We show that supersonic MHD turbulence yields a star formation rate (SFR) as low as observed in molecular clouds (MCs), for characteristic values of the free-fall time divided by the dynamical time, $t_{rm ff}/t_{rm dyn}$, the alfv{e}nic Mach number,
${cal M}_{rm a}$, and the sonic Mach number, ${cal M}_{rm s}$. Using a very large set of deep adaptive-mesh-refinement simulations, we quantify the dependence of the SFR per free-fall time, $epsilon_{rm ff}$, on the above parameters. Our main results are: i) $epsilon_{rm ff}$ decreases exponentially with increasing $t_{rm ff}/t_{rm dyn}$, but is insensitive to changes in ${cal M}_{rm s}$, for constant values of $t_{rm ff}/t_{rm dyn}$ and ${cal M}_{rm a}$. ii) Decreasing values of ${cal M}_{rm a}$ (stronger magnetic fields) reduce $epsilon_{rm ff}$, but only to a point, beyond which $epsilon_{rm ff}$ increases with a further decrease of ${cal M}_{rm a}$. iii) For values of ${cal M}_{rm a}$ characteristic of star-forming regions, $epsilon_{rm ff}$ varies with ${cal M}_{rm a}$ by less than a factor of two. We propose a simple star-formation law, based on the empirical fit to the minimum $epsilon_{rm ff}$, and depending only on $t_{rm ff}/t_{rm dyn}$: $epsilon_{rm ff} approx epsilon_{rm wind} exp(-1.6 ,t_{rm ff}/t_{rm dyn})$. Because it only depends on the mean gas density and rms velocity, this law is straightforward to implement in simulations and analytical models of galaxy formation and evolution.
The observed similarities between the mass function of prestellar cores (CMF) and the stellar initial mass function (IMF) have led to the suggestion that the IMF is already largely determined in the gas phase. However, theoretical arguments show that
the CMF may differ significantly from the IMF. In this Letter, we study the relation between the CMF and the IMF, as predicted by the IMF model of Padoan and Nordlund. We show that 1) the observed mass of prestellar cores is on average a few times smaller than that of the stellar systems they generate; 2) the CMF rises monotonically with decreasing mass, with a noticeable change in slope at approximately 3-5 solar masses, depending on mean density; 3) the selection of cores with masses larger than half their Bonnor-Ebert mass yields a CMF approximately consistent with the system IMF, rescaled in mass by the same factor as our model IMF, and therefore suitable to estimate the local efficiency of star formation, and to study the dependence of the IMF peak on cloud properties; 4) only one in five pre-brown-dwarf core candidates is a true progenitor to a brown dwarf.
