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We derive distributions of dust temperature and dust opacity across M31 at45" resolution using the Spitzer data. With the opacity map and a standard dustmodel we de-redden the Ha emission yielding the first de-reddened Ha map ofM31. We compare the emissions from dust, Ha, HI and H2 by means of radialdistributions, pixel-to-pixel correlations and wavelet cross-correlations. Thedust temperature steeply decreases from 30K near the center to 15K at largeradii. The mean dust optical depth at the Ha wavelength along the line of sightis about 0.7. The radial decrease of the dust-to-gas ratio is similar to thatof the oxygen abundance. On scales<2kpc, cold dust emission is best correlatedwith that of neutral gas and warm dust emission with that of ionized gas. Haemission is slightly better correlated with emission at 70um than at 24um. Inthe area 6kpc<R< 17kpc, the total SFR is ~0.3Msun/yr. The Kennicutt-Schmidt lawbetween SFR and total gas has a power-law index of 1.30+-0.05 in the radialrange of R=7-11kpc increasing by about 0.3 for R=11-13kpc. The lack of H2 inthe central region could be related to the lack of HI and the low opacity/hightemperature of the dust. Since neither SFR nor SFE is well correlated with thesurface density of H2 or total gas, other factors than gas density must play animportant role in the formation of massive stars in M31. The moleculardepletion time scale of 1.1 Gyr indicates that M31 is about three times lessefficient in forming young massive stars than M33.

Relating dust, gas and the rate of star formation in M31