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Draftversion January6,2010 PreprinttypesetusingLATEXstyleemulateapjv.11/10/09 MOLECULAR CLOUD EVOLUTION III. ACCRETION VS. STELLAR FEEDBACK Enrique Va´zquez-Semadeni, Pedro Col´ın, Gilberto C. Go´mez, and Alan W. Watson1 CentrodeRadioastronom´ıayAstrof´ısica,UniversidadNacionalAutónomadeMéxico,CampusMorelia,Apdo. Postal3-72,Morelia, 58089,México Draft version January 6, 2010 ABSTRACT We numerically investigate the effect of feedback from the ionizing radiation heating from massive 0 stars on the evolution of giant molecular clouds (GMCs) and their star formation efficiency (SFE), 1 which we treatas a time-dependent quantity. We considerthe GMCs’ evolutionfromtheir formation 0 2 by colliding warm neutral medium (WNM) streams up to advanced star-forming stages. We find, in agreement with our previous studies, that the star-forming regions (“clouds”) within the GMCs n are invariably formed by gravitational contraction, so that their internal non-thermal motions must a containasignificantcomponentofglobalconvergence,whichdoesnotopposegravity. Afteraninitial J periodofcontraction,thecollapsingcloudsbeginformingstars,whosefeedbackevaporatespartofthe 6 clouds’ mass, opposing the continuing accretion from the infalling gas. The competition of accretion against dense gas consumption by star formation (SF) and evaporation by the feedback, regulates ] A the clouds’ mass and energy balance, as well as their SFE. We find that, in the presence of feedback, the clouds attain levels of the SFE that are consistent at all times with observational determinations G for regions of comparable SF rates (SFRs). However, we observe that the dense gas mass is larger . in general in the presence of feedback, and that the total (dense gas + stars) is nearly insensitive to h the presence of feedback, suggesting that the total mass is determined by the accretion, while the p feedback inhibits mainly the conversion of dense gas to stars, because it acts directly to reheat and - o disperse the gas that is directly on its way to forming stars. r We find that the factor by which the SFE is reduced upon the inclusion of feedback is a decreasing t s function of the cloud’s mass, for clouds of size 10 pc. This naturally explains the larger observed a SFEs of massive-star forming regions. We also∼find that the clouds may attain a pseudo-virialized [ state,with avalue ofthe virialmass verysimilarto the actualcloudmass. However,this state differs 1 fromtruevirializationinthatthecloudsarethecenterofalarge-scalecollapse,continuouslyaccreting v mass,ratherthanbeingequilibriumentities. Finally,wealsocalculatethedensityprobabilitydensity 2 functions of the clouds, finding that they in general exhibit the bimodal shape characteristic of ther- 0 mally bistable flows, rather than a lognormal form, which is characteristic of isothermal flows. This 8 supports suggestions that low density, atomic gas pervades molecular clouds. We conclude that the 0 generalstate of star-formingregions is likely to be one of gravitationalcollapse,although most of the . mass in a GMC may be not participating of the instantaneous star-forming activity, as recently sug- 1 gested by Elmegreen (2007); that is, that SF is a spatially and temporally intermittent phenomenon, 0 with strong, localized bursts interspersed within much more quiescent gas. 0 1 Subject headings: interstellar matter – stars: formation – turbulence : v Xi 1. INTRODUCTION this case, the SFR of active star-forming sites may be largefor brief periods, and then halted by the very stars The evolution of giant molecular clouds (GMCs) is a r that have just been formed. a keyingredientinourunderstandingofthestarformation In the other scenario, the role of stellar feedback process. In particular, the low observed star formation is to drive turbulent motions within the GMC, which efficiency (SFE) at the scale of whole GMCs ( 2%; ∼ oppose its self-gravity, allowing it to remain in near Myers et al.1986)remainsatopicofstrongdebate,with hydrostatic equilibrium for times significantly longer there being two main competing scenarios that attempt than its free-fall time (t ) (Krumholz & McKee 2005; to explain it. These scenarios refer essentially to the ff Krumholz, Matzner, & McKee 2006; Li & Nakamura effectofthe stellarfeedback (mainly frommassivestars) 2006). In this case, the low efficiency of star formation on the star-forming clouds. One is the scenario that the wouldbeduetothedualroleofsupersonicturbulencein stars quickly disrupt their parent clouds by dispersal self-gravitatingclouds,ofopposingglobalcollapseofthe and/or photoionization, before the gaseous mass of the cloudwhilepromotinglocalcollapseofturbulentdensity cloud is completely converted to stars (e.g. Whitworth enhancements, which involve small fractions of the 1979; Elmegreen 1983; Franco, Shore & Tenorio-Tagle total cloud mass (Klessen, Heitsch & Mac Low 2000; 1994; Williams & McKee 1997; Va´zquez-Semadeni, Ballesteros-Paredes,& Klessen Hartmann, Ballesteros-Paredes& Bergin 2001). In 2003, see also the reviews by Mac Low & Klessen 2004; [email protected], [email protected],[email protected],[email protected] al. 2007). 