198OApJ. . .238. .158N 4 The AstrophysicalJournal,238:158-174,1980May15 © 1980.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. consider herehaveatypical mass~10M,radius molecular cloudsmaybeinhomogeneous indensity and possesscomplexvelocity fields.Thecloudsthatwe 0 Considerable evidencehasaccumulated thatdense © American Astronomical Society • Provided by theNASA Astrophysics Data System 78 _v 34- 78 45-3 - 3 macroscopic cloudmotionscannotaccountfortheenergeticsofmanymolecularclouds.Astellar in amorerarefiedinterclumpmedium.Observationsconstraintheclumpstoberam-pressure massive starsindarkclouds.RecentobservationsofahighspacedensityTTaurisome continuous recyclingoccursbetweenclumps,ICM,andlow-massstars.Wefindthatstar continuing accelerationbyprotostellarwinds,collapseisfollowedtheformationoflow-mass main-sequence starsprovideacontinuousdynamicinputintomolecularclouds.TheTTauri dark cloudsprovidethebasisforourassertionthathigh-velocitywindsfromtheselow-masspre- energy sourcemustevidentlybetapped,andinfraredobservationsindicatethatonecannotutilize structure ofdarkmolecularclouds.Itseemsclearthatthekineticandgravitationalenergyin clouds (withlifetimes~10-10yr)mustpossesssteeperclumpmassspectra(v>2).Inthiscase, the high-massend\_N(m)~ra;v<2],catastrophicburstsofstarformationoccur.Long-lived primarily duetotheadoptedformofclumpmassspectrum.Ifmostinclumpsisat the evolutionofmassspectrumclumpsisstudied.Wepostulatethatasaredriven the interclumpmediumleadstoexistenceoftwophases;adense,coldphase(clumpsdensity confined, butattherelativelylowMachnumbers,continuousleakageoccurs.Thismassinputinto winds sweepupshellsofgas,theintersectionsorcollisionswhichformdenseclumpsembedded not bedisruptedbytheformationoflow-massstars.However,initiationsequentialOBstar formation occursonarelativelyslowtimescale,comparabletothecloudlifetime.Thewill above theJeansmass,bothbycoalescenceandenhancementoframpressurethrough of density~10-10cmandtemperature~30K).Clumpcollisionsleadtocoalescence, circumstellar gas.Theshocked-windgasalsoprovidesan internalsourceofultravioletradiation stars thatgenerateadditionalprotostellarwinds.Wefindthiscontinuingfeedbackis longward of912Âthatisequivalenttoanaverageinterstellar fluxwithavisualextinction^4^æ4. confinement, broademissionwingsarepredominantly producedbythedensestclumps.Two presence ofdenseclumpscoexistingwithawarmer ICM.Becauseoftheram-pressure This mayleadtoasignificantdegreeofphoto-ejection moleculesandelectronsfrominterstellar notably [Oi]63¡imandm]88¿an,othermolecular lineemissionfromthedenseshocked such aneventwillleadtorapiddisruptionofthecloud. grains. further consequencesofthismodelarethatTTauristars shouldoccurfrequently(spacedensity formation islikelytomodifyseverelythecloudevolutionafter~10-10yr.Weenvisagethat Subject headings:interstellar:molecules—nebulae:general —stars:formation ~ 10-10cmandtemperatureK)awarm,morediffuse,interclumpmedium(ICM, > 10pcindarkclouds)andmaybedetectablebyobservations ofH,strongfar-infraredlines, CLUMPY MOLECULARCLOUDS:ADYNAMICMODELSELF-CONSISTENTLY 2 A newmodelisproposedwhichcanaccountforthelongevity,energetics,anddynamical Specific predictionsofourmodelarethathighspatialresolutionmappingwillrevealthe I. INTRODUCTION stars: pre-main-sequence REGULATED BYTTAURISTARFORMATION Department ofAstronomy,UniversityCalifornia,Berkeley Received 1978December11;accepted1979October31 Huygens Laboratory,UniversityofLeiden Colin Norman ABSTRACT Joseph Silk AND 158 43 more extensiveanddiffusemolecular cloudcomplexes in whichtheymaybeembedded. Weshallcitein§IIa number ofobservationsseveral differentmolecules ~ 2pc,andmeanmolecular hydrogendensity ~10cm~, andshouldbe distinguishedfromthe 198OApJ. . .238. .158N 17 _1 6 -3 167 that supportthehypothesismolecularcloudsare clumpy. Whiletheultimatetestofthishypothesis cloud model(Townes1976).Thesimplestsuch awaits thecompletionofhigh-resolutionmaps,it consists ofmanysmall,dense,coldclumpsimmersed pelling tojustifyexplorationoftheastrophysical appears thattheexistingevidenceissufficientlycom- emission-line widthsareduetosystematicmotions, plausibility andpossibleimplicationsofaclumpy in amorediffuseinterclumpmedium(ICM). rotation (Field1978a).Systematicmotionsmaybe widths areinfactasuperpositionofemissionprofiles and amorecomplexvelocityand/ordensitystructure inadequate toaccountfortheobservedlineprofiles, such asradialinfall(GoldreichandKwan1974)or This typeofmodelshouldbedistinguishedfromthe randomly inthecloudgravitationalpotentialwell. interpretation oftheemissionprofilesisthatline cloud modelcanbeconstructedifasuitabledynamical clumps, anddemonstratethataplausibleclumpy discuss indetailtheformationanddestructionof more phenomenologicalmodelsofmolecularclouds from anumberofcoldclumpsthataremoving and clumpsurvivalagainstdissipationcoalescence substructure asdiscussedhere. of energeticsandstabilitytocontaincomplexclumpy massive fragmentsmaywellbeconstrainedongrounds involving afewmassivefragments(Zuckermanand extended toconsiderwarmcloudscontainingOB by collisionsandinteractionwiththeICM.Weshall ber ofseriousproblems,notablyclumpcontainment is probablyrequired(Kwan1978). molecular clouds;manyofourargumentscanbe source ispresent.Weshallconcentratehereondark stars. Infact,anaturalsourceofcloudinhomogeneity Evans 1974;Elmegreen1978).However,individual typical radii~10cmifthereisnosignificantrelative stars withtheirimmediateenvironment.Weshallshow is availableviatheinteractionofembeddedTTauri shall hereafterrefertothisasthestaticmodel. motion 10pc(Cohen supports theuseofsuchamodel,atleastedges clouds (cf.Elias1978)lackthe sensitivitytopickupthe and ICMthenthebubbleswillhavearadius and theICMisatrest.Current infraredsurveysofdark velocity ofthegravitational potential wellofthecloud limiting caseiswherethestars haveacquiredthevirial hereafter refertothisasthemovingbubblemodel; numerous low-luminosity emission-line stars(with 10~ cmandbubble-bubblecollisionswilloccuron Alternative possibilitiesarethattheobserved In aclumpycloudmodel,themoststraightforward A multiplecloudletmodelclearlypossessesanum- © American Astronomical Society • Provided by theNASA Astrophysics Data System CLUMPY MOLECULARCLOUDS 78 12 characteristic L^1)whicharepresumablyem- note hereasimpleargumentdemonstratingthatT energy availableinTTauriwinds(velocityV)relative clumps, andthelower-densitymaterialinteriorto intersecting denseshellssweptupbythewindsform bedded withinthecloudsandthatarefoundinoptical Tauri stellarwindscanprovideanadequateenergy surveys oftheouterregionsclouds.The shells permeatestheICM. cloud, andismoreoverinausableform.Infact,the available inthegravitationalbindingenergyof supply. Thisstellarenergysourcecanexceedthat cloud mass.Thatanenergysourcewhichexceedsthat to thatincloudpotentialenergy(randomvelocityv)is available incloudgravitationalpotentialenergyis clouds. Itseemsclearthatanenergysourcetappingthe with infraredobservationsofmanydarkmolecular the solutionofferedbytheseauthorsrequires and Kwan(1974).Whileapplicableincertainregions, often