesystemradius,R,N,andluminosity,L, This ratiodependsprimarilyontherandomstellarvelocity,vjislogarithmicallyde- a particularvalueofI/Tr.Thenumericalvaluesthevariousphysicalquantitiesfor For asystemwithdifferentN,butthesameU/Tr,valuesofvandfwillnot each statewillbecomputedforasystemwithJV,thetotalnumberofstars,equalto10. 416 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 s œ Tr, issmallcomparedwiththecollisiontime,t.Thisphaseofstellarevaporation, c 8 s s œs c S c s c © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem This luminositymaybeincreasedsignificantlyby theradiationfromnewstars The luminosityL,ortotalpowerradiatedbytheliberatedgas,isgiven The firstevolutionaryphaseofagalacticnucleusisthatinwhichtherelaxationtime, We shallnotattempthereadetailedsolutionofequation(32)butinsteaddis- State 8 t/T cR Evolutionary StatesofaGalacticNucleuswith10Suns r/3 1.0 r u L =1.41X106/2¿ergs/sec. 05) WQfo (km/sec) 2100 1650 750 Vs Voo/V 0.21 0.58 .27 7725/2 TABLE 4 fiV mr 2 fl* ÇM 0.066 0.023 tc * .056 6 5 9 3.4X10 6.5X10 1.1X10 tw (years) R' (pc) 0.049 0.39 .080 (ergs/sec) 42.8 43.8: 39.3 log L (34) 19 66ApJ. . .143. .400S 4 3/4 42 3 collisions andformationof newstarswilltendtoincreaseNanddecreasew, increasingL attracting massneartothe nuclearcenterincreases.Inthesecondplace, the continual will tendtomoveintowardthemiddleandincrease theirrandomvelocitiesasthe potential becomesmorehighlynegativethere,and visincreased.Eventheoldstars preciable amountofthesystemmassbeginsto collect inthedisk,gravitational paper. actual luminositywillexceedlOergs/secfortwo reasons.Inthefirstplace,asanap- magnitude higherthaninstate2.Theevolutiontime isnowsubstantiallyshorter.The maximum velocitydispersionreachedduringthe evolutionarystagesdiscussedinthis equals thevaluesinthirdrowofTable4are computedforthisparticularstate. sume here,somewhatarbitrarily,thatthispeak luminosityisreachedwhenU/^Tr ning ofthisstage,butatalaterperiod,whenthasfallenappreciablybelowÇTr.Weas- some kineticenergyfromtheoldstars,peakluminositywilloccurnotatbegin- will stilloccurduringtheearlypartofthisstage,wherenewstarsareabsorbing nosity associatedwithcollisionalliberationofhotgaswilldieaway.Sincecontraction By theendofthisphase,collisionsathighrelativespeedswillhaveceased,andlumi- transformed toaflatteneddisk,andtherandomstellarvelocitieswillbemuchreduced. The randomstellarvelocityof2100km/seccomputed forstate3representsaboutthe essentially destroyedbycollisions.Inthisphasethestellarsystemwillbegradually ly havetimetogainkineticenergyandleavethediskbeforemostofolderstarsare s the timerequiredformuchofgasinstarstobecomeliberatedbycollisions.As this phasedevelops,collisionsliberategassorapidlythatthenewlyformedstarsscarce- stars, whilethegravitationalinteractionbetweenoldstarsandyoungis system. Itisforthisreasonthatgravitationalencountersbetweentheoldstarsand contraction ofthesystem. dominant processwhichtakesenergyawayfromtheoldstarsandleadstocontinuing ever, themassofgaswithindiskisarelativelysmallfraction disk, condenseintonewstars,andbecomemixedwithotherstars.Atanyonetime,how- ponent tothetotalluminosityofsystem. about fiftycollisions,andmostofthemasssystemwillbeliberated,fallto gas, evaluatedinequation(8),provideanegligiblelossofkineticenergyfromtheseold states 1and2theinfraredradiationfromdiskwouldcontributeasignificantcom- by afactor(l/4.8)or0.31,yieldingtemperatureofabout730°K.Atphasesbetween Actually, asthesystemcontractsbyafactor1/4.8,radiusofdiskwillcontract c In state2aluminousfluxofabout3X10ergsfromthediskwillrequiretemperature 4.9 betweenthetwostates,andLisincreasedbyafactor3X10. of 550°K,ifadisk0.1pcinradiusisassumedtoradiateasblackbodyfrombothsides. lision. AccordingtotheparameterslistedinTable4,radiusisreducedbyafactor which thecollisiontime,/,equalsÇTr,wherefisagainfractionalmasslostpercol- with tdf,thetimerequiredforasubstantialfractionofnuclearmasstobeliberated short comparedto¿/f.Theendofthisphasemaybetakenstate2inTable4, by collisions.Thusthenewstarswillcometoequipartitionwitholdinatime traction ofthenuclearradius,andanincreasev.Duringthisevolutionaryphase however, therelativelylowenergyoutputfromgaswillnotchangeappreciably evaporation islessandimportant,buttherelaxationtime,Tr,stillshortcompared likely, theenergyradiatedfromgasmayberelativelymoreimportant.Inanycase, No. 2,1966EVOLUTIONOFGALACTICNUCLEI417 temperature ofthegaseousdiskcomputedin§IIIabove. c c s © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Evidently theluminosityinthisstage,computed fromequation(34),isanorderof In thethirdphaseofevolution,followingstate2,relaxationtimeisgreaterthan During thecontractionofstate1to2eachstarpresentinitiallywillundergo