Gravito-turbulent bi-fluid protoplanetary discs

I. An analytical perspective to stratification

¹ Leibniz-Institut für Astrophysik Potsdam (AIP), Potsdam, Germany
² Università degli Studi di Milano, Dipartimento di Fisica, via Celoria 16, 20133 Milano, Italy

^★ Corresponding author: srendon@aip.de

Received: 13 December 2024
Accepted: 17 March 2025

Abstract

Context. In Class 0 and I as well as in the outskirts of Class II circumstellar discs, the self-gravity of gas is expected to be significant, which certainly impacts the disc vertical hydrostatic equilibrium. Notably, the contribution of dust, whose measured mass is still uncertain, could also be a factor in this equilibrium.

Aims. We aim to formulate and solve, approximately, the equations governing the hydrostatic equilibrium of a self-gravitating disc composed of gas and dust. Particularly, we aim to provide a fully consistent treatment of turbulence and gravity that almost symmetrically affects gas and dust. From an observational perspective, we study the possibility of indirectly measuring disc masses through gas layering and dust settling measurements.

Methods. We used analytical methods to approximate the solution of the 1D Liouville equation with additional non-linearities governing the stratification of a self-gravitating protoplanetary disc. The analytical findings were verified through numerical treatment, and their consistency was validated with a physical interpretation.

Results. For a constant vertical stopping time profile, we discovered a nearly exact layering solution valid across all self-gravity regimes for gas and dust. From first principles, we defined the Toomre parameter of a bi-fluid system as the harmonic average of its constituents’ Toomre parameters. Based on these findings, we propose a method to estimate disc mass through gas or dust settling observations. We introduce a generic definition of the dust-to-gas scale height that is applicable to complex profiles. Additionally, we identified new exact solutions useful for benchmarking self-gravity solvers in numerical codes.

Conclusions. The hydrostatic equilibrium of a gas-dust mixture is governed by their Toomre parameters and their effective relative temperature. The equilibrium we found could possibly be used for measuring disc masses, thus enabling a more thorough understanding of disc settling and gravitational collapse, and it will also improve the computation of self-gravity in thin disc simulations.