Spatially resolved star-formation relations of dense molecular gas in
  NGC 1068
  We analyse the influence of the dynamical environment on the star formation
(SF) relations of the dense molecular gas in the starburst (SB) ring of the
Seyfert 2 galaxy NGC 1068. We used ALMA to image the emission of the 1-0
transitions of HCN and HCO+ with a resolution of 56 pc. We also used ancillary
data of CO(1-0) at a resolution of ~100 pc, and CO(3-2) and its underlying
continuum emission at ~40 pc. These observations allow us to probe a range of
molecular gas densities (n(H2)~10$^{3-5}cm^{-3}$). The SF rate (SFR) is derived
from Pa$\alpha$ line emission imaged by HST/NICMOS. We analysed how SF
relations change depending on the choice of aperture sizes and molecular gas
tracer. The scatter in the Kennicutt-Schmidt relation is about a factor of two
to three lower for the HCN and HCO+ lines compared to CO(1-0) for a common
aperture. Correlations lose statistical significance below a critical spatial
scale $\approx$300-400 pc. The SF efficiency of the dense molecular gas
(SFEdense) shows a scattered distribution as a function of the HCN luminosity
(L'(HCN)) around a mean value of $\simeq0.01$Myr$^{-1}$. An alternative
prescription for SF relations, linking the SFEdense and the boundedness of the
gas measured by the parameter b$\equiv\Sigma$dense/$\sigma^2$, where
$\Sigma$dense is the dense molecular gas surface density and $\sigma$ the
velocity dispersion, resolves the degeneracy associated with the
SFEdense-L'(HCN) plot. We identify two branches in the SFEdense-b plot that
correspond to two dynamical environments in the SB ring, which are defined by
their proximity to the bar-ring interface region. This region corresponds to
the crossing of two density wave resonances, where an increased rate of
cloud-cloud collisions would favour an enhanced compression of molecular gas.
Our results suggest that galactic dynamics plays a major role in the efficiency
of the gas conversion into stars.
