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 ($\Sigma_{\mathrm{SFR}}$) in the brightest clump of
this galaxy to be $357^{+135}_{-85}\,\mathrm{M_{\odot}\,yr^{-1}\,kpc^{-2}}$,
over an area of $0.07\pm0.02\,\mathrm{kpc}^2$. Using the intensity-weighted
velocity of CO$\,$(5-4), we measure the turbulent velocity dispersion in the
plane-of-the-sky and find $\sigma_{\mathrm{v,turb}} =
37\pm5\,\mathrm{km\,s}^{-1}$ 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
$\Sigma_{\mathrm{SFR,KS}} = 52\pm17\,\mathrm{M_{\odot}\,yr^{-1}\,kpc^{-2}}$,
the single-freefall model by Krumholz, Dekel and McKee (KDM) predicts
$\Sigma_{\mathrm{SFR,KDM}} = 106\pm37\,\mathrm{M_{\odot}\,yr^{-1}\,kpc^{-2}}$.
In contrast, the multi-freefall (turbulence) model by Salim, Federrath and
Kewley (SFK) gives $\Sigma_{\mathrm{SFR,SFK}} =
491^{+139}_{-194}\,\mathrm{M_{\odot}\,yr^{-1}\,kpc^{-2}}$. 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.
