Star formation in M 33: the radial and local relations with the gas

Aims. In the Local Group spiral galaxy M 33, we investigate the correlation between the star formation rate (SFR) surface density, Sigma(SFR), and the gas density Sigma(gas) (molecular, atomic, and total). We also explore whether there are other physical quantities, such as the hydrostatic pressure and dust optical depth, which establish a good correlation with Sigma(SFR). Methods. We use the H alpha, far-ultraviolet (FUV), and bolometric emission maps to infer the SFR locally at different spatial scales, and in radial bins using azimuthally averaged values. Most of the local analysis is done using the highest spatial resolution allowed by gas surveys, 180 pc. The Kennicutt-Schmidt (KS) law, Sigma(SFR) proportional to Sigma(n)(gas) is analyzed by three statistical methods. Results. At all spatial scales, with H alpha emission as a SFR tracer, the KS indices n are always steeper than those derived with the FUV and bolometric emissions. We attribute this to the lack of Ha emission in low luminosity regions where most stars form in small clusters with an incomplete initial mass function at their high mass end. For azimuthally averaged values the depletion timescale for the molecular gas is constant, and the KS index is n(H2) = 1.1 +/- 0.1. Locally, at a spatial resolution of 180 pc, the correlation between Sigma(SFR) and Sigma(gas) is generally poor, even though it is tighter with the molecular and total gas than with the atomic gas alone. Considering only positions where the CO J = 1-0 line is above the 2-s detection threshold and taking into account uncertainties in Sigma(H2) and Sigma(SFR), we obtain a steeper KS index than obtained with radial averages: n(H2) = 2.22 +/- 0.07 (for FUV and bolometric SFR tracers), flatter than that relative to the total gas (n(Htot) = 2.59 +/- 0.05). The gas depletion timescale is therefore larger in regions of lower Sigma(SFR). Lower KS indices (n(H2) = 1.46 +/- 0.34 and n(H2) = 1.12) are found using different fitting techniques, which do not account for individual position uncertainties. At coarser spatial resolutions these indices get slightly steeper, and the correlation improves. We find an almost linear relation and a better correlation coefficient between the local Sigma(SFR) and the ISM hydrostatic pressure or the gas volume density. This suggests that the stellar disk, gravitationally dominant with respect to the gaseous disk in M 33, has a non-marginal role in driving the SFR. However, the tight local correlation that exists between the dust optical depth and the SFR sheds light on the alternative hypothesis that the dust column density is a good tracer of the gas that is prone to star formation.