2013A&A...549A.132P

A positive power-law trend between the local surface densities of molecular gas, Σ_gas, and young stellar objects, Σ_*, in molecular clouds of the solar neighbourhood has recently been identified. How it relates to the properties of embedded clusters, in particular to the recently established radius-density relation, has so far not been investigated. We model the development of the stellar component of molecular clumps as a function of time and initial local volume density. Our study provides a coherent framework able to explain both the molecular-cloud and embedded-cluster relations quoted above. 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 star formation efficiency per free-fall time, ε_ff. For the volume density profiles typical of observed molecular clumps (i.e. power-law slope ≃-1.7), our model gives a star-gas surface-density relation of the form Σ_*∝Σ_gas², which agrees very well with the observations. Taking the case of a molecular clump of mass M₀≃10⁴M_☉ and radius R≃6pc experiencing star formation during 2Myr, we derive what star formation efficiency per free-fall time matches the normalizations of the observed and predicted (Σ_*, Σ_gas) relations best. We find ε_ff≃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, owing to the locally high star formation efficiency in the inner part of star-forming regions, global star formation efficiency as low as 10% can lead to the formation of bound gas-free star clusters.