Testing star formation laws in a starburst galaxy at redshift 3 resolved with ALMA

Using high-resolution (sub-kiloparsec scale) submillimeter data obtained by ALMA, we analyze the star formation rate (SFR), gas content and kinematics in SDP 81, a gravitationally-lensed star-forming galaxy at redshift 3. We estimate the SFR surface density (Sigma(SFR)) in the brightest clump of this galaxy to be 357(-85)(+135)M(circle dot)yr(-1)kpc(-2), over an area of 0.07 +/- 0.02kpc(2). Using the intensity-weighted velocity of CO(5-4), we measure the turbulent velocity dispersion in the plane-of-the-sky and find sigma(v,turb) = 37 +/- 5kms(-1) for the star-forming clump, in good agreement with previous estimates along the line of sight. Our measurements of gas surface density, freefall time and turbulent Mach number reveal that the role of turbulence is vital to explaining the observed SFR in this clump. While the Kennicutt Schmidt (KS) relation predicts a SFR surface density of Sigma(SFR,KS) = 52 +/- 17M(circle dot)yr(-1)kpc(-2), the single-freefall model by Krumholz, Dekel and McKee (KDM) predicts Sigma(SFR,KDM) = 106 +/- 37M(circle dot)yr-1kpc(-2). In contrast, the multi-freefall (turbulence) model by Salim, Federrath and Kewley (SFK) gives Sigma(SFR,SFK) = 491(-194)(+139)M(circle dot)yr(-1)kpc(-2). Although the SFK relation overestimates the SFR in this clump (possibly due to the ignorance of magnetic field), it provides the best prediction among the available models. Finally, we compare the star formation and gas properties of this high-redshift galaxy to local star-forming regions and find that the SFK relation provides the best estimates of SFR in both local and high-redshift galaxies.