Simulating radiative feedback and star cluster formation in GMCs: II. Mass dependence of cloud destruction and cluster properties
MMon. Not. R. Astron. Soc. , 1–15 (2017) Printed 17 April 2018 (MN L A TEX style file v2.2)
Simulating radiative feedback and star cluster formation inGMCs: II. Mass dependence of cloud destruction andcluster properties
Corey S. Howard (cid:63) , Ralph E. Pudritz , , William E. Harris Department of Physics and Astronomy, McMaster University, 1280 Main St. W, Hamilton, ON L8S 4M1, Canada Origins Institute, McMaster University, 1280 Main St. W, Hamilton, ON L8S 4M1, Canada
13 March 2017
ABSTRACT
The process of radiative feedback in Giant Molecular Clouds (GMCs) is an importantmechanism for limiting star cluster formation through the heating and ionization of thesurrounding gas. We explore the degree to which radiative feedback affects early ( (cid:46) − M (cid:12) using the FLASH code. The inclusion of radiative feedback lowers the efficiency of cluster for-mation by 20-50% relative to hydrodynamic simulations. Two models in particular —5 × and 10 M (cid:12) — show the largest suppression of the cluster formation efficiency,corresponding to a factor of ∼
2. For these clouds only, the internal energy, a measure ofthe energy injected by radiative feedback, exceeds the gravitational potential for a sig-nificant amount of time. We find a clear relation between the maximum cluster mass,M cl,max , formed in a GMC of mass M
GMC ; M cl,max ∝ M . GMC . This scaling result sug-gests that young globular clusters at the necessary scale of 10 M (cid:12) form within hostGMCs of masses near ∼ × M (cid:12) . We compare simulated cluster mass distributionsto the observed embedded cluster mass function ( dlog ( N ) /dlog ( M ) ∝ M β where β =-1) and find good agreement ( β = -0.99 ± Key words: galaxies: star clusters: general – H ii regions – radiative transfer – stars:formation – methods: numerical – hydrodynamics
The formation of star clusters takes place within dense(n > cm − ) clumps of molecular gas embedded in GiantMolecular Clouds (GMCs) (Lada & Lada 2003; Mac Low& Klessen 2004; Bertoldi & McKee 1992; Kruijssen 2012).These clouds are supersonically turbulent and highly fila-mentary with the most massive clusters forming at the in-tersection of these filaments (Balsara et al. 2001; Banerjee &Pudritz 2006; Schneider et al. 2012; Kirk et al. 2013). Sincestar clusters are almost exclusively formed in GMCs, un-derstanding the processes that lead to their formation anddestruction is vital for a complete understanding of galaxyevolution over cosmic time.The properties of GMC within a galaxy — such as the (cid:63) E-mail: [email protected] mass and virial parameter (Solomon et al. 1987; Rosolowsky2007; Hernandez & Tan 2015; Howard et al. 2016) — varyfrom cloud to cloud. Within the Milky Way (MW), the typ-ical size of a GMC ranges from 50 pc to several hundredsof parsecs with masses in the range of ∼ − M (cid:12) (Fukui& Kawamura 2010a). More specifically, the mass distribu-tion of clouds within the inner disk of the MW follows apower law dN/dM ∝ M α where α ∼ -1.5 (Sanders et al.1985; Solomon et al. 1987; Rosolowsky 2005). The powerlaw index for the GMC mass distribution in other LocalGroup galaxies is found to be significantly steeper, rangingfrom -1.7 for the LMC to -2.5 for M33 (Blitz et al. 2007;Rosolowsky 2005).The mass of a GMC has a direct impact on a clus-ter that form within it. Both simulations (Fujii & PortegiesZwart 2015) and observations (Hughes et al. 2013) indicatea relation between the mass of a GMC (M GMC ) and the c (cid:13) a r X i v : . [ a s t r o - ph . GA ] J un C.S. Howard, R.E. Pudritz, & W.E. Harris maximum mass cluster (M c,max ) it produces of the formM c,max ∝ M . GMC . Based on the similarity of the mass scal-ing of GMCs and star clusters, Harris & Pudritz (1994) pro-posed that Globular Clusters (GCs) originated in SupergiantMolecular Clouds ( (cid:62) M (cid:12) ). Overall, these results suggestthat the massive stellar content should increase with GMCmass. This is indeed borne out in observations of the LMC(Kawamura et al. 2009; Fukui & Kawamura 2010b) whichshow that GMCs with large HII regions, indicating the pres-ence of massive stars, are typically more massive than GMCswith no, or low luminosity, HII regions.The overall conversion of molecular gas into stars, re-gardless of cloud mass, is an inherently inefficient process.Typical estimates of the star formation efficiency over thelifetime of an individual GMC in the MW range from 2-5%(Lada & Lada 2003; McKee & Ostriker 2007; Murray 2011).The question of what limits star formation in a GMCto such low values, and ultimately disrupts the cloud, isdebated. Both turbulence (Klessen et al. 2000; Bate et al.2003; Bonnell et al. 2008) and magnetic fields (Myers &Goodman 1988; Tilley & Pudritz 2007; Federrath & Klessen2012) can provide added pressure support against gravita-tional collapse and lower the star formation rate per freefalltime, but cannot completely disperse the GMC. Alterna-tively, feedback from newly-formed stars can both limit thestar formation efficiency and destroy the GMC via the inputof energy and momentum into the gas.The goal of this paper is to explore how cluster forma-tion and radiative feedback affect GMCs and ultimately starcluster properties. For this purpose, we present the resultsfrom a suite of simulations which examine the role of radia-tive feedback in 5 clouds ranging from 10 − M (cid:12) . The initialaverage density and the initial virial parameter are identicalfor all models in order to ensure all observed differences aredue solely to varying the mass.Stellar feedback comes in many forms — protostellarjets (Li & Nakamura 2006; Maury et al. 2009; Federrathet al. 2014), stellar winds (Dale & Bonnell 2008; Gattoet al. 2017), ionization/heating of the gas (Dale et al. 2005;Peters et al. 2010; Klassen et al. 2012), radiation pressure(Krumholz & Thompson 2012; Murray et al. 2010), and su-pernovae feedback (Rogers & Pittard 2013; Fierlinger et al.2016; Keller et al. 2014; Gatto et al. 2015; Walch & Naab2015). Of these mechanism, radiative feedback has been sug-gested as being most important during the early phases ofcluster formation, particularly in clusters which are host-ing massive star formation (Whitworth 1979; Matzner 2002;Murray et al. 2010; Dale et al. 2012; Bate 2012). The heat-ing and ionization of the gas surrounding star-forming clus-ters prevents further fragmentation, and expanding HII re-gions can drive further turbulence (Gritschneder et al. 2009;Boneberg et al. 2015). Direct radiation pressure from highenergy photons interacting with dust grains can also drivestrong outflows.Previous studies which examine the impact of radiativefeedback on both small (individual cluster) scales and large(entire GMC) scales show that the overall star formation ef-ficiency can be reduced (Dale et al. 2007; Peters et al. 2010;Dale et al. 2012; Bate 2012; Klassen et al. 2012; Walch et al.2012). In particular, the work of Dale et al. (2012) showedthat radiative feedback produces large scale HII regionswhich drive significant gas outflows from the