Feedback in Forming Star Clusters: The Mass-Radius Relation and Mass Function of Molecular Clumps in the Large Magellanic Cloud
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Feedback in Forming Star Clusters: The Mass-Radius Relation and Mass Function of MolecularClumps in the Large Magellanic Cloud
Angus Mok, Rupali Chandar, and S. Michael Fall Department of Physics & Astronomy, The University of Toledo, Toledo, OH 43606, USA Space Telescope Science Institute, Baltimore, MD 21218, USA (Received; Revised; Accepted)
Submitted to ApJABSTRACTWe derive the mass-radius relation and mass function of molecular clumps in the LargeMagellanic Cloud (LMC) and interpret them in terms of the simple feedback modelproposed by Fall, Krumholz, and Matzner (FKM). Our work utilizes the dendrogram-based catalog of clumps compiled by Wong et al. from CO and CO maps of sixgiant molecular clouds in the LMC observed with the Atacama Large Millimeter Array(ALMA). The Magellanic Clouds are the only external galaxies for which this typeof analysis is possible at the necessary spatial resolution ( ∼ R ∝ M α and dN/dM ∝ M β , with indices α = 0 . ± .
03 and β = − . ± . M (cid:12) (cid:46) M (cid:46) M (cid:12) and 10 M (cid:12) (cid:46) M (cid:46) M (cid:12) , respectively. With thesevalues of α and β for the clumps (i.e., protoclusters), the predicted index for the massfunction of young LMC clusters from the FKM model is β ≈ .
7, in good agreement
Corresponding author: Rupali [email protected] a r X i v : . [ a s t r o - ph . GA ] J a n Mok et al. with the observed index. The situation portrayed here for clumps and clusters in theLMC replicates that in the Milky Way. INTRODUCTIONStar clusters form in the dense subunits of giant molecular clouds (GMCs) known as clumps (Lada& Lada 2003; McKee & Ostriker 2007; Krumholz et al. 2019). Thus, the properties of clusters mustreflect the properties of clumps (as “initial” conditions) modified by the actions of star formationand stellar feedback within them. In this context, two of the most relevant statistical properties ofcluster and clump populations are the mass function and mass–radius relation, usually represented bypower laws: ψ ( M ) ≡ dN/dM ∝ M β and R ∝ M α . A comparison of the indices β and α for clustersand clumps should then tell us something about star formation and feedback. Fall, Krumholz, &Matzner (2010, hereafter FKM) developed this idea into a simple analytical model and applied it toobservations of clusters and clumps in the Milky Way. In a previous paper, we applied the modelto observations of clusters and GMCs in six nearby galaxies (Mok et al. 2020; see also Hughes et al.2013 for a similar study of M51). Here, we apply the FKM model to observations of clusters andclumps in the Large Magellanic Cloud (LMC) for the first time.The mass functions of clusters, clumps, and GMCs are related by the star formation efficiencies(SFEs) E clump ≡ M cluster /M clump and E GMC ≡ M cluster /M GMC . Most determinations of these massfunctions find power laws with indices β ≈ − ± . E clump ∼ E GMC ∼ ass-Radius Relation and Mass Functions of Molecular Clumps in the LMC E/M ∝ GM/R and
F/M ∝ GM/R ) and their mean surface and volume densities (Σ ∝ M/R and ρ ∝ M/R ). The first two ofthese are relevant in the present context because it is gravity that feedback must overcome to removethe gas from protoclusters. Estimates of the index of the mass–radius relation for clusters in multiplegalaxies range from α ≈ . α ≈ . . (cid:46) α (cid:46) . α ≈ .