1Also: Instituto de Astronom´ıa, Universidad Nacional Another controversy, related to the control of the AutónomadeMéxico,A.P.70-264,04510, México,D.F.,México SFE, refers to the nature of the motions originating the 2 Va´zquez-Semadeni et al. linewidths observed in GMCs and their substructure. (Va´zquez-Semadeni et al. 2007; Banerjee et al. 2009), The latter were initially proposed to correspond to but including a prescriptionfor stellar feedback mimick- gravitational contraction by Goldreich & Kwan (1974), ingionizationheatingfrommassivestars. Withthistool, but this suggestion was quickly deemed untenable by we investigate the effect of the feedback on the global Zuckerman & Palmer (1974), who noted that it would SFE of the evolving GMC, as well as the nature of the implytotalGalacticSFRsoftheorderofthetotalmolec- motions in the cloud, in a first effort to shed light on ular gas mass in the Galaxy ( 109M⊙) divided by the these issues. As we shall see, it turns out that the phys- ∼ typicalfree-falltime foraGMC ( 4Myr),or 250M⊙ ical conditions in the clouds differ significantly from the yr−1,anestimateroughlytwoord∼ersofmagnitu∼delarger “normal” picture, since accretion of gas from the warm than the observed Galactic SFR. Zuckerman & Evans diffusemediumisanintegralpartoftheclouds’dynamics (1974)thensuggestedthattheobservedlinewidthscould andevolution, and thus the clouds cannot be considered correspond instead to random, small-scale2 turbulent as isolated. motions, a notion that has prevailed until the present. The plan of the paper is as follows. In 2 we describe § However, a number of workers have recently advocated the numerical code, and the implementation of our star a return to the gravitational contraction picture, noting formation and stellar feedback prescriptions. In 3 we § that various observational properties of clouds and describethesimulations,andin 4wedescribetheresults § clumps can be well matched by models dominated by concerningthecontroloftheSFEbystellarfeedbackand gravitational contraction (e.g., Hartmann & Burkert the nature of the “clouds” themselves. Finally, in 6 we § 2007; Peretto, Hennebelle & André 2007; present a summary and some conclusions. Va´zquez-Semadeni et al. 2009). Moreover, the notion of completely random, small-scale turbulent 2. THENUMERICALMODEL motions appears difficult to reconcile with the recent 2.1. Heating and cooling realization that the principal component of the velocity The numericalsimulations used inthis work wereper- differenceswithincloudsandclumpsatallscalesappears formedusingthehydrodynamics+N-bodyAdaptiveRe- to be “dipolar”, indicative of coherent motions at the finementTreecodeART(Kravtsov et al.1997;Kravtsov scale of the whole cloud or clump (Heyer & Brunt 2007; 2003). Among the physical processes implemented in Brunt et al.2009). Ingeneral,severalstudies comparing ART, relevant for our physical problem, are the radia- simulations and observations have concluded that the tive heating and cooling of the gas, its conversion into motions in molecular clouds are consistent with large-, stars, ionization-like heating from stellar feedback, and rather than small-scale driving (Ossenkopf & Mac Low self-gravity,from both the stars and the gas. 2002; Brunt 2003; Padoanet al. 2009). We use heating (Γ) and cooling (Λ) functions of the If a return to the collapsing scenario is to be form considered, it is necessary to somehow avoid the Zuckerman & Palmer (1974) criticism of it. This is actually not so difficult because that criticism neglects Γ=2.0 10−26 erg s−1 (1) the internal structure of the GMCs. Recent numeri- × Λ(T) 1.184 105 cal studies of cloud formation by converging streams of =107exp − × warm neutral gas in the interstellar medium show that Γ (cid:18) T +1000 (cid:19) the clouds are born turbulent, due to one or more of 92 thethermal,thin-shell,andKelvin-Helmholzinstabilities +1.4 10−2√T exp − cm3. (2) × (cid:18) T (cid:19) (Hennebelle & Pérault 1999;Koyama & Inutsuka 2000; Koyama & Inutsuka 2002; Audit & Hennebelle 2005; Thesefunctionsarefitstothevariousheatingandcooling Heitsch et al.2005; Va´zquez-Semadeni et al. 2006). The processes considered by Koyama & Inutsuka (2000), as turbulence is subsonic with respect to the warm gas, givenby equation (4) of Koyama & Inutsuka (2002). As but supersonic with respect to the cold phase, imply- notedinVa´zquez-Semadeni et al.(2007,hereafterPaper inglargedensityfluctuationsinthelatter. Infact,ithas I), eq. (4) in Koyama & Inutsuka (2002) contains two recently been proposed that molecular clouds may actu- typographical errors. The form used here incorporates ally contain a warmer, atomic substrate in which colder the necesary corrections,kindly provided by H. Koyama molecularclumpsareembedded(Hennebelle & Inutsuka (2007, private communication). With these heating and 2006). Ineithercase,themolecularcloudcontainslarge, coolinglaws,thegasisthermallyunstableinthedensity nonlinear density enhancements in which the local free- range 1.n.10 cm−3 (cf. Paper I). falltime issignificantlyshorterthanthecloud’saverage. Thus, once the global collapse begins, the local clumps 2.2. Star formation and stellar feedback prescriptions maycomplete their collapsesearlierthanthe bulk ofthe cloud. Theycanthus beginformingstarsthatcanbegin In ART, star formation is modeled as taking place in their feedback action on the GMC before it completes the coldest and densest regions,defined by T <TSF and the bulk collapse. n > nSF, where T and n are the local temperature and In this paper, we present numerical simulations numberdensityofthegas,respectively,andnSF andTSF aimed at investigating this scenario, in which we use are respectively a density and a temperature threshold. the same cloud-formation setup of previous papers We set TSF = 9000 K, which is easily satisfied by all cells with density n , so in practice our SF condition SF 2 Zuckerman&Evans (1974) referred to these motions as “lo- depends on density only. cal”,andexplicitlydiscardedlarge-scalecoherentmotionssuchas A stellar particle of mass m∗ is placed in a grid cell gravitationalcontraction. where these conditions are simultaneously satisfied, and Molecular Cloud Evolution III. Accretion vs. Stellar Feedback 3 thismassisremovedfromthegasmassinthecell. There- the expense of a somewhat ad-hoc SF prescription. We after, the particle is treated as non-collisional, and fol- show a typical HII region in our simulation in Fig. 1. lows N-body dynamics. No other criteria are imposed. In each gas