The TW Hya Rosetta Stone Project II: Spatially resolved emission of
  formaldehyde hints at low-temperature gas-phase formation
  Formaldehyde (H$_2$CO) is an important precursor to organics like methanol
(CH$_3$OH). It is important to understand the conditions that produce H$_2$CO
and prebiotic molecules during star and planet formation. H$_2$CO possesses
both gas-phase and solid-state formation pathways, involving either UV-produced
radical precursors or CO ice and cold ($\lesssim 20$ K) dust grains. To
understand which pathway dominates, gaseous H$_2$CO's ortho-to-para ratio (OPR)
has been used as a probe, with a value of 3 indicating "warm" conditions and
$<3$ linked to cold formation in the solid-state. We present spatially resolved
ALMA observations of multiple ortho- and para-H$_2$CO transitions in the TW Hya
protoplanetary disk to test H$_2$CO formation theories during planet formation.
We find disk-averaged rotational temperatures and column densities of $33\pm2$
K, ($1.1\pm0.1)\times10^{12}$ cm$^{-2}$ and $25\pm2$ K,
$(4.4\pm0.3)\times10^{11}$ cm$^{-2}$ for ortho- and para-H$_2$CO, respectively,
and an OPR of $2.49\pm0.23$. A radially resolved analysis shows that the
observed H$_2$CO emits mostly at rotational temperatures of 30-40 K,
corresponding to a layer with $z/R\ge0.25$. The OPR is consistent with 3 within
60 au, the extent of the pebble disk, and decreases beyond 60 au to
$2.0\pm0.5$. The latter corresponds to a spin temperature of 12 K, well below
the rotational temperature. The combination of relatively uniform emitting
conditions, a radial gradient in the OPR, and recent laboratory experiments and
theory on OPR ratios after sublimation, lead us to speculate that gas-phase
formation is responsible for the observed H$_2$CO across the TW Hya disk.
