We study the star formation efficiency ( SFE ) in simulations and observations of turbulent , magnetized , molecular clouds .
We find that the probability density functions ( PDFs ) of the density and the column density in our simulations with solenoidal , mixed , and compressive forcing of the turbulence , sonic Mach numbers of 3 – 50 , and magnetic fields in the super- to the trans-Alfvénic regime , all develop power-law tails of flattening slope with increasing SFE .
The high-density tails of the PDFs are consistent with equivalent radial density profiles , \rho \propto r ^ { - \kappa } with \kappa \sim 1.5 – 2.5 , in agreement with observations .
Studying velocity–size scalings , we find that all the simulations are consistent with the observed v \propto \ell ^ { 1 / 2 } scaling of supersonic turbulence , and seem to approach Kolmogorov turbulence with v \propto \ell ^ { 1 / 3 } below the sonic scale .
The velocity–size scaling is , however , largely independent of the SFE .
In contrast , the density–size and column density–size scalings are highly sensitive to star formation .
We find that the power-law slope \alpha of the density power spectrum , P _ { \mathrm { 3 D } } ( \rho,k ) \propto k ^ { \alpha } , or equivalently the \Delta -variance spectrum of the column density , \sigma _ { \Delta } ^ { 2 } ( \Sigma, \ell ) \propto \ell ^ { - \alpha } , switches sign from \alpha \lesssim 0 for \mathrm { SFE } \sim 0 to \alpha \gtrsim 0 when star formation proceeds ( \mathrm { SFE } > 0 ) .
We provide a relation to compute the SFE from a measurement of \alpha .
Studying the literature , we find values ranging from \alpha = -1.6 to +1.6 in observations covering scales from the large-scale atomic medium , over cold molecular clouds , down to dense star-forming cores .
From those \alpha values , we infer SFEs and find good agreement with independent measurements based on young stellar object ( YSO ) counts , where available .
Our \mathrm { SFE } – \alpha relation provides an independent estimate of the \mathrm { SFE } based on the column density map of a cloud alone , without requiring a priori knowledge of star-formation activity or YSO counts .