Volume density thresholds for overall star formation imply mass–size thresholds for massive star formation

Abstract

We aim at understanding the massive star formation (MSF) limit m(r) = 870M_☉(r/pc)^(1.33) in the mass–size space of molecular structures recently proposed by Kauffmann & Pillai. As a first step, we build on the property that power-law density profiles for molecular clumps combined with a volume density threshold for the overall star formation naturally leads to mass–radius relations for molecular clumps containing given masses of star-forming gas. Specifically, we show that the mass m_(clump) and radius r_(clump) of molecular clumps whose density profile slope is −p and which contain a mass mth of gas denser than a density threshold ρ_(th) obeys the following: m_(clump) = m^(p/3)_(th) (^(4πpth) _(3-p))^((3-p)/3) r^(3-p)_(clump). In a second step, we use the relation between the mass of embedded clusters and the mass of their most massive star to estimate the minimum mass of the star-forming gas needed to form a 10-M_⊙ star. Assuming a star formation efficiency (SFE) of SFE ≃ 0.30, this gives m_(th,crit)≃150 M_⊙. In a third step, we demonstrate that, for sensible choices of the clump density index (p ≃ 1.7) and of the cluster formation density threshold (n_(th)≃ 10^4 cm^(−3)), the line of constant m_(th,crit)≃150 M_⊙ in the mass–radius space of molecular structures equates to the MSF limit for spatial scales larger than 0.3 pc. Hence, the observationally inferred MSF limit of Kauffmann & Pillai is consistent with a threshold in star-forming gas mass beyond which the star-forming gas reservoir is large enough to allow the formation of massive stars. For radii smaller than 0.3 pc, the MSF limit is shown to be consistent with the formation of a 10-M_⊙ star (m_(th,crit) ≃ 30 M_⊙ with SFE ≃ 0.3) out of its individual pre-stellar core of density threshold n_(th) ≃ 10^5 cm^(-3). The inferred density thresholds for the formation of star clusters and individual stars within star clusters match those previously suggested in the literature.

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