An analytic solution to measure the gas size in protoplanetary discs in the viscous self-similar scenario

TL;DR

This paper derives an analytical model to estimate the gas disc radius in viscously evolving protoplanetary discs using CO tracers, accounting for photo-dissociation and freeze-out effects, enabling efficient large-scale disc size predictions.

Contribution

It provides the first analytical solution for disc radius evolution in viscous protoplanetary discs based on CO observations, incorporating photo-dissociation and freeze-out effects.

Findings

CO abundance is reduced by about two orders of magnitude at photo-dissociation radius.

The analytical model can predict disc sizes efficiently without complex radiative transfer simulations.

Both photo-dissociation and freeze-out significantly influence observed disc sizes.

Abstract

In order to understand which mechanism is responsible for accretion in protoplanetary discs, a robust knowledge of the observed disc radius using gas tracers such as $^{12}$CO and other CO isotopologues is pivotal. Indeed, the two main theories proposed, viscous accretion and wind-driven accretion, predict different time evolution for the disc radii. In this Letter, we present an analytical solution for the evolution of the disc radii in viscously evolving protoplanetary discs using $^{12}$CO as a tracer, under the assumption that the $^{12}$CO radius is the radius where the surface density of the disc is equal to the threshold for CO photo-dissociation. We discuss the properties of the solution and the limits of its applicability as a simple numerical prescription to evaluate the observed disc radii of populations of discs. Our results suggest that, in addition to photo-dissociation, also freeze out plays an important role in setting the disc size. We find an effective reduction of the CO abundance by about two orders of magnitude at the location of CO photo-dissociation, which however should not be interpreted as the bulk abundance of CO in the disc. The use of our analytical solution will allow to compute disc sizes for large quantities of models without using expensive computational resources such as radiative transfer calculations.