The Origin of the Gaussian Initial Mass Function of Old Globular Cluster Systems

[Abridged] Evidence favouring a Gaussian initial globular cluster mass function has accumulated over recent years. We show that an approximately Gaussian mass function is naturally generated from a power-law mass distribution of protoglobular clouds by expulsion from the protocluster of star forming gas due to supernova activity, provided that the power-law mass distribution shows a lower-mass limit. As a result of gas loss, the gravitational potential of the protocluster gets weaker and only a fraction of the newly formed stars is retained. The mass fraction of bound stars ranges from zero to unity, depending on the local star formation efficiency $\epsilon$. Assuming that $\epsilon$ is independent of the protoglobular cloud mass, we investigate how such variations affect the mapping of a protoglobular cloud mass function to the resulting globular cluster initial mass function. A truncated power-law cloud mass spectrum generates bell-shaped cluster initial mass functions, with a turnover location mostly sensitive to the lower limit of the cloud mass range. We also show that a Gaussian mass function for the protoglobular clouds with a mean ${\rm log}m_G \simeq 6.1-6.2$ and a standard deviation $\sigma \lesssim 0.4$ provides results very similar to those resulting from a truncated power-law cloud mass spectrum, that is, the distribution function of masses of protoglobular clouds influences only weakly the shape of the resulting globular star cluster initial mass function. The gas removal process and the protoglobular cloud mass-scale dominate the relevant physics. Moreover, gas removal during star formation in massive clouds is likely the prime cause of the predominance of field stars in the Galactic halo.Comment: 24 pages, accepted for publication in MNRA

Early dynamical evolution of star cluster systems

Violent relaxation -- the protocluster dynamical response to the expulsion of its residual star forming gas -- is a short albeit crucial episode in the evolution of star clusters and star cluster systems. Because it is heavily driven by cluster formation and environmental conditions, it is a potentially highly rewarding phase in terms of probing star formation and galaxy evolution. In this contribution I review how cluster formation and environmental conditions affect the shape of the young cluster mass function and the relation between the present star formation rate of galaxies and the mass of their young most massive cluster.Comment: 8 pages, 5 figures. Invited talk. To appear in the proceedings of IAU Symp. 266 (Star clusters, Rio de Janeiro, Brazil, August 2009), eds. R. de Grijs and J. Lepin

On the origin of the radial mass density profile of the Galactic halo globular cluster system

We investigate what may be the origin of the presently observed spatial distribution of the mass of the Galactic Old Halo globular cluster system. We propose its radial mass density profile to be a relic of the distribution of the cold baryonic material in the protogalaxy. Assuming that this one arises from the profile of the whole protogalaxy minus the contribution of the dark matter (and a small contribution of the hot gas by which the protoglobular clouds were bound), we show that the mass distributions around the Galactic centre of this cold gas and of the Old Halo agree satisfactorily. In order to demonstrate our hypothesis even more conclusively, we simulate the evolution with time, up to an age of 15 Gyr, of a putative globular cluster system whose initial mass distribution in the Galactic halo follows the profile of the cold protogalactic gas. We show that beyond a galactocentric distance of order 2-3 kpc, the initial shape of such a mass density profile is preserved despite the complete destruction of some globular clusters and the partial evaporation of some others. This result is almost independent of the choice of the initial mass function for the globular clusters, which is still ill determined. The shape of these evolved cluster system mass density profiles also agrees with the presently observed profile of the Old Halo globular cluster system, thus strengthening our hypothesis. Our result might suggest that the flattening shown by the Old Halo mass density profile at short distances from the Galactic centre is, at least partly, of primordial origi

The mass function and dynamical mass of young star clusters: Why their initial crossing-time matters crucially

We highlight the impact of cluster-mass-dependent evolutionary rates upon the evolution of the cluster mass function during violent relaxation, that is, while clusters dynamically respond to the expulsion of their residual star-forming gas. Mass-dependent evolutionary rates arise when the mean volume density of cluster-forming regions is mass-dependent. In that case, even if the initial conditions are such that the cluster mass function at the end of violent relaxation has the same shape as the embedded-cluster mass function (i.e. infant weight-loss is mass-independent), the shape of the cluster mass function does change transiently {\it during} violent relaxation. In contrast, for cluster-forming regions of constant mean volume density, the cluster mass function shape is preserved all through violent relaxation since all clusters then evolve at the same mass-independent rate. On the scale of individual clusters, we model the evolution of the ratio between the dynamical mass and luminous mass of a cluster after gas expulsion. Specifically, we map the radial dependence of the time-scale for a star cluster to return to equilibrium. We stress that fields-of-view a few pc in size only, typical of compact clusters with rapid evolutionary rates, are likely to reveal cluster regions which have returned to equilibrium even if the cluster experienced a major gas expulsion episode a few Myr earlier. We provide models with the aperture and time expressed in units of the initial half-mass radius and initial crossing-time, respectively, so that our results can be applied to clusters with initial densities, sizes, and apertures different from ours.Comment: 14 pages, 10 figures, accepted for publication in MNRA

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

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 after gas expulsion. Specifically, 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 $\tau_{GExp}/\tau_{cross}$ 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 observations. Our modelling includes predictions about the evolution of high-mass cluster-forming cores, a regime not yet covered by the observations. An overview of various issues directly affected by the nature of the core mass-radius relation is presented (e.g. cluster mass function, galaxy star formation histories, globular cluster self-enrichment). 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. [Abridged]Comment: 14 pages, 7 figures, accepted to MNRA

Evidence for two populations of Galactic globular clusters from the ratio of their half-mass to Jacobi radii

