From Grains to Planetesimals: Les Houches Lecture

This pedagogical review covers an unsolved problem in the theory of protoplanetary disks: the growth of dust grains into planetesimals, solids at least a kilometer in size. I summarize timescale constraints imposed on planetesimal formation by circumstellar disk observations, analysis of meteorites, and aerodynamic radial migration. The infall of ~meter-sized solids in a hundred years is the most stringent constraint. I review proposed mechanisms for planetesimal formation. Collisional coagulation models are informed by laboratory studies of microgravity collisions. The gravitational collapse (or Safronov-Goldreich-Ward) hypothesis involves detailed study of the interaction between solid particles and turbulent gas. I cover the basics of aerodynamic drag in protoplanetary disks, including radial drift and vertical sedimentation. I describe various mechanisms for particle concentration in gas disks -- including turbulent pressure maxima, drag instabilities and long-lived anticylonic vortices. I derive a general result for the minimum size for a vortex to trap particles in a sub-Keplerian disk. Recent numerical simulations demonstrate that particle clumping in turbulent protoplanetary disks can trigger gravitational collapse. I discuss several outstanding issues in the field.Comment: 20 pages, 3 figures, to appear in the proceedings of the Les Houches Winter School "Physics and Astrophysics of Planetary Systems" (EDP Sciences: EAS Publications Series). Version 2 is the same paper, simply adds above publisher inf

Protoplanetary Disk Turbulence Driven by the Streaming Instability: Non-Linear Saturation and Particle Concentration

We present simulations of the non-linear evolution of streaming instabilities in protoplanetary disks. The two components of the disk, gas treated with grid hydrodynamics and solids treated as superparticles, are mutually coupled by drag forces. We find that the initially laminar equilibrium flow spontaneously develops into turbulence in our unstratified local model. Marginally coupled solids (that couple to the gas on a Keplerian time-scale) trigger an upward cascade to large particle clumps with peak overdensities above 100. The clumps evolve dynamically by losing material downstream to the radial drift flow while receiving recycled material from upstream. Smaller, more tightly coupled solids produce weaker turbulence with more transient overdensities on smaller length scales. The net inward radial drift is decreased for marginally coupled particles, whereas the tightly coupled particles migrate faster in the saturated turbulent state. The turbulent diffusion of solid particles, measured by their random walk, depends strongly on their stopping time and on the solids-to-gas ratio of the background state, but diffusion is generally modest, particularly for tightly coupled solids. Angular momentum transport is too weak and of the wrong sign to influence stellar accretion. Self-gravity and collisions will be needed to determine the relevance of particle overdensities for planetesimal formation.Comment: Accepted for publication in ApJ (17 pages). Movies of the simulations can be downloaded at http://www.mpia.de/~johansen/research_en.ph

Three-Dimensional Simulations of Kelvin-Helmholtz Instability in Settled Dust Layers in Protoplanetary Disks

As dust settles in a protoplanetary disk, a vertical shear develops because the dust-rich gas in the midplane orbits at a rate closer to true Keplerian than the slower-moving dust-depleted gas above and below. A classical analysis (neglecting the Coriolis force and differential rotation) predicts that Kelvin-Helmholtz instability occurs when the Richardson number of the stratified shear flow is below roughly one-quarter. However, earlier numerical studies showed that the Coriolis force makes layers more unstable, whereas horizontal shear may stabilize the layers. Simulations with a 3D spectral code were used to investigate these opposing influences on the instability in order to resolve whether such layers can ever reach the dense enough conditions for the onset of gravitational instability. I confirm that the Coriolis force, in the absence of radial shear, does indeed make dust layers more unstable, however the instability sets in at high spatial wavenumber for thicker layers. When radial shear is introduced, the onset of instability depends on the amplitude of perturbations: small amplitude perturbations are sheared to high wavenumber where further growth is damped; whereas larger amplitude perturbations grow to magnitudes that disrupt the dust layer. However, this critical amplitude decreases sharply for thinner, more unstable layers. In 3D simulations of unstable layers, turbulence mixes the dust and gas, creating thicker, more stable layers. I find that layers with minimum Richardson numbers in the approximate range 0.2 -- 0.4 are stable in simulations with horizontal shear.Comment: 33 pages, 11 figures (5 color, low-resolution versions), Submitted to The Astrophysical Journal, see http://www.physics.sfsu.edu/~barranco for higher resolution color figures and associated avi animation file

