HERSCHEL EXPLOITATION OF LOCAL GALAXY ANDROMEDA (HELGA). III. THE STAR FORMATION LAW IN M31**Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.

We present a detailed study of how the star formation rate (SFR) relates to the interstellar medium (ISM) of M31 at ∼140 pc scales. The SFR is calculated using the far-ultraviolet and 24 μm emission, corrected for the old stellar population in M31. We find a global value for the SFR of and compare this with the SFR found using the total far-infrared luminosity. There is general agreement in regions where young stars dominate the dust heating. Atomic hydrogen (H i) and molecular gas (traced by carbon monoxide, CO) or the dust mass is used to trace the total gas in the ISM. We show that the global surface densities of SFR and gas mass place M31 among a set of low-SFR galaxies in the plot of Kennicutt. The relationship between SFR and gas surface density is tested in six radial annuli across M31, assuming a power law relationship with index, N. The star formation (SF) law using total gas traced by H i and CO gives a global index of N = 2.03 ± 0.04, with a significant variation with radius; the highest values are observed in the 10 kpc ring. We suggest that this slope is due to H i turning molecular at ΣGas ∼ 10 M☉ pc−2. When looking at H2 regions, we measure a higher mean SFR suggesting a better spatial correlation between H2 and SF. We find N ∼ 0.6 with consistent results throughout the disk—this is at the low end of values found in previous work and argues against a superlinear SF law on small scales.