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.

We study the gas kinematics of a sample of six isolated gas-rich low surface brightness galaxies, of the class called ultra-diffuse galaxies (UDGs). These galaxies have recently been shown to be outliers from the baryonic Tully–Fisher relation (BTFR), as they rotate much slower than expected given their baryonic mass, and to have a baryon fraction similar to the cosmological mean. By means of a 3D kinematic modelling fitting technique, we show that the H i in our UDGs is distributed in ‘thin’ regularly rotating discs and we determine their rotation velocity and gas velocity dispersion. We revisit the BTFR adding galaxies from other studies. We find a previously unknown trend between the deviation from the BTFR and the exponential disc scale length valid for dwarf galaxies with circular speeds ≲ 45 km s−1, with our UDGs being at the extreme end. Based on our findings, we suggest that the high baryon fractions of our UDGs may originate due to the fact that they have experienced weak stellar feedback, likely due to their low star formation rate surface densities, and as a result they did not eject significant amounts of gas out of their discs. At the same time, we find indications that our UDGs may have higher-than-average stellar specific angular momentum, which can explain their large optical scale lengths.