Cluster-forming core mass-radius relation

The puzzle of the cluster-forming core mass-radius relation
and why it matters

Geneviève Parmentier and Pavel Kroupa
Argelander-Institut für Astronomie, Bonn Universität, Auf dem Hügel 71, D-53121 Bonn, Germany
Humboldt Fellow - E-mail:
Accepted Received ; in original form

We highlight how the mass-radius relation of cluster-forming cores combined with an external tidal field can influence infant weight-loss and disruption likelihood of clusters at the end of their violent relaxation, namely, when their dynamical response to the expulsion of their residual star-forming gas is over. Specifically, building on the cluster -body model grid of Baumgardt & Kroupa (2007), we study how the relation between the bound fraction of stars staying in clusters at the end of violent relaxation and the cluster-forming core mass is affected by the slope and normalization of the core mass-radius relation. Assuming mass-independent star formation efficiency and gas-expulsion time-scale and a given external tidal field, it is found that constant surface density cores and constant radius cores have the potential to lead to the preferential removal of high- and low-mass clusters, respectively. In contrast, constant volume density cores result in mass-independent cluster infant weight-loss, as suggested by some observations. These trends result from how core volume density and core mass scale with each other. Infant weight-loss is quantified for cluster-forming cores with either number density , or surface density , or radius  pc. Our modelling includes predictions about the evolution of high-mass cluster-forming cores (say ), a regime not yet covered by the observations. We show how, for a given external tidal field, the core mass-radius diagram constitutes a straightforward diagnostic tool to assess whether the tidal field influences the fate of clusters after gas expulsion.

An overview of various issues directly affected by the nature of the core mass-radius relation is presented. In relation with the tidal field impact, these are the evolution of the cluster mass function at young ages (i.e. over the first  Myr), and our ability to reconstruct the star formation history of galaxies from their cluster age distribution. Independently of the tidal field impact, the slope and/or normalization of the cluster-forming core mass-radius relation also influences the mass-metallicity relation of old globular clusters predicted by self-enrichment models, and the duration of cluster violent relaxation.

Finally, we emphasize that observational mass-radius data-sets of dense gas regions must be handled with caution as they may be the imprint of the molecular tracer used to map them, rather than reflecting cluster formation conditions.

stars: formation — galaxies: star clusters: general — galaxies: evolution — stars: kinematics and dynamics
pagerange: The puzzle of the cluster-forming core mass-radius relation and why it mattersReferencespubyear: 201?

1 Introduction

Modelling the early evolution of star cluster systems provides crucial insights into cluster formation physics through a comparison between predicted and observed correlations and distribution functions of individual cluster properties. Over the past 30 years, considerable efforts have been dedicated to modelling cluster violent relaxation, i.e. cluster evolution after residual star forming gas expulsion (e.g. Tutukov, 1978; Hills, 1980; Mathieu, 1983; Lada, Margulis & Dearborne, 1984; Kroupa, Aarseth & Hurley, 2001; Geyer & Burkert, 2001; Goodwin & Bastian, 2006; Baumgardt & Kroupa, 2007; Proszkow & Adams, 2009). In particular, cluster -body model grids (e.g. Baumgardt & Kroupa, 2007) allow to model entire star cluster systems while browsing the parameter space extensively. Initial conditions of star cluster systems constitute crucial ingredients of their time-evolution modelling, and the mass-radius relation of cluster-forming cores is therefore an issue at the forefront of the physics of both cluster formation and cluster evolution.

In an influential study of Galactic molecular clouds and of the density enhancements they contain, Larson (1981) find that molecular clouds and their cores are in approximate virial equilibrium and have a near constant surface density. These properties were then included by Harris & Pudritz (1994) in their model of formation of old globular clusters, whose birth sites are identified as the dense cores of ‘supergiant molecular clouds’ at the early protogalactic epoch. Constant surface density cluster-forming cores show a strong mass-radius relation , which contrasts with the absence of a clear mass-radius relation for gas-free star clusters, regardless of their age (Zepf et al., 1999; Larsen, 2004; Scheepmaker et al., 2007). Kroupa (2005) therefore suggests that cluster-forming cores themselves have uncorrelated masses and radii. Such an hypothesis has a bearing on the time-evolution of the mass function of clusters from the embedded phase to the end of violent relaxation. Owing to their deeper potential well, massive cores undergo more adiabatic gas expulsion and, therefore, retain a greater fraction of their stars. Kroupa & Boily (2002) show that this can account for the formation of a turnover in the cluster mass function at young ages. Baumgardt et al. (2008) and Parmentier et al. (2008) further that hypothesis and show that cluster-forming cores with constant radii indeed produce features (flattening or turnover) in the cluster mass function provided that the star formation efficiency (SFE), assumed to be mass-independent, is not higher than 30 per cent. These studies mostly aim at explaining the prominent and universal turnover characterizing the mass function of old globular clusters (see Ashman & Zepf, 1998, and references therein).