cloud. This is especially important in low mass ( ∼ M (cid:12) ) clouds. De-spite the production of these large features, the influence onstar formation efficiencies and rates was small. Their models,however, were limited to gravitationally bound clouds.Our own work (Howard et al. 2016) also showed thatthe inclusion of radiative feedback did reduce the efficiencyof cluster formation, but only by a maximum of ∼ M (cid:12) ) which,while present in the MW, are not typical of the average GMCas illustrated by the powerlaw mass distribution discussedabove. Moreover, since the properties of the population ofclusters formed in a GMC depends on its initial mass, theeffects of radiative feedback can possibly differ when consid-ering a spectrum of cloud masses.We evolve all models to ∼ × and 10 M (cid:12) clouds which have theefficiency of cluster formation reduced by approximately afactor of 2. We show that this is the result of a trade offbetween the energy injected by radiative feedback and thegravitational potential energy of the cloud. GMCs in thisparticular mass range are massive enough to form a popula-tion of massive stars but have a small enough gravitationalpotential to become unbound under the influence of radia-tive feedback.In Section 4, we compare our star formation rates andinitial cluster mass function to their observed counterparts.We find that the slope of our cluster mass function over therange of masses observed for embedded clusters is consistentwith observations only when radiative feedback is included.However, the combination of high SFRs at late times andstar formation efficiencies which range between 16 and 21%suggest that radiative feedback alone is not responsible forlimiting early star and cluster formation. Here, we provide a brief description of the numericalmethods employed in our simulations. For more detail, werefer the reader to Howard et al. (2016).We perform numerical simulations using the AdaptiveMesh Refinement (AMR) code FLASH (version 2.5) (Fryx-ell et al. 2000) which solves the compressible gas dynamicequations on a Eulerian grid. FLASH also includes modulesto treat self-gravity, radiative transfer, star formation, andcooling via molecules and dust.Gas cooling is treated using the methods from Banerjeeet al. (2006) which handles cooling via gas-dust interactions,H dissociation, and molecular line emission. The coolingrates for molecular line emission and gas-dust transfer are c (cid:13) , 1–15 MC Mass Dependence of Radiative Feedback Mass (M (cid:12) ) Radius (pc) Virial Parameter Initial Mach Number Resolution (pc) Particle Radius (pc) Radiative Threshold (M (cid:12) ) × × Table 1.
Summary of parameters for each simulation. Note that two simulations were completed for every entry in the table — oneincluding radiative feedback and one without radiative feedback. taken from Neufeld et al. (1995) and Goldsmith (2001), re-spectively.The hybrid-characteristics raytracing scheme developedby Rijkhorst et al. (2006), and expanded for astrophysicaluse by Peters et al. (2010), is used to treat radiative transfer.This scheme follows the propagation of ionizing and non-ionizing radiation and uses the DORIC routines (Frank &Mellema 1994; Mellema & Lundqvist 2002) to calculate theionization state of the gas. The DORIC routines considerhydrogen to be the only gas species when calculating theabsorption of ionizing photons. We adopt the temperaturedependent Planck mean opacities from Pollack et al. (1994)for non-ionizing radiation which were calculated for a mix-ture of gas, silicates, ices, and organics. The absorption ofnon-ionizing radiation acts as a source term when calculat-ing the temperature of the gas.Radiation pressure is included by adopting a single UVopacity of κ = 775 cm g − (Li & Draine 2001) which isscaled by the neutral fraction of the gas such that fully ion-ized regions have zero opacity. The radiative force per unitmass is calculated via, F = Lc e − τ uv πr (1)where c is the speed of light, L is the source luminosity, r is the distance between the source and the cell, and τ uv is the optical depth between the source and the cell calcu-lated using the raytracer. We note that the scattering andabsorption of infrared (IR) photons is not included in ourradiation pressure calculation. The trapping of IR photonsin high density regions would introduce an additional factorof τ IR — the optical depth to IR radiation — in Equation1. For typical MW cluster forming regions, this additionalcontribution is thought to be negligible (Murray et al. 2010).The formation of star clusters is represented by the sinkparticle methods from Federrath et al. (2010). A customsubgrid model is used to model star formation within theseclusters (see Howard et al. (2014) for a full description of thismodel). When a particle is formed above the adopted den-sity threshold of 10 cm − , which is based on observations ofstar-forming clumps (Lada & Lada 2003), its mass is dividedinto two components; mass available for star formation dur-ing this timestep, and the remaining gas mass (referred toas the ’reservoir’). The mass available for star formation isdrawn from the available gas reservoir and randomly dis-tributed into stars using a Chabrier (2005) IMF. The reser-voir gas is converted to stars with an efficiency of 20% perfreefall time, where the freefall time is taken to be 0.36 Myr(ie. the freefall time of gas at 10 cm − with a mean molec-ular weight of 2.14), and the IMF is sampled every tenth ofa freefall time to ensure cluster properties evolve smoothlyover time. The efficiency per freefall time was chosen to be consistent with observations of star-forming clumps whichare estimated to range from 10-30% (Lada & Lada 2003).The masses of all stars formed in each cluster arerecorded and analytic fits from Tout et al. (1996) are used toobtain each star’s temperature from its mass. We neglect theeffects of protostellar evolution and assume each star to beradiating as a blackbody at its corresponding temperature.The total luminosity of each star is calculated by integratingthe entire blackbody spectrum and the ionizing luminosityis calculated using the same method but only consideringphoton energies greater than 13.6 eV. The total ionizing lu-minosity of each cluster is then the sum of its constituentstars which is used by the radiative transfer scheme.We allow our cluster sink particles to merge under theconditions that they are separated by less than a particleradius, their relative velocities are negative, and they aregravitationally bound to one another. When a merger oc-curs, all mass (including both the stellar mass and reser-voir mass) is transferred to the more massive particle andthe smaller particle is deleted. The total number of clustersmay therefore either increase or decrease as the simulationsevolves.We employ a stellar mass threshold for our clusters, be-low which the clusters do not radiate. This was included inorder to reduce the computational time, since the radiativetransfer scheme is expensive. Clusters below this thresholdcontinue to form stars, accrete gas, and interact gravita-tionally with their surroundings, but they are not includedin the radiative transfer calculation. We discuss the specificthresholds we used for each simulation below. We simulate a suite of GMCs that have masses of 10 ,5 × , 10 , 5 × , and 10 M (cid:12) . Two simulations were com-pleted for each cloud mass — one with radiative feedbackincluded, and one without radiative feedback (ie. purely hy-drodynamic). The clouds are initially spherical, with a den-sity profile which is uniform in the inner half of the cloudand decreases as r − / in the outer half. A quadratic fit isapplied at the transition between these two profiles to en-sure the density is smooth and continuous. The radius ofeach cloud is chosen such that the average density is n =100 cm − .Each GMC is initially overlaid with a Burgers turbu-lent velocity spectrum, as in Girichidis et al. (2011), afterwhich the turbulence is not driven and allowed to decay.The strength of the turbulence varies between simulationsbut is chosen such that each cloud has the same initial virialparameter, α , defined by (Bertoldi & McKee 1992), α = 2 E kin | E grav | (2) c (cid:13)000
