The mechanisms governing the star formation rate in spiral galaxies are not yet clear. The nearby, almost face-on, and interacting galaxy M51 offers an excellent opportunity to study at high spatial resolutions the local star formation laws. In this first paper, we investigate the correlation of H2, HI, and total gas surface densities with the star forming activity, derived from the radio continuum (RC), along radial averages out to radii of 12kpc. We have created a complete map of M51 in 12CO 2-1 at a resolution of 450kpc using HERA at the IRAM-30m telescope. These data are combined with maps of HI and the radio-continuum at 20cm wavelength. The latter is used to estimate the star formation rate (SFR), thus allowing to study the star formation efficiency and the local Schmidt law. The velocity dispersion from CO is used to study the critical surface density and the gravitational stability of the disk. The critical gas velocity dispersions needed to stabilize the gas against gravitational collapse in the differentially rotating disk of M51 using the Toomre criterion, vary with radius between 1.7 and 6.8 km/s. Observed radially averaged dispersions derived from the CO data vary between 28 km/s in the center and 8 km/s at radii of 7 to 9 kpc. They exceed the critical dispersions by factors Q_gas of 1 to 5. We speculate that the gravitational potential of stars leads to a critically stable disk.
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