Particle Stirring in Turbulent Gas Disks: Including Orbital Oscillations

By A. Youdin, Y. Lithwick. 2007

449 citations recorded by the source corpus.

We describe the diusion 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 diusion of solids. The Schmidt number (ratio of gas to particle diusivity) is Sc 1 + (t stop) 2 , in terms of the particle stopping time tstop and the orbital frequency . The standard result, Sc = 1 +tstop=teddy, in terms of the eddy turnover time, teddy, is shown to be incorrect. The main dierence is that Sc rises quadratically, not linearly, with stopping time. Consequently, particles larger than 10 cm in protoplanetary disks will suer less radial diusion 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 diusion coecients will help interpret numerical simulations. We conrm 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 teddy > 1= with the strength of turbulent diusion held xed. We use two analytic techniques { the Hinze-Tchen formalism and the FokkerPlanck equation with velocity diusion { with identical results when the regimes of validity overlap. We include simple physical arguments for the scaling of our results. Subject headings: Disks, Planetary Formation, Solar Nebula

Subfields: Astronomy and Astrophysics, Computational Mechanics, Fluid Flow and Transfer Processes

View Paper: https://www.semanticscholar.org/paper/313b54ef6b64bf2c88606a7b4b01936166adc4b9

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Citations in the corpus

Rapid growth of gas-giant cores by pebble accretion (2012, background): “…particle scale height Hp is a balance between A32, page 3 of 13 midplane-directed gravity and turbulent diffusion parametrised by the coefficient δt (Youdin & Lithwick 2007), Hp H ≈ √ δt τf ≈ 0.01, (12) where we will make the approximation that this holds for the pebble size range we consider (τf…”
Dynamics of Protoplanetary Disks (2010, background): “Larger particles, that are imperfectly coupled aerodynamically to the turbulent motions within the gas, are expected to diffuse less readily than gaseous contaminants (Cuzzi, Dobrovolskis & Champney 1993; Youdin & Lithwick 2007).”
ZONAL FLOWS AND LONG-LIVED AXISYMMETRIC PRESSURE BUMPS IN MAGNETOROTATIONAL TURBULENCE (2008, background): “Youdin & Lithwick (2007) consider a mathematically similar problem: the stirring of particle velocities by aerodynamic coupling to turbulent gas.”
Dust Evolution and the Formation of Planetesimals (2016, background): “The ratio of the dust diffusivity to the gas diffusivity is commonly called the Schmidt number2 Sc which was shown to be (Youdin & Lithwick 2007) Sc = Dg Dd ' 1 + St2, (13) and it is commonly assumed that the gas diffusivity Dg equals the gas viscosity ν (but see Johansen & Klahr 2005; Fromang &…”
Pebble-isolation mass: Scaling law and implications for the formation of super-Earths and gas giants (2018, background): “Pinilla et al. (2016) considered turbulent mixing of dust particles, where the dust diffusivity follows the prescriptions by Youdin & Lithwick (2007), which depend on the Stokes number and the gas diffusivity (assumed to be equal to the disc viscosity).”
The Disk Substructures at High Angular Resolution Project (DSHARP). VII. The Planet–Disk Interactions Interpretation (2018, methodology): “The diffusion coefficient is related to the α parameter as in Youdin & Lithwick (2007) through the so-called Schmidt number Sc.”
Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits (2010, background): “…decouple and their scale height is decreasing with grain size, Hd Hg = √ αt St , for St > αt (25) (see e.g., Dubrulle et al. 1995; Schräpler & Henning 2004; Youdin & Lithwick 2007) where the Stokes number St is a dimensionless quantity which describes the dynamic properties of a suspended particle.”
Chemical enrichment of giant planets and discs due to pebble drift (2017, methodology): “For gas tracers we take Scφ = 1, while for dust we use φi = i and Sci = (1 + St2i ) 2 1 + 4St2i (11) (Youdin & Lithwick 2007).”
A simple model for the evolution of the dust population in protoplanetary disks (2012, background): “…for turbulent motion (see Ormel & Cuzzi 2007) Δu = √ 3αt St cs. (11) Assuming St < 1, the Epstein drag law, and a scale height of the dust distribution of (Youdin & Lithwick 2007) Hd = Hg √ αt St , (12) the growth time scale can be written as τgrow 1 Ωk , (13) where = Σd/Σg is the vertically…”
The effect of gas drag on the growth of protoplanets. Analytical expressions for the accretion of small bodies in laminar disks (2010, background): “For tL = Ω−1 the definitions agree (Youdin & Lithwick 2007). introduces two parameters: the velocity of the gas flow ζw and the coupling parameter St.”
Forming the cores of giant planets from the radial pebble flux in protoplanetary discs (2014, background): “Since the vertical scale height of particles depends on the strength of the turbulence in the disc through Hd ≈ H √ αt/τf (4) (Youdin & Lithwick 2007), the growth rate becomes independent of αt, Ṙ = √ 3 4 ρ0 ρ• Zcsτf (5) (Brauer et al. 2008).”
Separating gas-giant and ice-giant planets by halting pebble accretion (2014, methodology): “(3) Here we have taken for simplicity an MMSN model with a particle scale height given by Hp/H = √ αt/τ (Youdin & Lithwick 2007), where αt is the turbulent diffusion parameter.”
Forming Planetesimals in Solar and Extrasolar Nebulae (2009, methodology): “Youdin & Lithwick (2007) confirm that orbital dynamics introduces only modest corrections when to ∼ Ω −1 K .”
Planetesimal formation near the snowline: in or out? (2017, background): “…collective effects, Ngas is the number of gas particles, and Dp is the particle diffusivity, which is related to Dgas through the Schmidt number Sc (Youdin & Lithwick 2007): Dp = Dgas × Sc = Dgas 1 + τ2S · (23) Since in our models the stopping times are much smaller than unity, we set Dp equal to…”
The Disk Substructures at High Angular Resolution Project (DSHARP). VI. Dust Trapping in Thin-ringed Protoplanetary Disks (2018, background): “The diffusion coefficient for the dust is (Youdin & Lithwick 2007): Dd = D 1 + St2 (F7) We take D to be equal to the turbulent viscosity ν divided by the Schmidt number Sc, which we usually set to Sc = 1.”