Molecular cloud and star cluster mass functions

# From the molecular-cloud- to the embedded-cluster-mass function with a density threshold for star formation

Geneviève Parmentier
Argelander-Institut für Astronomie, Bonn Universität, Auf dem Hügel 71, D-53121 Bonn, Germany
Humboldt Fellow - E-mail: gparm@astro.uni-bonn.de
Accepted 2010 December 24. Received 2010 December 15; in original form 2010 September 28
###### Abstract

The mass function of molecular clouds and clumps is shallower than the mass function of young star clusters, gas-embedded and gas-free alike, as their respective mass function indices are and . We demonstrate that such a difference can arise from different mass-radius relations for the embedded-clusters and the molecular clouds (clumps) hosting them. In particular, the formation of star clusters with a constant mean volume density in the central regions of molecular clouds of constant mean surface density steepens the mass function from clouds to embedded-clusters. This model is observationally supported since the mean surface density of molecular clouds is approximately constant, while there is a growing body of evidence, in both Galactic and extragalactic environments, that efficient star-formation requires a hydrogen molecule number density threshold of .

Adopting power-law volume density profiles of index for spherically symmetric molecular clouds (clumps), we define two zones within each cloud (clump): a central cluster-forming region, actively forming stars by virtue of a local number density higher than , and an outer envelope inert in terms of star formation. We map how much the slope of the cluster-forming region mass function differs from that of their host-clouds (clumps) as a function of their respective mass-radius relations and of the cloud (clump) density index. We find that for constant surface density clouds with density index , a cloud mass function of index gives rise to a cluster-forming region mass function of index . Our model equates with defining two distinct SFEs: a global mass-varying SFE averaged over the whole cloud (clump), and a local mass-independent SFE measured over the central cluster-forming region. While the global SFE relates the mass function of clouds to that of embedded-clusters, the local SFE rules cluster evolution after residual star-forming gas expulsion. That the cluster mass function slope does not change through early cluster evolution implies a mass-independent local SFE and, thus, the same mass function index for cluster-forming regions and embedded-clusters, that is, . Our model can therefore reproduce the observed cluster mass function index .

For the same model parameters, the radius distribution also steepens from clouds (clumps) to embedded-clusters, which contributes to explaining observed cluster radius distributions.

###### keywords:
stars: formation — galaxies: star clusters: general — ISM: clouds — stars: kinematics and dynamics
pagerange: From the molecular-cloud- to the embedded-cluster-mass function with a density threshold for star formationLABEL:lastpagepubyear: 2011

## 1 Introduction

The star formation efficiency (SFE) achieved by star cluster gaseous precursors at the onset of residual star-forming gas expulsion is a crucial quantity since it influences the cluster dynamical response to gas expulsion significantly (the so-called violent relaxation; Hills, 1980; Geyer & Burkert, 2001; Baumgardt & Kroupa, 2007; Proszkow & Adams, 2009). Specifically, the SFE is tightly related to whether the cluster survives violent relaxation and, if it survives, what mass fraction of its stars it retains. The SFE being the ratio between the stellar mass of embedded-clusters and the initial gas mass of their precursor molecular clouds, the comparison of the mass functions of young star clusters and of molecular clouds holds the potential of highlighting whether the SFE varies with molecular cloud mass.

The mass function of giant molecular clouds (GMCs) in the Local Group of galaxies has an index - (Rosolowski, 2005; Blitz et al., 2006) (see also Fukui et al., 2008, for the case of the GMC mass function in the Magellanic Clouds). These GMCs, when compressed by the high pressure of violent star-forming environments, are expected to be the parent clouds of massive star clusters forming profusely in galaxy mergers and starbursts (Jog & Solomon, 1992; Jog & Das, 1996). The same slope - is also found for the mass function of density enhancements contained by GMCs – referred to as molecular clumps (Lada, Bally & Stark, 1991; Kramer et al., 1998; Wong et al., 2008). In quiescent disc galaxies such as the Milky Way, those are observed to be the progenitors of open clusters (Harris & Pudritz, 1994; Lada & Lada, 2003, and references therein).

In contrast to molecular clouds and clumps, the mass function of embedded and young clusters is, in most cases, reported to have an index (e.g. Zhang & Fall, 1999; Bik et al., 2003; Lada & Lada, 2003; Oey et al., 2004), which is steeper than the mass function of molecular structures. Given the uncertainties affecting both slopes, the significance of the difference remains uncertain. Elmegreen & Falgarone (1996) suggest that error-free measurements of GMC masses may bring the mass function slopes of young star clusters and GMCs in agreement. Conversely, one can consider that the slope difference is significant, which is the approach we adopt in this paper.

The question we set to answer is: what process of the physics of cluster-formation steepens the power-law mass function of molecular clouds and clumps from to ? The difference suggests that the SFE is a decreasing function of the cloud (clump) mass. Besides sounding counter-intuitive, a mass-varying SFE is necessarily conducive to mass-dependent cluster infant weight-loss since the fraction of stars remaining bound to clusters through violent relaxation is a sensitive function of the SFE (e.g. fig. 1 in Parmentier & Gilmore, 2007). This does not seem to be supported by observations of young star clusters, as their mass function slope is reported to remain invariant with time over their first 100 Myr of evolution (Kennicutt et al., 1989; McKee & Williams, 1997; Lada & Lada, 2003; Zhang & Fall, 1999; Oey et al., 2004; Dowell et al., 2008; Chandar et al., 2010).

