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, $t_{\rm ff}/t_{\rm dyn}$, 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.
