An ALMA Survey of H$_2$CO in Protoplanetary Disks
  H$_2$CO is one of the most abundant organic molecules in protoplanetary disks
and can serve as a precursor to more complex organic chemistry. We present an
ALMA survey of H$_2$CO towards 15 disks covering a range of stellar spectral
types, stellar ages, and dust continuum morphologies. H$_2$CO is detected
towards 13 disks and tentatively detected towards a 14th. We find both
centrally-peaked and centrally-depressed emission morphologies, and half of the
disks show ring-like structures at or beyond expected CO snowline locations.
Together these morphologies suggest that H$_2$CO in disks is commonly produced
through both gas-phase and CO-ice-regulated grain-surface chemistry. We extract
disk-averaged and azimuthally-averaged H$_2$CO excitation temperatures and
column densities for four disks with multiple H$_2$CO line detections. The
temperatures are between 20-50K, with the exception of colder temperatures in
the DM Tau disk. These temperatures suggest that H$_2$CO emission in disks is
generally emerging from the warm molecular layer, with some contributions from
the colder midplane. Applying the same H$_2$CO excitation temperatures to all
disks in the survey, we find that H$_2$CO column densities span almost three
orders of magnitude ($\sim 5 \times 10^{11} - 5 \times 10^{14}
\mathrm{cm}^{-2}$). The column densities appear uncorrelated with disk size and
stellar age, but Herbig Ae disks may have less H$_2$CO compared to T Tauri
disks, possibly because of less CO freeze-out. More H$_2$CO observations
towards Herbig Ae disks are needed to confirm this tentative trend, and to
better constrain under which disk conditions H$_2$CO and other oxygen-bearing
organics efficiently form during planet formation.