requiredhaspreviouslybeennotedbyGoldreich crossing time.Enhancedstarformationmayoccuron candidates. Tauri windsfromthelow-massstarsthatcanbe bol0 presence ofluminousOBstars,andisnotcompatible confines theclumps.SuchanincreaseinICMpressure time scalethatcanconsiderablyexceedthecloud present indarkcloudsseemthemostplausible reservoir ofstellarbindingenergyisrequired,andT £ (~0.01)istheratioofmassejectedinwinds undergo catastrophicenergyinputfromeithermech- provides positivefeedbackbyloweringtheJeansmass increase intheambientpressureofICMthat a morerapidtimescaleasconsequenceofsecular specific modeofstarformationonarelativelyslow and eventuallybecomeJeansunstable.Thisyieldsa In fact,weshallarguethatcollidingclumpscoalesce are notdisruptivebecauseofstrongpostshockcooling. on theorderofacloudcrossingtime),suchcollisions w fundamental assumptioninourmodelisthatchimp and thereforetriggeringfurtherstarformation.A formation intheouterregions andcontinuingTTauri anism is^10-10yr.Thefinalstagesinacloud’s coalescence andcompressionbytheICMleadsonlyto formation inthesurroundingmolecularcloudcom- triggers maybetheinitiationofsequentialOBstar trigger stimulatesmassivestarformation.Specific stabilized andheatedinthismanneruntilanexternal spectrum issatisfied,molecularcloudsaretherefore Provided thatacertainconditionontheclumpmass supernova blastwave:thetimescaleforacloudto plex byeitherspiraldensitywaveshocksoranearby lifetime wouldthereforefinditundergoingOBstar low-mass starformationanddrivesTTauriwinds. order toaccountforthe apparent noncoevalstar interesting tonotethatHerbig (1962)proposedthat,in tration ofourmodelisshown inFigure1.Itis star formationinthedense core.Aschematicillus- -10(^0.01) (UJtOOkms-)(Bkms"/^,(where While detailsofthismodelaredescribedin§III,we Although clump-clumpcollisionsoccur(typically 159 198OApJ. . .238. .158N 6b 8 - 3 160 6 6 shells fromICM(in~lOyr).Theisreplenishedbyleakingclumps~3x10yr),andshellintersectionsformnew~2 density ofTTauristarsin the Taurus-Aurigaregion periods of~10yr,beingterminatedbyclouddisrup- formation musthaveproceededindarkcloudsfor tion associatedwiththeonsetofOBstarformation. formation intheHyadesandPleiades,low-massstar may beextremevalues,althoughconsistentwith model forself-consistentlysustainingmolecularclouds electrons (viaphotoelectricemission). photons indarkcloudinteriors.TheresultingUVflux provides asmallbutsignificantinternalsourceofUV 912 Âproducedbythewind-drivenshockedgas be mentionedhere.TheUVradiationlongwardof T Tauristars.Moreover, some studieshaveeven current observationalconstraints.Theobservedspace are numerousdarkcloudswith nohithertoobserved only approaches~10pcin denseclumps,andthere surfaces, andmayalsoprovideasmallsourceoffree suffices toremovenewlyformedmoleculesfromgrain questioned theprevalenceof massoutflowinTTauri x 10yr).Coalescenceeventuallyleadstocollapseandclumpdestructionbylow-massstarformation(overseveralcrossingtimes,or~5 x 10yr). 6 Fig. 1.—Schematicrepresentationofclumpy-cloudfeedbackmodel.TTauristarsdrivewinds(over~10yr)andbubblessweepoutdense One importantconsequenceofourmodelmayalso It isimportanttorealizethattherequirementsofour © American Astronomical Society • Provided by theNASA Astrophysics Data System NORMAN ANDSILK - 81 12 1318 clouds. Themostcompellingargumentsinfavorof vational evidenceforclumpystructureinmolecular ular speciesdiscussedbelow. emission- andabsorption-linedataonthefivemolec- such apicturearisefromtheinterpretationof molecular cloudsinthe./ =2—►1and/0 moreover, moresensitiveinfraredsurveysarerequired intersecting bubblesamounttoonly~10Myr; to detectembeddedTTauristarsintheturbulentcores rotational transitionsofC O, Cand0 of densemolecularclouds. stars. However,thewindstrengthsnecessarytoform inferred densitiestypically an orderofmagnitude lower thanpreviousestimates basedonobservations 0 We shallattempttosummarizethecurrentobser- Plambeck andWilliams(1979) havestudied10 II. OBSERVATIONS a) CO Vol. 238 198OApJ. . .238. .158N 53 63 1313 13 1 1 6-317 3 clumped withdensity~3x10cm-.IntheOrion necessary toexcitethehighrotationaltransitionsof and linearsize~0.05pc. rotational transitionsweremadebyEvansetal. molecular cloudtheclumpingappearstobeonalinear HCN hasbeenusedbyMorris,Snell,andVanden their datawasamodelwithclumpdensity~10cm“ (1979), whoconcludedthatthebestexplanationof nation ofthelineratiosCO(2—►l)/l0) scale oflessthan0.2pc. B2 andOrionclouds,itappearsthattheemissionis Bout (1977)toprobeseveralclouds.ForboththeSgr the linearresolutionisrelatively low.Forexample,the ula isthattheCSclumpedandCOcomes Kwan (1978),whosuggestedthatapossibleexpla- show noobviousevidence for dumpiness.However, from theICM. and CS(3—►2)/CS(21)intheKleinman-Lowneb- Encrenaz, Falgarone,and Lucas 1975;Bok1956) favor aclumpymodelratherthansubthermalexcit- clump velocities~3kms"andinternalveloc- ation asanexplanationofthesmallfillingfactor. No. 1,1980 rotational, andmicrowavetransitionsofNH the COdatayieldanexcitationtemperature~70K. nebula isaclumpycloudmodelwithdensities his observationsofaregionintheKleinman-Low fine structureofthesizlowestmetastablerotational ities ~0.3kms". ments givesbeamfillingfactors~0.2,andthese hyperfine transitioninemissionandabsorption for beamfillingfactorsare~0.1-0.01.Theseauthors presence ofarelativelydiffusemediumwithdensity of othermolecularspecies.Thisclearlyindicatesthe deduced abeamfillingfactorof0.04.Furthermore, Combining bothemissionandabsorptionmeasure- against thestrongcontinuumsourceinDR21. found byMatsakisetal(1977)whoobservedthe (Schwartz etal1977)hasshownthatcanonicalvalues states ofNHintheKleinman-Lownebula,and authors presentamodelwithclumpsizes~0.2pc, Sweitzer (1978)hasalsofoundthatthebestmodelfitto 3 3 ~ 10cmandlinearsizescm.Inthisregion, ~(2-8) x10cm-. 3 53 The factthatahighdensity(>3x10cm-)is Recent observationsofOrionintheJ=3—►2^ Indirect evidenceforclumpingofCScomesfrom Star countsindarkclouds (Dickman1978; A surveyof21galacticregionsusingthehyperfine, Further evidencefordumpinessinNHhasbeen Barrett, Ho,andMyers(1977)analyzedthehyper- 12 3 © American Astronomical Society • Provided by theNASA Astrophysics Data System d) HCN c) HCO 3 2 b) NH 3 e) CS CLUMPY MOLECULARCLOUDS _1 3-8 67 _1 _7 -1 81 =m31 31/2 kinematic properties,arealsoclumpy.Weshallregard exceeds thetotalmassejectedbystar,adense will bestronglyradiative.Specifically,wemayconsider is thatifthewindvelocity<100kms, The subsequentevolutionresemblesthatofinterstellar shell formsandsnowplowsintothemolecularcloud. energy inputfromlow-massstarscansolvethese spatial resolutionintheTaurusdarkcloudattainedby three phases(detailsaregiveninAppendixA). bubbles (Weaveretal.1977).Oneimportantdifference for warmclouds,whereonehasembeddedmassive problems indarkclouds,thecorresponding type ofobservationprovidesanobviousconstrainton medium, n=/\0cm,MM/10 undecelerated expansionlastsuntilthebubbleexpands unsteady, laststypically10-10yrfora1Mstar, characteristic velocityV—(100-300)kmswith by ananalogoustwo-phasemodel. of darkcloudsasanextremecaseourmodel.If an explanationoftheenergetics,stability,andlifetime seems likelythatdarkclouds,whichhavesimilar that wehavecitedreferstowarmmolecularclouds.It any clumpycloudmodel. Bok (1956)workingdowntom=20was0.2pc.This to aradius be neglected,thebubblesexpandatconstantvelocity, star andtheambientmolecularcloud. values willprovidethebasisforourestimateof these stars(CohenandKuhi19796).Theseparameter and bolometricluminositiesareoforder(1-10)Lfor general phenomenonappearstobethatofawindat tions. Whileinstancesofinfallhavebeenfound,the is generallyaccompaniedbyhigh-velocitymassmo- stars aswelllow-massstars,canreadilyberesolved where nisthedensityofambientinterclump R =Vt,whereisthebubbleradiusattimet.The average mass-lossratesamountingtoM~10- outflow, whichrecentobservationssuggestmaybe this phaseis yr, andUoo=V/300kms\Thetimescalefor magnitude oftheinteractionzonebetweenTTauri ambient pressureisstillnegligible, anddeceleration 10-Myr- (Kuhi1964).Thedurationofthe 3lCM80 0 w v 0 lCM w w G h (^/4^icMH^w)^ 0.6(M/«F)yr. 2 83300 The TTauriphaseofpre-main-sequenceevolution Phase I.