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

We aim at understanding the massive star formation (MSF) limit m(r) = 870M(circle dot)(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 mclump and radius rclump of molecular clumps whose density profile slope is -p and which contain a mass mth of gas denser than a density threshold.th obeys the following: mclump = m p/ 3 rho(th) m(clump) = m(th)(p/3) (4 pi rho th/3-p)((3-p)/3)r(clump)(3 p). 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-circle dot star. Assuming a star formation efficiency (SFE) of SFE similar or equal to 0.30, this gives mth, crit similar or equal to 150M(circle dot). In a third step, we demonstrate that, for sensible choices of the clump density index (p similar or equal to 1.7) and of the cluster formation density threshold (n(th) similar or equal to 10(4) cm(-3)), the line of constant m(th, crit) similar or equal to 150M(circle dot) 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-circle dot star (m(th, crit) similar or equal to 30M(circle dot) with SFE similar or equal to 0.3) out of its individual pre-stellar core of density threshold n(th) similar or equal to 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.