A Kennicutt-Schmidt relation at molecular cloud scales and beyond

Using N-body/gasdynamic simulations of a Milky Way-like galaxy we analyse a Kennicutt-Schmidt relation, $\Sigma_{SFR} \propto \Sigma_{gas}^N$, at different spatial scales. We simulate synthetic observations in CO lines and UV band. We adopt the star formation rate defined in two ways: based on free fall collapse of a molecular cloud - $\Sigma_{SFR, cl}$, and calculated by using a UV flux calibration - $\Sigma_{SFR, UV}$. We study a KS relation for spatially smoothed maps with effective spatial resolution from molecular cloud scales to several hundred parsecs. We find that for spatially and kinematically resolved molecular clouds the $\Sigma_{SFR, cl} \propto \Sigma_{\rm gas}^N$ relation follows the power-law with index $N \approx 1.4$. Using UV flux as SFR calibrator we confirm a systematic offset between the $\Sigma_{\rm UV}$ and $\Sigma_{\rm gas}$ distributions on scales compared to molecular cloud sizes. Degrading resolution of our simulated maps for surface densities of gas and star formation rates we establish that there is no relation $\Sigma_{\rm SFR, UV} - \Sigma_{\rm gas}$ below the resolution $\sim 50$ pc. We find a transition range around scales $\sim 50-120$ pc, where the power-law index $N$ increases from 0 to 1-1.8 and saturates for scales larger $\sim 120$ pc. A value of the index saturated depends on a surface gas density threshold and it becomes steeper for higher $\Sigma_{gas}$ threshold. Averaging over scales with size of $>150$ pc the power-law index $N$ equals 1.3-1.4 for surface gas density threshold $\sim 5 M_\odot$pc$^{-2}$. At scales $>120$ pc surface SFR densities determined by using CO data and UV flux, $\Sigma_{\rm SFR, UV}/\Sigma_{\rm SFR, cl}$, demonstrate a discrepancy about a factor of 3. We argue that this may be originated from overestimating (constant) values of conversion factor, star formation efficiency or UV calibration used in our analysis.