—Whiletheamountofsweptupmasscan Most oftheobservationalevidencefordumpiness Once themasssweptoutbyTTauriwind Phase II.—Duringthisnext phase,theeffectof 12 R =(M/47mraVf) IICMH2 a) ObservationsofTTauriStars 2 = 7xlO^MJniV^cm, b) BubbleDynamics III. TTAURIBUBBLES 161 198OApJ. . .238. .158N 5 62 lsoth -1 -1 1/2 certain criticalvaluegivenby the bubbleinteriorwillbeapproximatelyadiabatic for theensuingevolution.Ifwindvelocityexceedsa occurs becauseoftheincreasedinertiadueto radius attimet=¿/10yrisgivenby(Weaveretal. swept-up mass.Wecandistinguishtwopossiblemodes and remainhot(at~10^3K).Inthiscase,theshell may beapproximatedbythegeometricmeanofR¡and At lowerwindvelocities,radiativecoolingprevailsin the bubbleinterior.Wenowfindthatradius the radiusofanundeceleratedbubble: It isclearthatpressureofthebubbleinteriorhaslittle estimates, weshallusethemoreconservativeradius effect ontheshellevolution.Insubsequentnumerical for radiisuchthatRj«RRjiVJv^,where¿;isthe 1977) sound velocityaheadofthebubble. 162 where theICMtemperatureT=r/30Kand Æ andcorrespondingshellvelocity.PhaseIIoccurs 5 emission atacharacteristic velocityof~15kms. that thewind-drivenshellmaybeobservablebyH The geometricalcrosssectionofabubbleisthen radius, namely, generating thewindshaveamotionv*=vj\kms shell isdeceleratedtovelocity~v. O ing phaseIIwithvelocity~(^ C). The observedradiusisconsistent withthesnowplow- pressure oftheICMdeterminesfinalbubble relative totheaveragebackgroundmedium,ram- may occurbeforeanygivenshell deceleratestovelocity describe thecouplingbetweenbubblesandambient drogen nearTTauri(Beckwithetal.1978)suggests important atradii ICM, thepressureofambientmediumbecomes gas. relative motionbetweentheTTauristarsand s 30iCM R 2 A s SW in An importantrecentobservationofmolecularhy- 2: Inthemovingbubblecase,whenstars 3. Bubbleintersection.In fact, shellintersection Phase HI.—Weshallconsidertwolimitingcasesto 1. Inthestaticbubblecase,wherethereisverylittle 2/15764/157 © American Astronomical Society • Provided by theNASA Astrophysics Data System = v =100^-MVkms",(1) soth12 w18 ^ ~UVJv) s R' -(RjVj) (T t 175/41/42 1/5í2l3 = 10-M«3í;300¿5cm.(3) = l0^-*Mn-vtcm.(2) 8 83O5 = 1x, 172 = 3x\0(MV/n)cm. 83003 Î/ 4«icm^H2w M NORMAN ANDSILK 67 -3 4-3 4-3 3- -1 -3 cloud tobe less thantheterminalbubblediameter,then static case,iftheaveragedistancebetweenTTauri’sis one candefineabubblefillingfactorformolecular bubbles willintersectsupersonically.Alternatively, maintain theclumpystructure?Evenifcloudis intersection isthengivenby The criticalspacedensityofTTauristarsforbubble of TTauristarswithhigh-velocitywinds.Inthe long, sincetheclumpswillcoalescewithinacrossing clumpy tobeginwith,theinhomogeneitywillnotlast clouds: cantheproposedbubbleintersectionorcol- pressure confinedbubblestocollideis lision modelinitiatetheclumpystructure,andcanit plausibility ofourtwo-phasemodelformolecular to aradiusÆintimescale~10-10yr,whichis occur withthefirstgenerationofTTauristars.Fora measured extinctionswehaveestimatedthemeangas parable tothebubblecrossingtimeofa3pccloudat where ni=(«/10pc).Thecollisiontimeiscom- problems canberesolvediftheTTaurispacedensityis densities intheregionsstudiedbyCohenandKuhi ICM densityislowerthanthemeandensity.From the dumpinessthantoinitiateitsimplybecause clumpy modelformaintainingstructurewhenthe will initiallybeconsiderablycooler(~10K)thanthat uniform cloudofdensity10cm,thetemperature (1979tf, 6)tobe~10cm.However,boththeabove ation canbeprovided.Itisclearlyeasiertomaintain time unlessasuitablemechanismforclumpgener- tersections isadequatesincebubblessweepoutshells the staticapproximationforestimatingbubblein- bubbles ploughintotheICM(«=10cm,T to acquirethevirialvelocity withrespecttothecloud. of thewarmerICM(20-30K).Sincebubbleradius velocity, sothatthebubbles areram-pressurecon- shorter thanorcomparabletothetimescaleforastar fined, wemaymakeanearly phaseofhigher(butnot ~ *;.Thisiscriticallydependentonthespacedensityn Even ifthestaticapproximation isinvalid,andwe 1 kms. assume thattheTTauristars arebornwiththevirial m r>0 >10 pc.Theinitiationofclumpystructurewill = 30K)isunchangedoncetheICMforms.Initially sT In themovingbubblecasetime,t^forram- There aretwocentralissuesinestablishingthe B co 1/2 az(nT) ,ourstaticestimateofÆinthe in 32-1 2l 1 «cru =[^(f)'/?,,,] f —1exp\_n{\)R^\.(4) = (n-jOrvJ 3m 3 = lO^^o/MgFjoo^pc“. IV. TWO-PHASEMODEL 4«icm«îh«* 2 nMv TM 300 'f*.!yr, Vol. 238 198OApJ. . .238. .158N 7-1 6 6 8-1 3- 5-3 56-3 1/37 w1/3 7 7 7 -12 _10 7 -2 10-2 8 9-12 No. 1,1980 assuming efficientsweepingoftheICM.From implausible) mass-lossrate~10~Myrovera period of>10yr.Inthiscase,themovingbubbles will collidein~10yr.Havinginitiatedtheclumpy even thelowmass-lossrateM~10“yrand this manner,thetwophasesareeasilymaintainedwith structure andensuingformationofthewarmerICMin clump Misinitiallycomparabletothemasssweptup the formationofdenseshellsswept-upmatter molecular cloudswillinevitablycollide.Thisleadsto an ICMdensityof10cm. intersecting bubbles.Thecharacteristicmassina and theimpliedvolumefillingfactoris~0.01(n/n). density n=(nJlOcm)(seebelowforderivation the aboveestimates.Therefore,adoptingaclump duced (fordensityn=10-10cm)aresimilarto where R=(fit«,-)“9x10/i“cmisthe (clumps) andofthemorediffuseICMformedby strophic ifstarformationoccurs.However,theaverage below thatclumpaggregationisnotnecessarilycata- result incoalescence.Aself-consistentmodeltherefore from considerationofclump-clumpcollisions(Field of theclumpdensity),sizeis~^it(^3/ci,5) observations discussedin§II,theclumpmassesde- moving ram-pressureconfinedbubbleinacollision by ashellatcollision: cloud lifetimetobei=^/S,where£isthe requires continuousclumpformation.Weshallargue time is dark cloudmaterialinthesolarneighborhood£ efficiency ofstarformation,thesurfacedensitycold, galactic star-formationratedoesconstrainthemean o This yieldsi=5x10( ln^n!Mq, T00 1 — t).lniQft3MQ T CLUMPY MOLECULARCLOUDS 7 76 -81 -3 9-1 7 2 1 -3 The durationofthepre-main-sequenceconvective phase forstarsofmass0.8,1.0,and1.25Mis2x10 observational constraintsofCohenandKuhi(1979a). yr, 1x10and6yr(Taam1978).Ifwe assume atotalefficiencyof10%,thenumbersuch driving TTauri-likewindsistherefore^(10- stars atanygiventimeshouldconsequentlybe~(5- estimate isconsistentwiththeobservedfrequencyn are stillintheirconvectivephasesandpresumably rate amountsto~10Myr.Highermass-loss viously estimatedforshellintersectiontooccur. convective phase.Recentobservationssuggestthat the durationofconvectivephase,thesestars 30)« pc.Giventheconsiderableuncertainties,this possess ahigh-velocitywindwhoseaveragemass-loss rates maybeattainedonlyoverafractionofthe T Tauriwindsprovideanaturalsourceofkinetic 1.5) per100Mofcloudmaterial. 