Summary of parameters for each simulation. Note that two simulations were completed for every entry in the table — oneincluding radiative feedback and one without radiative feedback. taken from Neufeld et al. (1995) and Goldsmith (2001), re-spectively.The hybrid-characteristics raytracing scheme developedby Rijkhorst et al. (2006), and expanded for astrophysicaluse by Peters et al. (2010), is used to treat radiative transfer.This scheme follows the propagation of ionizing and non-ionizing radiation and uses the DORIC routines (Frank &Mellema 1994; Mellema & Lundqvist 2002) to calculate theionization state of the gas. The DORIC routines considerhydrogen to be the only gas species when calculating theabsorption of ionizing photons. We adopt the temperaturedependent Planck mean opacities from Pollack et al. (1994)for non-ionizing radiation which were calculated for a mix-ture of gas, silicates, ices, and organics. The absorption ofnon-ionizing radiation acts as a source term when calculat-ing the temperature of the gas.Radiation pressure is included by adopting a single UVopacity of κ = 775 cm g − (Li & Draine 2001) which isscaled by the neutral fraction of the gas such that fully ion-ized regions have zero opacity. The radiative force per unitmass is calculated via, F = Lc e − τ uv πr (1)where c is the speed of light, L is the source luminosity, r is the distance between the source and the cell, and τ uv is the optical depth between the source and the cell calcu-lated using the raytracer. We note that the scattering andabsorption of infrared (IR) photons is not included in ourradiation pressure calculation. The trapping of IR photonsin high density regions would introduce an additional factorof τ IR — the optical depth to IR radiation — in Equation1. For typical MW cluster forming regions, this additionalcontribution is thought to be negligible (Murray et al. 2010).The formation of star clusters is represented by the sinkparticle methods from Federrath et al. (2010). A customsubgrid model is used to model star formation within theseclusters (see Howard et al. (2014) for a full description of thismodel). When a particle is formed above the adopted den-sity threshold of 10 cm − , which is based on observations ofstar-forming clumps (Lada & Lada 2003), its mass is dividedinto two components; mass available for star formation dur-ing this timestep, and the remaining gas mass (referred toas the ’reservoir’). The mass available for star formation isdrawn from the available gas reservoir and randomly dis-tributed into stars using a Chabrier (2005) IMF. The reser-voir gas is converted to stars with an efficiency of 20% perfreefall time, where the freefall time is taken to be 0.36 Myr(ie. the freefall time of gas at 10 cm − with a mean molec-ular weight of 2.14), and the IMF is sampled every tenth ofa freefall time to ensure cluster properties evolve smoothlyover time. The efficiency per freefall time was chosen to be consistent with observations of star-forming clumps whichare estimated to range from 10-30% (Lada & Lada 2003).The masses of all stars formed in each cluster arerecorded and analytic fits from Tout et al. (1996) are used toobtain each star’s temperature from its mass. We neglect theeffects of protostellar evolution and assume each star to beradiating as a blackbody at its corresponding temperature.The total luminosity of each star is calculated by integratingthe entire blackbody spectrum and the ionizing luminosityis calculated using the same method but only consideringphoton energies greater than 13.6 eV. The total ionizing lu-minosity of each cluster is then the sum of its constituentstars which is used by the radiative transfer scheme.We allow our cluster sink particles to merge under theconditions that they are separated by less than a particleradius, their relative velocities are negative, and they aregravitationally bound to one another. When a merger oc-curs, all mass (including both the stellar mass and reser-voir mass) is transferred to the more massive particle andthe smaller particle is deleted. The total number of clustersmay therefore either increase or decrease as the simulationsevolves.We employ a stellar mass threshold for our clusters, be-low which the clusters do not radiate. This was included inorder to reduce the computational time, since the radiativetransfer scheme is expensive. Clusters below this thresholdcontinue to form stars, accrete gas, and interact gravita-tionally with their surroundings, but they are not includedin the radiative transfer calculation. We discuss the specificthresholds we used for each simulation below. We simulate a suite of GMCs that have masses of 10 ,5 × , 10 , 5 × , and 10 M (cid:12) . Two simulations were com-pleted for each cloud mass — one with radiative feedbackincluded, and one without radiative feedback (ie. purely hy-drodynamic). The clouds are initially spherical, with a den-sity profile which is uniform in the inner half of the cloudand decreases as r − / in the outer half. A quadratic fit isapplied at the transition between these two profiles to en-sure the density is smooth and continuous. The radius ofeach cloud is chosen such that the average density is n =100 cm − .Each GMC is initially overlaid with a Burgers turbu-lent velocity spectrum, as in Girichidis et al. (2011), afterwhich the turbulence is not driven and allowed to decay.The strength of the turbulence varies between simulationsbut is chosen such that each cloud has the same initial virialparameter, α , defined by (Bertoldi & McKee 1992), α = 2 E kin | E grav | (2) c (cid:13)000 , 1–15 C.S. Howard, R.E. Pudritz, & W.E. Harris
Figure 1.
Density slices through the center of the simulation volume for the 10 (left), 10 (center), and 10 (right) M (cid:12) GMCs. Time,shown in the top left of each panel, increases from top to bottom. Cluster locations are projected onto this slice and shown by blackcircles. Note that the physical (xy) scales change with cloud mass (10x10 pc, 20x20 pc, and 40x40 pc from left to right).c (cid:13) , 1–15
MC Mass Dependence of Radiative Feedback Figure 2.
