From the molecular-cloud- to the embedded-cluster-mass function with a density threshold for star formation

Abstract

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.

High-mass star formation at high luminosities: W31 at >106 L⊙

“…Higher mass gas clumps may exhibit on average a lower SFE that would steepen the slope from the clump mass function to the IMF. Similar results were recently inferred by Parmentier (2011). In a similar direction, it is worth noting that Simon et al (2001) also inferred for their most luminous mini-starburst W49 the flattest clump mass function with a power-law index of 1.56, in very close agreement with our result for the high-luminosity region W31.…”

Star formation efficiency as a function of metallicity: from star clusters to galaxies

“…this happens when cores/clumps are detected at threshold densities of ≳ 10 5 cm −3 ; Dib & Kim 2007; Dib et al 2007a). The argument of the existence of a threshold density of ∼ 10 5 cm −3 for star formation as proposed by Dib et al (2007a) was also used by Parmentier (2011) who showed that this may well explain the difference between the slope of the mass function of low‐density clouds/clumps (∼−1.7) and that of the high‐density protocluster‐forming clumps (∼−2).…”

“…Several studies have established that star clusters form in dense (≳10 3 cm −3 ) clumps embedded in a lower density parental molecular cloud (e.g. Lada & Lada 2003; Lada, Lombardi & Alves 2010; Csengeri et al 2011; Parmentier 2011). Saito et al (2007) recently studied, using the C 18 O molecular emission line, a large sample of cluster‐forming clumps whose masses and radii vary between [15–1500] M ⊙ and [0.14–0.61] pc, respectively.…”

The dense gas mass fraction in the W51 cloud and its protoclusters

“…The dense gas mass fraction has been measured for various definitions of dense (Battisti & Heyer 2014;Wu et al 2005), but these definitions don't always correspond to the associated threshold (Kauffmann & Pillai 2010;Parmentier et al 2011;Parmentier 2011). Often, the dense gas threshold is observationally defined as the threshold to see a given molecule, which is sometimes incorrectly assumed to correspond to a fixed critical density for the molecule.…”