0 mass-loss ratesaslow10“Myrmaybemore energy forclumps.Clumpsareformedatsupersonic confine densecoldclumpswithparameterssimilarto and evolutionof,theclumpsICM. tained onlyfortimeslessthan^10yrduringamore typical. Highermass-lossratescanpresumablybesus- the residualmomentumofTTaurishells,sinceone n =«icm(^ci/^ci,s)vwhereistheclumpvelocityand ram-pressure confinement.Thisyieldsaclumpdensity CO (PlambeckandWilliams1979),theinferredICM ICM. Asshellintersectionandbreak-upoccur,the shell intersections,whichtypicallyoccuratvelocities those inferredpreviously.Consequently,weshalluse kinetic temperature(~50K)istoolowtothermally active phase. expects, inarelativecollision,someappreciable v ~3kms~.Theclumpinjectionvelocitiesreflect v^ istheinternalsoundvelocityinclump,whence From observationsoftherotationaltemperature ICM ispermeatedbyhigh-velocityTTauriwinds. T would thecollidingshellspossessequalandopposite 0 momentum). Thisalsoprovidesanaturalexplanation amount oftheinitialmomentumtoremain(onlyrarely ~ 10pcandwiththecriticalspacedensitypre- up materialinclumps,weareimplicitlytakingthe model incorporatesasignificantfractionoftheswept- 4 energy oftheclumps. driven shellsandusingthis toprovidethekinetic residual momentumandkineticenergyinthewind- of emission-linewidths(§VI).Notethatsinceour 0 0 parison oflinewidthsand kinetictemperaturesare d only oforder5-10.Atsuch low Machnumbers,ram- d ds 6-123 n =10a(i;/3kms)0.1kms/^ci,cm. The totalnumberofsuchstarsthatatanygiventime d3cls Our modelrequiresustopostulatethat,throughout T Tauriwindsprovidetheenergyinputinto We nowconsiderthedetailsofenergyinputinto, However, theMachnumbers inferredfromcom- a) ClumpFormationandConfinement 163 198OApJ. . .238. .158N -13 2 7 -61 Wicm7icm - pressure confinementisrelativelyinefficient,andleak- 164 age occursintheclumpwakesatapproximately kms and«o=%/10pc.Notethatclumps many (>30«^i837)clumps,wherev^=vß internal thermalvelocityofaclump.Thisleakage process constrainsaclump’slifetimetobeoftheorder clumps exceedsthenumberofTTauristars,clump Over thedurationofwindphase(t=i/10yr), can punchholesinbubbleshellsbutthatthese any givenstaticbubblewillthereforebeintersectedby of acloudcrossingtime,or will loseseveralpercentofitsmassasitcrossesthe in theshell.Since,aswenowshow,numberof will disappearasfurthersweepingofICMmaterialfills bubble (notethattheaverageinternalpressure clump wakesandwiththewindsinbubbleinteriors bubble ofradiusÆ,sufficienttointersectanother is similartothatintheICM:see§\Nbbelow).In inelastic interactionsbetweenwindsandclumpwakes bubble, is static modelthetimescaleforwindtosweepouta teristic valueofthewindpressureatbubblein- interaction providesaneffectivepressureintheICM provide amomentuminputintotheICMbecauseof at avelocitycharacteristicoftheclumpmotion.The Consequently, thewindmaysweepoutbubbles~10 whose magnitudemaybeestimatedtothecharac- strong radiativecooling.Therandomnatureofthe clump formationinourgeneralformalism,itisnot time is^10^yearsandconsequentlybubblecollisions Tauri star. decaying clumpsovertheTTauristar’slifetime.This times fromthemediumwhichhasbeenreplenishedby ra tersection, namely, clump isoftheordermassclumps crucial tothemodelandwechoosenotintroducean Although itispossibletoincorporatetheefficiencyof the efficiencyofclumpformationisorderunity. and resultingclumpformationwilloccur<10timesin raocX assumed that,uponbubbleintersectionorcollision, yields aclumpformationrateof10yrperT resupplying thebubble,andthusanapproximate additional freeparameter.Themassoftheswept-out the lifetimeofaTTauristar.Noteherethatwehave nr below. destruction, suchascollisions,arediscussedin§IVc form ofclumpdestruction.Otherforms steady statemaybeattainedifleakageisthedominant int ’ 1625 -RMVJ- =10^(10cm/^^oo"yr. 18 Clumps leavewakeswhichwillmixwithother For ram-pressureconfinedbubbles,thecollision © American Astronomical Society • Provided by theNASA Astrophysics Data System 423 3 = 1.5x10MF«,' cm-K.(5) x lOWO.OlpcXO.lkms-V^yr. 83OOrlo b) EnergyBalanceoftheICM MV W NORMAN ANDSILK 41/23192 use theanalyticapproximationofHollenbachand The majorcoolantisCOovertherelevanttemperature ultimately radiated.Theresultingheatinputis kinetic energydissipatedbyinelasticinteractionsis energy balanceandassumethattherandomizedwind cooling rateabove20K,namely, and densityrange.Linetrappingisimportant,we mass lossissubsonic,themotionofclump Note thatthemasssweptupbyclumpsisafactor this leadstoleakagelifetimesfortheclumpsof~¿, where n=2x10Tcm~,iV2xrm“ McKee (1979)toprovideanestimateoftheCO finement fortheclumps.Aswehavepreviouslynoted, (6), and(7)yieldstheapproximatesolution where thecloudcrossingtime timates ofICMparametersrequireram-pressurecon- and NistheICMcolumndensity.Combining(5), unaffected byleakage,andthespecificmomentumof tion: leakage,drag,andcollisions.Ourmodeles- motion isthatdragforcewiththeICMwillbe the clumpmaterialremainsunchanged. clump totheaveragecolumndensity(ofclumpsand proportional totheratioofcolumndensityin appreciable. Infact,thestoppinglengthforcloudis where <«>istheaverageclouddensity. ICM) acrossthecloud,or ^ci/^cu lessthanthemasslostbyleakage.Since cross to dragwiththeICMrequires thattheram-pressure To avoiddestructionbydeceleration andstoppingdue by sweepingupICMandcollisions withotherclumps. CÎcr confined clumpsbelargerthan acriticalsizegivenby and 3/2 To estimatetheICMtemperature,weuselocal These arethreepossiblemodesofclumpdestruc- However, oneconsequenceofthesupersonicclump This expressionallowsforbothclumpdeceleration A =3x10“Vr T ^MVvn co wc]T 2431 = 1x10~MF3t;3n.ergscm~s~(6) 8O7ilo 6 ¿cross =10(Æioud/3Pc)(3kms"7r)yr. ccl nN 30 4/31/37/9 cm 1 H(+T7~ i /1 5001 X (MFo)^3^7\10 3 7^ Rc\ouà(Bc\Rc\l(By^cXoVià)’ /3 2 Reloua V c) ClumpDestruction n \ 7? >l^cloudC^cUs/^cl)’ 3 pc/\MV- nr cl % R\ r 3 pc -1/3 ergs cm 1/3 "TUO ^5 -3 -1 c - 1/9 Vol. 238 , (7) No. 1, 1980 CLUMPY MOLECULAR CLOUDS 165 or equivalently, be denser than crossing time, this process could result in considerable

n > W radial clump mass segregation. Since these times are c\ 4 ICM^cloud/^cl * comparable, the effect is marginal and we shall not