The mass function of young star clusters in the present-day Universe is reported to be a featureless power-law of spectral index of about (e.g. Zhang & Fall, 1999; Lada & Lada, 2003), irrespective of the cluster age (say, 1, 10 or 100 Myr). This implies that cluster infant weight-loss (i.e. the gas-expulsion-driven cluster star-loss) is independent of the embedded-cluster mass. Recently, Fall, Krumholz & Matzner (2010) have estimated that the SFE required for a cluster-forming core to expel its gas via stellar feedback is mass-independent if cluster-forming cores have a constant surface density. Their model assumes that the gas expulsion time-scale in units of a core crossing time, , is constant. While preserving the shape of the cluster mass function at young ages, such a finding leaves unanswered the question of why young star clusters are deprived of a significant mass-radius relation if the mass-radius relation of their progenitors scales as .

In this contribution we are adding one more piece to this intriguing puzzle. Previous studies (Kroupa & Boily, 2002; Parmentier et al., 2008; Fall, Krumholz & Matzner, 2010) have ignored the influence that an external tidal field may exert upon clusters experiencing violent relaxation. Most cluster stars venturing beyond the cluster tidal radius become unbound field stars. 111Stars on highly excentric orbits may experience transient passages beyond the tidal radius and re-integrate into the cluster thereafter. However, such stars are expected to be rare. Therefore, as a cluster expands following gas expulsion, its infant weight-loss and its likelihood of disruption are partly governed by how deeply the embedded cluster sits within its limiting tidal radius, that is, how severe tidal overflow due to cluster expansion is. Goodwin (1997) performs -body simulations highlighting this effect: infant weight-loss of otherwise identical model clusters is stronger closer to the Galactic centre by virtue of the stronger tidal field and hence smaller cluster tidal radius (his fig. 3). Baumgardt & Kroupa (2007) quantify this effect by the ratio of the half-mass radius to the tidal radius of the embedded cluster. If , the cluster has much space in which to expand following gas expulsion, the tidal field impact is low and cluster infant weight-loss is solely driven by the SFE and the gas expulsion time-scale . In contrast, if , Baumgardt & Kroupa (2007) find that protoclusters are mostly disrupted. We refer to as the tidal field impact. As we shall see in Section 2, not only is it related to the external tidal field, it also depends on the embedded cluster mass and size and, therefore, to the cluster-forming core mass-radius relation. It is therefore crucial to quantify to what extent the mass-radius relation of cluster-forming cores combined to an external tidal field influences the early evolution of star cluster systems. In this introductory paper, we focus our attention on the fraction of stars which remains bound to their parent clusters at the end of violent relaxation.

The outline of the paper is as follows. Section 2 investigates how different cluster-forming core mass-radius relations (constant radius, constant volume density and constant surface density) influence the cluster bound fraction as a function of core mass. We also show how the mass-radius diagram of cluster-forming cores can be used to estimate whether an external tidal field influences cluster violent relaxation or not. To what extent mass-size data-sets of dense molecular gas regions can help us constrain the cluster-forming core mass-radius relation is the topic of Section 3. In Section 4, we comment about the importance of the core mass-radius relation regarding crucial issues such as cluster infant mortality/weight-loss as a function of cluster mass, and the reconstruction of the star formation history of galaxies based on their surviving clusters. We conclude in Section 5.

2 Core mass-radius relation and external tidal field

At the end of its violent relaxation (age  Myr; see Section 4.3), the mass of a star cluster is


with the mass of the cluster progenitor core and the fraction of stars remaining bound to the cluster at the end of violent relaxation. SFE is the ‘local’ star formation efficiency, namely, the mass fraction of gas turned into stars at the onset of gas expulsion.

Cluster infant weight-loss, , is a sensitive function of the SFE at the onset of gas expulsion and of the gas expulsion time-scale expressed in units of a cluster-forming core crossing-time, (e.g. Hills, 1980; Mathieu, 1983; Lada, Margulis & Dearborne, 1984; Geyer & Burkert, 2001). The higher the SFE, the slower the gas expulsion time-scale , the higher the bound fraction (SFE,).

Formally, the bound fraction depends on the ‘effective’ star formation efficiency (eSFE, Verschueren, 1990; Goodwin & Bastian, 2006; Goodwin, 2009), rather than on the local SFE. That is, . The eSFE incorporates how far from virial equilibrium the cluster is at the onset of gas expulsion. If the stars and gas are in virial equilibrium at the onset of gas expulsion as a result of, for instance, a several-crossing-time time-span between star formation and gas expulsion, the eSFE is simply the ‘local’ SFE and . This is the approach we adopt in this contribution since the assumption of virial equilibrium underpins the -body model grid of Baumgardt & Kroupa (2007) (and the vast majority of other studies dedicated to cluster gas expulsion). However, if the dynamical state of the newly-formed stars at the onset of gas expulsion is ‘cold’, then the is higher than the local SFE, and the bound fraction becomes larger than shown in Figs. 3, 5, 6 and 7 (and the opposite if stars are in a ‘hot’ dynamical state). Cluster models in which stars are not in virial equilibrium at gas expulsion have been investigated by Lada, Margulis & Dearborne (1984), Verschueren (1990) and Goodwin (1997). Note however that if the cold collapse at gas expulsion stems from stars forming out of a contracting pre-cluster core, this implies that the star-formation activity in the pre-cluster cloud core would need to be synchronised to occur within a time shorter than the core crossing-time and so is most likely (see Kroupa, 2008, for a discussion).

