A complete (12)CO 2-1 map of M 51 with HERA - I. Radial averages of CO, HI, and radio continuum

Context. 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. Aims. In this first paper, we investigate the correlation of H(2), HI, and total gas surface densities with the star forming activity, derived from the radio continuum ( RC), along radial averages out to radii of 12 kpc. Methods. We have created a complete map of M51 in (12)CO 2-1 at a resolution of 450 pc using HERA at the IRAM-30 m telescope. These data are combined with maps of HI and the radio-continuum at 20 cm 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 Sigma(SFR) proportional to Sigma(n)(gas). The velocity dispersion from CO is used to study the critical surface density and the gravitational stability of the disk. Results. The total mass of molecular material derived from the integrated (12)CO 2-1 intensities is 2 x 10(9) M(circle dot). The 3 sigma detection limit corresponds to a mass of 1.7 x 10(5) M(circle dot). The global star formation rate is 2.56 M(circle dot) yr(-1) and the global gas depletion time is 0.8 Gyr. HI and RC emission are found to peak on the concave, downstream side of the outer south-western CO arm, outside the corotation radius. The total gas surface density Sigma(gas) drops by a factor of similar to 20 from 70 M(circle dot) pc(-2) at the center to 3 M(circle dot) pc(-2) in the outskirts at radii of 12 kpc. The fraction of atomic gas gradually increases with radius. The ratio of HI over H(2) surface densities, Sigma(HI)/Sigma(H2), increases from similar to 0.1 near the center to similar to 20 in the outskirts without following a simple power-law. Sigma(HI) starts to exceed Sigma(H2) at a radius of similar to 4 kpc. The star formation rate per unit area drops from similar to 400 M(circle dot) pc(-2) Gyr(-1) in the starburst center to similar to 2 M(circle dot) pc(-2) Gyr(-1) in the outskirts. The gas depletion time varies between 0.1 Gyr in the center and 1 Gyr in the outskirts, and is shorter than in other non-interacting normal galaxies. Neither the HI surface densities nor the H(2) surface densities show a simple power-law dependence on the star formation rate per unit area. In contrast, Sigma(gas) and Sigma(SFR) are well characterized by a local Schmidt law with a power-law index of n = 1.4 +/- 0.6. The index equals the global Schmidt law derived from disk-averaged values of Sigma(gas) and Sigma(SFR) of large samples of normal and starburst galaxies. 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(-1). Observed radially averaged dispersions derived from the CO data vary between 28 km s(-1) in the center and similar to 8 km s(-1) 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.