Let/be the ratio njnicu. If the total mass in clumps is pursue it further here. comparable to the cloud mass, then / is the volume Other processes that must be considered in discuss- filling factor of the clumps. Only those clumps with ing the spectrum of clumps include sources and sinks radii Rá > \fRc\ovtá will survive for more than a cross- of clumps. We have already mentioned the clump ing time. For a given individual clump mass Mcl, the injection rate, and we shall argue that star formation clump density cannot exceed provides a sink when clump coalescence produces 6 3 3 3 masses above the Jeans mass. A highly simplified «cl < 10 (Mc1/10 M0)(3 pc/Æcloud) (0.02//) cm" . model for the evolution of the velocity-averaged clump These conditions are probably well satisfied by the mass spectrum 7V(ra, t) is given by the generalized observed clumps (§ II), and we expect that clumps will coagulation equation. Solutions for the evolution of survive for at least a crossing time. A/m, t) for mass-dependent collision cross sections but Hitherto we have considered clumps of equal mass. without any sources or sinks have recently been given However, the effects of clump-clump collisions, which by Silk and Takahashi (1979). We have extended this we have already incorporated in our estimate of the work to obtain a steady-state solution of the coagu- stopping time, will lead to the development of a mass lation equation with source, sink, and leakage (or spectrum. To demonstrate this, we note first that the accretion) terms for a mass-independent collision rate Mach numbers ^ 30 considered here will result in (Appendix B). coalescence because of the strong dissipation at molec- If the leakage over a collision time amounts to less ular cloud densities. A sufficient condition for coales- than Ml, where clumps are assumed to be injected at mass Ml, our result reduces to the well-known power- cence is that the shock thickness be much less than the 3/2 dimension of the clump (^0.1 pc). law solution (oc M~ if the collision rate is constant), To estimate the shock thickness, we make use of as found, for example, by Klett (1975). If leakage approximate cooling rates given for molecular clouds dominates, our solution indicates that the mass spec- by Hollenbach and McKee (1979). The thickness of trum becomes truncated at the high-mass end. the postshock layer, / , can be expressed as However, a power-law mass spectrum remains valid sh over a wide range of masses. We shall make use of this result below. L = GXifl,inc\ where g = 10“15 cm, y æ 4/3 (Shull and Hollenbach V. STAR FORMATION 1978), and Coalescence will eventually lead to clumps with masses greater than the Jeans mass, which for a clump Sfc/sh^O is E0nGVsh 3/2 4 3 1 2 Mj = 10(rcl/25 K) («cl/10 cm“ )“ / M0 . Here xt refers to the fractional abundance of the dominant coolant molecules (CO and H20 at tempera- The key assumption underlying our model (and in- tures <500K), «shcr (evaluated for n, T, and vs at ferred from the observed properties of dark clouds and postshock values) is the critical density above which T associations) is that when these fragments become collisional deexcitation becomes important, E0 is the Jeans unstable, only low-mass stars are formed. rotational constant, and A0 is the Einstein A- Massive star formation requires an external trigger, 6 -3 according to this viewpoint. The estimated Jeans mass coefficient. Typical values of «shcr are ~10 cm . Inserting appropriate numerical values, we obtain at is actually a factor of ~ 10 larger than our estimated n < «sh,cr initial clump mass, based on considerations of shell intersections. We therefore make the following in- 104 cm' 10- L = 2 x 1015 cm , ferences: T Tauri star formation may be self- sustaining if the swept-up matter forms clumps that eventually are Jeans unstable. If the clumps are below At high Mach number, or large compression the Jeans limit, they will survive several crossing times (^ > 10), one finds that there is an additional weak until significant mass aggregation can occur via col- dependence on the shock velocity, /sh oc It is lisions. Mass loss via leakage will tend to have a apparent that /sh « Æcl for the clump velocity range that stabilizing effect on the clumps and could possibly limit -1 is observed (¿;cl < lOkms ox JÍ < 30). This con- clump masses. This could conceivably account for the firms that clump-clump collisions will generally result preferential low-mass star formation that has pro- in coalescence. Note that the clump-clump collisions ceeded in dark clouds and T associations. It is conceiv- are highly inelastic and can result in loss of orbital able, given the model uncertainties, that the clumps are kinetic energy of the merged clumps. If the clump- initially gravitationally bound. While it does not seem clump collision time is less than, or of the order of, a possible on the basis of our model to decide this

© American Astronomical Society • Provided by the NASA Astrophysics Data System 198OApJ. . .238. .158N 166 times longerthanthedurationofTTauriphasefor ties indelayingcollapseofsuchclumpsfor>10free- that molecularcloudshavelifetimesareatleast10 question, theobservationalevidenceclearlysuggests a 1M©star.Weconcludefromthisthatitisreason- that theclumpsaremagneticallyorrotationallysup- and donotimmediatelyformstars.Onepossibilityis able toassumethatthenewlyformedclumpsarestable considerably exceedsafree-falltime.Inthiscase,ram- ported, andhavealifetimeagainstcollapsethat pressure confinementisnolongernecessaryand leakage willnotoccur.Collisionsandcoalescence should stillleadtothedevelopmentofamassspectrum (Silk andTakahashi1979). viewpoint thatclumpsareinitiallyunboundandram fall times.Henceweshallsubsequentlyadoptthe pressure confined,asissuggestedbyourinterpretation ing possibilityisthatlow-massstellarcorescouldbe more clumpswillbecomeJeansunstable.Aninterest- proceeds, theclumpmasswillincrease,andmore of themolecularcloudobservations.Ascoalescence clump andanotherinwhichstarformationisoccur- exposed andstabilizedbycollisionsbetweenone ficant effectinsomeclouds,butwedonotconsiderit low-mass starformationislongerthanthecharacteris- ring. Thiswilloccurfrequentlyonlyifthetimescalefor tic clump-clumpcollisiontimes.Sincethesetwotime winds drivefurtherstarformation.Theefficiencyof unstable clumpsformlow-massstarswhoseTTauri further here.Asstatedpreviously,weassumethat scales areroughlycomparable,thismaybeasigni- cloud disruptioncanoccur.Alternatively,TTauristar imately constant,whichisinturnrequiredtoyielda number ofTTauristarsatanygiventimebeapprox- the starformationisfixedbyrequirementthat efficiency, averagedoverthecloudlifetime,is~107- long cloudlifetime.Aswehaveshownearlier,this in effectivepressure(includingrampressure)ofthe tervene thatsuggestOBstarformationandsubsequent Over sufficientlylongtimescales,othereffectsin- coalescence andtheeffectivepressureincreaseof which incorporatesbothoftheseeffects(namely, simple theoreticalmodelfortherateofstarformation clumps, whichwenowgeneralize toincorporatethe ICM duetoTTauriwinds.Wehavedevelopeda formation mayexhaustthegasreservoirofcloud, principal results. here weoutlinetheassumptionsanddescribe ICM). DetailsofthistheoryaregiveninAppendixC; as willnowbedescribed. for