In addition to the SFE and gas expulsion time-scale, the bound fraction may also depend on the tidal field impact , i.e.:


an effect mapped by Baumgardt & Kroupa (2007) by means of cluster -body modelling. Stronger tidal field impacts lower the bound fraction . To provide a clear understanding of how the cluster-forming core mass-radius relation affects the bound fraction through the tidal field impact, in what follows, each simulation is assigned a given local SFE, a given gas expulsion time-scale and a given external tidal field. That way, any variation will necessarily result from varying the core mass-radius relation.

2.1 Fiducial model: SFE=0.33,

We adopt (Lada & Lada, 2003) and (Krumholz & Matzner, 2009, see also Section 4.1) as the local SFE and gas expulsion time-scale of our fiducial model. The parameter space in terms of SFE and is explored more widely later in this section.

At this stage, we note that Baumgardt & Kroupa (2007) model cluster gas expulsion as an exponential decrease with time of the cluster gas mass (see their eq. 3):


Therefore, , which they define as the gas-expulsion time-scale, is actually the e-folding time of the gas expulsion process. It corresponds to the time when the residual gas mass fraction is of its initial value. In all our models, we define the gas-expulsion time-scale rather as the time-scale over which the cluster expels the entirety of its residual gas. Prior to using the -body model grid of Baumgardt & Kroupa (2007), we therefore define (i.e. we multiply the ‘gas expulsion time-scale’ of Baumgardt & Kroupa (2007) by a factor 3) so that corresponds to a residual gas mass fraction of , i.e. the cluster is practically devoid of gas.

We subject all cluster-forming cores to the same external tidal field, which is that of an isothermal potential with a circular velocity at galactocentric distances of either or . This will allow us to assess how the strength of the external tidal field influences modelling outputs. The embedded-cluster tidal radius obeys:


where is the embedded cluster stellar mass and is the host galaxy mass enclosed within (Binney & Tremaine, 1994). is the gravitational constant.

Tidal overflow sets in if the embedded-cluster radius is larger than the tidal radius . Substituting with in Eq. 4, it follows that this equates with a volume density smaller than:


At , Eq. 5 gives , equivalent to an number density . At , these figures are or .

Cluster gaseous progenitors are denser than these limits by several orders of magnitude. Figure 1 shows mass-radius diagrams of molecular cores mapped with different tracers. The top panel shows radii and masses of cores mapped with the CO emission line, some of them displaying signs of star formation (see Section 3 for a detailed discussion). In contrast, the middle and bottom panels present mass-radius diagrams of molecular cores selected for their star formation activity, then mapped with higher density tracers: emission line and dust continuum emission. We will discuss these observations in greater detail in Section 3. For now, suffice is to say that, owing to their systematic star formation activity, molecular cores of the middle and bottom panels constitute better proxy of cluster gaseous progenitors than the cores of the top panel. Each panel also shows lines of constant volume number density (, dashed black lines) and of constant surface density (, dotted black lines) fitting the data. The mean number density ranges from for the CO cores (top panel), to (middle panel) and (bottom panel) for the molecular cores selected for their star formation activity. Observed molecular cores are thus denser than the tidal limit defined by Eq. 5 by about 4 orders of magnitude, i.e. they are ‘immune’ to galactic tides.

Following gas expulsion, however, gas-loss, infant weight-loss and spatial expansion decrease the density of clusters compared to that of their parent cores. The key-point our simulations aim to address is: in terms of cluster-forming core mass-radius relations, what conditions lead to tidal overflow for the expanded clusters and, therefore, to enhancement of infant weight-loss/mortality compared to what would be obtained for isolated clusters (i.e. no tidal field).

We test 6 different mass-radius relations: constant core surface density, constant core volume density, and constant core radius, each with two different normalizations. We parametrize the mass-radius relation by its slope and normalization :


Table 1 shows adopted and values, along with the corresponding surface densities, volume densities and radii. Models with constant core surface density (), constant core volume density () and constant core radius () are labelled , and , respectively. For each slope , we consider two normalizations , referred to as ‘compact’ or ‘loose’ model. The ‘loose’ and models are fits to the CO data with the slope imposed (dotted and dashed black lines in top panel of Fig. 1). The ‘compact’ and relations describe the data of molecular cores selected for their star formation activity, for which we adopt and (dotted and dashed blue lines with filled circles in middle and bottom panels, respectively). These densities are at the logarithmic mid-points between the data-fits of the middle and bottom panels of Fig. 1. As for the constant radius models, we adopt and . These are shown as the blue (‘compact’ model, middle panel) and black (‘loose’ model, top panel) solid lines in Fig. 1. In all forthcoming figures, , and models are depicted by dotted, dashed, and solid lines, respectively.

Compact 0.3pc
Loose 1.5pc
Table 1: Adopted mass-radius relations for cluster-forming cores, with their corresponding constant surface density (), volume density (), and radius ().
Figure 1: Radius vs. mass of dense molecular gas cores. Top panel: molecular cores mapped in CO emission (filled symbols). Surrounded filled symbols represent cores with detected star formation activity. Horizontal arrows depict the lowest-size cores in Aoyama et al. (2001), Saito et al. (1999) and Yonekura et al. (2005), based on their half-power beam width. Middle panel: molecular cores mapped in dust continuum emission (Müller et al., 2002; Faundez et al., 2004). Bottom panel: molecular cores mapped in (Shirley et al., 2003) and dust continuum emission (Müller et al., 2002; Fontani et al., 2005). Our loose  pc, and models are shown in the top panel as solid, dashed and dotted lines, respectively. The compact models  pc and are shown as the (blue) solid and dotted lines with filled circles in the middle panel. The compact model is depicted as the dashed (blue) line with filled circles in the bottom panel.