Zonal Flows and Long-Lived Axisymmetric Pressure Bumps in Magnetorotational Turbulence

We study the behavior of magnetorotational turbulence in shearing box simulations with a radial and azimuthal extent up to ten scale heights. Maxwell and Reynolds stresses are found to increase by more than a factor two when increasing the box size beyond two scale heights in the radial direction. Further increase of the box size has little or no effect on the statistical properties of the turbulence. An inverse cascade excites magnetic field structures at the largest scales of the box. The corresponding 10% variation in the Maxwell stress launches a zonal flow of alternating sub- and super-Keplerian velocity. This in turn generates a banded density structure in geostrophic balance between pressure and Coriolis forces. We present a simplified model for the appearance of zonal flows, in which stochastic forcing by the magnetic tension on short time-scales creates zonal flow structures with life-times of several tens of orbits. We experiment with various improved shearing box algorithms to reduce the numerical diffusivity introduced by the supersonic shear flow. While a standard finite difference advection scheme shows signs of a suppression of turbulent activity near the edges of the box, this problem is eliminated by a new method where the Keplerian shear advection is advanced in time by interpolation in Fourier space.Comment: Accepted for publication in Ap

Survival of the mm-cm size grain population observed in protoplanetary disks

Millimeter interferometry provides evidence for the presence of mm to cm size "pebbles" in the outer parts of disks around pre-main-sequence stars. The observations suggest that large grains are produced relatively early in disk evolution (< 1 Myr) and remain at large radii for longer periods of time (5 to 10 Myr). Simple theoretical estimates of the radial drift time of solid particles, however, imply that they would drift inward over a time scale of less than 0.1 Myr. In this paper, we address this conflict between theory and observation, using more detailed theoretical models, including the effects of sedimentation, collective drag forces and turbulent viscosity. We find that, although these effects slow down the radial drift of the dust particles, this reduction is not sufficient to explain the observationally determined long survival time of mm/cm-sized grains in protoplanetary disks. However, if for some reason the gas to dust ratio in the disk is reduced by at least a factor of 20 from the canonical value of 100 (for instance through photoevaporation of the gas), then the radial drift time scales become sufficiently large to be in agreement with observations.Comment: Accepted for publication in Astronomy and Astrophysic

Adding particle collisions to the formation of asteroids and Kuiper belt objects via streaming instabilities

Modelling the formation of super-km-sized planetesimals by gravitational collapse of regions overdense in small particles requires numerical algorithms capable of handling simultaneously hydrodynamics, particle dynamics and particle collisions. While the initial phases of radial contraction are dictated by drag forces and gravity, particle collisions become gradually more significant as filaments contract beyond Roche density. Here we present a new numerical algorithm for treating momentum and energy exchange in collisions between numerical superparticles representing a high number of physical particles. We adopt a Monte Carlo approach where superparticle pairs in a grid cell collide statistically on the physical collision time-scale. Collisions occur by enlarging particles until they touch and solving for the collision outcome, accounting for energy dissipation in inelastic collisions. We demonstrate that superparticle collisions can be consistently implemented at a modest computational cost. In protoplanetary disc turbulence driven by the streaming instability, we argue that the relative Keplerian shear velocity should be subtracted during the collision calculation. If it is not subtracted, density inhomogeneities are too rapidly diffused away, as bloated particles exaggerate collision speeds. Local particle densities reach several thousand times the mid-plane gas density. We find efficient formation of gravitationally bound clumps, with a range of masses corresponding to contracted radii from 100 to 400 km when applied to the asteroid belt and 150 to 730 km when applied to the Kuiper belt, extrapolated using a constant self-gravity parameter. The smaller planetesimals are not observed at low resolution, but the masses of the largest planetesimals are relatively independent of resolution and treatment of collisions.Comment: Version accepted for publication in A&