# Particle Stirring in Turbulent Gas Disks: Including Orbital Oscillations

**Authors:** Andrew N. Youdin, Yoram Lithwick  
**arXiv:** [0707.2975](https://arxiv.org/abs/0707.2975)  
**Date:** 2007-07-19  
**Summary page:** https://www.summarizepaper.com/en/paper/0707.2975v1/

## Key points

- Andrew N. Youdin and Yoram Lithwick's paper describes the diffusion and random velocities of solid particles in protoplanetary disks due to stochastic forcing by turbulent gas.
- The authors include the orbital dynamics of Keplerian disks, both in-plane epicycles and vertical oscillations, and obtain a new result for the diffusion of solids.
- They find that the Schmidt number (ratio of gas to particle diffusivity) is Sc = 1 + (Omega t_stop)^2, where Omega is the orbital frequency and t_stop is the particle stopping time.
- This differs from the standard result, Sc = 1 + t_stop/t_eddy (in terms of the eddy turnover time, t_eddy), which is shown to be incorrect.
- The authors predict that particles larger than approximately 10 cm in protoplanetary disks will suffer less radial diffusion and settle closer to the midplane.
- As a result, such a layer of boulders would be more prone to gravitational collapse.
- The authors' predictions of RMS speeds, vertical scale height, and diffusion coefficients will help interpret numerical simulations.
- The authors 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.
- They also find that the particle layer becomes thinner for t_eddy > 1/Omega with the strength of turbulent diffusion held fixed.
- To obtain their results, Youdin and Lithwick 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.
- They include simple physical arguments for scaling their results.
- Overall, this study provides important insights into how turbulence affects solid particles in protoplanetary disks and how these particles may evolve over time.

## Summary

In their paper "Particle Stirring in Turbulent Gas Disks: Including Orbital Oscillations," Andrew N. Youdin and Yoram Lithwick describe the diffusion and random velocities of solid particles in protoplanetary disks due to stochastic forcing by turbulent gas. The authors include the orbital dynamics of Keplerian disks, both in-plane epicycles and vertical oscillations, and obtain a new result for the diffusion of solids. They find that the Schmidt number (ratio of gas to particle diffusivity) is Sc = 1 + (Omega t_stop)^2, where Omega is the orbital frequency and t_stop is the particle stopping time. This differs from the standard result, Sc = 1 + t_stop/t_eddy (in terms of the eddy turnover time, t_eddy), which is shown to be incorrect. The main difference between these two results is that Sc rises quadratically with stopping time rather than linearly. The authors predict that particles larger than approximately 10 cm in protoplanetary disks will suffer less radial diffusion and settle closer to the midplane. As a result, such a layer of boulders would be more prone to gravitational collapse. The authors' predictions of RMS speeds, vertical scale height, and diffusion coefficients will help interpret numerical simulations. The authors 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. They also find that the particle layer becomes thinner for t_eddy > 1/Omega with the strength of turbulent diffusion held fixed. To obtain their results, Youdin and Lithwick 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. They include simple physical arguments for scaling their results. Overall, this study provides important insights into how turbulence affects solid particles in protoplanetary disks and how these particles may evolve over time.