Multi-wavelength observations of protoplanetary discs as a proxy for the gas disc mass

Recent observations of protoplanetary discs reveal disc substructures potentially caused by embedded planets. We investigate how the gas surface density in discs changes the observed morphology in scattered light and dust continuum emission. Assuming that disc substructures are due to embedded protoplanets, we combine hydrodynamical modelling with radiative transfer simulations of dusty protoplanetary discs hosting planets. The response of different dust species to the gravitational perturbation induced by a planet depends on the drag stopping time - a function of the generally unknown local gas density. Small dust grains, being stuck to the gas, show spirals. Larger grains decouple, showing progressively more axisymmetric (ring-like) substructure as decoupling increases with grain size or with the inverse of the gas disc mass. We show that simultaneous modelling of scattered light and dust continuum emission is able to constrain the Stokes number, ${\rm St}$. Hence, if the dust properties are known, this constrains the local gas surface density, $Σ_{\rm gas}$, at the location of the structure, and hence the total gas mass. In particular, we found that observing ring-like structures in mm-emitting grains requires ${\rm St} \gtrsim 0.4$ and therefore $Σ_{\rm gas} \lesssim 0.4\,\textrm{g/cm}^{2}$. We apply this idea to observed protoplanetary discs showing substructures both in scattered light and in the dust continuum.