Chondrules are millimeter-sized spherules that dominate primitive meteorites (chondrites) originating from the asteroid belt. The incorporation of chondrules into asteroidal bodies must be an important step in planet formation, but the mechanism is n
ot understood. We show that the main growth of asteroids can result from gas-drag-assisted accretion of chondrules. The largest planetesimals of a population with a characteristic radius of 100 km undergo run-away accretion of chondrules within ~3 Myr, forming planetary embryos up to Mars sizes along with smaller asteroids whose size distribution matches that of main belt asteroids. The aerodynamical accretion leads to size-sorting of chondrules consistent with chondrites. Accretion of mm-sized chondrules and ice particles drives the growth of planetesimals beyond the ice line as well, but the growth time increases above the disk life time outside of 25 AU. The contribution of direct planetesimal accretion to the growth of both asteroids and Kuiper belt objects is minor. In contrast, planetesimal accretion and chondrule accretion play more equal roles for the formation of Moon-sized embryos in the terrestrial planet formation region. These embryos are isolated from each other and accrete planetesimals only at a low rate. However, the continued accretion of chondrules destabilizes the oligarchic configuration and leads to the formation of Mars-sized embryos and terrestrial planets by a combination of direct chondrule accretion and giant impacts.
We present high-resolution computer simulations of dust dynamics and planetesimal formation in turbulence generated by the magnetorotational instability. We show that the turbulent viscosity associated with magnetorotational turbulence in a non-strat
ified shearing box increases when going from 256^3 to 512^3 grid points in the presence of a weak imposed magnetic field, yielding a turbulent viscosity of $alphaapprox0.003$ at high resolution. Particles representing approximately meter-sized boulders concentrate in large-scale high-pressure regions in the simulation box. The appearance of zonal flows and particle concentration in pressure bumps is relatively similar at moderate (256^3) and high (512^3) resolution. In the moderate-resolution simulation we activate particle self-gravity at a time when there is little particle concentration, in contrast with previous simulations where particle self-gravity was activated during a concentration event. We observe that bound clumps form over the next ten orbits, with initial birth masses of a few times the dwarf planet Ceres. At high resolution we activate self-gravity during a particle concentration event, leading to a burst of planetesimal formation, with clump masses ranging from a significant fraction of to several times the mass of Ceres. We present a new domain decomposition algorithm for particle-mesh schemes. Particles are spread evenly among the processors and the local gas velocity field and assigned drag forces are exchanged between a domain-decomposed mesh and discrete blocks of particles. We obtain good load balancing on up to 4096 cores even in simulations where particles sediment to the mid-plane and concentrate in pressure bumps.
We perform hydrodynamical simulations of the accretion of pebbles and rocks onto protoplanets of a few hundred kilometres in radius, including two-way drag force coupling between particles and the protoplanetary disc gas. Particle streams interacting
with the gas far out within the Hill sphere of the protoplanet spiral into a prograde circumplanetary disc. Material is accreted onto the protoplanet due to stirring by the turbulent surroundings. We speculate that the trend for prograde rotation among the largest asteroids is primordial and that protoplanets accreted 10%-50% of their mass from pebbles and rocks during the gaseous solar nebula phase. Our model also offers a possible explanation for the narrow range of spin periods observed among the largest bodies in the asteroid and trans-Neptunian belts, and predicts that 1000 km-scale Kuiper belt objects that have not experienced giant impacts should preferentially spin in the prograde direction.
We present three-dimensional numerical simulations of particle clumping and planetesimal formation in protoplanetary disks with varying amounts of solid material. As centimeter-size pebbles settle to the mid-plane, turbulence develops through vertica
l shearing and streaming instabilities. We find that when the pebble-to-gas column density ratio is 0.01, corresponding roughly to solar metallicity, clumping is weak, so the pebble density rarely exceeds the gas density. Doubling the column density ratio leads to a dramatic increase in clumping, with characteristic particle densities more than ten times the gas density and maximum densities reaching several thousand times the gas density. This is consistent with unstratified simulations of the streaming instability that show strong clumping in particle dominated flows. The clumps readily contract gravitationally into interacting planetesimals of order 100 km in radius. Our results suggest that the correlation between host star metallicity and exoplanets may reflect the early stages of planet formation. We further speculate that initially low metallicity disks can be particle enriched during the gas dispersal phase, leading to a late burst of planetesimal formation.
