Mid-plane sedimentation of large solid bodies in turbulent protoplanetary discs

We study the vertical settling of solid bodies in a turbulent protoplanetary disc. We consider the situation when the coupling to the gas is weak or equivalently when the particle stopping time tau(st) due to friction with the gas is long compared to the orbital time-scale Omega(-1). An analytical model, which takes into account the stochastic nature of the sedimentation process using a Folcker-Planck equation for the particle distribution function in phase space, is used to obtain the vertical scaleheight of the solid layer as a function of the vertical component of the turbulent gas velocity correlation function and the particle stopping time. This is found to be of the same form as the relation obtained for strongly coupled particles in previous work. We compare the predictions of this model with results obtained from local shearing box magnetohydrodynamic simulations of solid particles embedded in a vertically stratified disc in which there is turbulence driven by the magnetorotational instability. We find that the ratio of the dust disc thickness to the gas disc thickness satisfies H-d/H = 0.08 (Omega tau(st))(-1/2), which is in very good agreement with the analytical model. By discussing the conditions for gravitational instability in the outer regions of protoplanetary discs in which there is a similar level of turbulence, we find that bodies in the size range 50-600 m can aggregate to form Kuiper belt-like objects with characteristic radii ranging from tens to hundreds of kilometres.