The corresponding temperature slices to the panels shown in Figure 1. where E kin is the cloud’s total kinetic energy, and E grav isthe total gravitational potential energy. We have chosen aninitial virial parameter of 3 (ie. unbound) since it resultedin more realistic formation efficiencies compared to boundclouds in Howard et al. (2016). As shown in that work, the turbulence decays rapidly and becomes virialized at ∼ α .We use outflow boundary conditions for all simulations.The total mass in the simulation volume is therefore notconserved, and can decrease over time due to gas leaving c (cid:13) , 1–15 C.S. Howard, R.E. Pudritz, & W.E. Harris
Figure 3.
The total mass contained in clusters (top) and the totalnumber of clusters (bottom) in our five GMC models includingradiative feedback. Note that the total mass contained in clus-ters can decrease over time due to clusters leaving the simulationvolume, and the number of clusters can decrease both throughescaping clusters and merging events. the domain. This is relevant to the discussion that followsin the next Section.Since the radius, initial Mach number, resolution, par-ticle size (given by 2.5 times the smallest cell size), andthe threshold for radiating differ between clouds of differ-ent mass, we summarize these parameters in Table 1.
To visually compare the evolution of GMCs with differ-ent initial masses, we show density slices through the centerof the simulation volumes in Figure 1. The columns, fromleft to right, show GMC masses of 10 , 10 , and 10 M (cid:12) respectively. All simulations shown in Figure 1 include ra-diative feedback. The rows are plotted at different times,ranging from 1.5 to 5 Myr. The black dots represent the lo-cations of clusters which have been projected onto the sliceplane. The corresponding temperature slices are shown inFigure 2. It is very important to note that cloud sizes andsimulation boxes are very different for these three GMCs:10x10 pc, 20x20 pc, and 40x40 pc, respectively.The first row shows the state of the simulation shortlyafter the formation of the first clusters. The gas has alreadybroken up into filaments due to the turbulent nature of the Figure 4.
The merged mass fraction, defined as the total clustermass that has participated in at least one merging event dividedby the total mass contained in all clusters, versus time for the 5GMCs including radiative feedback. gas. The 10 M (cid:12) simulation has formed significantly moreclusters by this time, totaling 37 compared to the 7 that haveformed in the 10 M (cid:12) cloud. Despite clusters being present,they have not grown to high enough masses to influence theirenvironment via heating or ionization. This can be seen inthe first row of Figure 2 which shows that the majority ofthe gas still remains at 10 K, with ∼
300 K gas filling thelow density voids between filaments.As the simulation progresses to 2.5 Myr, the clusters inthe 10 M (cid:12) cloud have become sufficiently populated withmassive stars to begin ionizing their surroundings. This re-sults in a hot ( ∼ and 10 M (cid:12) clouds,in contrast, have not produced enough massive stars for ra-diative feedback to have any effects.At 3.75 Myr, Figure 2 clearly shows that radiative feed-back is active in all clouds. The 10 M (cid:12) and 10 M (cid:12) cloudsin particular show extended HII regions centered on a groupof massive clusters. The corresponding density images showthat radiative feedback is in the process of destroying thefilaments in the vicinity of these HII regions due to the ex-pansion of the hot gas which smears out overdense regions.The final panels of Figures 1 and 2 show marked differ-ences between the three simulations. The gas in the 10 M (cid:12) cloud is centrally condensed with the majority of clustersexisting in this central region. This allows these clusters tocontinue accreting from their surroundings.The 10 M (cid:12) cloud has been effectively destroyed byradiative feedback. The entire cloud is nearly fully ionizedand the resulting expansion of gas has caused a large fractionof the initial mass to leave the simulation volume. The cloudremains fully ionized after this point and the accretion of gasby the clusters has been halted.While there are large voids produced by HII regions inthe 10 M (cid:12) cloud, little mass loss has occurred. The clustersare also dispersed more evenly throughout the cloud, some of c (cid:13) , 1–15 MC Mass Dependence of Radiative Feedback Figure 5.
The total star formation rates (SFRs) for the 5 simulated GMCs which include radiative feedback. Note that this plot hasbeen smoothed for readability (see text). which are still actively accreting gas. Large scale filamentarystructures are still present at 5 Myr.To compare cluster formation across different clouds,we plot the total mass contained in clusters (top panel) andthe total number of clusters (bottom panel) in Figure 3.Since all GMCs were initialized with the same averagedensity and virial parameter, the onset of cluster formationis comparable, ranging from 0.39 Myr for the 10 M (cid:12) cloudto 0.59 Myr at 10 M (cid:12) . The clusters then rapidly grow inmass via gas accretion with the higher mass clouds contain-ing more mass in clusters at any given time, as expected. At5 Myr, the total mass contained in clusters, in order of lowestto highest initial cloud mass, is 4.1 × , 2.3 × , 2.3 × ,1.8 × , and 2.8 × M (cid:12) . Note that the total cluster massdoes not scale directly with the initial cloud mass. This willbe relevant to the discussion of formation efficiencies whichfollows.The number of clusters formed also does not scale di-rectly with the initial cloud mass. The numbers of clustersat 5 Myr in the 10 , 10 , and 10 M (cid:12) simulations are 11,22, and 199, respectively. Note that we allow our clusters tomerge so the number of clusters can decrease. Since mass isconserved in the merger, however, the total mass in clusterscannot decrease unless a cluster leaves the simulation vol-ume entirely which does not play a significant role. Only the5 × M (cid:12) and 10 M (cid:12) GMCs are still forming clusters insignificant numbers at the end of the simulation.Cluster merging plays a significant role in the growthof clusters. We demonstrate this in Figure 4 which plots themerged mass fraction versus time. We define the merged mass fraction as total amount of cluster mass that has par-ticipated in a merger event up until a given time divided bythe total mass contained in clusters at that same time. Amerged mass fraction of 0.5, for example, means that half ofthe mass contained in clusters has participated in at leastone merger.While there does not appear to be a trend with GMCmass, it is clear that significant numbers of mergers are oc-curring in all clouds. At ∼ within the clusters and not the formationof new clusters.As a product of our star formation subgrid model whichsamples the IMF to form new stars at prescribed intervals,there are timesteps in which no new stars formed and oth-ers which have a burst of star formation. We have thereforesmoothed these plots using a sliding average window to as-sist in readability. This leads to a highly variable SFR, par- c (cid:13)000
