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

Using high-resolution (sub-kiloparsec scale) data obtained by ALMA, we analyse the star formation rate (SFR), gas content, and kinematics in SDP 81, a gravitationally lensed starburst galaxy at redshift 3. We estimate the SFR surface density (Σ_SFR) in the brightest clump of this galaxy to be 357^{+135}_{-85} M_{⊙} yr^{-1} kpc^{-2}, over an area of 0.07 ± 0.02 kpc². Using the intensity-weighted velocity of CO (5-4), we measure the turbulent velocity dispersion in the plane of the sky and find σ_{v, turb} = 37 ± 5 km s^-1 for the clump, in good agreement with previous estimates along the line of sight. Our measurements of the gas surface density, freefall time, and turbulent Mach number allow us to compare the theoretical SFR from various star formation models with that observed, revealing that the role of turbulence is crucial to explaining the observed SFR in this clump. While the Kennicutt-Schmidt (KS) relation predicts an SFR surface density of Σ_{SFR, KS} = 52 ± 17 M_⊙ yr^-1 kpc^-2, the single-freefall model by Krumholz, Dekel, and McKee (KDM) predicts Σ_{SFR, KDM} = 106 ± 37 M_⊙ yr^-1 kpc^-2. In contrast, the multifreefall (turbulence) model by Salim, Federrath, and Kewley (SFK) gives Σ _{SFR,SFK} = 491^{+139 }_{-194} M_{⊙} yr^{-1} kpc^{-2}. Although the SFK relation overestimates the SFR in this clump (possibly due to the negligence of magnetic fields), it provides the best prediction among the available models. Finally, we compare the star formation and gas properties of this 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.