A Simple Law of Star Formation

We show that supersonic MHD turbulence yields a star formation rate (SFR) as low as observed in molecular clouds, for characteristic values of the free-fall time divided by the dynamical time, t _ff/t _dyn, the Alfvénic Mach number, {\cal M}_a, and the sonic Mach number, {\cal M}_s. Using a very large set of deep adaptive-mesh-refinement simulations, we quantify the dependence of the SFR per free-fall time, epsilon_ff, on the above parameters. Our main results are (1) that epsilon_ff decreases exponentially with increasing t _ff/t _dyn, but is insensitive to changes in {\cal M}_s, for constant values of t _ff/t _dyn and {\cal M}_a. (2) Decreasing values of {\cal M}_a (stronger magnetic fields) reduce epsilon_ff, but only to a point, beyond which epsilon_ff increases with a further decrease of {\cal M}_a. (3) For values of {\cal M}_a characteristic of star-forming regions, epsilon_ff varies with {\cal M}_a by less than a factor of two. We propose a simple star formation law, based on the empirical fit to the minimum epsilon_ff, and depending only on t _ff/t _dyn: epsilon_ff ≈ epsilon_windexp (- 1.6 t _ff/t _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.