5, but different normalizations(Larson 1981; Wong et al. 2011; Miville-Deschˆenes et al. 2017; Sun et al. 2018). Indices near α = 0 . R ∼ OBSERVATIONSWe use the published Wong et al. (2019) clump catalog derived from new and archival ALMAmaps of six widely separated GMCs in the LMC (called 30 Dor, PGC, N59C, A439, GMC104, andGMC1). The first two of these GMCs were observed in the CO(2-1) and CO(2-1) lines, while the
Mok et al. last four were observed in the CO(1-0) and CO(1-0) lines. Wong et al. smoothed all these mapsto a common angular resolution of 3 . (cid:48)(cid:48) ( ∼ . CO and CO lines toconstruct samples of clumps (Nayak et al. 2016, 2018; Naslim et al. 2018). However, because thesesamples are based on different resolutions, sensitivities, and selection criteria, we do not attempt tocombine them with the Wong et al. (2019) sample in the work presented here.The Wong et al. (2019) clump catalog is based on the dendogram analysis developed by Rosolowskyet al. (2008) and others. This procedure decomposes intensity maps into a nested hierarchy ofstructures known as leaves, branches, and trunks. The leaves are located at the local intensity peaksand contain no resolved substructure, the branches contain leaves and other branches, while thetrunks are the largest contiguous structures. Since the leaves, branches, and trunks are all subunitswithin GMCs, we refer to them collectively as clumps. The dendogram analysis is designed to reflectthe hierarchical structure of the interstellar medium (ISM), and its leaves are likely sites of clusterformation. It is a popular and proven method for analyzing intensity maps but not the only one(clumpfind being the main alternative). Figure 1 shows the CO intensity maps of the six GMCs,along with the dendogram-based leaves, branches, and trunks from the Wong et al. (2019) catalog.The Wong et al. (2019) catalog lists the masses M and radii R separately for clumps detected inthe CO lines (which traces bulk molecular gas) and in the fainter, optically thin CO lines. Themasses are estimated in three different ways: (1) M Lum from the measured CO luminosity and anassumed CO-to-H conversion factor , (2) M LTE from the measured CO and CO luminosities,the assumption of local thermodynamic equilibrium (LTE), and an assumed CO abundance ratio,and (3) M Vir from the measured radius and velocity dispersion and the assumption of virial equi-librium. The radii are taken to be the geometric means of the major and minor axes of the COcontours. Previous studies have found that the exact method used to measure the sizes of clumps M = α CO L CO , with α CO = 10 . M (cid:12) / (K km s − pc ) for CO(1-0) and α CO = 12 . M (cid:12) / (K km s − pc )) for CO(2-1) ass-Radius Relation and Mass Functions of Molecular Clumps in the LMC Figure 1. CO intensity maps (grayscale) of the six GMCs included in this study. The leaves (green) andbranches plus trunks (brown) are represented by ellipses with dimensions and orientations derived by Wonget al. (2019). The panels are all aligned with North up and East left with the same angular scale in bothdirections, indicated by the inset bars. Note the hierarchical arrangement of clumps within these GMCs.
Mok et al. is unimportant, with similar size-linewidth relations found when different segmentation algorithmswere applied to the same observations (Colombo et al. 2015).In Figure 2, we compare the three different mass estimates for the CO- and CO-detected clumps.The CO catalog (top panels) contains more clumps than the CO catalog (bottom panels), asexpected. For both catalogs, there is less scatter between M Lum and M Vir (right panels) than between M LTE and M Vir (left panels), likely because both CO and CO measurements are required for M LTE estimates, and the CO line is barely detected in many clumps. In any case, the correlations betweenthe three mass estimates are close to linear in both catalogs (except for M LTE vs. M Vir at low massesin the CO catalog). In the following, we repeat our analysis with all three mass estimates in boththe CO and CO catalogs as a guide to the uncertainties in our results.The masses of clumps in the Wong et al. (2019) catalog range from below 10 M (cid:12) to above 10 M (cid:12) ,while their radii range from below 1 pc to above 10 pc. The typical surface density, volume density,and free-fall time of the clumps are Σ = M/ ( πR ) ∼ M (cid:12) pc − , ρ = 3 M/ (4 πR ) ∼ M (cid:12) pc − ,and t ff = (3 π/ Gρ ) / ∼ yr, with only weak dependencies on mass (see Section 3.1 below).While these properties seem conducive to the onset of gravitational collapse and star formation, theclumps have not yet been systematically surveyed for direct evidence of recent star formation (youngstellar objects, etc). RESULTS3.1.
