Mon.Not.R.Astron.Soc.000,1–9(0000) Printed19January2011 (MNLATEXstylefilev2.2) Why are most molecular clouds not gravitationally bound? C. L. Dobbs1,2(cid:63), A. Burkert2,3 and J. E. Pringle4 1 Max-Planck-Institut fu¨r extraterrestrische Physik, Giessenbachstraße, D-85748 Garching, Germany 2 Universitats-Sternwarte Mu¨nchen, Scheinerstraße 1, D-81679 Mu¨nchen, Germany 1 3 Max-Plack fellow, Max-Planck-Institut fu¨r extraterrestrische Physik, Giessenbachstraße, D-85748 Garching, Germany 1 4 Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA 0 2 n 19January2011 a J 8 ABSTRACT 1 The most recent observational evidence seems to indicate that giant molecular clouds arepredominantlygravitationallyunboundobjects.Inthispaperweshowthatthisis ] anaturalconsequenceofascenarioinwhichcloud-cloudcollisionsandstellarfeedback A regulatetheinternalvelocitydispersionofthegas,andsopreventglobalgravitational G forces from becoming dominant. Thus, while the molecular gas is for the most part . gravitationally unbound, local regions within the denser parts of the gas (within the h clouds)dobecomeboundandareabletoformstars.Wefindthattheobservations,in p terms of distributions of virial parameters and cloud structures, can be well modelled - o provided that the star formation efficiency in these bound regions is of order 5 – 10 r percent. We also find that in this picture the constituent gas of individual molecular t s clouds changes over relatively short time scales, typically a few Myr. a [ 1 INTRODUCTION (e.g. Bertoldi & McKee 1992; Dib et al. 2007), where σ is v thelineofsightvelocitydispersionandRisameasureofthe 1 The belief that molecular clouds are gravitationally bound radiusofthecloud.Ifthecloudisinvirialequilibrium,then v objectshasledtoalong-standingprobleminstarformation. α=1and2T+W=0whereT isthekineticandW thegrav- 4 Ifcloudsarebound,andiftheycollapseinoforderafree-fall 1 itationalenergy,whereasT+W =0correspondstothezero time, then the rate of star formation should be around two 4 energyconfiguration.Thevalueofα issimplyameasureof orders of magnitude greater than what is observed (Zuck- 3 the ratio of kinetic to gravitational energies, and finding α erman & Evans 1974). To circumvent this there have been . both observationally and from simulations is highly uncer- 1 twomainapproaches.Firstly,onecanassumethatmolecular tain,dependingonthemassandradiusdeterminations,and 0 cloudsundergocollapseonlongertimescales,requiringthat 1 typicallydoesnotaccountformagneticfields,surfaceterms some process prevents, or at least slows their collapse. This 1 or projection effects (Ballesteros-Paredes 2006; Dib et al. could be magnetic fields and slow ambipolar diffusion (Shu : 2007;Shettyetal.2010).Thustheestimatedvalueofαfor v et al. 1987; Basu & Mouschovias 1994; Allen & Shu 2000; one cloud may not predicate its consequent evolution, but i Mouschoviasetal.2006;Shuetal.2007),ormicroturbulence X the distribution of α gives an indication of the importance (Krumholz & McKee 2005; Krumholz et al. 2006). How- of gravity in a population of clouds, and whether they are r ever turbulence can also promote collapse on small scales a predominantly bound or unbound. (Klessen et al. 2000; Mac Low & Klessen 2004), (includ- ingMHDturbulenceHeitschetal.2001;V´azquez-Semadeni Inarecentobservationalstudy,Heyeretal.(2009)pub- etal.2005;Elmegreen2007).Alternatively,inlightofrecent lished revised estimates of molecular cloud masses, sizes observationalevidence,onecanassumethatcloudsaretyp- and virial parameters from the previous seminal work by icallyshort-livedentities(Hartmannetal.2001;Elmegreen Solomon et al. (1987). Although they claim that molecular 2002,2007;Ballesteros-Paredesetal.2007)andpositmech- cloudsarevirialised,theirplotsseemtoindicatethatinfact anisms which prevent most of the gas forming stars, such mostofthecloudsareunbound(asalsofoundbyHeyeretal. as stellar feedback and magnetic fields (Va´zquez-Semadeni 2001).InFig.1(lowermiddlepanel)weplotthevalueofthe et al. 2005; Elmegreen 2007; Price & Bate 2008). However virial parameter taken from the clouds observed by Heyer if the cloud is not globally bound in the first place, as sug- et al. (2009). Heyer et al. (2009) suggest that the masses gested by recent observational evidence, we