theevolutionofmassspectrumN(m,t) with theJeansmass).Studies ofthecoagulation effects ofatime-dependentmass truncation(identified equation haveshownthat, in thelongtimelimit, effect ofimposingatruncation doesnotsignificantly o However, thereareconsiderabletheoreticaldifficul- Star formationmayalsobetriggeredbyanincrease Our startingpointisthecoagulationequation(Bl) © American Astronomical Society • Provided by theNASA Astrophysics Data System a) Low-MassStarFormation NORMAN ANDSILK 2 an 1/2 1/3 enables ustomakeuseoftheresultsAppendixB. modify theformofsolution(Klett1975).This The generalizedcoagulationequationthatweuseis regime (misthemaximumclumpmass)of injection rate.Wenotethattowardthehigh-mass Here T(t)isthe(generallytime-dependent)clump dominates overthecoagulationsourceterm(~N).In clump destruction,orequivalently,therateatwhich spectrum, theclumpcoagulationsinkterm(~A) any giventime.Wethenobtain this high-masslimit,itispossibletoevaluatetherateof if weadoptamass-independentcollisionratea.Useof Mo,cium dletM*(t)bethetotalmassinstarsat the massinclumpsisbeingtransformedintostarswith made useofthefactthatmassinclumpsand Takahashi 1979).Inderivingthisequationwehave derived slopeofthemassspectrum(Silkand an efficiencyrj.Firstdefinetheinitialclumpmass a morerealisticmassdependenceformodifiesthe interclump mediumisgivenatanytimeby M*(t) =—rj—N(m,i)mdm+Tm These twoequationsexplicitlysatisfymassconser- and clump accelerationandcorrespondinglyincreased vation. TheJeansmassmdecreaseswithtimeasthe protostellar windsprovideincreasedamountsof mal clumps,ocp\~,wherepistheconfin- ram-pressure confinementoftheclump.Forisother- ing rampressure.However,thepressureis u which injectwindenergyover acharacteristicdistance produced byenergyinputfrom thewind-drivenshells 0 0 P Æ oc~n~.Consequently, L 5 cumpdump intT d 1 1 =— M =,+jUc*->r clump0c!ump Micm ^o,icmiTdt+.(10) 213 Atop «iMV^R^Va-'n-rcc n, T Ul JiriT x /'mj(i) ^-^0,clump + T(t)ô(m—m). x N(m—m')dm' L / — ocN cc(m\ m—m')N(m) 0 (x(m, m')N(m)N(m)dm' Jo *1 í Jo f 1 Tdt —rj~ Vol. 238 (9) (8) No. 1, 1980 CLUMPY MOLECULAR CLOUDS 167 and we infer that this clump mass is in clumps at masses near raj, once triggering occurs as raj is lowered, star formation is strongly self-sustaining. The star-formation rate, ob- tained by solving equation (9), is (see eq. [C8]) where the subscript 0 denotes initial values. This ^X = (l (15) equation yields the desired feedback of star formation M* onto the cloud evolution, and equation (9) can now be 0 i solved. To proceed, we adopt the limiting form where = M^t = 0) and i k (mLlmM)(a.Noy . 7V(ra, t) that attains in the long-time limit, For v > 1, i is short compared to a mean collision time 1 where the actual value of v( > 1) can in principle be self- (aiV0) “ . We interpret this to indicate the occurrence consistently obtained by solving the coagulation equa- of a burst of rapid star formation. We now argue that tion (Bl). In practice, the uncertainty in choice of an the long lifetime required for dark clouds (§ III) appropriate form for the collision rate means that v effectively constrains v to exceed 2 because of the likely may lie in the range 1 < v < 3 (cf. Silk and Takahashi disruptive nature of the v < 2 star formation bursts. 1979). We shall argue that stability considerations can While we cannot rigorously substantiate this conjec- narrow the allowable range for v, and consider two ture it is definitely suggested by our results. Moreover, cases below. it is subject to observational test when high-resolution i) V > 2 maps of molecular clouds become available. In this case, with most of the mass in low-mass b) OB Star Formation clumps, coalescence effects dominate over the feed- back onto the Jeans mass, resulting in a slow mode of We have already argued that there are at least two star formation described by distinct modes of star formation, involving T Tauri and OB stars. We propose to incorporate OB star M^t) K r] T(t)dt. (12) formation into our model in the following manner. An Jo external trigger initiates massive star formation by accelerating the rate of clump coalescence. Two pos- This is valid at times greater than the maximum of two sible external triggers are available. They both yield time scales : the time scale to inject a mass in clumps characteristic time scales ~ 107-108 yr. comparable to the initial total mass in clumps, and a i) The time for a nearby, more-diffuse cloud either mean clump-clump collision time, or to be shocked directly or to undergo enhanced ac- ^^0 .clump cretion or coalescence of smaller clouds as a con- max (13) (T}m sequence of the passage of the spiral density wave, T thereby triggering Jeans instability with ensuing col- 7 8 Here raris the mean mass of a T Tauri star and is lapse, is ~ 10 -10 yr. Initiation of sequential OB star an average injection rate. Using the results of § Ilia, we formation in such a manner has been suggested by estimate that several authors (e.g., Roberts 1969; Woodward 1976). 5 1 3 Once initiated, sequential OB star formation will T= 10 nT l0 yr pc . continue to trigger collapse of molecular clouds along This yields the spiral arms (Elmegreen and Lada 1977). ii) A supernova shell moving at ~(30-100) km s-1 5 5 will suffice to induce collapse and therefore act as a tCT = max \0 ñ4.n r.io 5 trigger for OB star formation (Herbst and Assousa 1977). Assuming that the supernova rate (1/íSn) io5-5 yr . (14) proportional to the star-formation rate, one can show that the external trigger time scale is Thus at long times (^ 107 yr), the mass in stars grows ~^Sn( ^disk/^SNiO^Ob/^TC 7 3 linearly with time: pT æ constant. The characteristic = 4 x 10 (iSN/30yr)(Fdlsk/10" pc ) 1 star-formation time scale is ^M ^ (nT)~ , or a 3 3 0 ump x (10 pc /K ) clump recycling time. SNR Note that this solution corresponds to a balance x (Qíob/0.03) yr , between injection of new clumps and clump collapse where USNR is the volume occupied by a supernova (to form stars), with the mass in clumps at any given remnant moving with an expansion velocity of (30- time remaining approximately constant. It is this -1 100) km s , Fdisk is the volume of the disk, ¿OB is the continuous recycling of clumps that forms the basis of lifetime of an OB star, and Q is the mean angular our model. velocity of the gas. The final factor in this expression is ii) 7 < v < 2 due to the fact that the effective volume available for Rapid star formation is found in this case where triggering star formation corresponds to that of the most of the mass is in massive clumps. Because most of spiral arms.