We link the cluster-forming core radius and the embedded cluster half-mass radius with:


We adopt . Molecular cores have power-law density profiles , with the density index (Müller et al., 2002; Beuther et al., 2002). corresponds to a truncated isothermal sphere (). Shallower molecular cores (i.e. smaller density indices ) lead to larger values since they have a greater fraction of their mass in their outer layers. Larger values in turn lead to larger tidal field impacts and cores more sensitive to tidal overflow.

Combining Eqs. 4, 6 and 7 provides the tidal field impact :


thereby highlighting the influence of both the slope and normalization of the core mass-radius relation. Figure 2 depicts Eq. 8 for the various parameters (, ) of Table 1 for (top panel) and (bottom panel). For constant core radii (), more massive clusters sit more deeply within their tidal radii () and are thus more resilient to the external tidal field. Conversely, constant surface density cores () are conducive to more massive clusters being more prone to tidal overflow (). In case of constant volume density (), the tidal field impact is independent of the embedded-cluster mass (). Equation 8 can actually be rewritten as a function of the core number density as the sole core parameter. The core radius, mass and number density are related by:


Inserting Eq. 6 and Eq. 9 in Eq. 8, we obtain:


Higher ratios, and thus greater vulnerability to the tidal field, result either from a lower core density , thus larger (see loose vs. compact models in top panel of Fig. 2), or from a stronger tidal field, equivalent here to a smaller galactocentric distance (see compact models in top and bottom panels of Fig. 2).

Building on Fig. 2 and on Baumgardt & Kroupa (2007) -body model grid of clusters, which provides the fraction of stars bound to a cluster at the end of violent relaxation as a function of SFE, and , we obtain in Fig. 3 the relation between and . Model parameters are identical to those in Fig. 2 and the gas-expulsion time-scale is . Note the correlation between a low and a high in Fig. 2. In Fig. 3, lower normalizations or larger galactocentric distances result in larger bound fractions through a smaller tidal field impact. The bound fraction as a function of mass is constant when , increases when and decreases when . The latter illustrates – for the first time – a case where violent relaxation preferentially destroys high mass clusters.

Figure 2: Relation between the ratio of the half-mass and tidal radii of embedded clusters and the mass of their progenitor gas core. , and refer to cores with constant radius, constant volume density and constant surface density, respectively. The top and bottom panels consider cores exposed to the tidal field of an isothermal halo with a circular velocity at galactocentric distances of 8 and 4 kpc, respectively. As the loose models (open symbols/black curves) have a density lower than that of realistic cluster-forming cores, they are shown for illustrative purposes in the top panel only. See text for additional details.
Figure 3: Bound fraction of stars after cluster violent relaxation as a function of cluster-forming core mass for the model parameters adopted in Fig. 2 and a gas expulsion time-scale set to a crossing-time of the cluster-forming core, i.e. . Line- and symbol-codings as in Fig. 2. The thin dash-dotted horizontal (black) line in each panel indicates the bound fraction for the adopted SFE and in the absence of an external tidal field, i.e. .

As quoted earlier in this section, the compact mass-radius relations constitute a better proxy of cluster initial conditions than their loose counterparts since they fit data of molecular cores selected for their star formation activity (see Section 3 for details). Therefore, from now on, we focus most of our attention onto the compact models. When  kpc, the core mass-radius relation and the tidal field impact are essentially two disconnected issues up to . That is, regardless of the adopted ‘compact’ model (, or ), when . In this regime, the tidal field impact is weak and exposed clusters respond to the loss of their residual star-forming gas essentially as if there were no external tidal field (Baumgardt & Kroupa, 2007). As a result, the bound fraction is almost constant and independent of the adopted core mass-radius relation ( when , see top panel of Fig. 3). We remind the reader that the constancy of in this regime also stems from our hypotheses of constant SFE and constant (see Eq. 2). At masses higher than , however, the model on the one hand, and the and models on the other hand, show very different behaviours, with ratios and bound fractions increasingly different as the core mass increases. A smaller galactocentric distance (e.g. instead of ) increases further the contrast between the and models at high mass.

That the and models show such contrasting behaviours in the high mass regime in a strong tidal field, i.e. close to the galactic centre, demonstrates the importance of distinguishing between these two mass-radius relations. Actually, many spiral galaxies show a transition from being predominantly atomic in their outer regions to being predominantly molecular at their centres (Wong & Blitz, 2002). One may thus expect that closer to the galactic centre, the amount of dense molecular gas available to star formation is larger, thus implying that the cluster-forming core mass function is sampled up to a higher mass (size-of-sample effect; see also Weidner, Kroupa & Larsen, 2004). This in turn would lead to the formation of more massive embedded-clusters in stronger tidal-field-environments, that is, the regime where the and models lead to highly different final bound fractions of stars.

Figure 4: Mass-radius diagram for the dense molecular cores of the middle and bottom panels of Fig. 1, i.e. those selected for their star formation activity. Symbol-coding is as in Fig. 1. The dotted, dashed and solid (blue) lines are the adopted ‘compact’ , and models, respectively. The dash-dotted (black) lines are characterised by (from top to bottom) for and  kpc (top panel) and  kpc (bottom panel).

