Volumetric star formation laws of disc galaxies
  Star formation (SF) laws are fundamental relations between the gas content of
a galaxy and its star formation rate (SFR) and play key roles in galaxy
evolution models. In this paper, we present new empirical SF laws of disc
galaxies based on volume densities. Following the assumption of hydrostatic
equilibrium, we calculated the radial growth of the thickness of the gaseous
discs in the combined gravitational potential of dark matter, stars, and gas
for 12 nearby star-forming galaxies. This allowed us to convert the observed
surface densities of gas and SFR into the deprojected volume densities. We
found a tight correlation with slope in the range 1.3-1.9 between the volume
densities of gas (HI+H$_2$) and the SFR with a significantly smaller scatter
than the surface-based (Kennicutt) law and no change in the slope over five
orders of magnitude. This indicates that taking into account the radial
increase of the thickness of galaxy discs is crucial to reconstruct their
three-dimensional density profiles, in particular in their outskirts. Moreover,
our result suggests that the break in the slope seen in the Kennicutt law is
due to disc flaring rather than to a drop of the SF efficiency at low surface
densities. Surprisingly, we discovered an unexpected correlation between the
volume densities of HI and SFR, indicating that the atomic gas is a good tracer
of the cold star-forming gas, especially in low density HI-dominated
environments.