Mass-Radius Relation
In Figure 3, we present the mass-radius relation separately for clumps in the six GMCs from the M Lum mass estimates in the CO catalog. Evidently, these relations all have similar power-law form, R ∝ M α . We determine the best-fit values of α and their standard errors from the linear regressionlog R = α log M + constant for clumps with R ≥ .
72 pc (the completeness limit adopted by Wonget al. 2019) and log(
M/M (cid:12) ) ≥ .
75 (the completeness limit we adopt for the mass functions inSection 3.2). The results of these individual fits for the six GMCs are displayed at the upper rightof the panels in Figure 2. We obtain nearly identical results for the other mass estimates ( M LTE and M Vir ) and for the CO catalog. ass-Radius Relation and Mass Functions of Molecular Clumps in the LMC Figure 2.
Comparison of the three mass estimates M
Lum , M
LTE , and M
Vir for clumps in the CO and CO catalogs as indicated in the panel legends. Leaves are represented by green triangles and branches andtrunks by brown squares.
Mok et al.
The variations in α from one GMC to another ( ∼ .
04) appear to be several times larger than thesmall statistical uncertainty in each value of α (0.01-0.02). This may indicate that there are realvariations in the mass-radius relation of clumps among different GMCs. Wong et al. (2019) speculatethat analogous variations in the size-linewidth relation may be caused by different levels of stellarfeedback in the GMCs. Testing this conjecture will require a full census of the energy and momentumoutput of all the massive young stars in each GMC, a worthwhile project for the future. Alternatively,the apparent variations in α may simply reflect small systematic (non-statistical) errors, such as thoseinherent to low-density molecular tracers and to any method for identifying clumps.In the following, we present results only for the combined sample of clumps in all six GMCs. Thebest-fit values of α and their standard errors for the combined clump sample are listed in Table 1for each of the three mass estimates and for the CO and CO catalogs. These six indices are allremarkably similar, differing from each other only by the small statistical uncertainties ( ± . α = 0 .
36. To allow for systematic errors, at least roughly, as discussedabove, we adopt α = 0 . ± .
03. This value of α indicates that the volume densities of clumps arenearly independent of their masses ( α = 1 / R on M than forconstant surface density ( α = 1 / Table 1.
Index α of the Mass-Radius Relation, R ∝ M α Mass CO CO M Lum . ± .
01 0 . ± . M Vir . ± .
01 0 . ± . M LTE . ± .
01 0 . ± . R ≥ .
72 pc and log (
M/M (cid:12) ) ≥ . It is interesting to compare the mass-radius relation derived here for LMC clumps identified bydendogram analysis of CO maps with that derived for Milky Way clumps identified by various meth-ods. FKM found α = 0 . ± .
02 for a composite sample of clumps selected for their star-formation ass-Radius Relation and Mass Functions of Molecular Clumps in the LMC Figure 3.
Mass-radius relations of clumps in the six GMCs based on the M
Lum mass estimate and COcatalog. Leaves are represented by green triangles and branches and trunks by brown squares. The best-fitindices α of the mass-radius relations are given in the panel legends. The horizontal and vertical dashedlines show the adopted completeness limits at R = 0 .
72 pc and log (
M/M (cid:12) ) = 1 . activity and with measurements of CS, C O, or 1.2 mm dust emission from three independent sur-veys (Shirley et al. 2003; Fa´undez et al. 2004; Fontani et al. 2005). Our value of α for LMC clumpsis nearly identical to the FKM value for Milky Way clumps. Our result also falls within the range0 . (cid:46) α (cid:46) . R ∼ Mok et al. turbulence within the GMCs (Wong et al. 2019). Thus, even for the leaves alone, we rely on theindex α = 0 . ± .