cell that satisfies the above criteria a stel- 2.3. Refinement lar particle is formed with a mass equal to 50% of the Thenumericalboxisinitiallycoveredbyagridof1283 gasmasscontainedin the cell. Since the stellarparticles (zeroth level) cells. The mesh is subsequently refined as are more massive than a single star, each stellar parti- the matter distribution evolves. The maximum allowed cle shouldbe consideredasa small cluster,within which refinement level was set to four, so that high-density re- theindividualstellarmassesaredistributedaccordingto gions have an effective resolution of 20483 cells, with a some stellar initial mass function (IMF). minimum cell size of 0.125 pc. Cells are refined (halved Stellar particles inject thermal energy at a rate E˙ erg in linear size) if the gas mass within the cell is greater Myr−1 per star with mass greater than 8 M⊙ contained than0.32M⊙. Thatis,thecellisrefinedbyafactorof2 in the stellar particle. We assume a Miller & Scalo whenthedensityincreasesbyafactorof8,sothat,while (1979) IMF, implying that each stellar particle of mass refinementisactive,thegridcellsize∆xscaleswithden- 133 M⊙ produces one 8-M⊙ star. The energy is de- sity n as ∆x n−1/3. Once the maximum refinement positedinthecellinwhichthestellarparticleisinstanta- ∝ level is reached, no further refinement is performed, and neouslylocated,overatypicalOBstellarlifetime, which the cell’s mass can reach much larger values. In par- we assume to be 10 Myr. ticular, a stellar particle is formed when a fourth-level It is important to note that, although initially we ex- cell reaches a density n = 4 106 cm−3, or a mass of perimented with realistic values of E˙ based on the Ly- 243.5M⊙,againassuminSgFµ=1×.27(weusethisvaluebe- man continuum fluxes of stars with masses between 10 causewedonotfollowtheactualchemistry,andthuswe and20M⊙ (e.g.,Diaz-Miller et al.1998),wefoundthat, assumetheentireboxtobefilledwithatomichydrogen.) because all the energy is deposited in a single cell, and Thus, a stellar particle typically has a mass &122M⊙. the neighboring cells are heated exclusively by conduc- Note that, because we use only four levels of re- tion, rather than by radiative heating, the resulting HII finement, the largest densities arising in the simula- regions were not so realistic. Thus, we opted instead for tion are by far not sufficiently resolved according to the taking E˙ as a free parameter, and adjusting it until we “Jeans criterion” proposed by Truelove et al. (1997) for obtained realistic HII regions, with temperatures 104 adaptive-mesh codes. Specifically, at our stellar-particle K, diameters of a few parsecs, and expansion velo∼cities formationthresholddensityof4 106 cm−3,andassum- of a few tens of km s−1. ing a gas temperature of T 1×5 K at that density, we ∼ Note also that we resort to the common strategy of find that the Jeans length (using the adiabatic sound turning off the cooling in the cell where a stellar par- speed) is 0.031 pc, while the minimum cell size, at ∼ ticle is located, so that the cell can reach realistically 0.125 pc, is roughly 4 times larger. Thus, according to high temperatures. Otherwise, the cooling can dissipate thoseauthors,oneshouldexpectartificialfragmentation mostofthethermalenergydepositedinverydensecells. to occur in our simulations. However, we do not con- In the real ISM this does not occur because the stellar sider this to be a problem because we are not concerned heating is applied through ionization, so that the tem- here with the numbers and masses of the stellar parti- perature reached in the star’s immediate environment is cles formed in the simulation, but simply with the total independent of the medium’s local density. Instead, in amount of mass that goes into stars. the simulations, the cooling depends on the density, and the temperature resulting from the balance between the 3. THESIMULATIONS stellar heating and the cooling does depend on the den- We consider four simulations using the same setup as sity. This problem is avoided by turning off the cooling in Paper I, which represents the evolution of a region of in the cell where the stellar particle is located. Note 256pcperside,initiallyfilledwithwarmgasatauniform thatthis contradictsclaims that the need to turnoff the density ofn =1 cm−3 and atemperature T =5000K, 0 0 coolingcanbealleviatedsimplybyincreasingtheresolu- implying an adiabatic sound speed c = 7.4 km s−1 (as- s tion (e.g., Ceverino & Klypin 2009). We argue that this suming a mean particle mass µ = 1.27). The whole nu- problem can only be alleviated by performing radiation- mericalboxthuscontains5.25 105M⊙. Inthismedium, hydrodynamics simulations. In their absence, we con- we set up two streams movi×ng with the same speed sider that turning off the cooling is actually a better v = 5.9 km s−1 (corresponding to a Mach number of model of the effect of stellar feedback, because it allows inf 0.8withrespecttotheunperturbedmedium)inopposite mimicking the fact that the gas temperature in the HII senses along the x-direction. The streams have a radius regions is independent of the local density. of 32 pc and a length of 112 pc each, so that the total Finally, in order to further constrain the physical con- mass in the two inflows is 2.25 104M⊙. The flows col- ditions in the HII regions, we also impose a “ceiling” to × lide head on at the box’s center (see Fig. 1 of Paper I). thetemperatureinthecellcontainingthestellarparticle To the inflow velocity field we superpose a field of initial because otherwise,with the cooling off, the temperature low-amplitudeturbulentvelocityfluctuations,inorderto in the cellmight diverge. We set this “ceiling”to 106 K. trigger the instabilities in the compressedlayer. We cre- Although this procedure is mostly one of trial-and- ate this initial velocity fluctuation field with a new ver- error,weconsiderittobe the mostadequateoneforour sionofthespectralcodeusedinVazquez-Semadeni et al. purposes, since it is the HII regions that drive the tur- (1995)andPassotet al.