However, this contradiction is apparent only for it is worth keeping in mind that the SFE driving cluster violent relaxation is the mass fraction of gas turned into stars over the volume of gas forming stars. And this volume of star-forming gas may not coincide with the entire cloud (clump). In what follows, we refer to it as the cluster-forming region (CFRg). Its SFE is the local SFE and its mass function slope is . The invariance of the young cluster mass function slope at early time suggested by many observations demands a mass-independent local SFE. That is, the mass fraction of gas turned into stars by a CFRg is independent of its mass. This in turn implies that the slopes of the CFRg and embedded-cluster mass functions are identical: . Therefore, understanding the slope difference between the star cluster and molecular cloud (clump) mass functions equates with understanding why the CFRg mass function is steeper than the mass function of their host clouds (clumps), i.e. and . The difference suggests that the mass fraction of star-forming gas inside molecular clouds (clumps) is a decreasing function of the cloud (clump) mass. Besides, that the CFRg represents a fraction only of its host cloud (clump) allows us to define a global SFE, namely, the ratio between the mass in stars formed inside a molecular cloud (clump) and its initial gas mass. The global SFE is relevant to explaining the slope difference, but irrelevant for modelling cluster violent relaxation.

What could be the origin of a mass-varying mass fraction of star-forming gas inside molecular clouds (clumps)? In other words, why should the global SFE vary with the cloud (clump) mass such that ?

The mean surface density of GMCs in our Galaxy is about constant (fig. 8 in Blitz et al., 2006). This result is reminiscent of Larson’s seminal study (Larson, 1981) showing that molecular clouds have approximately constant mean column densities (see also Lombardi, Alves & Lada, 2010). On the other hand, star-forming regions are observed to be systematically associated with dense molecular gas, namely, with number densities of at least - (Müller et al., 2002; Gao & Solomon, 2004; Faundez et al., 2004; Fontani et al., 2005; Shirley et al., 2003; Wu et al., 2010). See also fig. 1 and section 3 in Parmentier & Kroupa (2010) for a discussion. Several studies have therefore suggested that star formation requires a gas volume (or number) density threshold (e.g. Evans, 2008; Wu et al., 2010; Lada, Lombardi & Alves, 2010).

In this contribution, we develop a model for a spherically symmetric molecular cloud (clump) with a power-law density profile which forms a star cluster in its central region. We demonstrate that if the mass-radius relation of CFRgs differs from that of the host-clouds (clumps), then the slopes of their respective mass functions are different too (i.e. ). This will be the case for molecular clouds of constant mean surface density hosting CFRgs of constant mean volume density. Applying the same model to the radius distribution, we will show that it can also contribute to explaining why the distribution of star cluster half-light radii is significantly steeper than the distribution of GMC sizes. Figure 1 summarises the different mass functions encompassed through the paper, along with their respective index and the various mass ratios relating them.

Note that this paper does not intend to explain the slope of the stellar initial mass function. That issue is addressed in Shadmehri & Elmegreen (2010) whose model successfully reproduces the Salpeter slope of for a population of pre-stellar cores exceeding a volume density threshold in a fractal cloud.

The outline of the paper is as follows. Section 2 summarises two types of evidence supporting the hypothesis of a constant mean volume density for CFRgs. One hinges on the early dynamical evolution of star clusters. The second is based on the mapping of star-forming regions with dense molecular gas tracers. In Section 3, we build a model relating the power-law mass function of CFRgs to the power-law mass function of their host clouds (clumps). We map how the slope difference varies as a function of the mass-radius relation and density profile of molecular clouds (clumps). In Section 4, we discuss the implications of our model. Specifically, we focus on the physically-motivated case of virialized pressure-bounded (i.e. constant mean surface density) clouds (clumps). Section 5 is the counterpart of Section 3 as it models the radius distribution of CFRgs in relation to that of their parent clouds (clumps). Our conclusions are presented in Section 6.

## 2 Constant mean volume density for cluster-forming regions (CFRgs)

The tidal field impact, namely, the ratio of the half-mass radius to the tidal radius of an embedded-cluster, quantifies how deeply a cluster sits within its tidal radius and hence its likelihood of experiencing tidal overflow as it expands in response to gas-expulsion. To satisfy the observed requirement of mass-independent cluster infant weight-loss, must be independent of the CFRg mass. Parmentier & Kroupa (2010) demonstrate that – for given local SFE, gas expulsion time-scale and external tidal field –, this constrain is robustly satisfied for CFRgs with constant mean volume density. This is because their half-mass radius and tidal radius scale alike with the embedded-cluster mass , namely, and . For constant volume density cluster progenitors, the tidal field impact is thus mass-independent. In contrast, constant surface density CFRgs lead to more massive clusters being more vulnerable to early destruction than their low-mass counterparts owing to a greater tidal field impact ( and ), while the opposite is true for constant radius CFRgs ( and ). Since observations suggest infant mortality/weight-loss to be mass-independent, the analysis performed by Parmentier & Kroupa (2010) lends strong support to the hypothesis that CFRgs have a constant mean volume density.

In our Galaxy, observational evidence for CFRgs of constant mean volume density is provided by the tight association between postsigns of star formation (/ sources, water masers, bipolar molecular outflows) and high density molecular gas, i.e. hydrogen molecule number densities (or mean volume densities ; Evans, 2008). Aoyama et al. (2001) and Yonekura et al. (2005) note that star formation in cores is often associated to the molecule, a tracer of molecular gas with (see bottom panel of Fig. 3 and Section 3).

