We probe the star formation properties of the gas in AzTEC-1 in the COSMOS field, one of the best resolved and brightest starburst galaxies at $z approx 4.3$, forming stars at a rate > 1000 $mathrm{M_{odot}},mathrm{yr^{-1}}$. Using recent ALMA observations, we study star formation in the galaxy nucleus and an off-center star-forming clump and measure a median star formation rate (SFR) surface density of $Sigma^{mathrm{nucleus}}_{mathrm{SFR}} = 270pm54$ and $Sigma^{mathrm{sfclump}}_{mathrm{SFR}} = 170pm38,mathrm{M_{odot}},mathrm{yr}^{-1},mathrm{kpc}^{-2}$, respectively. Following the analysis by Sharda et al. (2018), we estimate the molecular gas mass, freefall time and turbulent Mach number in these regions to predict $Sigma_{mathrm{SFR}}$ from three star formation relations in the literature. The Kennicutt-Schmidt (Kennicutt 1998, KS) relation, which is based on the gas surface density, underestimates the $Sigma_{mathrm{SFR}}$ in these regions by a factor 2-3. The $Sigma_{mathrm{SFR}}$ we calculate from the single-freefall model of Krumholz et al. 2012 (KDM) is consistent with the measured $Sigma_{mathrm{SFR}}$ in the nucleus and the star-forming clump within the uncertainties. The turbulence-regulated star formation relation by Salim et al. 2015 (SFK) agrees slightly better with the observations than the KDM relation. Our analysis reveals that an interplay between turbulence and gravity can help sustain high SFRs in high-redshift starbursts. It can also be extended to other high- and low-redshift galaxies thanks to the high angular resolution and sensitivity of ALMA observations.
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