Figure 4 is the mass-radius diagram of molecular cores from the middle and bottom panels of Fig. 1. We superimpose onto these data the adopted ‘compact’ mass-radius relations (blue dotted, dashed and solid blue lines). We also show lines of constant (black dash-dotted lines) for  kpc (top panel) and  kpc (bottom panel): (from top to bottom). Note that iso- lines are vertically shifted by in the bottom panel compared to the top one since (see Eq. 8). Note also that iso- lines are lines of constant volume density, as shown by Eq. 10.

For the galactocentric distances considered, the vast majority of the observed molecular cores are tidal-field resilient, i.e. . The observational data, however, occupy a limited mass range, with only a few cores more massive than . They therefore fail to probe the high-mass regime where we predict and models to respond very differently to gas expulsion through the tidal field.

Figure 4 allows us to understand Figs. 2 and 3 from another perspective. While the compact model has irrespective of core mass, a model is characterised by a decreasing mean volume density with increasing core mass. This equates with a greater tidal field impact for more massive cores (Eq. 10 and Fig. 4) and, thus, a lower final bound fraction of stars (Fig. 3). Conversely, cores with mass-independent radii ( model) increase their volume density with their mass, rendering less massive cores more prone to tidal overflow through a larger ratio.

One may argue that the reason why cluster infant weight-loss/mortality for the model is so prominently mass-dependent in Fig. 3 partly stems from the high adopted upper limit on the core mass range, i.e. . One should keep in mind, however, that such a large mass of dense molecular gas is needed to form a cluster of a few million solar masses, that is, with a mass comparable to that of the most massive old globular clusters and star clusters formed in galaxy mergers. For the model in Fig. 3, and lead to a cluster mass at the end of violent relaxation an order of magnitude lower than its progenitor core mass (see Eq. 1). As for the ‘compact’ model at , it prevents the formation of massive clusters since cores more massive than fail at forming bound clusters (=0), and cores give rise to bound clusters in mass only ( and ).

Figure 5: Bound fraction of stars at the end of violent relaxation as a function of the core mass. Top and middle panels: Same as Fig. 3 for slower gas expulsion (). Bottom panel: Same as top panel but for .

2.2 Exploring a wider parameter space

Figure 5 explores more widely the parameter space (SFE, ). Its top and middle panels are the counterparts of Fig. 3, with longer gas expulsion time-scales: . For slower gas expulsion, clusters retain a higher fraction of their stars because they are better able to adjust to the new gas-depleted potential they sit in. Besides, slower gas expulsion is conducive to smaller spatial expansion of the exposed cluster (see fig. 3 in Geyer & Burkert, 2001), thus to a higher mean density at the end of violent relaxation and greater resilience to the external tidal field. This is another channel through which the bound fraction of stars is increased compared to quicker gas expulsion. Compared to Fig. 3, the bound fractions in top and middle panels of Fig. 5 are increased by factors . At a galactocentric distance (top panel), this strongly dampens any dependence of the bound fraction on the core mass for the model (). But it also results in cluster infant weight-loss hardly compatible with observations since infant weight-loss is reported to range from 70 % (Bastian et al., 2005) to 90 % (Lada & Lada, 2003), that is, . The bottom panel of Fig. 5 illustrates the vs. relation for a lower SFE, namely, and the same gas-expulsion time-scale. In that case, cluster survival () requires . We will further discuss the consequences of these plots for the cluster mass function in Section 4.1. Note that the models of the middle and bottom panels behave almost similarly, that is, the combination of the weaker tidal field and lower SFE in the bottom panel compared to the middle one leads to a model degeneracy.

Figure 6: Top and middle panels: Final bound fraction of stars as a function of the core mass for the compact (top) and loose models (middle) in case of long gas expulsion (). Symbol-coding as previously: models, triangles; models, circles; models, squares. Bottom panel: Mass-radius relations for the loose and compact and models. The thick (black) dash-dotted line is the minimum number density molecular cores must have for their clusters not to be affected by tides in case of adiabatic gas expulsion with SFE=0.25 and  kpc. The vertical arrows actually indicate that the crossing of that density limit in the mass-radius diagram corresponds to a sharp increase of the bound fraction of stars.

2.3 Adiabatic gas expulsion

Adiabatic gas expulsion () allows an analytic analysis which illuminates these results further. The top and middle panels of Fig. 6 depict the evolution of with the core mass for the compact and loose models, respectively, for  kpc, and (i.e. the longest gas expulsion time-scale in the model grid of Baumgardt & Kroupa, 2007). Line- and symbol-codings are as in Figs. 3 and 5. For isolated clusters (), so long a gas expulsion time-scale when is conducive to , namely, no cluster infant weight-loss. In fact, in a tidal-field-free environment, adiabatic gas expulsion implies (Mathieu, 1983). If there is a strong enough tidal field, however, stars driven beyond the cluster tidal radius by its spatial expansion will get unbound and . In the framework of the adiabatic approximation, we can estimate what minimum number density the cluster progenitor core must have to prevent tidal overflow.