That star formation requires a volume density threshold is also supported by studies of the molecular gas content and star formation activity of external galaxies. Gao & Solomon (2004) obtain the ratio of 65 infrared galaxies, where is the galaxy-integrated infrared luminosity, and is the galaxy-integrated HCN J line luminosity. maps molecular gas with , while traces the star formation rate (SFR). From the near-constancy of , Gao & Solomon (2004) deduce that, on the average, galaxy-integrated SFRs scale linearly with their dense molecular gas content, from quiescent spirals to violent Ultra-Luminous Infra-Red Galaxies (ULIRGs). In contrast, the ratio , where the galaxy-integrated CO luminosity traces molecular gas with , is not on the average constant (see figs 1 and 2 in Gao & Solomon, 2004). That is, traces the global SFR of galaxies better than .

Using the same HCN J molecular tracer as Gao & Solomon (2004), Wu et al. (2005) map dense molecular clumps in Galactic GMCs. They find that the one-to-one correlation between and established by Gao & Solomon (2004) for entire galaxies also holds for individual dense molecular clumps. Gao & Solomon (2004) and Wu et al. (2005) therefore argue that the most relevant parameter for the SFR is the amount of dense molecular gas, namely, gas with . For instance, the high SFR of ULIRGs stems from their large content of molecular gas with densities comparable to that of molecular clumps in Galactic GMCs (see also Section 4.1). As pointed out by Evans (2008), these dense clumps provide the connection between star formation in the Milky Way and in other galaxies. These conclusions are reminiscent of the earlier study of Lada (1992). She find that, while the bulk of star formation in the Orion B molecular cloud is associated with gas with , the CO-traced gas is inert in terms of star formation. Lada, Lombardi & Alves (2010) achieve the same conclusion by comparing infrared extinction maps of local molecular clouds with their respective census of young stellar objects.

A theoretical prediction of a number density threshold for star formation is made by Elmegreen (2007) who note that, when , several microscopic effects enhance magnetic diffusion in the molecular gas, thereby significantly accelerating star formation (e.g. steeper density scaling for the electron fraction, modification of the coupling between dust grains and the magnetic field, sudden drop in the cosmic-ray ionization rate and hence in the ionization fraction) (see Elmegreen, 2007, his section 3.6).

That the bulk of star formation activity takes place in dense molecular gas regions characterised by a mean constant volume density is thus supported by both the analysis of the tidal field impact upon young clusters (Parmentier & Kroupa, 2010), and by the tight association observed between star formation and dense molecular gas. Given the uncertainties regarding the volume density threshold requested for star formation, calculations presented below are performed for two distinct cases: (e.g. Lada, Lombardi & Alves, 2010) and (e.g. Elmegreen, 2007).

## 3 From molecular clumps to high-density cluster-forming regions

Before going any further, a clarification of the terminology applied through this paper may be needed. The following nomenclature has taken root in the community: the word ‘core’ is now often restricted to the gaseous precursor of an individual star or of a small group of stars, while the term ‘clump’ is designated for regions hosting cluster formation. We will follow that terminology. The CFRg is the clump central region where active star formation takes place owing to a high enough volume/number density, i.e. . CFRg-related quantities are identified by the subscript ‘th’, where ‘th’ stands for (volume density) threshold. By virtue of the assumed spherical symmetry, a clump is assumed to contain one single forming-cluster. This constitutes a major difference between the present model and the model by Shadmehri & Elmegreen (2010) who consider the formation of many high-density regions in one single fractal cloud. Note that the present terminology implies that the expressions ‘cluster-forming cores’ used profusely in Parmentier et al. (2008), Parmentier & Fritze (2009) and Parmentier & Kroupa (2010) are now to be read ‘cluster-forming regions’. Although the subscript ‘clump’ is used systematically in the equations below, these equations can be applied indifferently to any spherical volume of molecular gas containing a star-forming region in its central zone. Should GMCs in galaxy starbursts and mergers be roughly spherical and forming each a massive star cluster in their centre, all equations developed in this paper can be applied to them.

The mass function of molecular clumps and clouds mapped in , or emission line is well-described by a power-law with (Kramer et al., 1998; Wong et al., 2008). This is shallower than the ‘canonical’ mass function of young star clusters for which .

To explain this difference in slope, our study rests on the clump (cloud) outer layers being inefficient at forming stars. Let us consider the clumps with masses and radii compiled in top and middle panels of Fig. 3. This mass-radius diagram is based on the data of Aoyama et al. (2001, the Orion B molecular cloud, triangles), Saito et al. (1999, the GMC toward HII regions S35 and 37, squares) and Yonekura et al. (2005, the  Carinae GMC, circles) (see also section 3 in Parmentier & Kroupa (2010) for additional details). By virtue of the molecular tracer used to map these clumps, their mean number densities sample the limited range . These density limits are shown as the dotted (black) lines in the middle panel of Fig. 3. Let us consider a particular clump of mass and radius  pc. Its mean number density suggests that it fails at forming stars if star formation actually requires a density threshold . Yet, molecular clumps show density gradients (Section 3.1) and the condition may be met in the clump inner regions. This in turn implies that the radius and mass of cluster gaseous progenitors are smaller than those of their host-clumps. In what follows, and are the mass and radius of the CFRg where . Conversely, clump outer layers are inefficient at forming stars owing to too low a volume density, i.e. . This situation is illustrated in Fig. 2.

The question we set to answer is: can the CFRg mass function and clump (cloud) mass function differ? As we shall see, it depends on the clump mass-radius relation and on the clump volume density profile.

