Vertical CO surfaces as a probe for protoplanetary disk mass and carbon depletion

As the sample of mid-inclination disks with measured CO emission surfaces grows, a fundamental unanswered question is how these vertical profiles connect to their host properties. This project aims to relate the vertical extent of protoplanetary disks as traced by $^{12}$CO $2-1$ to key stellar and physical parameters. In order to produce a result that is applicable towards an observational analysis, we benchmark our results with ALMA observations of CO emission from nineteen disks. We produce a grid of disk models using the physical-chemical code DALI, for a template T Tauri and Herbig star. Our models use an iterative solver to calculate the hydrostatic equilibrium equations and determine a physically-motivated density structure. Key stellar and disk parameters such as stellar luminosity and temperature, total disk mass, carbon abundance and critical radius are varied to determine their effect on the CO emitting surface. Each vertical profile is fitted by an exponentially tapered power-law and characterized by the $z/r$ value that represents the structure inwards of 80% of the tapering radius. The CO emission surface location is primarily determined by the disk mass ($M_d$) and the level of volatile carbon depletion. T Tauri and Herbig systems show different vertical profiles, with disks around T Tauri stars being more vertically extended. We derive a $z/r$-$M_d$ relationship, which for each stellar type has a degeneracy with the volatile carbon abundance. In order to reconcile total disk mass estimates from the characteristic $z/r$ and the values obtained based on dust continuum analysis, a volatile carbon depletion of 10-100 (with respect to the ISM) is needed for the majority of our sources. Our carbon depletion values are in agreement with previous literature estimates, highlighting the potential of this method to rapidly calculate key disk parameters.