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.