### 3.1 Model for cluster-forming regions and their host-clumps

Let us characterise molecular clumps (clouds) with the following properties:
(i) Their volume density profile obeys a power-law of slope

 ρclump(s)=kρs−p, (1)

with the distance from the clump centre and a normalizing factor. The assumption of spherical symmetry is supported by e.g. the 1.2-mm continuum observations of Beltran et al. (2006) who find the mean and median ratios of the full widths at half maximum of their clumps along the - and -axes, , to be 1.04 and 0.96, respectively. Power-law density profiles for molecular clumps are put forward by various studies, e.g. Heaton et al. (1993), Hatchell et al. (2000), Beuther et al. (2002), Fontani et al. (2002) and Müller et al. (2002). Estimates for the density index are found mostly in the range . We insist that, in what follows, expressions ‘constant volume density clumps’ or ‘constant surface density clumps’ do not imply that these clumps have a uniform volume or surface density. Rather, it means a population of clumps all characterized by the same mean surface or volume density.

(ii) The mass-radius relation of molecular clumps is quantified by its slope and normalization :

 rclump[pc]=χ(mclump[M\sun])δ. (2)

The combination of Eqs. 1 and 2 leads to the CFRg mass , i.e. the mass of the clump region where . Equation 3 provides as a function of the clump mass , radius and density index :

 mth=(3−p4πρth)(3−p)/pm3/pclumpr−3(3−p)/pclump. (3)

It is valid for density indices . Similarly, the radius of the spherical CFRg is:

 rth=(3−p4πρth)1/pm1/pclumpr−(3−p)/pclump. (4)

It immediately follows that the CFRg mean density is constant. It depends solely on the density index and density threshold :

 <ρth>=3mth4πr3th=33−pρth. (5)

In other words, the mass-radius relation of CFRgs obeys . This is a key-point for our forthcoming discussion about how clump and CFRg mass functions differ from each other. Equation 5 shows that for a truncated isothermal sphere (), the CFRg mean volume density is three times higher than the threshold . For shallower density indices, the density contrast is weaker, e.g. when .

To quantify CFRg masses, an estimate of (or ) is needed. In the middle panel of Fig. 3, open symbols marked with a black filled circle indicate clumps with detected star formation activity ( or source, or bipolar molecular outflow). Data come from tables 1-3 in Aoyama et al. (2001), tables 2 and 4 in Saito et al. (1999), and table 3 in Yonekura et al. (2005). These C clumps were also observed by Aoyama et al. (2001) and Yonekura et al. (2005) in the HCO emission line so as to detect gas. Filled symbols in top panel of Fig. 3 highlight clumps detected in -emission. Symbol size is proportional to the mass of detected in -emission. No data is provided by Saito et al. (1999). The bottom panel of Fig. 3 zooms in on the -data and the star formation activity detections for Orion B and Car [the S35/37 data of Saito et al. (1999) are ignored as they lack an -mapping]. The comparison between both types of data highlights the tight correlation between HCO-detected gas and star formation activity, a point already made by Aoyama et al. (2001) and Yonekura et al. (2005): 10 in 12 -detections also show signs of star formation activity, while 10 in 12 clumps with detected star formation activity also host an -detected region. Similarly, Higuchi et al. (2010) detect -emission in the clumps associated to embedded clusters studied by Higuchi et al. (2009) (see also Section 4.2). This suggests that star formation requires number densities of order (or ). We therefore adopt as the fiducial (see also Elmegreen, 2007). We will also present results for (Lada, Lombardi & Alves, 2010).

Note that the presence of clumps with mean densities and hosting star formation activity (middle panel of Fig. 3) does not imply that star formation can take place at so low number densities. Molecular clumps are characterized by density gradients and star formation is very likely confined to the clump deeper regions (Fig. 2) whose higher volume densities are revealed in HCO.

Solid (red) lines in the top panel of Fig. 3 are iso- lines in the vs.  space when and (Eq. 3). In our model, the total stellar mass inside a clump scales with the mass of its central CFRg rather than with the clump total mass . Given the CFRg (local) SFE, the stellar mass of the embedded-cluster is . The top panel of Fig. 3 also displays vectors along which , and increase by 2 orders of magnitude, with the average clump surface density. These vectors illustrate that the CFRg mass depends more sensitively on the average clump surface density than on , an effect depicted in Fig. 4. Its top and bottom panels show the mass of the high-density CFRg as a function of the total clump mass and of the average clump surface density , respectively. These relations are obtained by combining Eqs. 2 and 3, and their logarithmic slopes are quoted in their respective panels. Four models combining or , with or are presented. In all cases, the clumps are assumed to have a constant mean volume density, namely, , typical of clumps. As suggested by the vectors in the top panel of Fig. 3, the slope of the vs relation is steeper than its counterpart vs , by a factor . As an example, for clumps with a given mean volume density (), the ratio between both slopes is a factor of 3. The embedded-cluster stellar mass , thus also its luminosity, therefore depend more sensitively on than on . This explains straightforwardly why Aoyama et al. (2001) find the IRAS luminosity of their clumps ‘to be more strongly dependent on the average column density than on the total mass of the clump’ . In a forthcoming paper, we will model the infrared luminosity of molecular clumps as a way of probing their forming stellar content and compare model outputs with existing data-sets.

The lowest-mass Orion B clumps (with ) show neither sign of star formation nor -detection. The top panel of Fig. 3 shows that if , these clumps contain of high-density gas. So low a mass may result in neither (detected) star formation, nor detection.

### 3.2 From the clump mass function to the CFRg mass function

Having defined the properties of CFRgs and of their host-clumps, we are now ready to relate the CFRg mass function to the clump mass function. Our model explicitly assumes that the high-density CFRgs – the genuine sites of cluster formation – have a constant mean volume density (Eq. 5) and, thus, that their mass-radius relation scales as . In this section, we show that if the mass-radius relation of their host-clumps has a different slope (i.e. ), the slope of the CFRg mass function differs from that of the clump mass function too (i.e. ).

