Star formation scaling relations at ~100 pc from PHANGS: Impact of
  completeness and spatial scale
  Aims: The complexity of star formation at the physical scale of molecular
clouds is not yet fully understood. We investigate the mechanisms regulating
the formation of stars in different environments within nearby star-forming
galaxies from the PHANGS sample. Methods: Integral field spectroscopic data and
radio-interferometric observations of 18 galaxies were combined to explore the
existence of the resolved star formation main sequence (rSFMS), resolved
Kennicutt-Schmidt relation (rKS), and resolved molecular gas main sequence
(rMGMS), and we derived their slope and scatter at spatial resolutions from 100
pc to 1 kpc (under various assumptions). Results: All three relations were
recovered at the highest spatial resolution (100 pc). Furthermore, significant
variations in these scaling relations were observed across different galactic
environments. The exclusion of non-detections has a systematic impact on the
inferred slope as a function of the spatial scale. Finally, the scatter of the
$\Sigma_\mathrm{mol. gas + stellar}$ versus $\Sigma_\mathrm{SFR}$ correlation
is smaller than that of the rSFMS, but higher than that found for the rKS.
Conclusions: The rMGMS has the tightest relation at a spatial scale of 100 pc
(scatter of 0.34 dex), followed by the rKS (0.41 dex) and then the rSFMS (0.51
dex). This is consistent with expectations from the timescales involved in the
evolutionary cycle of molecular clouds. Surprisingly, the rKS shows the least
variation across galaxies and environments, suggesting a tight link between
molecular gas and subsequent star formation. The scatter of the three relations
decreases at lower spatial resolutions, with the rKS being the tightest (0.27
dex) at a spatial scale of 1 kpc. Variation in the slope of the rSFMS among
galaxies is partially due to different detection fractions of
$\Sigma_\mathrm{SFR}$ with respect to $\Sigma_\mathrm{stellar}$.