In the case of adiabatic gas expulsion, the radius multiplied by the mass is an adiabatic invariant: the cluster expands by a factor SFE after gas expulsion while its mass gets lower than the core mass by a factor SFE (Hills, 1980; Mathieu, 1983). The gas-free cluster density thus follows:


Therefore, for SFE=0.25 and adiabatic gas expulsion, an expanded cluster at experiences tidally-driven mass-loss (, Eq.5) if its parent core has , or . In contrast, clusters formed out of cores with are not or little affected by tides.

This density limit is shown as the thick dash-dotted (black) line in the vs.  diagram of Fig. 6 (bottom panel). In this diagram, the intersections of the density limit with lines of constant give the core mass above which constant surface density cores give rise to clusters significantly affected by tides. Similarly, the intersections with lines of constant gives the core mass below which clusters formed out of constant radius cores experience tidal overflow. These intersections are highlighted by upside-down open triangles in the bottom panel of Fig. 6.

Let us consider the loose model . Its intersection in the diagram with the density limit renders . This matches the core mass regime over which drops significantly for that particular core mass-radius relation (open triangles in the middle panel of Fig. 6), as indicated by the vertical dotted double-head arrow. corresponds to for which we expect little or no infant weight-loss. The middle panel of Fig. 6 indeed shows that over that mass regime. In contrast, leads to , for which we expect expanded clusters to be severely affected by tides. The middle panel of Fig. 6 indeed predicts for . A stronger tidal field (i.e. closer to the galactic centre) would lower the mass-limit at which decreases. Similarly, the intersection between the constant radius loose model  pc and yields , where is sharply increasing from 0 to 0.75, as indicated by the solid double-head arrow. As for the compact models, which better describe star cluster initial conditions, those are more resilient to tidally-driven mass-loss. The diagram confirms that must strongly decrease at for (see dotted upward arrows in top and bottom panels of Fig. 6).

We underline that the above-described effects take place even if all cluster-forming cores are located within a galaxy region over which the external tidal field does not vary markedly. In these models, the variations are solely driven by the core mass-radius relation.

Although unbound stars located beyond the cluster tidal radius linger around the cluster and may result in observed clusters with estimated radii larger than their tidal limit, observations of young clusters in the Small Magellanic Cloud show that this effect fades away by an age of about 30 Myr (Glatt et al., 2010).

3 What information can we extract from the observations of dense molecular gas regions?

In Fig. 1, we compile masses and radii of dense gas regions from the literature. The top panel shows results of mapping of dense gas regions in neighbouring giant molecular clouds (GMC). The middle and bottom panels are mass-radius diagrams of molecular cores selected for their star formation activity.

Three Galactic regions are encompassed by the top panel: Orion B (Aoyama et al., 2001), the GMC toward HII regions S35 and 37 (referred to S35/37 in what follows) (Saito et al., 1999) and the  Carinae GMC (Yonekura et al., 2005), at assumed distances of 400 pc, 1.8 kpc and 2.5 kpc, respectively. All observations were performed at the same resolution with the NANTEN telescope. The lower bound on resolved-core radii for each data-set as imposed by the NANTEN half-power beam width () is shown by dotted horizontal arrows. That CO cores in the  Carinae GMC are larger than in Orion B is thus a purely resolution-driven effect. CO emission traces molecular gas with number densities a few - , and the volume density range covered by these observations is thus an imprint of the CO tracer.

The and loose models of Section 2 are fits to these CO data. As we cautioned in Section 2, cores do not systematically host signs of star formation activity and, therefore, loose models may not trace actual cluster formation conditions. In Fig. 1, CO cores associated to one or more sources identified as a Young Stellar Object (YSO) candidate by Saito et al. (1999), Aoyama et al. (2001) or Yonekura et al. (2005) are circled. Open squares depict Carinae cores showing other signs of active star formation (e.g. YSO candidate from the point-source catalog or bipolar outflows).

CO cores were also observed by Aoyama et al. (2001) and Yonekura et al. (2005) in HCO emission. HCO emission traces molecular gas at , i.e. an order of magnitude higher than CO. They note a tight correlation between the presence of high-density HCO clumps and star formation activity. This suggests that star formation requires number densities of order () and hence that only the densest, presumably most inner, regions of CO cores form stars. Loose and models have therefore too large a normalization to emulate realistic cluster formation conditions. That is why we insisted in Section 2 that those models should be considered only for illustrative purposes, e.g. to show how model outputs respond to variations of the normalization of the core mass-radius relation. A related point worth being quoted here is that SFEs measured over the whole volume of cores (e.g. Higuchi et al., 2009, their table 3) are global SFEs and, as such, are not indicative of how an embedded cluster dynamically responds to gas expulsion. This is the local SFE, namely, the SFE estimated over the volume of gas forming the cluster, which matters when modelling cluster violent relaxation. Global SFEs averaged over whole cores constitute lower limits to their local counterparts. A low global SFE (say, 10 %) may be misleading in prompting us to conclude that a cluster will not survive its violent relaxation, even though the local SFE may be high enough for the cluster to retain a fraction of its stars. We will come back to this point in a forthcoming paper (Parmentier,, in prep).