We investigate the ratio between the half-mass radii r_h of Galactic globular clusters and their Jacobi radii r_J given by the potential of the Milky Way and show that clusters with galactocentric distances R_{GC}>8 kpc fall into two distinct groups: one group of compact, tidally-underfilling clusters with r_h/r_J<0.05 and another group of tidally filling clusters which have 0.1 < r_h/r_J<0.3. We find no correlation between the membership of a particular cluster to one of these groups and its membership in the old or younger halo population. Based on the relaxation times and orbits of the clusters, we argue that compact clusters and most clusters in the inner Milky Way were born compact with half-mass radii r_h < 1 pc. Some of the tidally-filling clusters might have formed compact as well, but the majority likely formed with large half-mass radii. Galactic globular clusters therefore show a similar dichotomy as was recently found for globular clusters in dwarf galaxies and for young star clusters in the Milky Way. It seems likely that some of the tidally-filling clusters are evolving along the main sequence line of clusters recently discovered by Kuepper et al. (2008) and are in the process of dissolution.Comment: 8 pages, 4 figures, MNRAS in pres

Volume Density Thresholds for Overall Star Formation imply Mass-Size Thresholds for Massive Star Formation

We aim at understanding the massive star formation (MSF) limit $m(r) = 870 M_{\odot} (r/pc)^{1.33}$ in the mass-size space of molecular structures recently proposed by Kauffmann & Pillai (2010). As a first step, we build on the hypothesis of a volume density threshold for overall star formation and the model of Parmentier (2011) to establish the mass-radius relations of molecular clumps containing given masses of star-forming gas. Specifically, we relate the mass $m_{clump}$, radius $r_{clump}$ and density profile slope $-p$ of molecular clumps which contain a mass $m_{th}$ of gas denser than a volume density threshold $\rho_{th}$. In a second step, we use the relation between the mass of embedded-clusters and the mass of their most-massive star to estimate the minimum mass of star-forming gas needed to form a $10\,M_{\odot}$ star. Assuming a star formation efficiency of $SFE \simeq 0.30$, this gives $m_{th,crit} \simeq 150 M_{\odot}$. In a third step, we demonstrate that, for sensible choices of the clump density index ($p \simeq 1.7$) and of the cluster formation density threshold ($n_{th} \simeq 10^4\,cm^{-3}$), the line of constant $m_{th,crit} \simeq 150 M_{\odot}$ in the mass-radius space of molecular structures equates with the MSF limit for spatial scales larger than 0.3\,pc. Hence, the observationally inferred MSF limit of Kauffmann & Pillai is consistent with a threshold in star-forming gas mass beyond which the star-forming gas reservoir is large enough to allow the formation of massive stars. For radii smaller than 0.3\,pc, the MSF limit is shown to be consistent with the formation of a $10\,M_{\odot}$ star out of its individual pre-stellar core of density threshold $n_{th} \simeq 10^5\,cm^{-3}$. The inferred density thresholds for the formation of star clusters and individual stars within star clusters match those previously suggested in the literature.Comment: 8 pages, 5 figs, accepted for publication in MNRA

Self-enrichment of Galactic halo globular clusters: stimulated star formation and consequences for the halo metallicity distribution

We explore the self-enrichment hypothesis for globular cluster formation with respect to the star formation aspect. Following this scenario, the massive stars of a first stellar generation chemically enrich the globular progenitor cloud up to Galactic halo metallicities and sweep it into an expanding spherical shell of gas. This paper investigates the ability of this swept proto-globular cloud to become gravitationally unstable and, therefore, to seed the formation of second generation stars which may later on form a globular cluster. We use a simple model based on a linear perturbation theory for transverse motions in a shell of gas to demonstrate that the pressures by which the progenitor clouds are bound and the supernova numbers required to achieve Galactic halo metallicities support the successful development of the shell transverse collapse. Interestingly, the two parameters controling the metallicity achieved through self-enrichment, namely the number of supernovae and the external pressure, also rule the surface density of the shell and thus its ability to undergo a transverse collapse. Such a supernova-induced origin for the globular cluster stars opens therefore the way to the understanding of the halo metallicity distributions. This model is also able to explain the lower limit of the halo globular cluster metallicity.Comment: 16 pages, 16 figures, MNRAS accepte

Abundance correlations in mildly metal-poor stars

Accurate relative abundances have been obtained for a sample of 21 mildly metal-poor stars from the analysis of high resolution and high signal-to-noise spectra. In order to reach the highest coherence and internal precision, lines with similar dependency on the stellar atmospheric parameters were selected, and the analysis was carried out in a strictly differential way within the sample. With these accurate results, correlations between relative abundances have been searched for, with a special emphasis on the neutron capture elements. This analysis shows that the r elements are closely correlated to the alpha elements, which is in agreement with the generally accepted idea that the r-process takes place during the explosion of massive stars. The situation is more complex as far as the s elements are concerned. Their relation with the alpha elements is not linear. In a first group of stars, the relative abundance of the s elements increases only slightly with the alpha elements overabundance until the latter reaches a maximum value. For the second group, the s elements show a rather large range of enhancement and a constant (and maximum) value of the alpha elements overabundance. This peculiar behaviour leads us to distinguish between two sub-populations of metal-poor stars, namely Pop IIa (first group) and Pop IIb (second group). We suggest a scenario of formation of metal-poor stars based on two distinct phases of chemical enrichment, a first phase essentially consisting in supernova explosions of massive stars, and a second phase where the enrichment is provided by stellar winds from intermediate mass stars. More specifically, we assume that all thick disk and field halo stars were born in globular clusters, from which they escaped, either during an early disruption of the cluster (Pop IIa) or, later, through an evaporation process (Pop IIb). Based on observations obtained at the European Southern Observatory, La Silla, Chile