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In the Local Group spiral galaxy M33, we investigate the correlation betweenthe star formation rate (SFR) surface density, Sigma_SFR, and the gas densitySigma_gas (molecular, atomic, and total). We also explore whether there areother physical quantities, such as the hydrostatic pressure and dust opticaldepth, which establish a good correlation with Sigma_SFR. We use the Ha,far-ultraviolet (FUV), and bolometric emission maps to infer the SFR locally atdifferent spatial scales, and in radial bins using azimuthally averaged values.Most of the local analysis is done using the highest spatial resolution allowedby gas surveys, 180 pc. The Kennicutt-Schmidt (KS) law, Sigma_SFR \propto(Sigma_gas)^n is analyzed by three statistical methods. At all spatial scales,with Ha emission as a SFR tracer, the KS indices n are always steeper thanthose derived with the FUV and bolometric emissions. We attribute this to thelack of Ha emission in low luminosity regions where most stars form in smallclusters with an incomplete initial mass function at their high mass end. Forazimuthally averaged values the depletion timescale for the molecular gas isconstant, and the KS index is n_H2 = 1.1 +- 0.1. Locally, at a spatialresolution of 180 pc, the correlation between Sigma_SFR and Sigma_gas isgenerally poor, even though it is tighter with the molecular and total gas thanwith the atomic gas alone. Considering only positions where the CO J=1-0 lineis above the 2-sigma detection threshold and taking into account uncertaintiesin Sigma_H2 and Sigma_SFR, we obtain a steeper KS index than obtained withradial 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). [abridged]

Star formation in M33: the radial and local relations with the gas