The Gravitational instability of solids assisted by gas drag: slowing by turbulent mass diffusivity

TL;DR

This paper investigates how turbulent mass diffusivity impacts the gravitational instability of solid particles in protoplanetary disks, revealing that turbulence significantly suppresses instability unless turbulence is extremely weak and particle density is high.

Contribution

It introduces the effect of turbulent mass diffusivity into the analysis of gravitational instability, showing its critical role in reducing instability growth rates and maximum turbulence levels.

Findings

Turbulent mass diffusivity significantly reduces instability growth rates.

Instability requires extremely weak turbulence and high solid-to-gas density ratios.

Radial drag opposes while azimuthal drag promotes instability.

Abstract

The Goldreich and Ward (1973) (axisymmetric) gravitational instability of a razor thin particle layer occurs when the Toomre parameter $Q_T \equiv c_p \Omega_0 / \pi G \Sigma_p < 1$ ($c_p$ being the particle dispersion velocity). Ward(1976,2000) extended this analysis by adding the effect of gas drag upon particles and found that even when $Q_T > 1$, sufficiently long waves were always unstable. Youdin (2005a,b) carried out a detailed analysis and showed that the instability allows chondrule-sized ($\sim 1 $ mm) particles to undergo radial clumping with reasonable growth times even in the presence of a moderate amount of turbulent stirring. The analysis of Youdin includes the role of turbulence in setting the thickness of the dust layer and in creating a turbulent particle pressure in the momentum equation. However, he ignores the effect of turbulent mass diffusivity on the disturbance wave. Here we show that including this effect reduces the growth-rate significantly, by an amount that depends on the level of turbulence, and reduces the maximum intensity of turbulence the instability can withstand by 1 to 3 orders of magnitude. The instability is viable only when turbulence is extremely weak and the solid to gas surface density of the particle layer is considerably enhanced over minimum-mass-nebula values. A simple mechanistic explanation of the instability shows how the azimuthal component of drag promotes instability while the radial component hinders it. A gravito-diffusive overstability is also possible but never realized in the nebula models.