(1995),modifiedtoruninparal- bulent motions in the dense gas in our simulations, and lel in shared-memory architectures. The simulations are so it is them that must have realistic properties, even at evolved for about 40 Myr. 4 Va´zquez-Semadeni et al. Fig.1.—Cross sections of the density (left panel), temperature (middle panel) and y-speed (right panel), shown onthe x-z plane, of a typicalisolatedHIIregion. Thescalebarnearthetopindicates lengthinparsecs. TABLE 1 Runparameters Run vrms Feedback name [kms−1] LAF0 1.7 off LAF1 1.7 on SAF0 0.1 off SAF1 0.1 on The collision nonlinearly triggers a transition to the cold phase, forming a turbulent, cold, dense cloud (Audit & Hennebelle 2005; Heitsch et al. 2005; Heitsch et al 2006; Va´zquez-Semadeni et al. 2006), con- sisting of a complex network of sheets, filaments, and Fig.2.—Viewinprojectionofthewholenumericalboxforsim- clumpsofcoldgasembeddedinawarmdiffusesubstrate ulationLAF1att≈31.64Myr. Theboxsizeis256pc. Thegreen (Audit & Hennebelle2005;Hennebelle & Inutsuka2006; dotsindicatestellarparticles. Thelightyellowspotsaretransient Hennebelle & Audit 2007). The largest cold structures dense cores, highlighted by saturation of the color table. In the may become gravitationally unstable and begin to col- electronicversion,thisfigureshowsananimationoftheentireevo- lutionof thesimulationuptot=40Myr. Records arespaced by lapse. Eventually, they proceed to forming stars, which atimeinterval∆t≈0.14Myr. thenheattheirenvironment,formingexpanding“HIIre- gions” that tend to disperse the clumps. collision (Rosas-Guevara et al. 2010) shows that the co- In the simulations reported here, we vary only two herence of the collapse may be lost in the presence of parameters: the amplitude of the initial velocity fluc- stronger initial velocity fluctuations, and the SFE is de- tuations and whether the stellar feedback is on or off. creased. In such cases, smaller clouds appeared to be We consider a “large-amplitude” (LA) and a “small- lessstronglygravitationallybound,withtheeffectofde- amplitude”(SA)casefortheinitialvelocityfluctuations, creasing the SFE. This feature also happens in our LA for which the three-dimensional velocity dispersions are v 1.7 km s−1 and v 0.1 km s−1, respectively. simulations,inwhichthecloudformedbytheinitialflow rms ∼ rms ∼ wasmuchmore fragmentedandscatteredoverthe simu- We thus employ a mnemonic nomenclature for the runs lationvolume. Asaresult,SFalsooccursinamuchmore using the acronyms LA or SA, followed by F0 or F1, in- scattered manner, and the SFEs are in general smaller dicating that feedback is off or on, respectively. Table 1 in the LA runs than in their SA counterparts. summarizes the runs considered in the paper. However, in general a common pattern is followed by 4. RESULTS allsimulations: thetransonicconvergingflowsinthedif- fuse gas induce a phase transition to the cold phase of 4.1. Evolution of the simulations the atomic gas, which is highly prone to gravitational The simulations performed here behave very similarly instability. This can be seen as follows. The thermal to previous simulations with similar setups, performed pressure at our initial conditions is 5000 K cm−3. From with other codes. In particular, our SA runs are very Fig. 2 of Paper I, it can be seen that the thermal bal- similar to run L256∆v0.17 in Paper I and the run pre- ance conditions of the cold medium at that pressure are sented by Banerjee et al. (2009). The main feature of n 130 cm−3, T 40 K. At these values, the Jeans ∼ ∼ these runs is that, because the initial fluctuations are length and mass are 7 pc and 640M⊙, respectively. ∼ ∼ very mild, the flow collision creates a large, coherent These sizes and masses are easily achievable by a large “pancake” of cold, dense gas, which is able to undergo fraction of the cold gas structures, which can then pro- gravitational collapse as a whole. This results in the ceed to gravitationalcollapse and form stars. Moreover, formation of a dense, massive, and turbulent region at theensembleoftheseclumpsmayalsobegravitationally the site where the global collapse finally converges,with unstable as a whole, the likelihood of this being larger physical properties similar to those of high-mass star for greater coherence of the large-scale pattern. forming regions (Va´zquez-Semadeni et al. 2009). How- Regions of active star formation form in both sets of ever, a recent study varying the parameters of the flow simulations by the gravitational merging of pre-existing Molecular Cloud Evolution III. Accretion vs. Stellar Feedback 5 Fig. 4.— Cross-section view through Clouds 1 (top panel) and 2 (bottom panel) at t ∼ 35 Myr in simulation LAF1. The plane of view is located at x = 100 pc for Cloud 1 and at x = 150 pc for Cloud 2. The dots represent the stellar particles. In the electronic edition, these figures show animations of the evolution ofbothcloudsfromt=23tot=40Myr. Fig.3.— Cross-section view through the Central Cloud in run SAF1 at t≈ 33 Myr, at which time it has grown to a mass of ∼ 3×104M⊙. Theplaneoftheimageisshownfromaninclinedlineof sightforbetterperspective. Thecloudisseentocontainnumerous HIIregionsmixedwithdenseregions. Intheelectronicversion,this figureshowsananimationofthebuild-upofthiscloud,illustrating how itformsbythecontinued accretionof infallingmaterialfrom the globally collapsing GMC. In the animation, note that many of the infalling clumps form stellar particles before reaching the 1000 center, and are disrupted by the local stellar feedback. However, oncetheCentralCloudisfullyassembled,itresistsdispersal,and formsstarsatahighrate. 