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 198OApJ. . .238. .158N -4 -3 2 7-21 20- -1 _ 1/3 8 17 210 1 32/-1 Salpeter 1972).Inourmodel, wecalculatethatthe compilation ofUVfluxes),theproducedbywind difficulty inunderstandinghow moleculesareremoved This hasseveralimportantimplications. ratio oftheincidentUVflux tothemoleculeflux from grainsurfacesindark clouds(Watsonand interior ofamolecularcloud(specificallyatA>4). shocks is~10>,andwillbedominantwithinthe radiation field. cm, theLafluxcouldprovidedominantUV model parameters.However,ifnisdecreasedto10 We concludethattheLafluxwillbeappreciablein consequences fortheclouddensityandionization major observationalimplications. dense shells.Inparticular,photonswithwavelengths to ourmodel.Inthissectionwebrieflydescribethe ecules fromgrainmantles. There isanoutstanding <0)^2 x10cms(seeDraine1978forarecent immediate vicinityoftheshocksonlyforadopted is ~10-100(foracolumndensityof4x10cm). diffusion willbemoreimportantthanfrequency the meannumberofscatteringswillbeatleastt.At volume. Theopticaldepthatlinecentertbetween probably notbeabletofillalargefractionofthecloud cloud interior,sincethefillingfactorisoforderunity. flux throughtheshock.ThisyieldsaUV(912- velocity of~100kms,thefluxbetween912and longward of912Âwillprovideaninternalsource redistribution, andconsequentlythemeannumberof visual extinctionofonlyx=0.4«3«magift While thisradiationwillbeexcludedfromclump balance. AquantitativeestimateoftheUVfluxcanbe significant inputintothesurroundingslowlymoving amount offar-UVradiationthatcanprovidea sources amountsto~3x10.Comparisonwiththe be themajorfar-UVconstituent.However,Lawill amounts to~10(912-1500).Infastershocks,Lawill including absorbedLymancontinuumphotons, intershock spacing,~4x10cm,isequivalenttoa interiors, itwillpermeatetheentireICM.Themean and McKee(1979),fromwhichweinferthatatashock UV radiationforthemolecularcloudthatmayhave La, assumingdestructionbyabsorptiondustgrains, scatterings shouldincreaseastandbeoforder10- such largeopticaldepths,itseemslikelythatphoton La transfercalculationsofAdams(1971)indicatesthat obtained fromtherecentshockcalculationsofShull v 10. Thissuggeststhattheeffectiveopticaldepthfor 1500) æ5x10Mflcmsthroughoutthe 1500 Âamountstoapproximately207oftheparticle 168 lCM 0 0 vr iouv 0 = 2.5^.AsignificantLafluxisalsoproduced,which, 8rio o Since theaverageinterstellarflux(over912-1500Â) i) TheUVfieldcanphotoejectnewlyformedmol- Shocked TTauriwindswillgenerateasubstantial There areanumberofobservationsthatrelevant © American Astronomical Society • Provided by theNASA Astrophysics Data System VI. OBSERVATIONALIMPLICATIONS a) UltravioletRadiationField NORMAN ANDSILK 3- 4 6 mwic2 M -2 core. Whethertheclumpscollapse orareblownoutof with onlythemoremassiveclumpsremainingin clumps canbeexpressedasacriticalmassintheform critical conditionforwind-drivenoutflowofthe the coredependsonwhether m>or/10cm.Ifthewindstrengthrises core willbegravitationallybound. Conversely,theless for characteristicparameters, theformerconditionis above acertaincriticalvalue,allclumpswillhave satisfied, thoseclumpsmassive enoughtoremaininthe condition onthecriticalcolumndensityofaclump: to remaininthecloudisthatforceexertedby mation inthecloudcore.Theconditionforclumps energetic windgeneratedbytheongoingstarfor- gravitational forcebindingittothecloud.Thisyieldsa wind actingtopushaclumpoutwardislessthanthe layers, drivenoutofthecloudbyasufficiently the core.Clumpswilltendtoaccumulateinouter ture, withmostoftheprotostellaractivityconfinedto -1 lar moleculesindarkcloudinteriorswillbe~10that ionization intheICM(compared,e.g.,tothatinferred ejection couldaccountforasignificantdegreeof This isasufficientlyhighratethatphotoelectron rates, F,foundbyDraine(1978),weobtain(adopting where ^isthedensityofanabundantmolecular et al1972),namely,<10yr.Thisiscomparableto in theunshieldedinterstellarmedium(<100yr;Stief from studiesofdeuteratedspeciesindarkclouds grains. Utilizingtheaveragephotoelectricemission (or lessthan)thechemicalequilibriumtimescale crit53 3 an effectivegraincrosssectionperatomof6 of freeelectronsviaphotoelectricejectionfromsolid ecules canbeejected. species (e.g.,CO).Forayieldoforder0.1-1(Watson incident ongrainsurfacesis sociation rateintheinteriorsofdarkclouds. (Iglesias 1978),andprovidesasignificantphotodis- and Salpeter1972),anappreciablefractionofmol- [Guelin etal1977]). crk =60[MV3«r,<3>-y(j4”3)o, pe 801m x 10cm) (/>(912-1500)(^) *20(10- iii) Thephotodissociationtimefortypicalinterstel- Suppose thatthecloudpossessesacore-halostruc- ii) TheUVradiationfieldcanalsoprovideasource b) TheEffectofWindPressureonClumps -13 r =3X10%cms. peCM 21- = l-0MVn(')~cm, 83OOTlo3 2 \6nG) (n}m H2 3 \nM T Vol. 238 198OApJ. . .238. .158N 2 _1 -1 4are No. 1,1980 massive clumps(andthismeansmostofthein -17 2-_1 15 21 necessarily beexpectedtooccur. will havethegreatestdensitybecauseofram- emission linewings.Themostrapidlymovingmaterial will havehighinitialvelocitiesandcontributetothe velocity. Notethattheprecedingcalculationishighly pressure confinement:specificallynocv.Thehigher moving atsupersonicvelocities,^3kms. superposition ofanumbernarroweremissionlines clumps maybeslowedandtrappedinthecloudhalo. significant velocitiesinitiallygreaterthantheescape our model)willbeblownoutofthecoreandacquire flattopped lineprofilethanotherwiseexpected. This effectmayresultintheproductionofabroader greater opticaldepthforlinesformedintheseclumps. densities inthehigher-velocityclumpsmayleadto taken intoaccountandalsobecauseinitiallyescaping idealized, sincemorerealisticdragforcesmustbe T Tauristarsaredistributedthroughoutthedark anisms discussedin§IIIc,thenewlyformedclumps (width ~0.1-0.3kms)fromindividualclumps initially shockedifthepostshockmatterisin- be producedintheregionswhereTTauriwindsare our model,self-reversalofthelineprofilesshouldnot Because ofthedecelerationanddestructionmech- characteristic ofTTauristarsobservedindarkclouds other moleculesinthevicinityofTTauribubble velocity is amount ofmatterthatcouldbemovingatveryhigh homogeneous. Aroughestimateofthefractional Because theICMismoreorlessuniformlyheatedin cloud interiors. shocks. Thiscouldprovideausefulprobeofwhether cX dark cloudmodelsinwhichhotgrainsheatthegas infrared fluxesarepredicted.Ontheotherhand,in (~l-10L; CohenandKuhi1979a),relativelylow vations maythereforeprovideadiscriminantbetween large as~3x10Lrequired.Infraredobser- (Evans, Blair,andBeckwith1977)infraredfluxesas the twoheatingmechanisms. total fluxfromasingleTTauri windinO1(63.1¿un) results ofShullandMcKee(1979),wefindthatthe are predictedfromtheshockedwind.Applying and Oin(88.2¿un)atadistance Dis o o 7(88.2) =(8x10)M(200 pc/7))ergscms. 