As a first step, this can be understood by obtaining the clump mass fraction occupied by the dense central CFRg. Combining Eqs. 2 and 3, we obtain this mass ratio

 mthmclump=(3−p4πρthχ3)(3−p)/pm[(3−p)(1−3δ)]/pclump (6)

as a function of normalization and slope of the clump mass-radius relation (Eq. 2), of the slope of the clump density profile (Eq. 1), of the clump mass , and of the volume density at the edge of the CFRg. With the normalization of Eq. 2, and are in units of and , respectively.
For constant volume density clumps (), this mass fraction is independent of

 mthmclump=(3−p4πρthχ3)(3−p)/p (7)

and the clump and CFRg mass function slopes are thus alike. For constant surface density clumps (), this mass fraction

 mthmclump=(3−p4πρthχ3)(3−p)/pm−(3−p)/(2p)clump (8)

is a decreasing function of (since ). Therefore, we predict a CFRg mass function steeper than the clump mass function. For constant radius clumps (), the CFRg mass fraction obeys

 mthmclump=(3−p4πρthχ3)(3−p)/pm(3−p)/pclump. (9)

It is an increasing function of , rendering the CFRg mass function shallower than the clump mass function. These effects are expected since, for constant surface density clumps, the mean volume density decreases with increasing mass, while the opposite is true for constant radius clumps.

Let us now quantify these effects in detail and let us consider a population of clumps whose mass distribution is a power-law of slope :

 dN=kclumpm−β0clumpdmclump. (10)

To derive the CFRg mass function

 dN=kthm−βthdmth, (11)

we need to relate the CFRg mass to the clump mass . Equation 6 straightforwardly leads to:

 mth=(3−p4πρthχ3)(3−p)/pm[3−3δ(3−p)]/pclump. (12)

The combination of Eqs. 10 and 12 leads to the power-law mass function of CFRgs:

 dN=kclump(4πρthχ33−p)(3−p)(1−β0)/[3−3δ(3−p)] p3−3δ(3−p)m−βthdmth. (13)

with obeying:

 β=pβ0−(p−3)(1−3δ)3−3δ(3−p) (14)

Figure 5 shows in dependence of for 6 distinct cases: clumps with constant surface density (), constant volume density () and constant radius (), combined to two density indices: (isothermal spheres) and . As we saw above, leads to , while constant clump surface density (radius) increases (decreases) compared to . Shallower clump density profiles (open symbols in Fig. 5) result in a greater contrast between the clump and CFRg mass function slopes, that is, is greater for smaller density index . This effect is further quantified in Fig. 6 which depicts as a function of for a given spectral index of the clump mass function. is conducive to . When , the difference between the clump and CFRg mass function slopes vanishes, irrespective of the clump mass-radius relation. In contrast, the smaller the density index , the steeper (shallower) the CFRg mass function when ().

Figure 7 presents the outcome of Monte-Carlo simulations performed to compare CFRg mass functions to their ‘parent’ clump mass function. In each panel, the latter is depicted as the upper black line with asterisks. The same six combinations of and as previously are considered and symbol/colour-coding is identical to Fig. 5. Adopted normalizations for the clump mass-radius relation (Eq. 2) correspond to the best-fits of the data in Fig. 3 with slope imposed. Those were obtained by Parmentier & Kroupa (2010, their table 1). We remind those clump mass-radius relations below for the sake of clarity. Fitting a constant surface density relation onto the data leads to :

 rclump=0.04m1/2clump, (15)

equivalent to . A constant volume density fit results in :

 rclump=0.11m1/3clump, (16)

or .

As discussed in Section 3.1, two distinct volume density thresholds are considered: (top panel of Fig. 7), and (bottom panel of Fig. 7).

Not only do shallower clump density profiles lead to greater , they are also conducive to lower normalizations of the CFRg mass function compared to the clump mass function. This effect stems from a smaller clump mass fraction achieving the volume density threshold for lower . The horizontal shift between the clump and CFRg mass functions also depends on and (see Eq. 6). The closer to the clump mean volume density is, the greater the clump mass fraction contained within the CFRg and the smaller the horizontal shift between the clump and CFRg mass functions. When the whole clump is at a density of at least , clump and CFRg mass functions coincide (e.g. when , and , see lines with triangles in bottom panel of Fig. 7).

That CFRgs hosted by constant surface density clumps () have a steeper mass spectrum than their parent clumps is a highly interesting result since the same is observed for gas-embedded clusters and young gas-free clusters () as compared to GMCs and their dense gas clumps (). We discuss this issue in Section 4.

### 3.3 ‘Global’ and ‘local’ SFEs

That only a limited region of a molecular clump may form a star cluster means that the SFE relevant to model cluster violent relaxation must be defined properly. The dynamical response of a cluster to the expulsion of its residual star-forming gas is governed in part by the mass fraction of gas turned into stars within the volume of gas forming stars and at the onset of gas expulsion111In this contribution, we assume that stars and gas in the forming-cluster are in virial equilibrium at gas expulsion onset. This implies that the gas mass fraction turned into stars within the CFRg equates with the effective SFE (eSFE, Goodwin, 2009). That is, the CFRg (or local) SFE plays a key-role in the early evolution of star clusters. For detailed discussions of this assumption, see Kroupa (2008) and Goodwin (2009).. We refer to this SFE as the ‘local’ SFE , namely, the CFRg mass fraction eventually turned into stars. Conversely, an SFE averaged over the whole molecular clump – hereafter ‘global’ SFE – constitutes a lower limit only to the local SFE.