already amelio- of their clouds are underestimated by a factor of 2 due to ratethedisparitybetweentheexistenceofshort-livedmolec- non-LTEeffectsandCOabundancevariations(seealsocal- ular clouds and the global star formation rate. culations by Glover et al. (2010) and Shetty et al. (2010)) The virial parameter of a molecular cloud is usually within a cloud, so we have doubled their cloud masses to defined as produce the panel in Fig. 1 (lower middle panel). It is evi- dentthatmostoftheobservedcloudshavevirialparameters 5σ2R largerthanunity,indicatingthatmostcloudsarenotgravi- α= v (1) tationallybound.Evenifwetakeα>2,wherebytheclouds GM (cid:13)c 0000RAS 2 C. L. Dobbs, A. Burkert and J. E. Pringle are strictly unbound, this still leaves 50 per cent unbound 2 SIMULATIONS clouds. Similar distributions of α are also found for exter- Thecalculationspresentedhereare3DSPHsimulationsus- nal GMCs, including those in M31 (Rosolowsky 2007, from a sample of 105−6 M clouds), and the GMCs detected in inganSPHcodedevelopedbyBenz(Benzetal.1990),Bate (cid:12) (Bateetal.1995)andPrice(Price&Monaghan2007).The several nearby galaxies by Bolatto et al. (2008). codeusesavariablesmoothinglength,suchthatthedensity InarecentpaperBallesteros-Paredesetal.(2010)sug- ρandsmoothinglengthharesolvediterativelyaccordingto gest that GMCs are undergoing hierarchical gravitational Price&Monaghan(2007),andthetypicalnumberofneigh- collapse, whereby the collapse occurs on scales from indi- bours for a particle is ∼ 60. Artificial viscosity is included vidual cores to the whole cloud. However it is not neces- to treat shocks, with the standard values α = 1 and β = 2 sarythatthecloudshouldbegloballygravitationallybound. (Monaghan1997).Inallthecalculationspresentedhere,the Simulationsofunbound,turbulent,GMCsnaturallyleadto gas is assumed to orbit in a fixed galactic gravitational po- localised star formation, rather than spread over the entire tential. The potential includes a halo (Caldwell & Ostriker cloud (Clark et al. 2005, 2008). This then naturally leads 1981), disc (Binney & Tremaine 1987) and 4 armed spiral to a low star formation efficiency. Recently, Bonnell et al. component (Cox & Go´mez 2002). The gas particles are ini- (2010)performedcalculationsofanunbound104 M cloud, (cid:12) tiallysetupwitharandomdistribution,andassignedveloc- andshowedthatstellarclustersforminboundregionsofthe itiesaccordingtotherotationcurveofthegalacticpotential cloud.Theinternalkinematicsofthesecloudscouldbedue with an additional 6 km s−1 velocity dispersion. The rota- to cloud-cloud collisions or large scale flows (Bonnell et al. tion curve is flat across most of the disc, with a maximum 2006; Dobbs & Bonnell 2007; Klessen & Hennebelle 2010), velocity of 220 km s−1. and/or stellar feedback (e.g. Mac Low & Klessen 2004 and Wepresentresultsfrom4differentcalculations,assum- references therein). marised in Table 1. Run A was already presented in Dobbs Most numerical work has tended to focus on calculat- (2008), and is more simplistic than Runs B, C and D. The ing the virial parameters of clumps within giant molecular totalgasmassis5×109 M inRunA,and2.5×109 M in (cid:12) (cid:12) clouds(Dibetal.2007;Shettyetal.2010),which,sincethey Runs B, C and D, and corresponds to one or two per cent are the sites of star formation in GMCs, are more likely to of the total mass of the galaxy. The surface density of the be bound. However in recent simulations of a galactic disk, MilkyWayisabout12M pc−2(itis10M pc−2inWolfire (cid:12) (cid:12) it has been possible to identify individual GMCs and de- etal.2003excludinghelium)thusalittlehigherthanRuns termine their virial parameters (Dobbs 2008; Tasker & Tan B, C and D. The mass resolution is 1250 M for Run A, (cid:12) 2009). These results show that the virial parameter, α (see and 2500 M for Runs B, C and D. (cid:12) Section 2.2) typically lies in the range of around 0.2 to 10. ForRunA,weallocateparticlesatradiibetween5and Inthispaper,weaddressthequestionofhowmolecular 10kpc.Thegasisassumedtobeatwophasefluid.Thein- cloudscanremainunbound.Pringleetal.