100 smaller-scale clumps, which, altogether, form a larger- scale GMC. Figure 2 shows a whole-box image of the SAF0 SAF1 density field in run LAF1, in projection. In the electronic version of the paper, this figure shows an animation of this run from t = 0 to t 40 Myr, illustrating ≈ theentireevolutionofthe simulation,fromtheassembly ofthe cloud, to its advancedstar-formingepochs. Inthe 1000 animation, subsequent “records” are separated by time intervals ∆t 0.14 Myr. ≈ IntheSAruns,thelargeststar-formingregionformsin 100 thecenterofthesimulation,duetothecoherentcollapse 0 10 20 30 400 10 20 30 40 oftheentiresheet-likecloudformedbythecollision. This t(Myr) t(Myr) wasthe regionshownin Va´zquez-Semadeni et al.(2009) toexhibitphysicalconditionstypicalofactualhigh-mass Fig.5.— Evolution of the dense gas mass and the stellar mass for the SA simulations, without (left panels, run SAF0) and with star-formingregions. Werefertothisregionas“theCen- (rightpanels,runSAF1)feedback. Thesolidlinesrefertothetotal tral Cloud”. Figure 3 shows a view of this region at massesinthecomputationalbox,whilethedottedlinesrefertothe t 33Myr,a time atwhich the centralcloudhas grown massesinacylinderwithlengthanddiameterof10parsecswithits to≈a mass of nearly 3 104M⊙ in run SAF1 (cf. Fig. 5). axisalongthex-direction,andwhichcontains theCentralCloud. × InthecaseoftheLAruns,becausestarformationoccurs in a much more scattered fashion, we study two of the box, and the dotted, short-dashed, and long-dashed lines regions exhibiting the strongest star formation activity, representthemassescontainedincylindersoflengthand neitherofwhichislocatedatthe centerofthe numerical diameter 10, 20, and 30 pc, respectively, enclosing the box. These are shown in Fig. 4, and we refer to them as clouds. We do this because the clouds have verycompli- Cloud 1 and Cloud 2. cated morphologies,with filaments that extend out over tens of parsecs and connecting with other clouds (Fig. 4.2. Effect of stellar feedback on the SFE and on the 4), thus making it virtually impossible to fully enclose clouds’ evolution the “clouds” in any given cylindrical volume. It is seen from Figs. 5-7 that the inclusion of feedback Our main interest in this contribution is the effect of (right panels)causesthe dense gasmasstobe larger and the feedback on the efficiency of the star formation pro- the stellar mass to be smaller than in the case without cess, and the identification of the mechanism through feedback in general, even though the total cloud mass which this effect is accomplished. Figure 5 shows the (dense gas+ stars)inthe simulationsis nearlythe same evolution of the dense gas mass and the stellar mass in inboththecaseswithandwithoutfeedback(Fig.8). As these simulations, with (right panels) and without (left a consequence, the instantaneous SFE, defined as panels)feedback. Thesolidlinesrefertothetotalmasses in the computational box, while the dotted lines refer to M∗(t) SFE(t)= , (3) the masses in the Central Cloud. Figures 6 and 7 show Mdense(t)+M∗(t) the corresponding plots for Cloud 1 and Cloud 2. Here, thesolidlinesrepresentthemassesforthefullsimulation where M (t) is the mass of the dense (n&50 cm−3) dense 6 Va´zquez-Semadeni et al. 1000 1000 100 100 LAF0 LAF1 LAF0 LAF1 1000 1000 100 100 0 10 20 30 400 10 20 30 40 0 10 20 30 400 10 20 30 40 t(Myr) t(Myr) t(Myr) t(Myr) Fig.6.— Evolution of the dense gas mass and the stellar mass Fig.7.— Evolution of the dense gas mass and the stellar mass forCloud 1in the LA simulations, without (left panels) and with forCloud 2inthe LA simulations, without (left panels) and with (right panels) feedback. The solid lines refer to the total masses (right panels) feedback. The solid lines refer to the total masses in the computational box. The other lines refer to the masses in in the computational box. The other lines refer to the masses in cylindersoflengthanddiameter10pc(dottedlines),20pc(short- cylindersoflengthanddiameter10pc(dottedlines),20pc(short- dashed lines),and30pc(long-dashed lines). dashed lines),and30pc(long-dashed lines). gasinthe simulation,andM∗(t)is the stellarmass,nat- plitude of the initial turbulent fluctuations. beginfigure urally decreases upon the inclusion of feedback in both sets of simulations(Fig. 9). Note thatin eq. (3) we have In order to compare the SFEs in our simulated clouds explicitly written out the time dependence of the quan- with those of real molecular clouds, it is convenient to tities involved. note that regions of low-mass star formation, such as The SFE is seen to be reduced by a larger factor ( ∼ Taurus (Goldsmith et al. 2008) or the Chamaeleon II 10 )in the case of the LA runs, in which the collapse is × dark cloud (Spezzi et al. 2008) generally have low SFEs less focused and less massive, than in the case of the SA ( 1-5%), while cluster forming cores have SFE 30- ones ( 3 ), in which the opposite is true. In addition, ∼ ∼ ∼ × 50% (Lada & Lada 2003). Thus, we can check whether inFig.10weshowtheevolutionoftheSFEatthelevelof theclouds. TheleftpanelshowstheevolutionoftheSFE the SFEs and SFRs of our three clouds follow the same forthe CentralCloudintheSAruns. Themiddle panels trend. Figure12showstheevolutionoftheSFRsforour three clouds. From these plots, we find that the average showthe correspondingplotsforCloud1andCloud2in the LAF0 run (without feedback), and the right panels SFR ofthe CentralCloud startingfrom t=32 Myr (the time atwhicha large,roughlystationarySFR sets in) is show the SFEs in the LAF1 run (with feedback). Again we see a trend for the less massive cloud (Cloud 1) to SFR 1450 M⊙ Myr−1, while for Clouds 1 and 2, we sufferagreaterreductionofitsSFE(byafactorof 20, hfind iSF∼R 30 and 60 M⊙ Myr−1 during their entire from 60-90% to 3-4%) than the more massiv∼e one star-fhorminig∼stages, respectively. From here, and using (Clou∼d2,byafacto∼rof 10,from 70-80%to 7-8%). the instantaneous SFE at t = 40 Myr from Fig. 10, we We discuss the possible∼causes for∼this mass-de∼pendent can then plot the SFE versus the SFR. This is shown effect of the feedback in Sec. 4.3.2. in Fig. 13, in which both the SFR and the SFE have ThefactorbywhichtheSFEisreducedupontheinclu- been multiplied by an extra factor of 0.5, to respresent sion of the feedback in the simulations is plotted versus the fact that the efficiency within the stellar particles, the system’s mass inFig. 