How the local SFE (), global SFE (), and CFRg mass fraction connect the embedded-cluster mass function , the CFRg mass function and the clump mass function is summarised in Fig. 1.

Using Eq. 6, it is straightforward to relate these two SFEs:

 ϵglobal=meclmclump=meclmthmthmclump=ϵlocalmthmclump =ϵlocal(3−p4πρthχ3)(3−p)/pm[(3−p)(1−3δ)]/pclump. (17)

Given a density threshold , a lower normalisation is conducive to higher-density clumps, larger clump mass fractions with and, therefore, higher global SFEs.

Figure 8 shows Eq. 17 for the same (, , , ) sets as in Fig. 7, with identical colour- and symbol-codings. The adopted local SFE is . It is shown as a horizontal dotted (black) line in both panels. For a weak external tidal field impact, this SFE ensures that the cluster survives violent relaxation, even if the gas expulsion time-scale is much shorter than a CFRg crossing-time (i.e. explosive gas expulsion; see fig. 1 in Parmentier & Gilmore, 2007). Yet, Fig. 8 illustrates that the global SFE measured over an entire clump can be significantly smaller than and misleadingly suggests that the embbeded-cluster is not to survive violent relaxation. Therefore, small SFEs for clumps reported in the literature (e.g. Higuchi et al., 2009, their table 3) do not necessarily imply that embedded-clusters get disrupted after gas expulsion.

Besides, one should also keep in mind that an observed SFE may be low because the CFRg is still in the process of building up its stellar content. This trend is observed by Higuchi et al. (2009) who mapped in -emission 14 molecular clumps associated to embedded-clusters. They define a sequence A-B-C of clump morphology (their table 3). In Type-A clumps, the cluster is associated with the peak of emission. In contrast, clusters hosted by Type-C clumps are located at a cavity-like -emission hole, which demonstrates that gas dispersal has started in Type-C clumps. Higuchi et al. (2009) therefore conclude that the morphology sequence A-B-C equates with an evolutionary sequence, with Type-A and Type-C corresponding to the least and most evolved clumps, respectively. In further support of their scenario, they find a trend for the global SFE to increase along the sequence A-B-C. The key-point to keep in mind here is that cluster violent relaxation depends on the local SFE – i.e. within the CFRg – at the onset of gas expulsion.

Figure 8 illustrates that, under the assumption of a constant , is clump-mass dependent when . As this is the local SFE which rules cluster violent relaxation, the potential dependence of the global SFE on the clump mass is in itself not conducive to mass-dependent effects during violent relaxation. If the local SFE, gas-expulsion time-scale in units of a CFRg crossing-time (Parmentier et al., 2008) and tidal field impact (Parmentier & Kroupa, 2010) are CFRg-mass-independent, the slope of the cluster mass function through violent relaxation does not change, regardless of whether the global SFE is clump-mass-dependent or not.

The cases illustrated in Fig. 8 assume that one clump hosts one CFRg, as depicted in Fig. 2. The structure of GMCs in spiral galaxies, however, is more complex as one GMC hosts several clumps, possibly strung out on a filament, each hosting a star-forming or cluster-forming region. For instance, let us consider the case of a GMC hosting 10 dense clumps characterized by a constant mean surface density (or ) and density index . That surface density is characteristic of clumps showing signs of star-formation activity (see top panel of Fig. 3). The random sampling of a clump mass function with slope and mass lower limit shows that those clumps totalize on the average of molecular gas. Assuming a star formation density threshold of (i.e. at the logarithmic midpoint between the two values tested through this paper), the total mass in dense star-forming gas represents only one tenth of the total clump mass, that is, . Over the scale of the whole GMC, the filling factor for the star-forming dense gas is thus . Assuming , the GMC global SFE is , a value typical for Galactic GMCs (Duerr, Imhoff & Lada, 1982).

In galaxy starbursts and mergers, GMCs get compressed by the high pressures pervading such violent environments. This results in high volume densities through most of the GMC volume. In Section 4.1, we will speculate that such a GMC is roughly spherical with a smooth density profile and the birth site of one single massive cluster in its central regions. That is, the relations established in this section remain valid and are transposed to the case of a GMC.

## 4 Model Consequences

Figures 5, 6 and 7 demonstrate that the mass distribution from molecular clumps (clouds) to CFRgs steepens () if clumps (clouds) have a constant mean surface density. We remind here that, in our model, CFRgs have a constant mean volume density by virtue of the assumed volume density threshold for star formation (Eq. 5). The mean density index found by Müller et al. (2002) for the star-forming regions they map in dust-continuum emission is . Combined to for the clump mass function, this is conducive to for the CFRg mass function (see Fig. 6). Other realistic values of (see references in Section 3.1) lead to steepenings ranging from when (an effect probably undetectable amidst data noise) to when . These values bracket the ‘canonical’ mass function slope of young star clusters (), provided that the ‘local’ SFE is mass-independent (i.e. ). The key-point to investigate now is whether molecular clumps and clouds hosting CFRgs have a constant mean surface density. We consider two distinct cases: GMCs and their dense clumps. In quiescent spirals such as our Galaxy, open clusters are observed to form within dense clumps in GMCs. In galaxy starbursts and mergers, compressed GMCs are likely the individual birth-sites of massive star clusters forming profusely in these violent star-forming environments.

