1. INTRODUCTION2. SIMULATIONS3. PARTICLE CLUMPING4. COLLISION SPEEDS5. SELF-GRAVITY6. CONCLUSIONSREFERENCESThe Astrophysical Journal, 704:L75–L79, 2009 October 20 doi:10.1088/0004-637X/704/2/L75C2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.PARTICLE CLUMPING AND PLANETESIMAL FORMATION DEPEND STRONGLY ON METALLICITYAnders Johansen1, Andrew Youdin2, and Mordecai-Mark Mac Low31Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, Netherlands; [email protected] Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada3Department of Astrophysics, American Museum of Natural History, 79th Street at Central Park West, New York, NY 10024-5192, USAReceived 2009 August 14; accepted 2009 September 10; published 2009 September 30ABSTRACTWe present three-dimensional numerical simulations of particle clumping and planetesimal formation in pro-toplanetary disks with varying amounts of solid material. As centimeter-size pebbles settle to the mid-plane,turbulence develops through vertical shearing and streaming instabilities. We find that when the pebble-to-gascolumn density ratio is 0.01, corresponding roughly to solar metallicity, clumping is weak, so the pebble den-sity rarely exceeds the gas density. Doubling the column density ratio leads to a dramatic increase in clumping,with characteristic particle densities more than 10 times the gas density and maximum densities reaching severalthousand times the gas density. This is consistent with unstratified simulations of the streaming instability thatshow strong clumping in particle-dominated flows. The clumps readily contract gravitationally into interactingplanetesimals on the order of 100 km in radius. Our results suggest that the correlation between host star metallicityand exoplanets may reflect the early stages of planet formation. We further speculate that initially low-metallicitydisks can be particle enriched during the gas dispersal phase, leading to a late burst of planetesimal formation.Key words: diffusion – hydrodynamics – instabilities – planetary systems: protoplanetary disks – solar system:formation – turbulence1. INTRODUCTIONThe concentration of particles to high spatial densities pro-motes the formation of planetesimals, the super-kilometer scalebuilding blocks of planets. Drag forces on pebbles and rocks indisks lead to spontaneous particle clumping (Goodman & Pindor2000). The discovery of a linear streaming instability (Youdin &Goodman 2005) shows that clumping is a robust consequenceof particles drifting in and gas flowing out in disks with someradial pressure support (Nakagawa et al. 1986). Johansen &Youdin (2007) studied the nonlinear saturation of the streaminginstability, neglecting vertical gravity and self-gravity. Thosesimulations showed that groups of boulders accelerate the gasaround them toward the Keplerian velocity, reducing the radialdrift locally and leading to temporary concentrations of boulders(see also Johansen et al. 2006).Chiang (2008) and Barranco (2009) recently performed three-dimensional (3D) simulations of vertical shear instabilities inKeplerian disks in the single fluid limit where particles and gashave exactly the same velocities. These studies confirmed expec-tations that mid-plane turbulence develops when the Richardsonnumber Ri 1. While perfect coupling is a good approximationfor small grains, it cannot include vertical settling or in-planestreaming motions.In this Letter, we present 3D simulations of the motion of gasand pebbles in sub-Keplerian disks, including vertical gravityand particle sedimentation. Thus, we can study the combinedeffect of vertical shearing and streaming instabilities as particlesself-consistently settle toward—and are stirred from—the mid-plane. We exclude external sources of turbulence, includingmagnetorotational instabilities (which can actually promoteclumping; see Johansen et al. 2007; Balsara et al. 2009).Our hydrodynamical simulations offer a first approximationto dead zones with low ionization (Sano et al. 2000) whereturbulent surface layers drive only weak motions in the mid-plane (Fleming & Stone 2003; Oishi et al. 2007).In this non-magnetized limit, we investigate the clumpingof smaller particles than those considered in Johansen et al.(2007), which increases the likelihood of coagulation up to theinitial sizes. We find that clumping of pebbles in the mid-planelayer increases sharply above a threshold mass fraction of solidsroughly consistent with solar metallicity. Thus, planetesimalformation may help explain the high probability of finding giantexoplanets around stars rich in heavy elements (Gonzalez 1997;Santos et al. 2001; Fischer & Valenti 2005; Johnson & Apps2009).2. SIMULATIONSWe perform 3D hybrid simulations. They model gas on a fixedgrid and solids with superparticles, each representing a swarm ofactual particles. We solve the standard shearing sheet dynamicalequations for a frame rotating at the Keplerian frequency Ω at afixed orbital distance r from the star. The axes are oriented suchthat x points radially outward, y points in the orbital direction,while z points vertically out of the disk. The gas is subject to aglobal radial pressure gradient that reduces the gas orbital speedby Δv=−0.05cs≈ 25 m s−1(Chiang & Goldreich 1997). Thesound speed cs, gas scale height Hg= cs/Ω, and mid-plane gasdensity ρ0are the natural units of the simulation.The motion of gas and particles are coupled throughmomentum-conserving drag forces with particle friction timeτf. Our dynamical equations are identical to those of Youdin& Johansen (2007), with the addition of a vertical gravitationalacceleration gz=−Ω2z affecting both gas and particles.The superparticles are evenly distributed in mass andnumber into four bins of normalized friction time Ωτf=0.1, 0.2, 0.3, and 0.4. These friction times are characteristic ofcompact solids with radius ap≈ 3, 6, 9, 12 cm at r = 5AUin the Minimum Mass Solar Nebula (Weidenschilling 1977a,1977b; Hayashi 1981). Rescaling to r = 10 AU yields ap≈ 1–4 cm. We colloquially refer to this range of particle sizes aspebbles to contrast with larger Ωτf≈ 1.0 boulders. The totalpebble mass is fixed by setting the pebble-to-gas column densityratio Zp=Σp/Σg, where Σp and Σg are the mean particleand gas column densities, taking into account that most
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