(2001)arguedthat terstellarmediumhastwoisothermalcomponents,onecool ifmolecularcloudsareshort-lived,withlifetimescomparable andonewarm.Weomitthermalconsiderationsandsothere toafewtensofMyr,thentheymustbeformedfromalarge isnotransitionbetweenthetwophases;thecoolgasremains reservoir of dense interstellar gas, which may or may not coolandthewarmgasremainswarm,throughout.Thecool itself be molecular. Dobbs et al. (2006) has shown that the and warm gas comprise equal mass in the simulations. The formation of the global structure of molecular gas (clouds, gashasinitialscaleheightsof150and400pcinthecoldand spurs etc.) does not in itself require self-gravity, but that warmcomponentsrespectively,butthesedecreaseto20-100 formationcancomeaboutforentirelykinematicreasons.In pc and 300 pc with time. The mean smoothing length is 40 thispaperwetaketheseideasastepfurtherandattemptto pc. In Run A, we also include a magnetic field, such that model the observed properties of molecular clouds. We self the plasma β of the cold gas is 4. The magnetic field is im- consistentlyfollowtheevolutionofcloudsinagalacticdisc, plemented using Euler potentials as described in Dobbs & taking into account cloud collisions and cloud dispersal by Price (2008). energy input from stellar feedback. The clouds we consider In the remaining calculations (B, C and D), we inves- areofsizetensofparsecs,weareunabletoresolveverysmall tigate the effect of stellar feedback. In these cases we allow clouds.Theparticularpropertieswetrytomatcharetheob- theISMtoexhibitamultiphasenaturefrom20Kto2×106 served distribution of the virial parameter α, the shapes of K. The cooling and heating of the ISM is calculated as de- the clouds and their internal structures. We find that these scribed in Dobbs et al. (2008). Apart from feedback from properties can be matched simply by assuming that those star formation, heating is mainly due to background FUV, regionswithinmolecularcloudsthatbecomeself-gravitating whilstcoolingisduetoavarietyofprocessesincludingcolli- are able to form stars at some small efficiency (5 – 10 per sional cooling, gas-grain energy transfer and recombination cent)whichgivesrisetofeedbackintheformofinputofen- on grain surfaces. The gas initially lies within a radius of ergyandmomentum(Section2).Thusifsayonlyaround10 10 kpc, and has an initial temperature of 7000 K. The im- percentofacloudisboundatanyonetime,andthoseparts plementation of stellar feedback will be described in detail form stars at around 10 per cent efficiency, the problem of inaforthcomingpaper,butasimpledescriptionisincluded the two order of magnitude difference in the star formation here. The gas is assumed to form stars when a number of rate identified by Zuckerman & Evans (1974) can be over- conditionsaremet,i)thedensityisgreaterthan250cm−3, come(seeSection3).Wedemonstratethatwiththissimple ii) the gas flow is converging, iii) the gas is gravitationally assumption, those structures which would be identified as bound(withinasizeofabout20pc,or3smoothinglengths), molecular clouds are, for the most part, globally unbound, iv) the sum of the ratio of thermal and rotation energies to with properties giving a reasonable match to the data. the gravitational energy is less than 1, and v) the total en- (cid:13)c 0000RAS,MNRAS000,1–9 Why are most molecular clouds not gravitationally bound? 3 ergy of the particles is negative (see Bate et al. 1995). If alltheseconditionsaremet,weassumethatstarformation takesplace;thereisnoprobabilisticelementinourcalcula- tion. We do not however include sink particles, instead we deposit energy in the constituent particles. We present cal- culations with star formation efficiencies, (cid:15), of 1, 5 and 10 percent(RunsB,CandDrespectively).Thismeansthatof themassthatsatisfiestheabovecriteria,afraction(cid:15)ofthe molecular gas contained therein is assumed to form stars instantaneously and to provide an energy input (approxi- mately1/3thermaland2/3kineticenergy)of1051 ergsper 160M ofstarsassumedtoform.1 Thisenergyinput,com- (cid:12) bined with our cooling and heating prescription, leads to a multiphaseISM.InthecaseofRunC((cid:15)=5percent),from 150 Myr onwards approximately one third of the gas lies in the cold, unstable and warm regimes. 