7(63.1) =(6x10)M(200 pc//))ergscms, 8 8 _ In ourmodel,thebroadprofilesaredueto i) Extremehigh-velocitywings(<100kms^may ii) OnemayexpectexcitationofH,NHand iii) Becauseofthelowbolometricluminosities iv) Substantialfluxesoffar-infraredlineradiation 23 © American Astronomical Society • Provided by theNASA Astrophysics Data System 3 F<«>w 1 MR^rbr wH2 d) OtherPredictions c) LineProfiles 7 5 X1(T ^300^3) M gRig CLUMPY MOLECULARCLOUDS n T,lO * -8_1 3 4_3 3- 45-3 - 3 v 78 1 78 clouds. Thekeyobservationalinputhascomefroma cular cloudsthatastellarenergysourceisoften clear fromconsiderationsoftheenergeticsmole- recent surveyofTTauristars,inwhichCohenand molecular lines. required tomaintaintheinferredenergyoutputin Kuhi (1979a)foundahighspacedensity(n we drawtheplausibleinferencethatahighdensityof inate thepossibilityofembeddedmassivestars.Thus low bolometricluminositystarsintheTTauriphase mean mass-lossrate(~10Myr)overthe molecular clouds.Conservativeassumptionsaboutthe are present,whoseconvectivelydrivenwinds conserving wind-drivenshellsthatmayberadiatively duration ofthepre-main-sequenceconvectivephase supplying theenergyanddynamicalinputinto cooled andintersecttoformdenseram-pressure prime couplingisbyapproximatelymomentum- confined clumps.AttherelativelylowMachnumbers suffice togiveanadequatesourceofmomentum.The consequence ofTTauristardensitiesn>10pc~in continuously replenishestheICM.Inthismannera two-component mediumisgenerated,consistingof and leakagefromthewakesofmovingclumps attainable, ram-pressureconfinementisinefficient, molecular cloudswith~10cm.Typicalclump interclump medium.Shellintersectionisaninevitable dense, coldclumpsembeddedinadiffuseandwarmer T coalescence. Wehavesolvedthegeneralizedcoagu- with density~10cmandtemperature~30K. parameters aredensity~10-10cm,temperature lation equationfortheclumpmassspectrumwith ^ 10pc)ofTTauristarsinseveraldarkclouds.Itis low-mass starsdriveTTauriwinds,theclumpandstar cence eventuallydrivesclumpsJeansunstable,and o results infurtherlow-massstarformation.Sincethese the clumpmassspectrumissufficientlysteep.If sources, sinks,andleakage.Wepostulatethatcoales- clump massspectrumN(m)ocra~,werequirev>2in formation processesareself-sustainingprovidedthat T creases becauseofenhancedprotostellarwindactivity, order forthestar-formationratetoincreasesecularly. this furtherlowerstheJeansmassandleadstoa star-formation raterisesandtherampressurein- If v<2,catastrophicstarformationoccurs.Asthe rate andobservedstar-formationefficiency,wecan inferred fortypicalmolecularclouds(~10-10yr) runaway effect.Becauseofthelongmeanlifetimes effectively constrainvtobe>2. from constraintsonthemeangalacticstar-formation ~ 10K,velocity1-5kms“,embeddedintheICM catastrophic influenceonthe cloud,andresultinrapid estimated timescaleof~10 -10 yr.Thiswillhavea disruption. Theresultingstellar associationwillbe formation willoccurfora typicalcloudwithinan We havedevelopedanewmodelformolecular Infrared observations(e.g.,Elias1978)oftenelim- Clump collisionsarehighlyinelasticandleadto Disruption byinitiationof sequentialOBstar VII. SUMMARYANDCONCLUSIONS 169 198OApJ. . .238. .158N 21 The apparentcoincidencebetweenemission-linewidths clumps, whichinturnissufficienttoaccountforthe natural consequenceofourmodel. and youngstellarassociationvelocitydispersionsisa born withavelocitydispersionsimilartothatofthe observed emission-linewidthsinmolecularclouds. following. with awarmer,morediffuseICM. should revealdumpinessonscales>0.1pccoexisting confinement. densest clumps,becauseoftheram-pressure wind-driven shells. dark cloudinteriors.PossiblesignaturesareHand 170 ward of912Â),amountingtotheaverageunshielded atomic andmolecularlineemissionfromtheshocked interstellar fluxat^—4,intheinteriorsofdark clouds. Thismayaffectcloudchemistryandionization photoejection ratefromgrains. balance byincreasingphotodissociationratesandthe the nearfuture.Whilemanyofourargumentsare 2 where R=(MVJ4npv)andM|+Mt.Hereistheinitialmassofshell is subsequently takentobezero.Thisequationneglectstheworkdoneonshellbyinternalpressureof with solutionR—Vt+oqIna.Usingtheboundary conditionsR=0ati0,wesettheconstantsoq shocked windgasinteriortothebubble.Wederiveacriterionforitsvelocityin(c)below.InphaseI,wehave 13 A firstintegralis where e=R^Vj/R^v, m0sshelinitialmiiM w2 shell>>47ipR/3and«,soequation(Al)becomes 0m The specificpredictionsofourmodelincludethe i) Highspatialresolutionmapsofmolecularclouds ii) Broademissionwingsareassociatedwiththe iii) TTauristarsshouldbeembeddedthroughout 2 iv) TherewillbeaninternalsourceofUV(long- Several ofthesepredictionsshouldbeverifiablein The equationofmotionforabubbleshellradiusRexpandingintoanambientmediumdensitypis 0 a =0.InthesnowplowingphasesIIandIII,whereM «4npRß,wehaveforequation(Al) © American Astronomical Society • Provided by theNASA Astrophysics Data System 2 shellQ 6„ 2 10 10 fortypicalparameter values.Thequadraturesolutionis EVOLUTION OFBUBBLESHELLS dt d R =v< WRo) .frcpo M, c f xdx_2v t s dt d 2 32 ^ R)=WRu dt \4np 0 NORMAN ANDSILK d /3Mt•»„? (RR) =3v(-R. m 2(-f 3 /Ru 3 2 a) RadiativeWinds + R (£ +lx APPENDIX A R} =W(Ru 3 -81 -3 x) convectively drivenwinds«M>~10Myr“) clouds isashighrecentobservationssuggest, fact thatthemolecularcloudobservershouldbearin based onplausiblespeculations,thereisoneoverriding served inonlyafewregions.Futurestudiesmustconfirm conservative assumptionsaboutmass-lossratesvia mentum sourceindarkclouds.However,itcannotbe formation, andprovideasignificantenergymo- overemphasized thatourmodelrequiresaspace inevitably indicatethatthesewindscollide,driveclump mind. IfthespacedensityofTTauristarsindark clouds ifsupersoniclinewidthsandlongcloudlife- that correspondstothehighestdensityhithertoob- the presenceofembeddedTTauri-likestarsindark times aretobeexplainedbymeansofmomentumand energy inputfromlow-masspre-main-sequencestars. density ofTTauristarsindarkclouds(>10pc) 0 Jong, L.Kuhi,C.McKee,andG.Wallerstein.This Dr. G.B.FieldandhelpfuldiscussionswithDrs. research hasbeensupportedinpartbytheNATO Baud, A.Cheung,M.Cohen,D.Hollenbach,T.de NGR 05-003-578. Scientific AffairsDivisionandbyNASAundergrant 1/2 ^ ~R™’ We acknowledgestimulatingcorrespondencewith Ru 2 R), 1/2 Vol. 238 (Al) (A2) (A4) (A3) (A5) 198OApJ. . .238. .158N 1/2 57 No. 1,1980 and for6«1,thisgives according toequation(2). where t=(f)(2)ÆAv where thesubscript“sw”referstoshockedwindandcoolingfunctionAcanbeapproximatedas over thetemperaturerange2x10K0)reducestotheresultgivenbyKlett(1975). we integrateovermassandobtain As discussedinthetext, the long-timelimitathigh-massend, The solutionforN(m,t)isfoundbydefining Starting fromthegeneralizedcoagulationequation, © American Astronomical Society • Provided by theNASA Astrophysics Data System <7V>(^ o=(^o+—)+ ^*(0 —~>1N(m,t)mdm+7m,=—r]rhN(m¡)m¡ —ti s N(m, t—>oo)= Ls = 0,ifm(s, 0= + t N =(—Itanh 0 m—-dm =-km' L 11/2 1/2 Vol. 238 jOit. (B2) (B6) (B5) (B3) (B7) (B4) (Cl) (02) (03) 198OApJ. . .238. .158N 1/3 No. 1,1980CLUMPYMOLECULARCLOUDS In addition,wemakeuseofthemass-conservationequation(10)forM,andobtain where Using therelationraj=KM*givenbyequation(11),wefindanevolutionfornamely, There arethreedistinctcases: where T(t)isassumedtohaveaconstantvalueT. and where Adams, T.F.1971,Astr.Ap.,12,280. clump .1979/?,Ap.J.{Letters),227,LI05. Bash, F.1979,Ap.J.,233,524. Barrett, A.H.,Ho,P.T.P.,andMyers,C.1977,Ap.J.{Letters), Cohen, M.,andKuhi,L.V.1979a,Ap.J.Suppl.,41,743. Beckwith, S.,Gatley,I.,Matthews,K.,andNeugebauer,G.1978, Bok, B.J.1956,Ap.J.,61,309. Dickman, R.L.1978,Ap.J.Suppl.,37,407. Draine, B.T.1978,Ap.J.Suppl.,36,595. Elmegreen, B.G.1978,Proc.WorkshoponMassiveMolecular Elias, J.H.1978,Ap.J.,223,859. This limitingcase(v=2)hasanintermediatebehaviorbetweenv>2and<2. Elmegreen, B.G.,andLada,C.J. 1977,Ap.J.,214,725. or Encrenaz, P.J.,Falgarone,E.,and Lucas, R.1975,Astr.Ap.,44,73. 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