11, both for the full amountof which are created at a density of n = 4 106 cm−3 in × densegasinthesimulations,andforeachofthecloudswe our simulations, is still smaller than unity. We take 0.5 havebeenconsidering. Weseethattwosetsofpointsare as a representative value. clearly defined in this plot, one for the clouds, and one We see that the Central Cloud has values of the for the simulations. In both cases, however, the trend SFR and the SFE comparable to those of cluster- of larger reduction factor at smaller dense gas mass is forming cores (Lada & Lada 2003). Specifically, clearly observed, although, at the level of simulations, Va´zquez-Semadeni et al. (2009) estimated an SFR & we see that their masses are not very different. Thus, 250M⊙ Myr−1 for the Orion Nebula Cluster (ONC). in this case the different reduction factors must include This calculation used an estimated age spread of the a contribution from the larger degree of fragmentation ONC of . 2 Myr (Hillenbrand 1997), and Tobin et al. occurring in the LA simulations due to the larger am- (2009)’s result of there being 1613 stars in the ONC. Molecular Cloud Evolution III. Accretion vs. Stellar Feedback 7 Taking this number as a proxy for the total stellar production of this region, and a mean stellar mass of 0.3 M⊙ (Hillenbrand & Carpenter 2000), this implied a total stellar mass of 500M⊙. On the other hand, ∼ Krumholz & Tan(2007)quoteatotalstellarmassinthe ONCof 4600M⊙(Hillenbrand & Hartmann1998)and ∼ anage spreadof 3 Myr (Tan et al. 2006), implying an SFR 1530M⊙ ∼Myr−1. These values bracket the SFR ∼ we measured for our Central Cloud. Concerning the SFE, Fig. 13 shows that the Central Cloud has SFE 10%, which is comparable to that of ∼ theOrionAcloud(Carpenter2000),inwhichthe OMC- 1 clump is contained. Thus, our Central Cloud may be compared to the Orion A cloud, and its dense core, dis- cussedinVa´zquez-Semadeni et al.(2009),iscomparable to the OMC-1 clump. Ontheotherhand,Clouds1and2areseeninFig.13to haveSFRscomparabletothoseoflow-massstar-forming clouds, 10-100 M⊙ Myr−1, such as Taurus, Cha II or ∼ Lupus, in which the SFE is known to be lower, 1-4% ∼ (Spezzi et al. 2008). This compares very well with the SFEswereportforourClouds1and2inthesamefigure. We conclude that Clouds 1 and 2 are good models of low-mass star-forming regions, while the Central Cloud is a good model of a massive SF region, as discussed in Va´zquez-Semadeni et al. (2009). 4.3. The physical processes acting on the clouds 4.3.1. Cloud “destruction” Fig.8.—Evolutionofthetotal(densegas+stars)massforthe four simulations. TheSA simulations areshown inthe top panel, Up to the 40-Myr time to which we have evolved our whiletheLArunsareshowninthebottompanel. Theblack, solid simulations,the three largeclouds(the CentralCloudin lines refer to simulations with feedback and the red, dotted lines runSAF1andClouds1and2inrunLAF1)donotshow represent runs without it. The colors are shown in the electronic anyinstancesofthe densegasmassreversingitsincreas- versiononly. ingtrendandbeginningtodecreaseduetothefeedback, as can be seen in Figs. 5 through 7. Apparently, the HII region-like feedback we use is unable to overwhelm 1 the largegravitationalpotential well of these clouds and their enveloping “atomic” gas reservoirs. Instead,completedestructionseemstobeabletooccur in small clumps. This can be seen in the animation corresponding to Fig. 3, in which various small clumps are seentobedestroyedbytheirstellarproducts. Threepar- ticularly conspicuous ones are, first, the one that forms 0.1 a stellar particle at the very starting frame (record 179) of the animation, slightly above and to the right of the screen’s center. Next, another stellar particle appears in the clump almost at the left border of the frame, about 2/3 of the way from bottom to top, at record 187. Fi- nally, a third particle appears at record189, slightly be- 0.01 lowandtotherightofthescreen’scenter,astheresultof the collisionof twoclumps. Inall ofthese cases,a single stellar particle is formed ( 120M⊙), and the clump is LAF0 ∼ LAF1 destroyed. Itisworthnotingthatactually,theexpansion SAF0 ofthe HII regionsformedproducesnew clumpsfromthe SAF1 materialcollectedaroundit,butthesenewclumpseither 0.001 disperse, or simply do not form new stars. Thus, we conclude, similarly to 20 25 30 35 40 Krumholz, Matzner, & McKee (2006), that small t(Myr) clouds (“clumps”) are rapidly destroyed, while large clouds may survive for longer times. However, our Fig.9.—Evolutionoftheinstantaneousstarformationefficiency clouds exhibit a fundamental difference with respect to (SFE), as defined ineq. (3), inthe fullsimulationboxinthefour the model considered by those authors, namely that the runs. clouds are accreting in general. In the next section we 8 Va´zquez-Semadeni et al. Fig.10.— Evolution of the instantaneous star formation efficiency (SFE) in the dense clouds in the simulations. The left panel shows the SFE for the Central Cloudof the SA runs,with and without feedback. Themeasurements referto acylindrical box witha diameter andalengthof10pc. Themiddle panelsshowtheSFEforClouds1and2intheLAF0simulation(withoutfeedback), forthreedifferent cylindrical boxes, of length and diameter indicated in the labels. The gaps in the curves for the smaller cylindrical boxes correspond to timeswhenthestellarparticlesmigrateoutofthem,andnonewparticleshavebeenformed. TherightpanelsshowtheSFEforClouds1 and2intheLAF1simulation(withfeedback). Fig.11.