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 b
eyond 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.
Hydromagnetic stresses in accretion discs have been the subject of intense theoretical research over the past one and a half decades. Most of the disc simulations have assumed a small initial magnetic field and studied the turbulence that arises from
the magnetorotational instability. However, gaseous discs in galactic nuclei and in some binary systems are likely to have significant initial magnetisation. Motivated by this, we performed ideal magnetohydrodynamic simulations of strongly magnetised, vertically stratified discs in a Keplerian potential. Our initial equilibrium configuration, which has an azimuthal magnetic field in equipartion with thermal pressure, is unstable to the Parker instability. This leads to the expelling of magnetic field arcs, anchored in the midplane of the disc, to around five scale heights from the midplane. Transition to turbulence happens primarily through magnetorotational instability in the resulting vertical fields, although magnetorotational shear instability in the unperturbed azimuthal field plays a significant role as well, especially in the midplane where buoyancy is weak. High magnetic and hydrodynamical stresses arise, yielding an effective $alpha$-value of around 0.1 in our highest resolution run. Azimuthal magnetic field expelled by magnetic buoyancy from the disc is continuously replenished by the stretching of a radial field created as gas parcels slide in the linear gravity field along inclined magnetic field lines. This dynamo process, where the bending of field lines by the Parker instability leads to re-creation of the azimuthal field, implies that highly magnetised discs are astrophysically viable and that they have high accretion rates.
This document contains refereed supplementary information for the paper ``Rapid planetesimal formation in turbulent circumstellar discs. It contains 15 sections (S1.1 -- S1.15) that address a number of subjects related to the main paper. We describe
in detail the Poisson solver used to find the self-potential of the solid particles, including a linear and a non-linear test problem (S1.3). Dissipative collisions remove energy from the motion of the particles by collisional cooling (S1.4), an effect that allows gravitational collapse to occur in somewhat less massive discs (S1.7). A resolution study of the gravitational collapse of the boulders is presented in S1.6. We find that gravitational collapse can occur in progressively less massive discs as the grid resolution is increased, likely due to the decreased smoothing of the particle-mesh self-gravity solver with increasing resolution. In S1.10 we show that it is in good agreement with the Goldreich & Ward (1973) stability analysis to form several-hundred-km-sized bodies, when the analysis is applied to 5 AU and to regions of increased boulder column density. S11 is devoted to the measurement of random speeds and collision speeds between boulders. We find good agreement between our measurements and analytical theory for the random speeds, but the measured collision speeds are 3 times lower than expected from analytical theory. Higher resolution studies, and an improved analytical theory of collision speeds that takes into account epicyclic motion, will be needed to determine whether collision speeds have converged. In S1.12 we present models with no magnetic fields. The boulder layer still exhibits strong clumping, due to the streaming instability, if the global solids-to-gas ratio is increased by a factor 3. Gravitational collapse occurs as readily as in magnetised discs.
The initial stages of planet formation in circumstellar gas discs proceed via dust grains that collide and build up larger and larger bodies (Safronov 1969). How this process continues from metre-sized boulders to kilometre-scale planetesimals is a m
ajor unsolved problem (Dominik et al. 2007): boulders stick together poorly (Benz 2000), and spiral into the protostar in a few hundred orbits due to a head wind from the slower rotating gas (Weidenschilling 1977). Gravitational collapse of the solid component has been suggested to overcome this barrier (Safronov 1969, Goldreich & Ward 1973, Youdin & Shu 2002). Even low levels of turbulence, however, inhibit sedimentation of solids to a sufficiently dense midplane layer (Weidenschilling & Cuzzi 1993, Dominik et al. 2007), but turbulence must be present to explain observed gas accretion in protostellar discs (Hartmann 1998). Here we report the discovery of efficient gravitational collapse of boulders in locally overdense regions in the midplane. The boulders concentrate initially in transient high pressures in the turbulent gas (Johansen, Klahr, & Henning 2006), and these concentrations are augmented a further order of magnitude by a streaming instability (Youdin & Goodman 2005, Johansen, Henning, & Klahr 2006, Johansen & Youdin 2007) driven by the relative flow of gas and solids. We find that gravitationally bound clusters form with masses comparable to dwarf planets and containing a distribution of boulder sizes. Gravitational collapse happens much faster than radial drift, offering a possible path to planetesimal formation in accreting circumstellar discs.
