The mass function of molecular clouds and clumps is shallower than the mass function of young star clusters, gas-embedded and gas-free alike, as their respective mass function indices are β0≃ 1.7 and β★≃ 2. We demonstrate that such a difference can arise from different mass–radius relations for the embedded clusters and the molecular clouds (clumps) hosting them. In particular, the formation of star clusters with a constant mean volume density in the central regions of molecular clouds of constant mean surface density steepens the mass function from clouds to embedded clusters. This model is observationally supported since the mean surface density of molecular clouds is approximately constant, while there is a growing body of evidence, in both Galactic and extragalactic environments, that efficient star formation requires a hydrogen molecule number density threshold of nth≃ 104−5 cm−3.
Adopting power-law volume density profiles of index p for spherically symmetric molecular clouds (clumps), we define two zones within each cloud (clump): a central cluster-forming region, actively forming stars by virtue of a local number density higher than nth, and an outer envelope inert in terms of star formation. We map how much the slope of the cluster-forming region mass function differs from that of their host clouds (clumps) as a function of their respective mass–radius relations and of the cloud (clump) density index. We find that for constant surface density clouds with density index p≃ 1.9, a cloud mass function of index β0= 1.7 gives rise to a cluster-forming region mass function of index β≃ 2. Our model equates with defining two distinct star formation efficiencies (SFEs): a global mass-varying SFE averaged over the whole cloud (clump), and a local mass-independent SFE measured over the central cluster-forming region. While the global SFE relates the mass function of clouds to that of embedded clusters, the local SFE rules cluster evolution after residual star-forming gas expulsion. That the cluster mass function slope does not change through early cluster evolution implies a mass-independent local SFE and, thus, the same mass function index for cluster-forming regions and embedded clusters, that is β=β★. Our model can therefore reproduce the observed cluster mass function index β★≃ 2.
For the same model parameters, the radius distribution also steepens from clouds (clumps) to embedded clusters, which contributes to explaining observed cluster radius distributions.