The total star formation rates (SFRs) for the 5 simulated GMCs which include radiative feedback. Note that this plot hasbeen smoothed for readability (see text). which are still actively accreting gas. Large scale filamentarystructures are still present at 5 Myr.To compare cluster formation across different clouds,we plot the total mass contained in clusters (top panel) andthe total number of clusters (bottom panel) in Figure 3.Since all GMCs were initialized with the same averagedensity and virial parameter, the onset of cluster formationis comparable, ranging from 0.39 Myr for the 10 M (cid:12) cloudto 0.59 Myr at 10 M (cid:12) . The clusters then rapidly grow inmass via gas accretion with the higher mass clouds contain-ing more mass in clusters at any given time, as expected. At5 Myr, the total mass contained in clusters, in order of lowestto highest initial cloud mass, is 4.1 × , 2.3 × , 2.3 × ,1.8 × , and 2.8 × M (cid:12) . Note that the total cluster massdoes not scale directly with the initial cloud mass. This willbe relevant to the discussion of formation efficiencies whichfollows.The number of clusters formed also does not scale di-rectly with the initial cloud mass. The numbers of clustersat 5 Myr in the 10 , 10 , and 10 M (cid:12) simulations are 11,22, and 199, respectively. Note that we allow our clusters tomerge so the number of clusters can decrease. Since mass isconserved in the merger, however, the total mass in clusterscannot decrease unless a cluster leaves the simulation vol-ume entirely which does not play a significant role. Only the5 × M (cid:12) and 10 M (cid:12) GMCs are still forming clusters insignificant numbers at the end of the simulation.Cluster merging plays a significant role in the growthof clusters. We demonstrate this in Figure 4 which plots themerged mass fraction versus time. We define the merged mass fraction as total amount of cluster mass that has par-ticipated in a merger event up until a given time divided bythe total mass contained in clusters at that same time. Amerged mass fraction of 0.5, for example, means that half ofthe mass contained in clusters has participated in at leastone merger.While there does not appear to be a trend with GMCmass, it is clear that significant numbers of mergers are oc-curring in all clouds. At ∼ within the clusters and not the formationof new clusters.As a product of our star formation subgrid model whichsamples the IMF to form new stars at prescribed intervals,there are timesteps in which no new stars formed and oth-ers which have a burst of star formation. We have thereforesmoothed these plots using a sliding average window to as-sist in readability. This leads to a highly variable SFR, par- c (cid:13)000 , 1–15 C.S. Howard, R.E. Pudritz, & W.E. Harris
Figure 6.
Left: The cluster particle formation efficiency ( (cid:15) cl , defined as the total mass in cluster particles divided by the GMC’s initialmass) for our 5 RHD simulations (shown by solid lines) and the 5 HD simulations (shown by dashed lines). Right: Identical to the leftpanel except the star formation efficiency ( (cid:15) ∗ , total mass of stars within clusters divided by the initial GMC mass) is plotted. ticularly at late times when there are many clusters formingstars at staggered times.All curves show a sharp rise in SFR following the on-set of cluster formation. For the 10 , 5 × , and 10 M (cid:12) GMCs, the SFR levels out to approximately constant valuesof 6 × − , 3 × − , and 5 × − respectively. These valuesare consistent with a SFR that scales directly with the initialGMC mass, assuming similar density structures, as found inobservations of local GMCs by Lada et al. (2010).The other two GMC models (5 × and 10 M (cid:12) ) in-stead show a SFR rate which decreases at late times. Asshown in Howard et al. (2014), which examined the proper-ties of our adopted subgrid model for star formation, a de-creasing SFR is indicative of a population of clusters whichhave completely stopped accreting and are simply using upthe rest of their gaseous reservoir. The images for the 10 M (cid:12) GMC in Figures 1 and 2 are consistent with this picturesince they demonstrate that the cloud has been almost fullyionized and destroyed by 5 Myr. This suggests the impactradiative feedback has on the formation and evolution ofclusters is stronger in these clouds. We compare the effectsof radiative feedback between cloud models below.
To understand the role of radiative feedback in GMCs withdifferent initial masses, we computed a grid of complemen-tary simulations which have radiative transfer turned off. Wewill refer to simulations with radiative feedback included as ”RHD” (Radiation Hydrodynamics) simulations and ”HD”(Hydrodynamics) simulations are those with radiative feed-back not included.How much the efficiency is suppressed when includ-ing radiative feedback is still debated. Howard et al. (2016)showed that it depends on the initial gravitational bound-edness of the molecular cloud, as measured by the virialparameter. Here, we find that radiative feedback does in-deed limit star cluster formation but, more importantly, thestrength of this suppression depends on the cloud’s initialmass.We show this in Figure 6, which plots the cluster parti-cle formation efficiency ( (cid:15) cl ) and the star formation efficiency( (cid:15) ∗ ) for both the RHD simulations, shown by the solid lines,and the HD simulations, shown by the dashed lines. We de-fine (cid:15) cl as the total mass in cluster particles divided by theinitial cloud mass. The star formation efficiency, (cid:15) ∗ , is de-fined as the total mass of stars within the clusters dividedby the initial cloud mass. Note that an entire cluster’s massis not necessarily only in stars, but can also be part of thegas reservoir which is available for future star formation.When comparing the RHD and HD simulations, we seethat star and cluster formation start at similar times andevolve identically for ∼ (cid:15) cl in the HDand corresponding RHD runs begin to diverge, with the HDsimulations having the higher efficiency in all cases. Thistrend continues to the end of the simulation and the differ-ence between the HD and RHD formation efficiencies grows.Choosing a time of 5 Myr to compare (cid:15) cl , the efficiencies in c (cid:13) , 1–15 MC Mass Dependence of Radiative