### 4.1 GMCs as cluster-forming sites

Pressure-bounded clouds in virial equilibrium have a constant mean surface density, provided that the external pressure is about constant for all clouds (see e.g. Harris & Pudritz, 1994):

 rcloud∝m1/2cloudP−1/4ext. (18)

GMCs in our Galaxy occupy a narrow range in mass surface density with (fig. 8 in Blitz et al., 2006). More recently, Heyer et al. (2009) found that, on the average, (see also Lombardi, Alves & Lada, 2010). Given the pressure characterising the Galactic disc, this is about the surface density expected for virialized gas clouds in pressure equilibrium with their environment. The cloud external pressure and cloud surface density are related through (Harris & Pudritz, 1994):

 ΣGMC=0.5(Pextk)1/2M\sun.pc−2. (19)

The pressure in the Galactic disc is (Blitz & Rosolowski, 2004), leading to (or ). This is in excellent agreement with the estimate of Heyer et al. (2009). A surface density equates with the mass-radius relation , shown as the solid (red) line with open squares in Fig. 9.

The mean number density of Galactic GMCs spans the range (see Fig. 9). This is far too low for them to experience overall efficient star formation. The situation in galaxy starbursts and mergers is drastically different, however, because external pressures there are several orders of magnitude higher than in the Galactic disc. Assuming that GMCs in these violent star-forming environments remain in virial equilibrium, Ashman & Zepf (2001) note that the corresponding high external pressures () compress GMCs such that their radii become a few parsecs, similar to those of globular clusters (see solid line with filled circles in Fig. 9: for ). They thus propose that compressed GMCs are the gaseous precursors of massive star clusters formed in starbursts and mergers. If the mean surface density of GMCs in mergers and starbursts is about constant, as it is the case for Galactic GMCs, and if their internal structure resembles what is depicted by Fig. 2, then the mass functions of CFRgs and embedded-clusters are steeper than the GMC mass function, with the embedded-cluster index depending on the GMC density index as shown by the (red) solid line with filled circles of Fig. 6.

Ashman & Zepf (2001) also propose that the high pressures and densities achieved in GMCs in starbursts and mergers are conducive to high SFEs. This is in exact agreement with our scenario of denser molecular clumps (clouds) having higher global SFEs (i.e. in Eq. 17, a lower leads to a higher ). Figure 9 shows the mass-radius relations of constant surface density clouds bounded by pressures of (characteristic of the Galactic centre, Spergel & Blitz, 1992; Jog & Das, 1996) and (characteristic of galaxy mergers, Jog & Solomon, 1992). The compression of GMCs by these high external pressures raise their volume densities, rendering them closer or even similar to what is observed for Galactic CFRgs, namely, . As a result, the GMC mass fraction with increases and so does the GMC global SFE. For , the normalization of the mass-radius relation is . Combined with (constant surface density), (to get with , see Fig. 6 and Eq. 14) and a star formation density threshold , this leads to a dense gas mass fraction of (Eq. 8):

 mthmGMC=6.9×(mGMCM\sun)−0.3. (20)

That is, GMCs of masses and have  % and  % of their mass at a density higher than . With a local SFE of 35 %, this corresponds to global SFEs of  % and  %, respectively. In comparison, the overall SFE in Galactic GMCs is of the order of  %.

That the mass fraction of dense gas in GMCs is a decreasing function of the GMC mass also implies that the most massive GMCs are not necessarily the largest providers of newly formed stars, despite them containing most of the molecular gas (when , clouds more massive than contain  % of the total gas mass for a GMC mass range  - ). For the parameters adopted here, it is easy to show that each decade of GMC mass contributes an equal fraction of the total mass in dense gas. Actually, the amount of dense gas contained within the GMC mass range obeys and is thus constant for any given logarithmic mass interval, as expected for a CFRg mass function with .

We emphasize that, in our scenario, the high pressures characteristic of starbursts and mergers do not modify the mass-radius relation of CFRgs. This still equates to a few -, similar to CFRgs in the Milky Way disc. What high-pressures of violent star-forming environments do modify compared to the quiesent environment of disc spirals is the mass fraction of molecular gas that is dense enough to form stars. This mass fraction is much higher in e.g. ULIRGs than in quiescent spirals, thereby increasing galaxy star formation rates and infrared luminosities (Gao & Solomon, 2004).

### 4.2 Molecular clumps as cluster-forming sites

CO mapping (emission-lines , or ) of molecular cloud structures reveal power-law mass spectra of index , from a fraction of a solar mass up to (Heithausen et al., 1998; Kramer et al., 1998; Hara et al., 1999; Wong et al., 2008). Since the same index also describes GMC data (Rosolowski, 2005; Blitz et al., 2006; Fukui et al., 2008), the mass range over which holds covers more than 6 orders of magnitude.

Alike to GMCs, the clump mass function is thus shallower than the young cluster mass function. Can we correct for this effect by arguing that -mapped clumps have a constant surface density, as we have done for GMCs? In what follows, we consider the data of Fig. 3 to illustrate the issue.

The mass-radius relation of CO clumps is – in essence – one of constant volume density () since their observed volume density corresponds to that needed to excite emission (see middle panel of Fig. 3 in Section 3). For , our model predicts no steepening of the clump-to-CFRg mass functions (, Fig. 5), and it thus seems that we cannot explain why open clusters have . However, we are interested in cluster-forming molecular clumps, and the middle panel of Fig. 3 suggests that their mass-radius relation may actually be steeper than .

The top panel of Fig. 10 shows the same data completed with the clump sample of Higuchi et al. (2009, their table 3). Their 14 molecular clumps display signs of star formation activity since they were selected based on their association with an embedded-cluster. In a follow-up study, Higuchi et al. (2010) detect -line emission in all but one of them. This strengthens the key-hypothesis of our model that star formation and high-density gas () are tightly related. Star-forming clumps (34 in 62 objects) are highlighted by filled symbols. For the sake of clarity, they are also shown in the bottom panel of Fig. 10.

