Particle Stirring in Turbulent Gas Disks: Including Orbital Oscillations

We describe the diffusion and random velocities of solid particles due to stochastic forcing by turbulent gas. We include the orbital dynamics of Keplerian disks, both in-plane epicycles and vertical oscillations. We obtain a new result for the diffusion of solids. The Schmidt number (ratio of gas to particle diffusivity) is Sc = 1 + (Omega t_stop)^2, in terms of the particle stopping time, t_stop, and the orbital frequency, Omega. The standard result, Sc = 1 + t_stop/t_eddy, in terms of the eddy turnover time, t_eddy, is shown to be incorrect. The main difference is that Sc rises quadratically, not linearly, with stopping time. Consequently, particles larger than ~ 10 cm in protoplanetary disks will suffer less radial diffusion and will settle closer to the midplane. Such a layer of boulders would be more prone to gravitational collapse. Our predictions of RMS speeds, vertical scale height and diffusion coefficients will help interpret numerical simulations. We confirm previous results for the vertical stirring of particles (scale heights and random velocities), and add a correction for arbitrary ratios of eddy to orbital times. The particle layer becomes thinner for t_eddy > 1/Omega, with the strength of turbulent diffusion held fixed. We use two analytic techniques -- the Hinze-Tchen formalism and the Fokker-Planck equation with velocity diffusion -- with identical results when the regimes of validity overlap. We include simple physical arguments for the scaling of our results.