03 derived for the full range of clump masses and sizes (above the completenesslimits). As noted above, this is consistent with the mass-radius relation of clumps in the Milky Way,which are resolved down to R ∼ . Mass Function
We derive the mass function of LMC clumps from the leaves of the Wong et al. (2019) dendogramanalysis. There are two justifications for this. (1) Because the leaves are defined by local peaks inthe intensity maps, they are the most likely sites of cluster formation. (2) Because the branchesand trunks contain the leaves in a nested hierarchy, including them would lead to inconsistencies inaccounting for the numbers and masses of clumps and hence their mass function. We stress that, inthe absence of a complete theory of clump and cluster formation, there is no unique or even generallyaccepted choice of exactly which ISM structures to identify as protoclusters. As we show here, thechoice of dendogram leaves leads to a consistent picture of cluster formation.In Figure 4, we present the mass functions of leaves in the combined GMC sample with the threemass estimates in the CO catalog. These have similar power-law form, dN/dM ∝ M β , above ouradopted completeness limit at log ( M/M (cid:12) ) = 1 .
75. Below this limit, the mass functions begin to fallsignificantly below the extrapolated power laws (by a factor of two in the case of M Vir ). We derive thebest-fit values of β and their standard errors from maximum-likelihood fits to the unbinned massesabove the completeness limit (see Mok et al. (2019) for details of the method). These are listed inTable 2 for the three mass estimates and both the CO and CO catalogs. The fitted values of β are not sensitive to the adopted lower mass limit so long as it lies near or above log ( M/M (cid:12) ) = 1 . β = − .
79 and β = − .
81, respectively. Thus, weadopt β = − . ± . α (Section 3.1).There are two previous determinations of the mass function of clumps in 30 Dor, one of the GMCsincluded in the present study, from the same ALMA data analyzed by Wong et al. (2019). The first isbased on clumps identified by clumpfind in the CO map (Indebetouw et al. 2013) and the second on ass-Radius Relation and Mass Functions of Molecular Clumps in the LMC Figure 4.
Mass functions of leaves in the CO catalog for the M
Lum , M
LTE , and M
Vir mass estimates inequal logarithmic bins. The vertical normalizations have been shifted for clarity. The diagonal lines aremaximum-likelihood fits of power laws to the unbinned masses with best-fit indices β listed in Table 2. Thevertical dashed line shows the adopted completeness limit at log( M/M (cid:12) ) = 1 .
75 (see text). clumps identified by clumpfind in the 1.3 mm and 3.2 mm dust continuum maps (Brunetti & Wilson2019). We plot these mass functions in Figure 5, along with the one derived from clumps identifiedas dendogram leaves in 30 Dor in the Wong et al. (2019) CO catalog. Evidently, the high-mass endof these functions (log (
M/M (cid:12) ) ≥
3) have similar power-law form with β ≈ − Mok et al.
Table 2.
Index β of the MassFunction, dN/dM ∝ M β Mass CO CO M Lum − . +0 . − . − . +0 . − . M Vir − . +0 . − . − . +0 . − . M LTE − . +0 . − . − . +0 . − . Maximum-likelihood fits for leaves with log (
M/M (cid:12) ) ≥ . We can make a more definitive comparison between the mass function derived here for LMC clumpsidentified as dendogram leaves of CO maps and that for Milky Way clumps identified by variousmethods. There are numerous studies of clumps in the Milky Way (e.g Mu˜noz et al. 2007; Wonget al. 2008; Schlingman et al. 2011; Pekruhl et al. 2013; Urquhart et al. 2014; Moore et al. 2015).The mass functions of clumps derived in these studies all have indices in the range − . ≤ β ≤ − . β = − . ± .
1, identical to our result for LMC clumps. INTERPRETATIONWe now check for consistency between the FKM feedback model, the results of the previous section,and the observed mass function of young clusters in the LMC. The FKM model is based on theassumption that stars will continue to form in a protocluster until they have injected enough energyor momentum to remove the remaining gas, thus ending further star formation and fixing the SFE.This condition, applied to protoclusters of different masses, then determines the mass function ofthe resulting clusters (with index β cluster ) in terms of the mass function and mass–radius relation oftheir antecedent clumps (with indices β clump and α clump , respectively). In the analytical FKM model,the SFE depends on the masses of the protoclusters mainly through their surface densities, a resultconfirmed by recent hydrodynamical simulations (Grudi´c et al. 2018; Kim et al. 2018).Feedback in protoclusters potentially involves non-linear combinations of protostellar outflows,main-sequence winds, photoionized gas, radiation pressure on dust, and supernovae. However, thegas removal in this complex realistic situation is likely bracketed by that in the simple idealized ass-Radius Relation and Mass Functions of Molecular Clumps in the LMC Figure 5.