2.1 Locating clouds We identify clouds using the same method as described in Figure2.ThegascolumndensityisshownforRunC((cid:15)=5per cent)atatimeof200Myr.Densegas,correspondingtotheclouds Dobbs (2008). We apply a clumpfinding algorithm, which located in the analysis presented here, predominantly lies along simply divides the simulation into a grid, and locates cells the arms, and spurs which extend from the arms into interarm over a given surface density. Adjacent cells which exceed regions. this criterion are grouped together and labelled as a cloud. Clouds which contain less than 30 particles are discarded, thus clouds in Runs B, C and D are at least 7.5×104 M unbound clouds, where local gravitational collapse is pre- (cid:12) (and clouds in Run A 3.75×104 M ). The mean number ventedbymagneticpressure.Withminimalstellarfeedback (cid:12) of particles in a cloud is ∼ 85 for Runs A, B and C. The (RunB,(cid:15)=1percent),weobtainmanymoreboundclouds, properties of the clumps reflect the total, rather than the particularly at higher masses. This clearly disagrees with molecular gas, but we would typically expect these clumps the observations. For both Runs C and D ((cid:15) = 5 and 10 to exhibit high molecular fractions. For most of the re- % respectively), we find that the clouds are predominantly sults we present, we chose a surface density criterion of 100 unbound, and the distributions in the virial parameter, α M pc−2 ∼4×1022 cm−3.Changingthiscriterionhaslittle are in agreement with the observations. This can be seen (cid:12) effect on the distribution of the virial parameter, it merely visually and is confirmed by comparing the distributions of reduces or increases the number of clouds selected. αusingtheKStest(seealsoFig.1).Giventheuncertainties For Runs C and D ((cid:15)=5, 10 per cent), we are able to indeterminingα,ifwerequireα>2foranunboundcloud, run the simulation for sufficiently long (300 Myr) that we about half of the clouds in Runs C and D are unbound. cancalculatecloudpropertieswhenthesystemhasroughly Thereislittlechangeinthefractionofunboundcloudswith reachedequilibrium.HoweverforRunsA(highsurfaceden- mass (and therefore resolution) in these calculations, with sity, no feedback) and B (1 % efficiency) we are limited by the exception of Run D, where feedback is responsible for the high surface densities reached by a large fraction of the preventing bound, higher mass clouds. gas. Toillustratetheglobalstructureofthediscinourmod- els,thecolumndensityofthegasinRunC((cid:15)=5percent) 2.3 Evolution of individual clouds isshownatatimeof200MyrinFig.2.Thedensegasisar- The evolution of an individual cloud is often very complex, rangedintocloudsalongthespiralarmsandspursextending involving collisions, fragmentation and dispersion by feed- from the arm to interarm regions. back.Moreoverthegaswhichconstitutesacloudcanchange on relatively short timescales. Thus studying the evolution of individual clouds, and establishing why they never, or 2.2 Virial parameter rarelybecomegravitationallyboundisnotstraightforward. In this Section we illustrate this behaviour by studying the In determining the virial parameters for our clouds, we cal- natureanddevelopmentofindividualGMCsinthedifferent culate α as shown in Eqn. 1, where σ is the line of sight v calculations. velocitydispersionandR isdefinedastheradiusofacircle with the equivalent area of the cloud. This corresponds to that used by Heyer et al. (2009). We take bound clouds as 2.3.1 Cloud development in Run A having α<1. Inthecasewithmagneticfields(RunA),wefindlargely In Fig. 3, we highlight the contribution of collisions to the internalvelocitydispersionsofcloudsinRunA.Weshowa collisionbetweentwocloudsinRunA(whichincludesmag- 1 ThiscorrespondstoaSalpeterIMFwithlimitsof0.1and100 netic fields), where small scale gravitational collapse does M(cid:12). not occur. After the collision between the two clouds, the (cid:13)c 0000RAS,MNRAS000,1–9 4 C. L. Dobbs, A. Burkert and J. E. Pringle Run Surfacedensity ISM β (cid:15) No.particles Timechosento (M(cid:12) pc−2) percent locateclouds(Myr) A 20 Twophaseisothermal 4 N/A 4x106 130 B 8 Multiphase(20-2×106 K) ∞ 1 106 110 C 8 Multiphase(20-2×106 K) ∞ 5 106 200 D 8 Multiphase(20-2×106 K) ∞ 10 106 200 Table1.Thedifferentcalculationsperformedaredescribedabove.RunAwaspresentedinDobbs(2008).Forthistwophaseisothermal calculation,halfthegasliesinthewarm(104K)phasewhilsthalfiscold(100K).