Closed-form expressions for particle relative velocities induced by turbulence

By C. Ormel, J. Cuzzi. 2007

321 citations recorded by the source corpus.

In this note we present complete, closed-form expressions for random relative velocities between colliding particles of arbitrary size in nebula turbulence. These results are exact for very small particles (those with stopping times much shorter than the large eddy overturn time) and are also surprisingly accurate in complete generality (that is, also apply for particles with stopping times comparable to, or much longer than, the large eddy overturn time). We note that some previous studies may have adopted previous simple expressions, which we find to be in error regarding the size dependence in the large particle regime.

Subfields: Astronomy and Astrophysics, Instrumentation

View Paper: https://www.semanticscholar.org/paper/002cbc82e0196ecad28770476e2c68f778c26dac

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Citations in the corpus

Trapping dust particles in the outer regions of protoplanetary disks (2011, background): “14 (for a detailed discussion about turbulent relative velocities see Ormel & Cuzzi (2007)).”
Particle Stirring in Turbulent Gas Disks: Including Orbital Oscillations (2007, methodology): “, 1991) and analytical fits (Ormel and Cuzzi, 2007) to the Völk et al.”
Ring shaped dust accumulation in transition disks (2012, background): “For particles with St ∼ 1, the maximum turbulent relative velocity is given by (Ormel & Cuzzi 2007) ∆u2max ' 3 2 αStc2s , (3) where α is the turbulence parameter (Shakura & Sunyaev 1973).”
Breaking through: The effects of a velocity distribution on barriers to dust growth (2012, methodology): “…velocity vrms P(v | vrms) = √ 54 π v2 v3rms exp ( −3 2 v2 v2rms ) · (3) For the turbulent relative velocity, we use the closed-form expressions derived by Ormel & Cuzzi (2007), which for small particles (St 1, here corresponding to m 106 g) can be approximately written as vrms = √ 9/2 × α St…”
Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits (2010, methodology): “(40)) is due to the jump in relative velocities as the stopping time of particles above this size exceeds the shortest eddy turn-over time (see Ormel & Cuzzi 2007).”
Forming the cores of giant planets from the radial pebble flux in protoplanetary discs (2014, background): “In the so-called intermediate regime (Ormel & Cuzzi 2007), the collision velocities are parametrized by ∆vt = √ 3αtτfcs, (3) with cs the sound speed, αt the turbulent viscosity parameter and τf the Stokes number (the product of the friction time of the particle and the Keplerian frequency ΩK).”
Planet formation bursts at the borders of the dead zone in 2D numerical simulations of circumstellar disks (2009, background): “…problem is that such bodies have Send offprint requests to: wlyra@astro.uu.se poor sticking properties and a low threshold velocity for fragmentation (Chokshi et al. 1993), such that collisions between them usually lead to destruction rather than growth (Benz 2000; Sirono 2004; Ormel & Cuzzi 2007).”
Close-in planetesimal formation by pile-up of drifting pebbles (2016, background): “(3) simplifies to τgrowth ≈ 1 Z ·ΩK (7) under the assumptions that the collisions are driven by turbulence, in which case the impact speed ∆v ≈ √ 3αtSt · cs (Ormel & Cuzzi 2007), the dust grains are in the Epstein regime (see Eq.”
Formation of Jupiter using opacities based on detailed grain physics (2010, methodology): “Perri and Cameron (1974) then constructed an adiabatic, hydrostatic equilibrium model of a protoplanet with a core and a gaseous envelope, extending outward to the Hill radius.”
Towards initial mass functions for asteroids and Kuiper Belt Objects (2010, background): “Meanwhile, we focus on a different accretion pathway, that also relies on turbulence but acts on particles much smaller than a meter, which are excited to small relative (collision) velocities well below the disruption threshold (Cuzzi and Hogan, 2003; Ormel and Cuzzi, 2007), and have sizes directly”
A Three-dimensional View of Turbulence: Constraints on Turbulent Motions in the HD 163296 Protoplanetary Disk Using DCO+ (2017, background): “In general, smaller α leads to weaker collisional velocities (Ormel & Cuzzi 2007) and subsequently to enhanced grain growth, as has been inferred in the disk around HD 163296 based on the submillimeter spectral index (Natta et al. 2004; Isella et al. 2007; Guidi et al. 2016).”
The Disk Substructures at High Angular Resolution Project (DSHARP). V. Interpreting ALMA Maps of Protoplanetary Disks in Terms of a Dust Model (2018, background): “(A6) Here, mp is the proton mass µ = 2.3 the mean molecular mass in atomic units, ya ' 1.6 (Ormel & Cuzzi 2007), σH2 ' 2× 10−15 cm2 the atomic hydrogen cross section.”
Dust Evolution and the Formation of Planetesimals (2016, methodology): “The classical picture was introduced by Völk et al. (1980) and allowed Ormel & Cuzzi (2007) to derive closed form expressions that are currently widely used.”
A simple model for the evolution of the dust population in protoplanetary disks (2012, background): “However, the initial stages of particle growth involve different turbulent regimes (see Ormel & Cuzzi 2007), which lead to faster growth for higher values of αt.”
Fluffy dust forms icy planetesimals by static compression (2013, methodology): “We use the closed formula of the turbulence model (Ormel & Cuzzi 2007) and assume the turbulent parameter αD = 10−3, except for the strong turbulence case, where αD = 10−2.”
WIND-DRIVEN ACCRETION IN PROTOPLANETARY DISKS. I. SUPPRESSION OF THE MAGNETOROTATIONAL INSTABILITY AND LAUNCHING OF THE MAGNETOCENTRIFUGAL WIND (2013, background): “The random gas velocities plays an important role on the collision velocity between dust grains that would impede grain growth (Ormel & Cuzzi 2007).”
Dust coagulation and fragmentation in molecular clouds. I. How collisions between dust aggregates alter the dust size distribution (2009, methodology): “Ormel & Cuzzi (2007) approximated the framework of Völk et al. (1980) and Markiewicz et al. (1991) to provide closed-form expressions for the relative motion between two solid particles.”
ELECTRIC CHARGING OF DUST AGGREGATES AND ITS EFFECT ON DUST COAGULATION IN PROTOPLANETARY DISKS (2009, background): “The relative velocity induced by turbulence is given by Δuturb ≈ usmall tsmall Δtstop, (57) where usmall and tsmall are the characteristic velocity and turnover time of the smallest turbulent eddies, respectively (Weidenschilling 1984; Ormel & Cuzzi 2007).”