It has recently been suggested that high-density clusters have stellar age distributions narrower than that of the Orion Nebula Cluster, indicating a possible trend of narrower age distributions for denser clusters. We show this effect to likely aris
e from star formation being faster in gas with a higher density. We model the star formation history of molecular clumps in equilibrium by associating a star formation efficiency (SFE) per free-fall time, eff, to their volume density profile. Our model predicts a steady decline of the star formation rate (SFR), which we quantify with its half-life time, namely, the time needed for the SFR to drop to half its initial value. Given the uncertainties affecting the SFE per free-fall time, we consider two distinct values: 0.1 and 0.01. For isothermal spheres, eff=0.1 leads to a half-life time of order the clump free-fall time, tff. Therefore, the age distributions of stars formed in high-density clumps have smaller full-widths at half-maximum than those of stars formed in low-density clumps. When eff=0.01, the half-life time is 10 times longer. We explore what happens if the star formation duration is shorter than 10tff, that is, if the half-life time of the SFR cannot be defined. There, we build on the invariance of the shape of the young cluster mass function to show that an anti-correlation between clump density and star formation duration is expected. Therefore, regardless of whether the star formation duration is longer than the SFR half-life time, denser molecular clumps yield narrower star age distributions in clusters. Published densities and stellar age spreads of young clusters actually suggest that the time-scale for star formation is of order 1-4tff. We conclude that there is no need to invoke the existence of multiple cluster formation mechanisms to explain the observed range of stellar age spreads in clusters.
A positive power-law trend between the local surface densities of molecular gas, $Sigma_{gas}$, and young stellar objects, $Sigma_{star}$, in molecular clouds of the Solar Neighbourhood has been identified by Gutermuth et al. How it relates to the pr
operties of embedded clusters, in particular to the recently established radius-density relation, has so far not been investigated. In this paper, we model the development of the stellar component of molecular clumps as a function of time and initial local volume density so as to provide a coherent framework able to explain both the molecular-cloud and embedded-cluster relations quoted above. To do so, we associate the observed volume density gradient of molecular clumps to a density-dependent free-fall time. The molecular clump star formation history is obtained by applying a constant SFE per free-fall time, $eff$. For volume density profiles typical of observed molecular clumps (i.e. power-law slope $simeq -1.7$), our model gives a star-gas surface-density relation $Sigma_{star} propto Sigma_{gas}^2$, in very good agreement with the Gutermuth et al relation. Taking the case of a molecular clump of mass $M_0 simeq 10^4 Msun$ and radius $R simeq 6 pc$ experiencing star formation during 2 Myr, we derive what SFE per free-fall time matches best the normalizations of the observed and predicted ($Sigma_{star}$, $Sigma_{gas}$) relations. We find $eff simeq 0.1$. We show that the observed growth of embedded clusters, embodied by their radius-density relation, corresponds to a surface density threshold being applied to developing star-forming regions. The consequences of our model in terms of cluster survivability after residual star-forming gas expulsion are that due to the locally high SFE in the inner part of star-forming regions, global SFE as low as 10% can lead to the formation of bound gas-free star clusters.
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-depe
ndent 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.
To understand how systems of star clusters have reached their presently observed properties constitutes a powerful probe into the physics of cluster formation, without needing to resort to high spatial resolution observations of individual cluster-fo
rming regions (CFRg) in distant galaxies. In this contribution I focus on the mass-radius relation of CFRgs, how it can be uncovered by studying the gas expulsion phase of forming star clusters, and what the implications are. I demonstrate that, through the tidal field impact upon exposed star clusters, the CFRg mass-radius relation rules cluster infant weight-loss in dependence of cluster mass. The observational constraint of a time-invariant slope for the power-law young cluster mass function is robustly satisfied by CFRgs with a constant mean volume density. In contrast, a constant mean surface density would be conducive to the preferential destruction of high-mass clusters. A purely dynamical line-of-reasoning leads therefore to a conclusion consistent with star formation a process driven by a volume density threshold. Developing this concept further, properties of molecular clumps and CFRgs naturally get dissociated. This allows to understand: (i) why the star cluster mass function is steeper than the molecular cloud (clump) mass function; (ii) the presence of a massive star formation limit in the mass-size space of molecular structures.
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.
We use Monte-Carlo simulations, combined with homogeneously determined age and mass distributions based on multi-wavelength photometry, to constrain the cluster formation history and the rate of bound cluster disruption in the Large Magellanic Cloud
(LMC) cluster system. We evolve synthetic star cluster systems formed with a power-law initial cluster mass function (ICMF) of spectral index $alpha =-2$ assuming different cluster disruption time-scales. For each of these disruption time-scales we derive the corresponding cluster formation rate (CFR) required to reproduce the observed cluster age distribution. We then compare, in a Poissonian $chi^2$ sense, model mass distributions and model two-dimensional distributions in log(mass) vs. log(age) space of the detected surviving clusters to the observations. Because of the bright detection limit ($M_V^{rm lim} simeq -4.7$ mag) above which the observed cluster sample is complete, one cannot constrain the characteristic disruption time-scale for a $10^4$ M$_odot$ cluster, $t_4^{rm dis}$ (where the disruption time-scale depends on cluster mass as $t_{rm dis} = t_4^{rm dis} (M_{rm cl} / 10^4 {rm M}_odot)^0.62$), to better than $t_4^{rm dis} ge 1$ Gyr. We conclude that the CFR has increased from 0.3 clusters Myr$^{-1}$ 5 Gyr ago, to a present rate of $(20-30)$ clusters Myr$^{-1}$. For older ages the derived CFR depends sensitively on our assumption of the underlying CMF shape. If we assume a universal Gaussian ICMF, then the CFR has increased steadily over a Hubble time from $sim 1$ cluster Gyr$^{-1}$ 15 Gyr ago to its present value. If the ICMF has always been a power law with a slope close to $alpha=-2$, the CFR exhibits a minimum some 5 Gyr ago, which we tentatively identify with the well-known age gap in the LMCs cluster age distribution.
