Stellar age spreads in clusters as imprints of cluster-parent clump densities

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 arise 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.