To better constrain cluster formation conditions, we gather in the middle and bottom panels of Fig. 1 masses and radii of dense molecular cores selected for their star formation activity (either sources or water masers). Mapping of star-forming cores in the CS J emission line has been performed by Shirley et al. (2003, columns 3 and 5 of their table 5; filled squares in the bottom panel of Fig. 1). Mapping of star-forming cores in dust-continuum emission has been performed by Faundez et al. (2004, their table 1), Fontani et al. (2005, radii and masses from their tables A.5 and A.6, respectively) and Müller et al. (2002, columns 2 and 3 of their table 4). They are depicted as the (blue) -symbols and asterisks and (black) open circles in the middle and bottom panels of Fig. 1. The radius of cores is defined as that of the contour at half-maximum of the CS or dust-continuum emission. The core mass is the mass enclosed within that radius. These cores have (volume and surface) densities significantly higher than those inferred from the data, as indicated by the lines of constant and constant in middle and bottom panels. Müller et al. (2002) provide an alternative definition of the masses and radii of their surveyed cores (columns 4 and 5 of their table 4), which we show as the (black) filled circles in the middle panel of Fig. 1. This sequence neatly defines a line of constant volume density, with a mean number density .

It is important to realize that this result stems from how core masses and radii are estimated, however. From the radial density profiles of the star-forming regions they observe, Müller et al. (2002) obtain the radius where . Their core mass is the gas mass enclosed within that radius. Defining core masses and radii that way necessarily results in a sequence of constant volume density. As such, this core mass-radius relation may constitute a measurement-imprint rather than a genuine imprint of the cluster-formation physics.

These various examples show that to infer the mass-radius relation of cluster-forming cores observationally is not a straightforward task. Results heavily depend on the tracer and/or the method used to map them. In Sections 2 and 4 realistic cluster initial conditions are described by mass-radius relations representative of the dense molecular cores selected for their star formation activity. We refer to them as the ‘compact’ , and models. They are shown as the (blue) dotted, solid and dashed lines with filled-circles in the middle and bottom panels of Fig. 1. Their constant surface density, radius and volume number density are ,  pc and , respectively. These volume and surface densities are at the logarithmic midpoints of the fits to the data in Fig. 1 middle and bottom panels (black dashed and dotted lines).

We emphasize that our core mass-radius relations (Eq. 6 and Table 1) relate the total mass of cores to their outer radius. In that sense, our relations are not directly comparable to the dust-continuum and CS emission data of Shirley et al. (2003), Faundez et al. (2004) and Fontani et al. (2005), who define the core as the region enclosed within the FWHM contour. Assuming an isothermal sphere density profile for the cores, the volume and surface densities within the half-mass radius are 4 and 2 times, respectively, higher than the volume and surface densities averaged over the whole core (since for an isothermal sphere). That is, the mean volume and surface densities within the half-mass radius of our compact and models are fully comparable to the mean densities within the FWHM contour of the data of Fig. 1 bottom panel. Note also that the adopted number density is not significantly different from the number density characterizing HCO-traced molecular gas () which is closely associated to star formation activity in cores (Aoyama et al., 2001; Yonekura et al., 2005).

Physically, a mean density of --) for cluster formation may result from the associated efficient decay of turbulence, leading such dense gas cores to undergo gravitational collapse and form star clusters (Klessen, 2003). We also note that leads to , implying that all cluster stars due to become unbound owing to gas expulsion have crossed the tidal radius boundary by an age of at most 15 Myr (see fig. 4 in Parmentier, 2009, see also Section 4.3).

4 From cluster early evolution to galaxy star formation histories: consequences

Section 2 shows that the combination of the cluster-forming core mass-radius relation with an external tidal field can contribute to determining how much infant weight-loss clusters experience and whether infant weight-loss is mass-independent or not. In this section, we survey a few topics which are directly influenced by the core mass-radius relation, either in relation to the tidal field impact (Sections 4.1 and 4.2), or independently of it (Sections 4.3 and 4.4).

4.1 The shape of the young cluster mass function

Most observational evidence gathered so far shows that the shape of the post-violent relaxation cluster mass function mirrors that of the embedded cluster mass function (Kennicutt et al., 1989; McKee & Williams, 1997; Lada & Lada, 2003; Zhang & Fall, 1999; Oey et al., 2004; Dowell et al., 2008, but see Anders et al. (2007) for the case of a bell-shaped young cluster luminosity function). That is, cluster infant weight-loss appears to be mass-independent. As Figs. 3 and 5 show, under the assumption of constant, hence mass-independent, SFE and gas expulsion time-scale , cluster-forming cores characterised by a constant volume density () constitute the most robust way of achieving mass-independent cluster infant weight-loss. In contrast, in the case of constant surface density, clusters formed out of massive cores may be preferentially destroyed (see Fig. 3), and the shapes of the post-violent-relaxation cluster mass function and core mass function may differ substantially (see fig. 4 in Parmentier, 2010).

Our result is at odds with that derived by Fall, Krumholz & Matzner (2010), following which mass-independent infant weight-loss requests near-constant surface density cores (). In the case of constant volume density cores, they find that the needed to clean the cluster of its residual star-forming gas is an increasing function of the core mass. This is conducive to less-massive clusters experiencing greater infant weight-loss and, thus, to a cluster mass function shallower than the embedded-cluster mass function. Our result and theirs stem from two utterly different approaches, however. Our model rests on how the tidal field impact varies with the core mass , under the assumptions of constant SFE and constant gas expulsion time-scale . Their model rests on the amount of stellar feedback required to clear an embedded cluster of its residual gas, neglecting the tidal field impact and assuming constant .