(cid:15)isthestarformationefficiencyinthecalculationswith feedback(seetextfordetails).ForRunsCandD,thetimewelocatethecloudsisnotimportantastheyhavereachedanapproximate equilibriumstate. ! = 5% n o cti a r f e v ati ul m u C Figure 1. The distribution of the virial parameter (α) is plotted with mass for clouds identified in Run A (top left, with magnetic fields),inthecalculationswithfeedbackadoptingefficienciesof1,5and10%,andtheHeyeretal.(2009)data(lowermiddle).Wefind apopulationofpredominantlyunboundclouds,inroughagreementwiththeobservations,forthemodelswherelocalisedgravitational collapse is limited by magnetic fields (Run A), or gravitational collapse occurs but there is a realistic level of stellar feedback (Run C, top right, Run D, lower left). There are many more bound clouds for the case with a very low level of stellar feedback (Run B, top middle).Inthefinalpanel(lowerright),thecumulativefractionofcloudswithagivenαisshownfortheHeyerdata(dotted)andfor the clouds from Run C, with 5 per cent efficiency (solid line). The KS test confirms that the distributions of α from the observations andsimulationsmatch,givingP =0.11andP =0.21forRunsCandDrespectively. velocitydispersionincreases,whichmeansthevirialparam- in the cloud for around 10 Myr, independent of the mag- eteralsoincreases.Theincreaseinthevelocitydispersionis netic field. This demonstrates that the energy input from prolonged because the clouds contain substantial substruc- cloud-cloud collisions can be comparable to, or even exceed ture – the merging of this substructure is seen in the mid- theenergydissipated,i.e.acollisioncanleadtoanincrease dle panel. Thus the collisions between clouds are not really in the global virial parameter. The generation of random dissipative (as stated in Dobbs et al. (2006)); rather the velocities is analogous to that presented in previous work energy is transferred to the internal motions of the clouds. (Bonnell et al. 2006; Dobbs & Bonnell 2007), and relies on We also simulated the cloud interaction shown in Fig. 3 in the assumption that the ISM is clumpy on all scales. isolation, and at higher resolution, without magnetic fields or feedback. This confirmed that the collision induces ran- dom large scale motions which prevent widespread collapse (cid:13)c 0000RAS,MNRAS000,1–9 Why are most molecular clouds not gravitationally bound? 5 Figure 4. The evolution of a cloud from Run C (with 5 % effi- ciencystellarfeedback)isshown,at189(top),198(second),200 (third) and 202 (fourth panel) Myr. The cloud is formed by the mergerofsmallerclumps.Stellarfeedbackevents(forexamplethe cross in the second panel) then alter the shape of the cloud and finally result in the separation of the cloud into several separate clumps.Separateclumps,(aspickedoutbytheclumpfindingalgo- Figure 3. The evolution of a cloud from Run A is shown at rithm),areshownsimplyindifferentcolours,buttheconstituent timesof123(top),132(second)and141(thirdpanel)Myr.The particlesarenotallthesameatdifferenttimes.Forexampleonly evolutionofthemass(inunitsof106M(cid:12)),velocitydispersionand 2/3oftheparticlesinthecloudat198Myrareinthecloudshown virial parameter are shown on the lowest panel. The two clouds at 200 Myr. The right hand panels show column density images inthetoppanelmerge,andthereisasubsequentincreaseinboth of the clumps and their surroundings. The white boxes indicate thevelocitydispersionandα.Theyformasingleclumpofmass thesizeoftheregionsshownonthelefthandpanels.Fig.5shows 7×105 M(cid:12).Theinitialclumpsarecolouredblueandred,sothe theevolutionofα,themassandthevelocitydisperions,σ,ofthe constituent particles can be traced at later times. The particles cloud,andtheconstituentclumpswhichformedthecloud. colouredgreenareparticlespresentinthecloudsat132and141 Myr,whichwerenotpresentintheclumpsat123Myr. (cid:13)c 0000RAS,MNRAS000,1–9 6 C. L. Dobbs, A. Burkert and J. E. Pringle Figure 5. The evolution of α (green, solid), mass (red, dashed in units of 106 M(cid:12)) and σ (blue, dotted) of the clouds shown inFig.4.Cloud-cloudinteractionshavesomeroleindetermining thedynamicsofthecloud,andmaintainingσ,butσisdominated by stellar feedback. The different lines at 189 Myr (and the line at 193 Myr) correspond to several clumps merging to form one largecloud(193Myrto200Myr).After200Myr,thecloudagain splitsupintomultiplecomponents,duetofeedback. 