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

Context. As the sample of mid-inclination disks with measured CO emission surfaces continues to grow, a fundamental question that has gone unanswered is how these vertical profiles are connected to their host properties.Aims. This work is aimed at relating the vertical extent of protoplanetary disks as traced by 12CO 2-1 to key stellar and physical parameters. We have benchmarked our results with ALMA observations of CO emission from nineteen disks to produce a result that is applicable to observational analyses.Methods. We produced 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 the stellar luminosity and temperature, total disk mass and carbon abundance 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.Results. The CO emission surface location is primarily determined by the disk mass (Md) and the level of volatile carbon depletion. T Tauri and Herbig systems show different vertical profiles, with disks around T Tauri stars shown to be be more vertically extended. We derived a z/r-Md relationship (which has a degeneracy) for each stellar type, with the volatile carbon abundance. To reconcile the total disk mass estimates from the characteristic z/r and the values obtained from the dust continuum analysis, we find that a volatile carbon depletion of 10-100 (with respect to the interstellar medium) 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.