Gravity or turbulence? VII. The Schmidt-Kennicutt law, the star formation efficiency, and the mass density of clusters from gravitational collapse rather than turbulent support

We explore the Schmidt-Kennicutt (SK) relations and the star formation efficiency per free-fall time ($ε_{\rm ff}$), mirroring observational studies, in numerical simulations of filamentary molecular clouds undergoing gravitational contraction. We find that (a)~collapsing clouds accurately replicate the observed SK relations for galactic clouds and (b)~$ε_{\rm ff}$ is small and constant in space and in time, with values similar to those found in local clouds. We propose that this constancy arises from the similar radial scaling of the free-fall time ($τ_{\rm ff}$) and the internal mass in density structures with spherically-averaged density profiles near $r^{-2}$. We additionally show that (c)~the star formation rate (SFR) increases rapidly in time; (d)~the low values of $ε_{\rm ff}$ result from evaluating $τ_{\rm ff}$ and the characteristic star-formation time scale over different time intervals, combined with the increasing SFR, and (e)~the fact that star clusters are significantly denser than the gas clumps from which they form is a natural consequence of the rapidly increasing SFR, the continuous replenishment of the star-forming gas by the accretion flow, and the near $r^{-2}$ density profile induced by the collapse. Finally, we argue that interpreting $ε_{\rm ff}$ as an efficiency is problematic since it is not bounded by unity, and because the gas mass in clouds evolves. Instead, we propose that viewing $ε_{\rm ff}$ as the ratio of the actual SFR to the gas free-fall rate. In summary, our results show that the SK relation, the low values of $ε_{\rm ff}$, and the mass density of stellar clusters arise naturally from gravitational contraction.