2.3.2 Run C: a multiple cloud interaction with feedback InFig.4weshowtheevolutionandinteractionofamultiple set of clouds in Run C ((cid:15) =5 per cent). A single cloud was selectedatatimeof198Myr,andthecloudswhichcontain thesameparticleswereidentifiedatearlierandlatertimes. Wefindthatthecloudidentifiedat198Myrisformedfrom themergersofseveralsmallerclouds.Inthefirstpanel(189 Myr) we can identify 5 separate clouds. As these merge to produceacloudof2×106M some9Myrlater,theeffectsof (cid:12) stellar feedback can be seen for example in the cloud in the Figure 6. The evolution of a second cloud from Run C, with second panel (a bubble blown out by feedback is indicated 5%efficiencystellarfeedback.UnlikethecloudshowninFig.4, by the cross). Feedback plays a large role in shaping the thiscloudistoomassivetobedisruptedbyfeedbackorcollisions cloud, and regulating the dynamics. The clouds, as picked andbecomesincreasinglymoremassive,andboundwithtime.At out by the clumpfinding algorithm (left hand plots) show 160Myr(toppanel),thecloudisstillfilamentary,andmarginally much more filamentary structures compared to the clouds unbound. By 200 Myr (lower panel), the cloud contains no fila- in Run A (Fig. 3). By 202 Myr (4th panel), stellar feed- mentarystructureandisstronglybound.Therighthandpanels backhassucceededblowingawaythetoppartofthecloud, showcolumndensityimagesoftheregionscontainingtheclouds and splitting the cloud apart. Over the course of the plots on the left, the white boxes indicating the sizes of the left hand shown(13Myr),thereare5supernovaeeventsinthecloud. plots. The lowest panel shows the evolution of α, mass (in units In Fig. 5 we show the corresponding evolution of α and the of106 M(cid:12))andσ forthecloud. velocity dispersion. It can be seen that the velocity disper- sionismaintainedatabout6kms−1,andthevirialparame- 2.3.3 Run C: the evolution of an isolated massive cloud ter,α,aboveunitythroughout.Thusinthiscase,amultiple cloud collision together with feedback (from small regions For the calculation, with 5 % efficiency feedback (Run C), ofthecloudwhichbecomeboundandallowstarformation) the timescales for the majority of clouds to merge and be- maintains the unbound nature of the cloud as a whole. come disrupted are relatively short, of order several Myr. ItcanalsobeseenfromFig.4(righthandpanels)that The exceptions are two longlived bound clouds, which have the clouds we identify are part of a larger region of dense massesof3×106M and5×106M respectively.Theseare (cid:12) (cid:12) gas,whichishundredsasopposedtotensofparsecsinsize. thehighmasspointsseeninFig.1(toprightplot).Weshow Inagalaxywithahighmoleculargasfractionsuchafeature theevolutionofthe5×106 M cloudinFig.6overaperiod (cid:12) wouldcorrespondtoaGiantMolecularAssociation(GMA). of 40 Myr. The top panel shows the cloud at a time of 160 Whilst these regions are still unbound in our calculations, Myr, when the cloud is clearly irregular in shape. By 200 they appear to have a longer lifetime than the GMC sized Myr,thecloudhasamuchmoreregular,quasi-sphericalap- clouds. pearanceandisnotfilamentaryinanyway.Thiscloudfinds (cid:13)c 0000RAS,MNRAS000,1–9 Why are most molecular clouds not gravitationally bound? 7 itself in between the spiral arms, and does not undergo any significant interactions with other clouds after 160 Myr. In thelowerpanel,weplottheevolutionofα,thevelocitydis- persionandthemass.Weseethatthecloudiscontinuingto accretematerial,andgrowinmass,becomingsteadilymore bound. Feedback (with (cid:15)=5 per cent) is insufficient to dis- ruptthecloud,althoughfeedbackdoesmaintainaconstant velocitydispersionof∼6kms−1.Itislikelythatthiscloud would eventually form a bound stellar cluster, though we do not attempt to follow this in our calculation. In Run B ((cid:15)=1 per cent) there are many more clouds which display this behaviour. 