To illustrate that both approaches are not irreconcilable, let us first consider the case of energy-driven feedback of Fall, Krumholz & Matzner (2010). The rate at which massive stars deposit energy in the cluster-forming core gas is proportional to the core stellar mass, that is, with a proportionality coefficient. The energy input accumulated over the gas expulsion time-scale is thus . Fall, Krumholz & Matzner (2010) derive the core SFE by equating the total energy input to the critical value needed to expel the intra-cluster gas, that is, the gas binding energy . Introducing the core crossing-time , it thus follows:


Since , with a unit-dependent proportionality constant:


Finally, introducing the core mass-radius relation (Eq. 6):


Neglecting the coefficient which matters little as long as , we see that, depending on the slope of the core mass-radius relation, the product increases with the core mass as (), (), and (). The greater dependence of on the core mass as decreases stems from the depth of the core potential well being itself a steeper function of the core mass for shallower core mass-radius relations.

Figure 7: Bound fraction of stars at the end of violent relaxation as a function of the gas expulsion time-scale expressed in units of a core crossing-time for the quoted in the key and a weak tidal field impact . Based on the -body model grid of Baumgardt & Kroupa (2007). The gas expulsion time-scale is defined as three times the -folding time of Baumgardt & Kroupa (2007) (see Eq. 3)

Equation 14 can now be applied to two limiting cases: either a mass-independent SFE, or a mass-independent gas expulsion time-scale in units of a core crossing-time. A constant SFE is the approach adopted by Baumgardt et al. (2008), which we come back to below.

If, on the other hand, is core-mass-independent, Eq. 14 leads to


and we have recovered eq. (1a) of Fall, Krumholz & Matzner (2010). Their result suggests that mass-independent cluster infant weight-loss demands . That is, compared to cores with constant volume density and constant radius, constant surface density cores introduce the smallest mass-dependence for cluster-infant weight-loss. Yet, Eq. 2 shows that the bound fraction of stars at the end of violent relaxation does not depend on SFE and only. It also depends on the tidal field impact which, as we demonstrate in Section 2, can introduce a strong core-mass-dependence for . A mass-independent tidal field impact requests constant volume density cores, but those are characterised by a mass-dependent SFE to expel residual star-forming gas: (Eq. 15). In that case and for constant gas expulsion time-scale (as assumed in Eq. 15), is a sharply increasing function of the SFE. Figure 1 in Baumgardt & Kroupa (2007) shows that when and when , with a threshold value dependent on and . For instance, explosive gas expulsion () and no tidal field impact () renders . The transition from to as the SFE increases beyond the threshold is conducive to the formation of features in the cluster mass function (flattening and turnover), in conflict with most observations of young star clusters in the present-day Universe.

As a brief summary before heading further: under the assumption , results in a mass-independent SFE (Eq. 15), but a mass-dependent tidal field impact (Eq. 8). Conversely, results in a mass-independent tidal field impact , but a mass-dependent SFE. These modelling results are to be contrasted with the observations of power-law mass functions for young star clusters which demand that all 3 parameters – SFE, and – weakly depend on the core mass to ensure that the bound fraction does not depend significantly on (Eq. 2).

The approach adopted by Baumgardt et al. (2008) can help us solve this intriguing conundrum. Instead of assuming a mass-independent in Eq. 14, Baumgardt et al. (2008) adopt a mass-independent SFE. Equation 14 thus becomes:


While in Eq. 15, the larger energy-input required to clear the gas out of more massive cores arises from a higher SFE, in Eq. 16, it is obtained by integrating the energy input over a longer gas expulsion time-scale . Results obtained for based on Eq. 16 are at first glance similar to those obtained for the SFE based on Eq. 15. A mass-independent gas expulsion time-scale requires , which leads to mass-dependent tidal field impact. Constant volume density cores (), needed to reproduce a mass-independent tidal field impact, are conducive to mass-dependent . There is a major difference with the SFE-varying approach of Eq. 15, however. Whether the mass-varying of Eq. 16 induces a mass-dependent depends very much on the range of involved. Actually, the -body simulations of Baumgardt & Kroupa (2007) show that for , the bound fraction of stars stays about constant. That is, it is doable to get even though is an increasing function of the core mass, as long as .

To assess this issue in more detail, let us derive the normalizing factor in Eq. 16. Building on models of deposition of stellar feedback energy, Baumgardt et al. (2008) derive the gas expulsion time-scale (in units of Myr) as a function of the core half-mass radius, core mass and SFE (their eq. 14 which we reproduce below for the sake of clarity):


Equation 17 can be combined with the core crossing-time and with a core mass-radius relation to infer as a function of the sole core mass . Using eq. 6 in Baumgardt & Kroupa (2007) for the core crossing-time and the core mass-radius relation of our compact model (Eq. 6 with and ), we obtain:


This equation is valid for constant volume density cores with the normalization .

Figure 7 presents the bound fraction in dependence of the gas expulsion time-scale (bottom x-axis, based on the -body model grid of Baumgardt & Kroupa, 2007) and of the core mass (top x-axis, based on Eq. 18). The adopted tidal field impact is weak, namely, , as we find for the compact model in Fig. 2. One can see that SFE of - leads to a constant bound fraction up to , and to an increase by a factor of over the high mass range