  Spitzer Space Telescope and Herschel Space Observatory imaging of M31 is
used, with a physical dust model, to construct maps of dust surface density,
dust-to-gas ratio, starlight heating intensity, and PAH abundance, out to
R=25kpc. The global dust mass is M_d=5.4x10^7Msol, the global dust/H mass ratio
is M_d/M_H=0.0081, and the global PAH abundance is <q_PAH>=0.039. The dust
surface density has an inner ring at R=5.6kpc, a maximum at R=11.2kpc, and an
outer ring at R=15.1kpc. The dust/gas ratio varies from M_d/M_H=0.026 at the
center to ~0.0027 at R=25kpc. From the dust/gas ratio, we estimate the ISM
metallicity to vary by a factor ~10, from Z/Zsol=3 at R=0 to ~0.3 at R=25kpc.
The dust heating rate parameter <U> peaks at the center, with <U> approx 35,
declining to <U> approx 0.25 at R=20kpc. Within the central kpc, the starlight
heating intensity inferred from the dust modeling is close to what is estimated
from the stars in the bulge. The PAH abundance reaches a peak q_PAH=0.045 at
R=11.2kpc. When allowance is made for the different spectrum of the bulge
stars, q_PAH for the dust in the central kpc is similar to the overall value of
q_PAH in the disk. The silicate-graphite-PAH dust model used here is generally
able to reproduce the observed dust spectral energy distribution across M31,
but overpredicts 500um emission at R=2-6kpc, suggesting that at R=2-6kpc, the
dust opacity varies more steeply with frequency (with beta approx 2.3 between
200 and 600um) than in the model
