Cold water emission cannot be used to infer depletion of bulk elemental oxygen [O/H] in disks

We re-examine the constraints provided by Herschel Space Observatory data regarding cold water emission from protoplanetary disks. Previous disk models that were used to interpret observed water emission concluded that oxygen (O/H) is depleted by at least 2 orders of magnitude if a standard, interstellar gas/dust mass ratio is assumed in the disk. In this work, we use model results from a recent disk parameter survey and show that most of the \textit{Herschel} constraints obtained for cold water (i.e. for transitions with an upper energy level $E_\mathrm{up}<200$ K, where the bulk of the disk water lies) can be explained with disk models adopting ISM-like oxygen elemental abundance (i.e. O/H=$3.2\times10^{-4}$) and the canonical gas/dust mass ratio of 100. We show that cold water vapor is mainly formed by photodesorption of water ice at the interface between the molecular layer and the midplane, and that its emission is relatively independent of the main disk properties like the disk gas mass and gas/dust mass ratio. We find that the abundance of water vapor in the outer disk is set by photoprocesses and depends on the (constant) vertical column density of water ice needed to attenuate the FUV photon flux, resulting in roughly constant emission for the parameters (gas mass, dust mass, disk radius) varied in our survey. Importantly, water line emission is found to be optically thick and hence sensitive to temperature more than abundance, possibly driving previous inferences of large scale oxygen depletion.