Feedback the lowest to highest mass clouds are 43%, 29%, 23%, 35%,and 28%. At the same time, (cid:15) ∗ ranges from 16% to 21%.Note that these values are higher than the measured valuesfrom GMC observations (eg. Lada & Lada (2003); McKee &Ostriker (2007); Murray (2011)) which suggests that whileradiative feedback does lower (cid:15) ∗ relative to HD runs, otherpieces of physics such as stellar winds are required to lowerthese values further.Both (cid:15) cl and (cid:15) ∗ are the highest for the lowest mass(10 M (cid:12) ) GMC. This is consistent with the results fromOchsendorf et al. (2017) who found evidence of a decreasing (cid:15) ∗ with increasing cloud mass in the LMC. While we repro-duce their results for the 10 M (cid:12) cloud, we do not see clearevidence for a trend with increasing GMC mass.It is clear from Figure 6 that radiative feedback playsa stronger role in suppressing star and cluster formation insome clouds more than others. To make this clear, we plotthe fractional reduction in efficiencies when including radia-tive feedback in Figure 7. This Figure shows that differencein formation efficiencies is largest for the 5 × and 10 M (cid:12) GMCs. Focusing on these two simulations at 5 Myr, the dif-ference between (cid:15) cl for the HD and RHD run is 27% and 18%for initial masses of 5 × and 10 M (cid:12) GMCs, respectively.This corresponds to approximately a factor of two reductionin both cases. The inclusion of radiative feedback in the 10 ,5 × , and 10 M (cid:12) GMCs reduced (cid:15) cl by 21%, 40%, and33% relative to the HD simulations, respectively.This is consistent with the density and temperature vi-sualizations discussed in Figures 1 and 2. It was clear fromthose images that the 10 M (cid:12) simulation is more globallyimpacted by radiative feedback than the other two cases,as evidenced by the nearly fully ionized simulation volume.In contrast, the 10 and 10 M (cid:12) GMCs showed small HIIregions which may stop the accretion onto local clusters butnot the entire population.These results suggest there is something unique hap-pening in clouds between 5 × - 10 M (cid:12) which make themmore susceptible to radiative feedback effects. We proposethat clouds lower than this mass are not able to form enoughmassive stars and therefore cannot completely ionize thecloud. Indeed, the 10 M (cid:12) did not produce any O-starsthroughout its evolution. On the other hand, clouds abovethis mass range are capable of producing O-stars but havetoo much gas mass, and therefore a higher column density toionizing radiation, to be fully ionized during the early stagesof cluster formation. The overall gravitational potential ofthese massive clouds is also deeper, meaning they are harderto unbind overall.We can demonstrate this balance between gravity andthe energy injected by radiative feedback by comparing thetotal gravitational potential energy to the total internal en-ergy of the gas at any given time. The internal energy iscalculated from the gas temperature and ionization fractionand is therefore a proxy for the energy injected by radiation.We plot the ratio of the total internal energy to the totalgravitational potential energy in Figure 8.All models start with a ratio less than 1, indicating thatgravitational potential energy dominates during early times.At approximately 3.5 and 3.8 Myr, the ratio rises above onefor the 10 M (cid:12) and 5 × GMCs, respectively. This indi-cates that the amount of radiation being injected into thegas is sufficient to unbind the cloud globally, resulting in the larger suppression of cluster formation under the influenceof radiative feedback in these clouds. While internal energydoes dominate over gravitational potential energy for the10 M (cid:12) GMC, it only does so at late times and thereforedoes not influence the early stages of cluster formation assignificantly. In contrast, the more massive models (5 × and 10 M (cid:12) ) are always dominated by gravitational poten-tial energy.These results explain why, in Figure 6, (cid:15) cl begins to flat-ten around 3 Myr for the 5 × and 10 M (cid:12) clouds. As alarger volume of gas becomes hot and ionized, the formationof new clusters, and the accretion onto existing clusters, issuppressed. A similar result is not seen (cid:15) ∗ due to our sub-grid model. Since we do not allow unused gas to leave theclusters, star formation can proceed regardless of whetheraccretion is still taking place. As shown in Howard et al.(2014), cluster masses are typically dominated by the reser-voir of gas, especially at early times.The varying strength of radiative feedback may have im-portant implications for the growth and evolution of GMCsif we assume they form through a bottom up process, suchas spiral arm induced collisions, as suggested by Dobbs& Pringle (2013). Our results indicate that once a cloudreaches ∼ × M (cid:12) , it should be destroyed via radiativefeedback. This may act as a bottleneck for the growth ofGMCs and could be partly responsible for their observedmass distribution.The cluster formation efficiency is essentially a normal-ized measure of how much mass is present in clusters at anygiven time. To understand how this mass is distributed, wealso need to know the total number of clusters. We examinehow the number of clusters is affected by radiative feedbackin Figure 9. This is similar to the plot shown in the previ-ous section, except the results from the HD simulations areincluded as dashed lines. We see that the early evolution ofthe RHD and HD simulations are similar, but at late timesthere are more clusters present in the RHD cases. This willimpact the distribution of cluster masses. Since the HD sim-ulations have more mass contained in clusters (as illustratedin Figure 6) but fewer clusters overall, the average clustermass will be higher than cases which include radiative feed-back. Taking, for example, the 5 × M (cid:12) cloud, the finalaverage cluster mass is decreased from 2046 M (cid:12) to 468 M (cid:12) when including radiative feedback. The mass function of star clusters have been characterizedobservationally. As discussed in Fall & Chandar (2012), themass function for embedded clusters (Lada & Lada 2003)and extragalactic clusters taken from the Magellanic Clouds,M83, M51, and Antennae are all consistent with a powerlawmass distribution of the form dlog(N) ∝ M β dlog(m) where β ∼ -1. Here, we compare the mass functions of our simulatedclusters to these results.The cluster mass functions for the 10 , 10 , 10 M (cid:12) GMCs are shown in Figure 10. The data is plotted at 5Myr, corresponding to the approximate end of the 10 M (cid:12) simulation. The mass values represent only the stellar mass c (cid:13) , 1–15 C.S. Howard, R.E. Pudritz, & W.E. Harris
Figure 7.
