Forming planetesimals by Gravitational instability. I. the role of the Richardson number in triggering the kelvin-Helmholtz instability

Abstract

Gravitational instability (GI) of a dust-rich layer at the midplane of a gaseous circumstellar disk is one proposed mechanism to form planetesimals, the building blocks of rocky planets and gas giant cores. Self-gravity competes against the Kelvin-Helmholtz instability (KHI): gradients in dust content drive a vertical shear which risks overturning the dusty subdisk and forestalling GI. To understand the conditions under which the disk can resist the KHI, we perform three-dimensional simulations of stratified subdisks in the limit that dust particles are small and aerodynamically well coupled to gas, thereby screening out the streaming instability and isolating the KHI. Each subdisk is assumed to have a vertical density profile given by a spatially constant Richardson number Ri. We vary Ri and the midplane dust-to-gas ratio μ0 and find that the critical Richardson number dividing KH-unstable from KH-stable flows is not unique; rather, Ricrit grows nearly linearly with μ0 for μ0 = 0.3-10. Plausibly, a linear dependence arises for μ0 < 1 because in this regime the radial Kepler shear replaces vertical buoyancy as the dominant stabilizing influence. Why this dependence should persist at μ0 > 1 is a new puzzle. The bulk (height-integrated) metallicity is uniquely determined by Ri and μ0. Only for disks of bulk solar metallicity is Ricrit ≈ 0.2, which is close to the classical value. Our empirical stability boundary is such that a dusty sublayer can gravitationally fragment and presumably spawn planetesimals if embedded within a solar metallicity gas disk ∼4× more massive than the minimum-mass solar nebula; or a minimum-mass disk having ∼3× solar metallicity; or some intermediate combination of these two possibilities. Gravitational instability seems possible without resorting to the streaming instability or to turbulent concentration of particles. © 2010. The American Astronomical Society. All rights reserved.