— Reduction factor of the SFE at 40 Myr upon inclusion of the feedback in the simulations, both for the three clouds aswellasfortheentiremassofdensegasinthesimulations. The errorbarsinthe massesindicate the rangeof values taken bythe systems once their initial assembly has finished (see Figs. 5-7). The error bars in the reduction factor for Clouds 1 and 2 denote the maximum and minimum values of the ratio SFE(w/o feedback)/SFE(w/feedback), usingtheSFEdataforthethreecylinder sizesof10,20and30pc. now discuss this feature. 4.3.2. Accretion vs. feedback One crucial feature in all our simulations is that the clouds are accreting material from the surrounding diffuse medium. This is fundamentally different frommod- elsinwhichthecloudsareisolatedentities,inroughbal- ance between their self-gravity and the turbulent pressure, possibly driven by the stellar feedback. The accre- tioncompetes withSFandstellarfeedbackinregulating Fig.12.—Evolutionofthestellarparticleformationrate, aver- agedover2-Myrintervals,oftheCentralCloud(toppanel),Cloud the cloud’s mass and coherence, with important conse- 1 (middle panel) and Cloud 2 (bottom panel). For this plot we quences. First of all, this implies that simple observa- onlyusethe10-pccylindricalvolumesforClouds1and2. Because tional estimates of the SFE in GMCs based on measur- ourstellarparticlesformatadensityofn=4×106 cm−3,actual starformationratesshouldbeafactorof2-3lower. Gapsindicate ing the stellar mass and dividing it by the cloud’s mass periodsoverwhichnonewstarsareformed. may be failing to take into account the additional “raw material” for SF contained in the part (or the whole) of the atomicenvelopeofthe cloudsthatwilleventuallybe incorporated into the GMC. Second, the competition between feedback and accre- Molecular Cloud Evolution III. Accretion vs. Stellar Feedback 9 factthatstellarfeedbackislocalized,whiletheaccretion isextended, andmoresoforgreatermassofthe globally gravitationally unstable region that will form the cloud. Thus, as we observe in the animation corresponding to Fig. 3, the low-mass fragments that are undergoing SF whileontheirwaytothecentralsiteoftheglobalcollapse inrunSAF1areeasilydestroyedbytheirstellaractivity. Instead,themassiveCentralCloudisnotdestroyed,asit continues to accrete mass from large distances at a pace that outweighs the local destruction by stellar feedback. Clouds 1 and 2 in run LAF1, which do not involve such an extended collapse, are intermediate cases, in which the clouds are not destroyed by the feedback, but the latter is more efficient in reducing the SFE. Thus, we can conclude that the efficacy of the feedback in destroying the cloud is maximal when its region of influence is comparable to the spatial extent of the Fig.13.—SFEversusstellar-particleformationrateforthethree infall, and is progressively reduced as the latter involves cloudsinthesimulations,usingthedatafromFigs.10and12. The progressivelylarger coherence lengths. values of both the SFR and the SFE have been multiplied by a factor of 0.5, representative of the still-lower-than-unityefficiency 4.4. Physical conditions in the clouds withinourstellarparticles. Here we consider the global physical properties of the tion may explain our observation from Sec. 4.2 that clouds. We postpone a discussion of the properties of cases with feedback are characterized by larger dense individual clumps within the clouds for a future study, gas masses and smaller stellar masses than their coun- to be performed at higher resolution. One such study, terparts without feedback. The smaller stellar mass is including magnetic fields, although not including stellar not surprising, as the obvious effect of stellar feedback feedback, has recently been presented by Banerjee et al. is to reheat the cold, collapsing star-forming gas, thus (2009). reducing the SFR. However, the larger cold gas mass in the presence of feedback is indeed surprising, since both 4.4.1. Density PDFs gas consumption by SF and the “ionization” by stellar feedback act to reduce the dense gas mass. Our result It is important to determine the physical conditions implies that the rate of dense gas consumption by SF in our clouds in order to assess their degree of realism. far outweighs its rate of destruction by stellar feedback, One basic diagnostic is the probability density function so that the net effect of reducing the SFR is to allow a (PDF)ofthedensityfield. Althoughitiswellestablished larger amount of dense gas to be collected by the accre- thatin isothermalflows the density PDFtakes a lognor- tion. This scenario is supported by Fig. 8, which shows mal form (Va´zquez-Semadeni 1994; Padoan et al. 1997; the total (dense gas + stars) in the clouds in the two Passot& Va´zquez-Semadeni 1998; Nordlund & Padoan setsofsimulations. Itisseenthatthetotalcloudmassis 1999; Ostriker et al. 1999, 2001; Federrath et al. 2008), nearly the same with and without feedback, suggesting for non-isothermal flows the expectation is in gen- thatthetotal cloud mass is mainly determined bythe ac- eral different, with a near-power-law tail develop- cretion, whiletheratioofdensegastostellar massseems ing at high densities for flows softer than isother- to be mainly determined by the feedback. mal (Passot & Va´zquez-Semadeni 1998; Scalo et al. These results moreover support the suggestions by 1998; Nordlund & Padoan 1999; Gazol et al. 2005; Elmegreen (2007) and Va´zquez-Semadeni et al. (2009) de Avillez & Breitschwerdt 2004, although see, e.g., that the majority of the gas in a GMC is not partici- Wada & Norman 2001, 2007), and a bimodal form aris- pating of the SF process at any given time. The latter ingforthermallybistableflows(Va´zquez-Semadeni et al. authors suggested that this is how local regions of SF 2000; Gazol et al. 2005; Audit & Hennebelle 2005, see