FORMING PLANETESIMALS BY GRAVITATIONAL INSTABILITY. II. HOW DUST SETTLES TO ITS MARGINALLY STABLE STATE

Dust at the midplane of a circumstellar disk can become gravitationally unstable and fragment into planetesimals if the local dust-to-gas ratio μ0 ≡ ρd/ρg is sufficiently high. We simulate how dust settles in passive disks and ask how high μ0 can become. We implement a hybrid scheme that alternates between a one-dimensional code to settle dust and a three-dimensional shearing box code to test for dynamical stability. This scheme allows us to explore the behavior of small particles having short but non-zero stopping times in gas: 0 < tstop≪ the orbital period. The streaming instability is thereby filtered out. Dust settles until Kelvin-Helmholtz-type instabilities at the top and bottom faces of the dust layer threaten to overturn the entire layer. In this state of marginal stability, μ0 = 2.9 for a disk whose bulk (height-integrated) metallicity Σd/Σg is solar—thus μ0 increases by more than two orders of magnitude from its well-mixed initial value of μ0,init = Σd/Σg = 0.015. For a disk whose bulk metallicity is 4× solar (μ0,init = Σd/Σg = 0.06), the marginally stable state has μ0 = 26.4. These maximum values of μ0, which depend on the background radial pressure gradient, are so large that gravitational instability of small particles is viable in disks whose bulk metallicities are just a few (≲4) times solar. Our result supports earlier studies that assumed that dust settles until the Richardson number Ri is spatially constant. Our simulations are free of this assumption but provide evidence for it within the boundaries of the dust layer, with the proviso that Ri increases with Σd/Σg in the same way that we found in Paper I. Because increasing the dust content decreases the vertical shear and increases stability, the midplane μ0 increases with Σd/Σg in a faster than linear way, so fast that modest enhancements in Σd/Σg can spawn planetesimals directly from small particles.