Gravito-turbulent bi-fluid protoplanetary discs: 1. An analytical perspective to stratification

TL;DR

This work presents an analytical framework for the vertical stratification of gravito-turbulent protoplanetary discs containing gas and dust. By solving coupled vertical hydrostatic equations with self-gravity under a slab approximation and incorporating turbulence-induced pressure and diffusion, the authors derive exact and approximate bi-fluid solutions across Keplerian to strongly self-gravitating regimes. A key result is that the bi-fluid Toomre parameter naturally appears as a harmonic average, Q_bi-fluid^{3D} = (1/Q_g^{3D} + 1/Q_d^{3D})^{-1}, and that a robust, general definition of dust-to-gas scale height Hd/Hg^{sg} can be constructed for complex profiles. They also provide practical observational pathways to estimate disc masses from stratification, and deliver new exact solutions to benchmark self-gravity solvers, with implications for 2D SG simulations and the onset of dust-driven gravitational instabilities.

Abstract

Context. In Class 0/I and the outskirts of Class II circumstellar discs, the self-gravity of gas significantly affects the disc's vertical hydrostatic equilibrium. The contribution of dust, whose measured mass is still uncertain, could influence 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, affecting almost symmetrically gas and dust. Observationally, 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 findings were verified numerically and validated through 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, applicable to complex profiles. We also identified new exact solutions for benchmarking self-gravity solvers in numerical codes. Conclusions. The hydrostatic equilibrium of a gas/dust mixture is governed by their Toomre parameters and effective relative temperature. This equilibrium could be used for measuring disc masses, improving our understanding of disc settling and gravitational collapse, and enhancing the computation of self-gravity in thin disc simulations.

Figures (7)

• Figure 1: Normalised gas density profile for various Toomre parameters ($Q_g$) comparing the numerical solution of Eq. \ref{['Eq: reduced eq. SG']} with the Spitzer solution.
• Figure 2: Vertical profile of gas for different Toomre's parameters when the dust component is disregarded. Specifically, we compare the numerical solution of Eq. \ref{['Eq: reduced eq. SG']} with the BL model and our model Eq. \ref{['Eq: stratification single fluid']}.
• Figure 3: Vertical profile of dust density for different gas Toomre's parameters, $Q_g$, and relative dust temperature, $\xi$. Specifically, we compare the numerical solution with our model Eq. \ref{['Eq: No const. stopping time, Bertin, Lodato stratification']}. Additionally, we added the limiting cases of Fromang & Nelson profile, valid in absence of SG, and the exact solution that we derived (Eq. \ref{['Eq: No const. stopping time, exact high gas mass']}), valid for massive gas discs.
• Figure 4: Comparison between the numerical solution of Eq. \ref{['Eq: constant stopping time, SG of gas and dust']} and our approximated solution (Eq. \ref{['Eq: stratification gas and dust']}) for different Toomre parameters of gas ($Q_g^{3D}$), dust ($Q_d^{3D}$) and relative effective gas-to-dust temperature ($\Tilde{\xi}$). Left column: Vertical profile of gas density. Right column: L2 norm error map in $\log_{10}$ scale.
• Figure 5: Self-gravitating differential rotation in a marginally gas-stable disc ($Q_g^{3D}=0.5$) for various dust Toomre parameters ($Q_d^{3D}$) and relative temperatures ($\Tilde{\xi}$).