2.4 The constituent gas of the clouds Figure 7. The percentage of gas which still lies in the same cloud after 10 Myr is plotted versus the cloud’s mass for clouds In the current paradigm of molecular cloud formation and identified in Run C (with stellar feedback and (cid:15) = 5 per cent). evolution, GMCs are assumed to be bound objects which Thispercentageiscalculatedbylocatingcloudsat2timeframes, consist of essentially the same gas for the duration of their 10 Myr apart. We search for the constituent gas particles of a lifetimes. In Fig. 7 we take all the clouds at a given time givencloudinthecloudspresent10Myrlater.Sometimesthere in Run C ((cid:15) = 5 per cent), and plot the percentage of gas maybemorethanonecloudcontainingtheparticlesofanearlier which remains in a given cloud after 10 Myr. cloud(andinsomeinstancesnoclouds!),inwhichcaseweselect Inallcaseswefindthattheconstituentgasoftheclouds the cloud which has the maximum of the particles the same as changes on timescales of < 10 Myr. We find that about the cloud at the earlier time frame. In 35 per cent of cases the 50 per cent of clouds are completely dispersed within 10 cloudsarecompletelydisrupted,whilstthemedianamountofgas remaininginthecloudis22percent. Myr. A substantial fraction of clouds are shortlived, either dispersing to lower densities, merging with other clouds to producemoremassiveclouds,orsomecombinationofthese 1.5and2.WealsoshowinFig.8thedistributionsofaspect processes.Thereareafewcloudswhichsubstantiallyretain ratios for the calculations with low feedback Run B (pre- their identity over a period of >10 Myr. dominantly bound clouds), and the higher feedback case, This highlights that generally the constituent gas in Run C (predominantly unbound clouds). GMCs is likely to change on timescales of Myr. This may In Run C (centre panel), with feedback efficiency of 5 mean that discussing clouds lifetimes, which are thought to per cent, the distribution of aspect ratios has reached an be20-30Myr(Leisawitzetal.1989;Kawamuraetal.2009), equilibriumstate,andisfoundtobesimilartoobservations, maynotmakesense.Acloudseenafter30Myrmaynotbea with a peak at about 1.5. In Run B (left panel), with 1 % counterparttoanycloudpresentatthecurrenttime.Inour efficiency,equilibriumhasyettobeachieved,andmoreand calculations this is only true for the most massive clouds. morecloudshaveaspectratiosofaroundunitywithincreas- Thus we see that the clouds tend to display a variety ing time. As we would expect, the virialised clouds tend to of behaviours. The relatively low mass GMCs undergo fre- haveaspectratiosclosetounity.Theprominentpeakatas- quent collisions, are readily disrupted, and α will change pect ratios of unity does not agree with the observations, accordingly. Even if the cloud becomes bound, it may un- reconfirming our previous conclusions that the simulations dergo another dynamical interaction on a short timescale, with efficiencies of 5 – 10 per cent are in best agreement and become unbound. In contrast the more massive clouds with the observations. undergo less dramatic behaviour. There are relatively few clouds of this mass, so they very rarely undergo collisions with objects of a similar size, whilst above a certain mass they are not so easily torn apart by feedback. 3 DISCUSSION In this paper we have addressed the recent observational evidence that most molecular clouds within the Galaxy are 2.5 The shapes of clouds notgravitationallybound.Thisevidencecontrastswiththe From our calculations with different levels of feedback, we original claims of Solomon et al. (1987), long since propa- obtain distributions of clouds which are predominantly un- gatedintothefield,thatmolecularcloudsareboundandin bound(RunsCandD,with5and10%efficiency)orbound virialequilibrium.Theideathatmolecularcloudsarebound (RunB,with1%efficiency).Wehavedemonstratedthatthe entitieshasalsobeentakenasastartingpointformanythe- observations are most likely fit by a distribution of mainly ories of star formation. unbound clouds. Of course, for star formation to take place it is neces- In addition, we find that, the bound clouds are much sary that some parts of a cloud be self-gravitating and able moreregular,sphericallyshaped,whilsttheunboundclouds toundergocollapse.Whattheobservationsseemtoindicate haveveryirregularshapes.Kodaetal.