In contrast to all clumps, those hosting star-formation occupy a band of narrow surface density, of lower and upper limits and (dotted (red) lines), respectively. Their mean surface density is , or . Therefore, the mass-radius relation of cluster-forming clumps is steeper () than the mass-radius relation of the whole clump sample (). In turn, this steepens the CFRg mass function compared to the clump mass function, resulting in if the clump density index is

This effect can be quantified further by fitting straightlines to the data. We have performed robust fits (Press et al., 1992), namely, fits in which the absolute value of the deviation is minimized. We consider two cases, the -data being either (Eqs. 21 and 25) or (Eqs. 22 and 26). The comparison of both fits provides a more realistic estimate of the actual uncertainties than fitting vs. (or vice-versa) alone.

Fitting all clumps (top panel of Fig. 10) gives:

 log(rC18O)=0.33log(mC18O)−0.96,  Δ=0.08, (21)
 log(mC18O)=2.41log(rC18O)+2.90,  Δ=0.21, (22)

the latter equating with:

 log(rC18O)=0.41log(mC18O)−1.20, (23)

Taking the mean slopes and intercepts of Eqs. 21 and 23 gives:

 log(rC18O)=0.37log(mC18O)−1.08. (24)

Performing the same robust fits onto clumps hosting forming-clusters (bottom panel of Fig. 10) provides:

 log(rC18O)=0.36log(mC18O)−1.08,  Δ=0.08, (25)
 log(mC18O)=1.92log(rC18O)+3.03,  Δ=0.18, (26)

the second equation being equivalent to:

 log(rC18O)=0.52log(mC18O)−1.58. (27)

Averaging slopes and intercepts of Eqs. 25 and 27 leads to:

 log(rC18O)=0.44log(mC18O)−1.33. (28)

Excluding clumps failing at displaying evidence of star formation indeed steepens the clump mass-radius relation, although the effect is mild (compare Eqs. 24 and 28). It is mostly driven by the exclusion of the low-mass clumps in Orion B () whose undetected (non-existent ?) star-formation activity may stem from a dearth of high-density () molecular gas, as discussed at the end of Section 3.1. A better handling of the mass-radius relation of cluster-forming clumps, compared to that of clumps in general, would require data covering a larger mass range.

Conversely, one may expect cluster-forming molecular clumps with an observed mass function to be in a regime of constant mean volume density (since constant volume density does not alter the mass function slope, i.e. when ; see Fig. 5). Several dust continuum studies report clump mass functions with indices . However, few of them are characterized by a significant number of clumps more massive than, say, , that is, a mass regime appropriate for star cluster progenitors rather than individual star progenitors.

Rathborne et al. (2006) find a mass function slope of for a sample of dust clumps with masses and mapped in millimeter continuum. The corresponding clump mass-diameter diagram (their fig. 8) shows a significant scatter and no clear-cut mass-size relation. That the vast majority of clumps are denser than is the only firm conclusion one can reach. More studies of that type are needed before drawing a conclusion. We encourage authors of such surveys to publish the clump radius distribution and clump mass-radius diagram in addition to the clump mass function, especially if the clump mass upper limit reaches several and beyond. This implies to include star-forming regions more distant from the Sun than a few kpc. In that respect, it should be kept in mind that clump mass and radius estimates depend on the assumed clump distance ( through the clump angular diameter; , see eq. 1 in Rathborne et al. (2006)). It may be of interest to test how the scatter of a mass-radius diagram responds to varying the clump distance accuracy.

Finally, we note for the sake of completeness that the existence of a link between the slope of the clump mass function and the clump volume density was put forward by Reid & Wilson (2005), who quote that ”a possible explanation for the apparently real discrepancy between the CO spectral line and dust continuum mass functions is that the dust maps trace denser clumps than the CO line maps”. [But see Muñoz et al. (2007) for a counter-argument following which steep mass functions inferred by some dust continuum studies are an artifact created by the clump mass upper limit.]

## 5 From the clump radius distribution to the cluster-forming region radius distribution

Similarly to what we have done in Section 3.2 to relate the clump and CFRg mass functions, we now infer the radius distribution of CFRgs from that of their parent clumps.

Let us consider a population of clumps whose radius distribution is a power-law of slope :

 dN=lclumpr−x0clumpdrclump. (29)

If clump masses and radii are correlated (i.e. in Eq. 2), then the slope of the radius distribution of clumps is determined by the slope of their mass-radius relation and the slope of their mass function. To show that, we use Eq. 2 to replace as a function of and in Eq. 29. This leads to the clump mass function with its slope a function of and :

 dN∝r−x0clumpdrclump∝m−[1+δ(x0−1)]clumpdmclump. (30)

It then follows:

 δ=β0−1x0−1. (31)

To derive the radius distribution of the dense central CFRgs:

 dN=lthr−xthdrth, (32)

we need to derive the CFRg radius as a function of the clump radius . By combining Eqs 2 and 4, we obtain:

 rth=(3−p4πρthχ1/δ)1/pr[(p−3)δ+1]/(δp)clump (33)

where and are the normalization and slope of the clump mass-radius relation (Eq. 2), is the clump density index (Eq. 1), and is the volume density at the edge of the CFRg. Combining Eq. 29 and Eq. 33 leads to the CFRg radius distribution:

 pδ(p−3)δ+1r−xthdrth, (34)

where the slope obeys:

 −x=−(x0p−3)δ+1(p−3)δ+1. (35)