The largefluxofenergyintothediskwillincreasetemperaturetheresignificantly. Further evolutionbeyondstate1,withtfallingbelowTr,willproducefurthercon- c 19 66ApJ. . .143. .400S f 2 8 8 45 galactic nucleibecomesbetterunderstoodtheoretically,wecanhopetodiscoverwhether However, anypositiveidentificationoftheseobservedsuperluminoussourceswiththe flattened disk,producesaluminositythatisrathersimilartoofthequasi-stellar frequent collisions.Ifthediskwerecomposedofsmallsolidparticles,likeSaturn’srings, of yearsinthefuture. servational evidenceaccumulatesonthequasi-stellarsourcesandasevolutionof Foundation grantNSF-GP579. the systemsenvisagedhereareexistingnoworwhethertheirappearanceismanybillions evolving galacticnucleiconsideredherewouldbeprematureatpresent.Asmoreob- varies inverselywithA,thetotalnumberofstars,whileforlessmassivenucleidura- sources, providedthatthetotalmassofnucleusisingeneralneighborhood sibly theformationofindividualmassivestars,assuggestedoriginallybyHoyleand disk itisevenmoredifficulttovisualizehowmaterialcanexistinanysteadystate.Pos- it couldexistinthisstatewithoutmuchchangeforalongtime.However,isnotclear of thisevolution,whenthegasliberatedfromstarsformsnewthatremainina evolution ofacompactgalacticnucleusmaynow,perhaps,bevisualized.Thefinalphase that thisparticularstatewillbeapproachedbythegalacticnucleus.Atcenterof can approach,thoughformationofmultiplestarsystemsseemslikelyasaresultthe density ishigh,butsincetherandomvelocityofnewlyformedstarsrelativelylow, Hoyle, F.,andFowler,W.A. 1963,M.N.,125,169. Hénon, M.1960,Ann.dap., 23, 668. tion oftheluminositypeakbecomesrathershort,sincethistimeintervalvariesasA. Ambartsumian, V.A.1938,Ann.LeningradStateUniv.,No. 22(“Astr.Series,”Issue4). Fowler (1963),maygreatlyincreasetheradiatedfluxduringthisstage. the releaseofenergywillbemuchless.Itisnotclearwhatfinalstage,ifany,system terminate theevolutionofsystem.Collisionswillstillcontinuetooccur,since cussion oftheseproblemswouldbebeyondthescopepresentpaper. requirements whichatheoryofenergygenerationingalacticnucleishouldsatisfy.Dis- are tobecomparedobservationallywithexistingradiosources.Burbidge,and emission areclearlyrequirediftheevolvinggalacticnucleiconsideredtheoreticallyhere sion ofparticleaccelerationtorelativisticenergies,eitherbystellarshockwaves(Woltjer Burbidge, G.R.,E.M.,andSandage,A.R.1963, Rev.Mod.Phys.,35,947. Burbidge, G.R.1965,ProceedingsofThirteenthSolvayCongress. Bergh, S.vanden.1965,A./.,70,124. Sandage (1963)havegivenadetailedsurveyofthemanyobservationalandtheoretical interplay ofmagneticfieldsandsuprathermalparticleswiththegasdynamics.Adiscus- uncertain. peak luminosityforagalacticnucleusof10suns,thoughclearlythisvalueismost ties inthecondensinggas,thermalconductivitybetweengasanddisk, hot liberatedgasandbythediskwilldependonsuchprocessesasdensityirregulari- Gold, T.,Axford,W.I.,and Ray,E.C.1965,Quasi-stellarSourcesandGravitational Collapse, ed.I. in equation(35)somewhat.Weinferthataluminosityof10ergs/secismorelikely Caughlan, G.R.,andFowler,W.A.1962,Ap./.,136,453. 10 mo.Moremassivenucleiwouldyieldalowerpeakluminosity,sinceLingivenstate of 10000°Kormore.Thedetailedspectralcompositiontheradiationemittedby 1964) orbycollidinginterstellargasclouds,andananalysisoftheresultingsynchrotron 418 LYMANSPITZER,JR.,ANDWILLIAMC.SASLAWVol.143 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem This workmadeuseofcomputerfacilitiessupported inpartbyNationalScience While thelaterdetailsofevolutionarecompletelyobscure,earlyandintermediate Robinson, A.E.Schild,and L.Schucking(Chicago:UniversityofChicagoPress),p. 93. Contraction oftheentiresystemintoaflatteneddiskstarsdoesnotbyanymeans During thisperiodofpeakluminositythetemperaturediskmayreachavalue REFERENCES No. 2, 1966 EVOLUTION OF GALACTIC NUCLEI 419 King, I. 1958a, A.J., 63, 109. ■ , 19586, ibid., p. 114. . 1960, ibid., 65, 122. Michie, R. W. 1964, Ann. Rev. of Astr. and Ap., 2, 49. Milford, S. N. 1961, A.J., 66, 49. Miller, R. H., and Parker, E. N. 1964, Ap. J., 140, 50. Safranov, V. 1960, Doklady Akad. Nauk, U.S.S.R., 130, 53 (Ann. d'ap., 23, 979). Saslaw, W. 1965, thesis, Princeton University. Schmidt, M. 1965, Proceedings of Thirteenth Solvay Congress. Schwarzschild, M. 1958, Structure and Evolution of the Stars (Princeton, N.J. : Princeton University Press). Spiegel, E. A. 1964, private communication. Spitzer, L. 1940, M.N., 100, 396. — . 1942, Ap. J., 95, 329. . 1958, ibid., 127, 17. . 1965, Proceedings of Thirteenth Solvay Congress. Spitzer, L., and Härm, R. 1958, Ap. J., 127, 544. Von Hoerner, S. 1957, Ap. /., 125, 451. Woltjer, L. 1964, Nature, 201, 803.

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