Galactocentric variation of the gas-to-dust ratio and its relation with
  metallicity
  (Abridged) Context: The assumption of a gas-to-dust mass ratio (\gamma) is a
common approach to estimate the basic properties of molecular clouds, such as
total mass and column density of molecular hydrogen, from (sub)mm continuum
observations of the dust. In the Milky Way a single value is used at all
galactocentric radii, independently of the observed metallicity gradients. Both
models and extragalactic observations suggest that this quantity increases for
decreasing metallicity Z, typical of the outer regions in disks, where fewer
heavy elements are available to form dust grains.
  Aims: We aim to investigate the variation of the gas-to-dust ratio as a
function of galactocentric radius and metallicity, to allow a more accurate
characterisation of the quantity of molecular gas across the galactic disk, as
derived from observations of the dust.
  Methods: Observations of the optically thin C\$^{18}\$O (2-1) transition were
obtained with the APEX telescope for a sample of 23 massive and dense
star-forming regions in the far outer Galaxy (galactocentric distance greater
than 14 kpc). From the modelling of this line and of the spectral energy
distribution of the selected clumps we computed the gas-to-dust ratio and
compared it to that of well-studied sources from the ATLASGAL TOP100 sample in
the inner galactic disk.
  Results: The gradient in gas-to-dust ratio is found to be 0.087 dex/kpc (or
equivalently \gamma\ proportional to Z\$^{-1.4}\$). The dust-to-metal ratio,
decreases with galactocentric radius, which is the most common situation also
for external late-type galaxies. This suggests that grain growth dominates over
destruction. The predicted gas-to-dust ratio is in excellent agreement with the
estimates in Magellanic clouds, for the appropriate value of Z.