The fractional reduction of (cid:15) cl (left) and (cid:15) ∗ (right) when including radiative feedback into a simulation, relative the HDformation efficiencies. contained in each cluster and therefore do not include theunused gas reservoir.As the initial mass of the GMC increases, the total num-ber of clusters formed also increases (see Figure 3) and sothe plot is consequently more populated. The cluster massdistributions also shift to higher masses as the initial GMCmass increases. The previous results suggest a relation between the max-imum mass cluster produced in a star-forming event andthe mass of the GMC out of which it forms. This relationhas been found in both observations (Hughes et al. 2013)and simulations (Fujii & Portegies Zwart 2015) to take theform M c,max ∝ M . GMC , where M c,max is the maximum masscluster that forms out of a GMC of mass M
GMC .We plot the maximum cluster mass obtained from our 5RHD GMC models in Figure 11, shown by the filled circles.We plot the relation at two times — 3 Myr (gold) and 5 Myr(black). Here, the maximum cluster mass includes only itsstellar mass and not the unused reservoir of gas.At 5 Myr, we find a relation between the maximumcluster mass and the host GMC mass given by, M c,max ∝ M . ± . GMC . (3)While this does not agree with the relation above, it isroughly consistent with the relation between the maximum mass star formed in a given cluster proposed by observa-tions from Pflamm-Altenburg et al. (2007) and the theoret-ical model of Elmegreen (2002) that have powerlaw indicesof 0.67 and 0.74, respectively. This seems to suggest thatthere may be self-similar star formation processes rangingfrom GMC masses down to protostellar core masses. In or-der to verify this claim, a fully consistent simulation of aGMC which resolves the formation of individual stars, incombination with a cluster finding algorithm, would be re-quired.Using the HD simulations at 5 Myr instead, we find asteeper slope of 0.93 and all points are higher than theirRHD counterparts. Radiative feedback is clearly limitingthe growth of the most massive clusters regardless of ini-tial GMC mass. The separation between the HD and RHDmaximum cluster masses is, however, more pronounced forthe larger GMCs. This is likely due to the large populationof massive stars in these clusters that can more effectivelyheat and ionize their surroundings and suppress further gasaccretion.Figure 11 shows that the slope of these distributionshardly vary with time. At 3 Myr, the slopes of the RHDand HD simulation are 0.85 and 0.97 (compared to 0.81 and0.92 at 5 Myr), respectively. The intercept, however, doeschange from 3 to 5 Myr due to the growth of the stellarpopulations in these clusters. The separation between theHD and RHD clusters is also less pronounced compared to5 Myr because radiative feedback has not been active for aslong.The insensitivity of the slope with time is likely due to c (cid:13) , 1–15 MC Mass Dependence of Radiative Feedback Figure 8.
The ratio between the total internal energy of the gasto the global gravitational potential energy of the cloud. A lowerratio suggests a higher suppression of cluster and star formation.
Figure 9.
The number of clusters as a function of time (as seenin Figure 3) including both RHD (solid) and HD (dashed) simu-lations.
Figure 10.
The cluster mass function for the 10 (left), 10 (cen-ter), 10 (right) M (cid:12) GMCs, plotted at 5 Myr.
Figure 11.
The maximum mass cluster produced in our 5 RHDmodels (circles) and the 5 HD simulations (triangles) as a functionof the initial cloud mass. The results are plotted at 3 Myr (gold)and 5 Myr (black). The slope of the distributions are shown at thetop of the plot and are colored based on the times they represent.Fits to the HD data are shown by the dashed lines, and fits tothe RHD data are shown by solid lines. two reasons. Firstly, we are plotting the clusters with thelargest stellar content and therefore the highest luminosity.For the RHD simulations, these clusters significant affecttheir local surroundings and suppress their own growth insimilar ways. Secondly, our subgrid model for star formationprescribes the rate at which stars form in the clusters. Oncethe most massive clusters accrete a significant amount ofgas, the total stellar mass will increase at the same rate.We can estimate the mass of the host GMCs out ofwhich Globular Clusters (GCs) ought to form by extrapo-lating our relation to larger cluster masses. We note that wehave not yet completed any GMC simulations greater than10 M (cid:12) and, as shown in this work, the effects of radia-tive feedback are a clear function of cloud mass. It is there-fore possible that the relation displayed in Figure 11 doesnot extend to higher masses. Assuming it does, a GMC of ∼ × M (cid:12) is required in order to form a GC of mass 10 M (cid:12) . This is consistent with Harris & Pudritz (1994) who ar-gued that Supergiant molecular clouds ( > M (cid:12) ) are thehosts to GC formation. c (cid:13) , 1–15 C.S. Howard, R.E. Pudritz, & W.E. Harris
Figure 12.
The combined cluster mass function obtained from all5 RHD simulations. The relative abundance of each parent cloudhas been included via the GMC mass function in order to com-pare directly to observations. Fits to the mass range of observedembedded clusters (solid) and the high mass regime (dashed) arealso included.
To make an accurate comparison with the observed clustermass function, we need to consider the relative number ofGMCs with different mass. The powerlaw index for the GMCmass distribution in the inner Milky Way is approximately-1.5 (Sanders et al. 1985; Solomon et al. 1987; Rosolowsky2005). Taking the GMC mass distribution to be dNdM ∝ M − . (4)in the range 10 - 10 M (cid:12) , we combine the cluster datafrom our 5 RHD GMCs at 5 Myr, weighted by the relativenumbers of the clouds in which they were bornIn Figure 12 we show the resulting, computed clustermass function that arises from the Milky Way cloud massfunction. This can then be compared to the observed massdistribution which is a collection of distinct clusters in form-ing in different regions. Note that we only include the stellarmass of each cluster when producing this distribution. TheFigure shows that the cluster mass function peaks at ∼ (cid:12) corresponding to a small stellar group. The peak clus-ter mass is on the same order of magnitude as the 50 M (cid:12) turnover found in Lada & Lada (2003).We make a further comparison to the results ofLada & Lada (2003) who measured the embedded clus-ter mass function of nearby star-forming regions and found dlog ( N ) /dlog ( M ) ∝ M β , where β ∼ -1. The cluster massfunction has been measured for extragalactic clusters (seeFall & Chandar (2012) for comprehensive overview) and thesame functional form is also found which, in some galax- ies, extends to > M (cid:12) clusters. Embedded clusters of thismass are not seen in the MW.Motivated by the observational data and a cluster massfunction which appears consistent with a broken powerlaw,we provide two fits to our data — one covering the range ofobserved, embedded clusters in the MW (solid line), and onefor the higher mass clusters (dashed line). We only includecluster masses greater than 10 M (cid:12) in the calculation.Fitting over the range of embedded clusters in the MW— 1 (cid:54) log(M/M (cid:12) ) (cid:54) β = -0.99 ± β = -2.82 is found for cluster masses greaterthan ∼ (cid:12) . One reason for the steeper slope at highmasses is that the largest GMCs — the source of the mostmassive clusters — are not disrupted at the end of the 5 Myrsimulation. This suggests that these clusters will continue togrow and accrete gas, leading to more clusters populatingthe high mass end of the distribution. We plot the SFRs of our individual clusters versus the clustermass at various times in Figure 13. We only show the 10 ,10 and 10 M (cid:12) for clarity.We have over plotted the results from Lada (2010),shown by black squares, who measured the SFRs of lo-cal star forming regions by counting Young Stellar Objects(YSOs) and adopting a star formation timescale to estimatethe SFR. In order to make an accurate comparison to theseresults, we adopt the same model parameters as Lada (2010)and estimate the SFR via recently formed stars. We directthe reader to Howard et al. (2016) which plotted the SFRsin the same way and contains more detail about this pro-cedure. We also include the results from Heiderman et al.(2010) who also measured the SFRs of local regions, someof which are also included in the Lada (2010) dataset.Our simulated SFRs agree well with the observed valuesat early times, particularly with the Lada (2010) results. At3 Myr, the low to intermediate mass clusters are still consis-tent with the measured SFRs, but the high mass clusters areoverproducing stars. This is strong evidence that radiativefeedback alone is not sufficient for limiting the SFR (andtherefore the SFE) since these high mass clusters have thehighest ionizing luminosity and should be influencing theirsurroundings the strongest.The trend of high SFRs extends to all mass regimes past4 Myr. While the slope of the SFR-mass relation is consis-tent with the observations, the normalization is not. This isalso true for the 10 M (cid:12) GMC which, as shown in Section3.2, had a large reduction in (cid:15) cl when including radiativefeedback and a globally decreasing SFR at late times. Thisalso supports the claim that other feedback mechanisms,such as stellar winds, are required to explain the SFRs ofyoung, nearby star-forming regions (Gatto et al. 2017).We note that high SFRs may also be due, in part, to ouradopted subgrid model for star formation. We do not include c (cid:13) , 1–15 MC Mass Dependence of Radiative Feedback Figure 13.
