Probing the protoplanetary disk gas surface density distribution with 13CO emission

Context. How protoplanetary disks evolve is still an unsolved problem where different processes may be involved. Depending on the process, the disk gas surface density distribution Σ_gas may be very different and this could have diverse implications for planet formation. Together with the total disk mass, it is key to constrain Σ_gas as function of disk radius R from observational measurements.
Aims: In this work we investigate whether spatially resolved observations of rarer CO isotopologues, such as ¹³CO, may be good tracers of the gas surface density distribution in disks.
Methods: Physical-chemical disk models with different input Σ_gas(R) were run, taking into account CO freeze-out and isotope-selective photodissociation. The input disk surface density profiles were compared with the simulated ¹³CO intensity radial profiles to check whether and where the two follow each other.
Results: For each combination of disk parameters, there is always an intermediate region in the disk where the slope of the ¹³CO radial emission profile and Σ_gas(R) coincide. In the inner part of the disk, the line radial profile underestimates Σ_gas, as ¹³CO emission becomes optically thick. The same happens at large radii where the column densities become too low and ¹³CO is not able to efficiently self-shield. Moreover, the disk becomes too cold and a considerable fraction of ¹³CO is frozen out, thus it does not contribute to the line emission. If the gas surface density profile is a simple power-law of the radius, the input power-law index can be retrieved within a 20% uncertainty if one choses the proper radial range. If instead Σ_gas(R) follows the self-similar solution for a viscously evolving disk, retrieving the input power-law index becomes challenging, in particular for small disks. Nevertheless, we find that the power-law index γ can be in any case reliably fitted at a given line intensity contour around 6 K km s^-1, and this produces a practical method to constrain the slope of Σ_gas(R). Application of such a method is shown in the case study of the TW Hya disk.
Conclusions: Spatially resolved ¹³CO line radial profiles are promising to probe the disk surface density distribution, as they directly trace Σ_gas(R) profile at radii well resolvable by ALMA. There, chemical processes like freeze-out and isotope-selective photodissociation do not affect the emission, and, assuming that the volatile carbon does not change with radius, no chemical model is needed when interpreting the observations.