(2006) carried outa howeveristhatitisnotnecessaryforthecloudasawholeto study of Galactic molecular clouds and determined the as- bedominatedbygravity.Withthisinmind,wepresenthere pect ratios of the clouds. Their results are shown in Fig. 8, simulationsoftheISMinagalaxywithafixedspiralpoten- and indicate the most common aspect ratios are between tial. By including simple heating and cooling of the gas, we (cid:13)c 0000RAS,MNRAS000,1–9 8 C. L. Dobbs, A. Burkert and J. E. Pringle Figure 8. The distribution of aspects ratios of the clouds is shown for the models with 1 % efficiency feedback (Run B, left), and 5 % efficiency feedback (Run C, centre). The distribution of aspect ratios for Galactic clouds is shown on the right (Koda et al. 2006). The clouds for the 5 % efficiency case (centre) reasonably match the observations,although even in this case our clouds are slightly more peaked towards low aspect ratios than the observations. The distribution does not change with time, once equilibrium has been established.With1%efficiency(left),thedistributionevolvestoastrongpeakat1,indefinitecontradictiontotheobservations. are able to identify those parts of the ISM which are dense Ballesteros-Paredesetal.(2010),isthatcollisionsandfeed- enough to represent molecular gas, and so are able to iden- back play a much more important role. tify what would be observed as molecular clouds. We allow Insummary,theideathatallmolecularcloudsaregrav- the parts of those clouds which are sufficiently dense and itationally bound entities is neither observationally viable, sufficiently gravitationally bound to notionally form stars. nor theoretically necessary. It is no real surprise that most Becausesuchcloudsaregenerallyhighlyinhomogeneousen- molecular clouds identified in the Galaxy are globally un- tities, within them there will be some regions (in our simu- bound,andthattherestareatmostonlymarginallybound. lationstypicallyrepresentingonly<30percentofthemass) whicharegravitationallybound,andwithinwhichstarfor- mationtakesplace.Thisstarformationistakentomanifest 4 ACKNOWLEDGMENTS itself as an input of energy and momentum into the sur- rounding gas. The global galactic star formation rate is in We thank a number of people who read through a draft accordance with the results of Kennicutt (1998). of this paper, provided many helpful comments and high- lighted issues which required clarity: Rob Kennicutt, Lee Using this simple, and highly idealised, input physics Hartmann, Mark Heyer, Bruce Elmegreen, Fabian Heitsch, we are able to reproduce both the observed distribution of Javier Ballesteros-Paredes. We thank Ralf Klessen for pro- virial parameters of molecular clouds in the Galaxy (with viding a useful referee’s report. CLD also thanks Jin Koda most of the clouds being unbound) and also the observed for providing the data for Figure 8, right panel, and Ian distribution of cloud shapes (in terms of their aspect ra- Bonnellforhelpfuldiscussions.TheresearchofA.B.issup- tios). We find that the velocity dispersions within clouds ported by a Max Planck Fellowship and by the DFG Clus- are maintained not just by feedback from star formation ter of Excellence “Origin and Structure of the Universe”. but also by collisions between non-homogeneous clouds (cf. Thecalculationspresentedinthispaperwereprimarilyper- Dobbs&Bonnell2007).Howeverwithno,orlittlefeedback, formedontheHLRB-II:SGIAltix4700supercomputerand thecloudsarepredominantlyboundandquasi-spherical(as Linux cluster at the Leibniz supercomputer centre, Garch- foundinRunBandbyTasker&Tan2009),indisagreement ing. Run A was performed on the University of Exeter’s with observations. SGI Altix ICE 8200 supercomputer. Some of the figures We also find that the constituent material of a typi- in this paper were produced using SPLASH (Price 2007), cal cloud only remains within that cloud for a timescale of a visualization tool for SPH that is publicly available at around 10 Myr. Thus for timescale much longer than this, http://www.astro.ex.ac.uk/people/dprice/splash. the concept of a cloud lifetime is no longer meaningful. We note that the properties and lifetimes of clouds de- pend somewhat on the size scales considered. Above some REFERENCES surface density threshold, we would expect to start select- ing bound regions within a GMC, and therefore we would Allen A., Shu F. H., 2000, ApJ, 536, 368 obtainahigherfractionofboundobjects.Wehavenotcon- Ballesteros-Paredes J., 2006, MNRAS, 372, 443 sidered the properties of larger GMAs either. The fraction Ballesteros-Paredes J., Hartmann L. 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