mayhaveaveryhighspecificSFR( (10Myr)−1),while also the review by Va´zquez-Semadeni 2009). ∼ that of their whole parent GMCs may be much smaller Figure 14 shows the density PDFs for the whole sim- ( (300 Myr)−1; McKee & Williams 1997). Because it ulation box of the SA runs top panels and of the LA is∼injectedbythenewlyformedstars,thestellarfeedback runs bottom panels. The right panels show the entire actspreferentiallyongasthatisabouttoformstarsnext. density range, while the left panels show the PDF for This allows an efficient suppression of SF, through only the dense gas (n 100 cm−3) only. The whole-range ≥ a modest fraction of the total available dense gas being PDFsexhibitthebimodalitytypicalofthermallybistable destroyed. That is, if SF is a highly localized process, flows, although the high-density tail is seen to exhibit thenitispossibleto achievesignificantreductionsofthe an excess over the power-law in both the cases with SFRwithonlyamodestreductionofthetotalamountof and without feedback. This is probably due to the ac- dense gas, by targeting the destruction precisely at the tion of self-gravity (Klessen 2000; Dib & Burkert 2005; star-forming gas. Va´zquez-Semadeni et al. 2008). In addition, the cases This mechanism may also explain the trend observed with feedback show a slight excess over the cases within Sec. 4.2, that the SFE is more strongly reduced by out it at densities 103 . n . 105 cm−3, probably due the feedback in cases where the collapsing gas mass is to the formation of compressed regions by the expand- smaller. Thismaybeunderstoodasaconsequenceofthe ing “HII regions”. The high-density PDFs, being just 10 Va´zquez-Semadeni et al. Fig.14.— Density PDFs for the entire simulation boxes of the Fig.15.—DensityPDFsforthethreemaincloudsinthe simu- SA runs (top panels) and the LA runs(bottom panels). Theright lations: theCentralCloudintheSAruns(left panel),andClouds panelsshowthePDFsfortheentirerangeofdensities,whiletheleft 1 (middle panel) and 2 (right panel) in the LA runs. The PDFs panelsshowthePDFsofthedensegasonly(n≥100cm−3). The fortheCentralCloudarecomputedinacylinderwithlengthand red, dotted lines refer to the simulations without feedback, while diameter equal to 10 pc, while the PDFs for Clouds 1 and 2 are theblack,solidlinesshowrepresentthesimulationswithfeedback. computed incylinders oflengthand diameter equal to10, 20and the n > 100 cm−3 tail of the whole-range PDFs, only 30 pc. Red lines indicate cases without feedback, and are dis- placeddownwardsbyafactorof10forbetterviewinginthecases show more detail of the very-high density gas, up to the ofClouds1and2. Blacklinesindicatecaseswithfeedback. star-formingdensity of 4 106 cm−3, but with the same × shape of the curves as in the whole-range PDFs. is shown in Fig. 17. The solid lines show the density- In order to see the PDFs at the locations of the weighted value, while the dotted lines show the volume three main clouds we have studied (the Central Cloud weighted value. Black lines denote cases with feedback, in the SA runs and Clouds 1 and 2 in the LA and red lines correspond to cases without it. runs), we show in Fig. 15 their corresponding density For the CentralCloud andCloud 2,which are the two PDFs. It is noteworthy that the PDFs again show a mostmassive ones, the density-weightedvelocity disper- roughly power-law shape at high densities over three sion, which highlights the dense gas, decreases upon the to four orders of magnitude in density, in agreement inclusion of feedback. Without feedback, in the Cen- with the expectation for softer-than-isothermal flows tral Cloud, it reaches very high values at the end of (Passot & Va´zquez-Semadeni1998;Nordlund & Padoan the run ( 15 km s−1, corresponding to Mach numbers 1999). This is at odds with results from numerical sim- M 50-∼75), which in fact are significantly larger than s ulations of turbulent isothermalgas in closedboxes, but typi∼cal values for cloud complexes of comparable mass then again our clouds are not isothermal. Instead, they (M & 104M⊙; e.g., Dame et al. 1986; Rathborne et al. are characterized by an effective polytropic equation of 2009). Instead,intherunincludingfeedback,∆ reaches fsotartne P&∝10ρ0γeffc,mw−i3th, aγseffcavnarbyiengsefernomin0F.8igt.o1n6e.arRlyeaulniinty- vmauluchesb∼ett5e-r6akgmrese−m1en(rtmwsitMhatcyhpincuamllybeorb,sMersve∼dv v2a0l)u,eisn. terstellar molecular gas is not expected to be exactly Onthe otherhand,the volume-weightedvelocitydisper- isothermal,either(Scalo et al.1998;Spaans & Silk2000; sion, which highlights the less dense gas, is seen in Fig. Jappsen et al.2005). Moreover,ourcloudsareimmersed 17 to increase upon the inclusion of feedback. In Clouds in a diffuse, warmer medium which, as proposed, e.g., 1 and 2, which are less massive and more scattered than by Li & Goldsmith (2003) and Hennebelle & Inutsuka theCentralCloud,∆ doesnotachieveexceedinglylarge v (2006), infiltrates the clouds to some degree. Thus, the values in any case. density PDFs for the volumescontaining ourclouds also These results can be understood as a consequence of exhibit the bimodal shape characteristic of thermally the fact that the dense gas acquires its largest velocities bistable flows, and have low-density tails that extend in the case of free-fall collapse. However, the collapse down to the WNM regime, contrary to what happens flowis dismantledinits final(fastest) stagesby the stel- in isothermal simulations. larfeedback,sothatitisthehigh-velocitydensegasthat ispreferentiallydestroyedbythe feedback. Onthe other 4.4.2. Velocity dispersions and virial masses hand, this gas becomes warm, diffuse, high-velocity ex- Finally, we investigate the global velocity dispersion panding gas, which is the one highlighted by the volume (∆ ) in the Central Cloud and Clouds 1 and 2. This weighting. These results again reinforce the notion that v