Relating dust, gas, and the rate of star formation in M 31

Aims. We investigate the relationships between dust and gas, and study the star formation law in M 31. Methods. We have derived distributions of dust temperature and dust opacity across M31 at 45 '' resolution using the Spitzer data. With the opacity map and a standard dust model we de-reddened the Ha emission yielding the first H alpha map of M 31 corrected for extinction. We compared the emissions from dust, H alpha, HI, and H-2 by means of radial distributions, pixel-to-pixel correlations, and wavelet cross-correlations. We calculated the star formation rate and star formation efficiency from the de-reddened Ha emission. Results. The dust temperature steeply decreases from 30K near the center to 15K at large radii. The mean dust optical depth at the Ha wavelength along the line of sight is about 0.7. The radial decrease in the dust-to-gas ratio is similar to that of the oxygen abundance. Extinction is nearly linearly correlated with the total gas surface density within limited radial intervals. On scales <2 kpc, cold dust emission is best correlated with that of neutral gas, and warm dust emission with that of ionized gas. The H alpha emission is slightly better correlated with emission at 70 mu m than at 24 mu m. The star formation rate in M 31 is low. In the area 6 kpc < R < 17 kpc, the total SFR is similar or equal to 0.3 M-circle dot yr(-1). A linear relationship exists between surface densities of SFR and H-2. The Kennicutt-Schmidt law between SFR and total gas has a power-law index of 1.30 +/- 0.05 in the radial range of R = 7-11 kpc increasing by about 0.3 for R = 11-13 kpc. Conclusions. The better 70 mu m-H alpha than 24 mu m-H alpha correlation plus an excess in the 24 mu m/70 mu m intensity ratio indicates that other sources than dust grains, e. g. those of stellar origin, contribute to the 24 mu m emission. The lack of H-2 in the central region could be related to the lack of HI and the low opacity/high temperature of the dust. Since neither SFR nor SFE is well correlated with the surface density of H-2 or total gas, other factors than gas density must play an important role in the formation of massive stars in M 31. The molecular depletion time scale of 1.1 Gyr indicates that M 31 is about three times less efficient in forming young massive stars than M 33.