Mass functions of clumps in the 30-Dor GMC from three independent catalogs: gas/dendrogram(Wong et al. 2019), gas/clumpfind (Indebetouw et al. 2013), and dust/clumpfind (Brunetti & Wilson 2019).The vertical normalizations have been shifted for clarity. The diagonal lines are maximum-likelihood fits ofpower laws to the unbinned masses with best-fit indices β = − . +0 . − . (gas/dendogram), β = − . +0 . − . (gas/clumpfind), and β = − . +0 . − . (dust/clumpfind). The vertical dashed line shows the adopted com-pleteness limit of the shallowest survey at log( M/M (cid:12) ) = 3 . cases in which the feedback is energy driven or momentum driven, corresponding to minimum andmaximum radiative losses, respectively. The predictions of the FKM model in these limiting cases4 Mok et al. are: β cluster = 2( β clump + α clump − − α clump ) (energy driven) (1) β cluster = 2 β clump + α clump − − α clump ) (momentum driven) (2) . We compute the index of the cluster mass function from these equations with the clump parametersderived in the previous section: α clump = 0 . ± .
03 and β clump = − . ± .
1. The results are β cluster = − . ± .
08 (energy driven) and β cluster = − . ± .
10 (momentum driven). Evidently,the predicted β cluster is not sensitive to whether the feedback in protoclusters is closer to the energy-driven or momentum-driven limits.There have been several determinations of the mass function of young clusters in the LMC, leadingto β cluster = − . ± . β cluster = − . ± .
05 by maximum-likelihood fit of a pure power law to the unbinned masses ofLMC clusters with M ≥ M (cid:12) and τ ≤
10 Myr (Mok et al. 2020), i.e., the same method we usehere to derive β clump . This result is based on the Hunter et al. (2003) catalog, which covers most ofthe LMC, with revised mass and age estimates from Chandar et al. (2010). For momentum-drivenfeedback, the value of β cluster predicted by the FKM model agrees perfectly with the observed value.The model predicts a slightly shallower mass function for energy-driven feedback, but the agreementwith observation is still satisfactory given the uncertainties (with ∆ β cluster = 0 . ± . β cluster forthe cluster and GMC populations in six nearby galaxies (Mok et al. 2020). For five of these galaxies,we found β cluster ≈ β GMC ≈ − ± .
3) and good agreement with the FKM model. The MilkyWay has β cluster ≈ β clump ≈ β GMC ≈ − ± . β GMC ≈ − . β cluster ≈ β clump ≈ − . ass-Radius Relation and Mass Functions of Molecular Clumps in the LMC β cluster of thecluster mass function in terms of the observed indices β clump and α clump of the clump mass functionand mass-radius relation. In this model, the properties of the clumps are regarded as “initial condi-tions” for the clusters. In a more complete theory, the formation and properties of the clumps wouldalso need to be explained. A variety of idealized analytical models and hydrodynamical simulationssuggest that the observed indices β clump ≈ − . α clump ≈ . CONCLUSIONSIn summary, we have derived the mass-radius relation and mass function of clumps in thedendrogram-based catalog compiled by Wong et al. (2019) from ALMA CO and CO maps ofsix GMCs in the LMC. The Magellanic Clouds are the only external galaxies for which this type ofanalysis is possible at the necessary spatial resolution ( ∼ R ∝ M α , with index α =0 . ± .
03 over the mass range 10 M (cid:12) (cid:46) M (cid:46) M (cid:12) . This is very similar to the mass-radiusrelation of clumps in the Milky Way.6 Mok et al.
2. The mass function of LMC clumps has power-law form, dN/dM ∝ M β , with index β = − . ± . M (cid:12) (cid:46) M (cid:46) M (cid:12) . This is very similar to the massfunction of clumps in the Milky Way.3. With these values of α and β for the clumps, the predicted index for the mass function of youngLMC clusters from the FKM model is β ≈ − .