The SFR of individual cluster particles at various times for the 10 , 10 , and 10 M (cid:12) GMCs. The squares are the observationalresults from Lada (2010) and the black triangles are a similar data set from Heiderman et al. (2010). feedback on scales smaller than our cluster particles, and soany accreted gas will inevitably be converted to stars over along enough timescale. We refer the reader to Section 2.1 ofHoward et al. (2016) for a detailed discussion of this point.If our cluster SFRs are artificially high, the total lu-minosity will also be too high. This means that the impactradiative feedback has on each cloud, as discussed above,should be considered an upper limit. This lends further sup-port for the need of other feedback mechanisms during theearly phases of cluster formation since the maximum sup-pression of (cid:15) cl relative to HD simulations was approximatelya factor of two. We examine the early phases of cluster formation and therole of radiative feedback in a suite of GMCs that havemasses in the range of 10 − M (cid:12) . To isolate the role of GMCmass, we use the same initial density and virial parameteracross clouds. Sink particles are used to represent the forma-tion of a cluster and a custom subgrid model is used for starformation within the clusters. The properties of the stellarpopulation in each cluster is tracked and, combined with a raytracing radiative transfer scheme, is used to compute theradiative feedback.The main result of this work is that the strength of ra-diative feedback depends on the initial GMC mass. The frac-tional reduction in the cluster formation efficiency, (cid:15) cl , whenincluding radiative feedback is the largest for the 5 × and10 M (cid:12) GMCs. Both of these models had (cid:15) cl reduced by afactor of ∼ M (cid:12) )showed a only ∼
20% reduction in (cid:15) cl .The variation in the impact radiative feedback has onthe cluster and star formation efficiencies is attributed to thebalance between how much radiation energy is absorbed bythe GMC and the gravitational potential energy of the cloud.The smallest GMC is not massive enough to form a popu-lation of massive stars and therefore cannot effectively limitearly star and cluster formation. The highest mass objects,on the other hand, do produce massive stars but their corre-sponding gravitational potential is too large for the cloud tobe globally unbound. The regime between these two limits(5 × to 10 M (cid:12) ) balances these two effects, leading to alarger suppression in (cid:15) cl and (cid:15) ∗ . We have shown this by plot-ting the ratio of the internal energy, a proxy for the amountof absorbed radiation energy injected by the star-forming c (cid:13) , 1–15 C.S. Howard, R.E. Pudritz, & W.E. Harris clusters, to the gravitational potential energy of the cloud.This ratio exceeds one, indicating the cloud has been glob-ally unbound, for the 5 × and 10 M (cid:12) clouds at 3.5 and3.8 Myr. respectively. The higher mass GMCs always havea ratio below 1, indicating that gravity dominates, and theratio for the 10 M (cid:12) only exceeds 1 at late times at whichpoint the majority of star and cluster formation has alreadyoccurred.The other important conclusions of this work can besummarized as follows: • The cluster formation efficiency ( (cid:15) cl ) and the star for-mation efficiency ( (cid:15) ∗ ) vary significantly across different massGMCs. At 5 Myr, (cid:15) cl is 43%, 29%, 23%, 35%, and 28% forthe 10 , 5 × , 10 , 5 × , and 10 M (cid:12) GMCs, respectively.At the same time, (cid:15) ∗ ranges from 16%-21%. • The high SFEs found in all models, even when includ-ing radiative feedback, suggests that other forms of feedback,such as stellar winds, are required to limit early star forma-tion in GMCs (Gatto et al. 2017). This is further supportedby the comparison between our clusters and the SFRs oflocal star-forming regions. We find good agreement with ob-served SFR-mass relation at early times, but by ∼ • We produced an initial cluster mass function by com-bining the results from all RHD simulations weighted bythe Galactic GMC mass function. The resulting slope of thepowerlaw distribution ( dlog ( N ) /dlog ( M ) ∝ M β ) over therange of embedded cluster masses in the MW (log(M/M (cid:12) ) < β = -0.99 ± (cid:12) ) = 3.3 is attributed to the 5 Myr timescaleof our simulations. The most massive GMCs are not yet dis-rupted and, given more time, will fill out the high mass endof the distribution. • The limiting of cluster growth by radiative feedback isalso supported by the relation between the host GMC mass(M
GMC ) and the maximum mass cluster it forms (M c,max ).Using the RHD data, we find that M c,max ∝ M . GMC . TheHD simulations always form higher mass clusters and therelation is instead M c,max ∝ M . GMC . The steeper slope forthe HD simulations indicates that the largest clusters in thehighest mass GMCs are more strongly limiting their growthvia radiative feedback compared to those formed in smallerclouds.
ACKNOWLEDGMENTS
We thank the anonymous referee for useful suggestions.C.S.H. acknowledges financial support provided by the Nat-ural Sciences and Engineering Research Council (NSERC)through a Postgraduate scholarship. R.E.P. and W.E.H. aresupported by Discovery Grants from the Natural Sciencesand Engineering Research Council (NSERC) of Canada. TheFLASH code was in part developed by the DOE supportedAlliances Center for Astrophysical Thermonuclear Flashes(ASCI) at the University of Chicago. Computations were performed on the gpc supercomputer at the SciNet HPCConsortium. SciNet is funded by: the Canada Foundationfor Innovation under the auspices of Compute Canada; theGovernment of Ontario; Ontario Research Fund - ResearchExcellence; and the University of Toronto.
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