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 (MCs), for characteristic values of the free-fall time divided by the dynamical time, $t_{\mathrm{f}\mathrm{f}}\mathrm{/}{t}_{\mathrm{d}\mathrm{y}\mathrm{n}}$, the alfv'{e}nic Mach number, ${\mathcal{M}}_{\mathrm{a}}$, 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, ${ϵ}_{\mathrm{f}\mathrm{f}}$, on the above parameters. Our main results are: i) ${ϵ}_{\mathrm{f}\mathrm{f}}$ decreases exponentially with increasing $t{\rm ff}/t_{\rm dyn}$,butisinsensitivetochangesin${\cal M}{\rm s}$, for constant values of $t_{\mathrm{f}\mathrm{f}}\mathrm{/}{t}_{\mathrm{d}\mathrm{y}\mathrm{n}}$ and ${\mathcal{M}}_{\mathrm{a}}$. ii) Decreasing values of ${\mathcal{M}}_{\mathrm{a}}$ (stronger magnetic fields) reduce ${ϵ}_{\mathrm{f}\mathrm{f}}$, but only to a point, beyond which ${ϵ}_{\mathrm{f}\mathrm{f}}$ increases with a further decrease of ${\mathcal{M}}_{\mathrm{a}}$. iii) For values of ${\mathcal{M}}_{\mathrm{a}}$ characteristic of star-forming regions, ${ϵ}_{\mathrm{f}\mathrm{f}}$ 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 ${ϵ}_{\mathrm{f}\mathrm{f}}$, and depending only on $t_{\mathrm{f}\mathrm{f}}\mathrm{/}{t}_{\mathrm{d}\mathrm{y}\mathrm{n}}$: $\epsilon_{\rm ff} \approx \epsilon_{\rm wind} \exp(-1.6 ,t_{\rm ff}/t_{\rm dyn})$. 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.