Kiloparsec-scale Variations in the Star Formation Efficiency of Dense
  Gas: the Antennae Galaxies (NGC 4038/39)
  We study the relationship between dense gas and star formation in the
Antennae galaxies by comparing ALMA observations of dense gas tracers (HCN,
HCO$^+$, and HNC $\mathrm{J}=1-0$) to the total infrared luminosity
($\mathrm{L_{TIR}}$) calculated using data from the \textit{Herschel} Space
Observatory and the \textit{Spitzer} Space Telescope. We compare the
luminosities of our SFR and gas tracers using aperture photometry and employing
two methods for defining apertures. We taper the ALMA dataset to match the
resolution of our $\mathrm{L_{TIR}}$ maps and present new detections of dense
gas emission from complexes in the overlap and western arm regions. Using OVRO
CO $\mathrm{J}=1-0$ data, we compare with the total molecular gas content,
$\mathrm{M(H_2)_{tot}}$, and calculate star formation efficiencies and dense
gas mass fractions for these different regions. We derive HCN, HCO$^+$ and HNC
upper limits for apertures where emission was not significantly detected, as we
expect emission from dense gas should be present in most star-forming regions.
The Antennae extends the linear $\mathrm{L_{TIR}-L_{HCN}}$ relationship found
in previous studies. The $\mathrm{L_{TIR}-L_{HCN}}$ ratio varies by up to a
factor of $\sim$10 across different regions of the Antennae implying variations
in the star formation efficiency of dense gas, with the nuclei, NGC 4038 and
NGC 4039, showing the lowest SFE$_\mathrm{dense}$ (0.44 and 0.70
$\times10^{-8}$ yr$^{-1}$). The nuclei also exhibit the highest dense gas
fractions ($\sim 9.1\%$ and $\sim7.9\%$).
