Rapid planetesimal formation in turbulent circumstellar disks (original) (raw)
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Supplementary Information for``Rapid planetesimal formation in turbulent circumstellar discs
2007
This document contains refereed supplementary information for the paper "Rapid planetesimal formation in turbulent circumstellar discs". It contains 15 sections ( §1.1 - §1.15) that address a number of subjects related to the main paper. Some of the subjects are highlighted here in the abstract. 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 ( §1.3). Dissipative collisions remove energy from the motion of the particles by collisional cooling ( §1.4), an effect that allows gravitational collapse to occur in somewhat less massive discs ( §1.7). A resolution study of the gravitational collapse of the boulders is presented in §1.6. 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 §1.10 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. §11 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 §1.12 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.
Protostellar discs formed from turbulent cores
Monthly Notices of the Royal Astronomical Society, 2010
We investigate the collapse and fragmentation of low-mass, trans-sonically turbulent pre-stellar cores, using smoothed particle hydrodynamics simulations. The initial conditions are slightly supercritical Bonnor-Ebert spheres, all with the same density profile, the same mass (M O = 6.1 M) and the same radius (R O = 17 000 au), but having different initial turbulent velocity fields. 400 turbulent velocity fields have been generated, all scaled so that the mean Mach number is M = 1. Then, a subset of these (in total 11 setups), having a range of net angular momenta, j, has been evolved. The evolution of these turbulent cores is significantly different from the collapse of a rigidly rotating core. It is not strongly correlated with j. Instead, it is moderated by the formation of filamentary structures due to converging turbulent flows. A high fraction (9 out of 13, ∼69 per cent) of the individual protostars forming from turbulent cores are attended by resolved (R ≥ 10 au) protostellar accretion discs, but only a very small fraction (1 out of 9, ∼11 per cent) of these discs is sufficiently cool and extended to develop non-linear gravitational instabilities and fragment. Protostars with discs show two distinct growth modes. They initially grow by direct gravitational collapse, followed by subsequent disc accretion.
Formation of Giant Planets by Fragmentation of Protoplanetary Disks
Science, 2002
The evolution of gravitationally unstable protoplanetary gaseous disks has been studied using three dimensional smoothed particle hydrodynamics (SPH) simulations with unprecedented resolution. We have considered disks with initial masses and temperature profiles consistent with those inferred for the protosolar nebula and for other protoplanetary disks. We show that long-lasting, self-gravitating protoplanets arise after a few disk orbital times if cooling is efficient enough to maintain the temperature close to 50 K. The resulting bodies have masses and orbital eccentricities remarkably similar to those of observed extrasolar planets.
The Astrophysical Journal, 2004
We carry out a large set of very high resolution, three dimensional smoothed particle hydrodynamics (SPH) simulations describing the evolution of gravitationally unstable gaseous protoplanetary disks. We consider a broad range of initial disk parameters. Disk masses out to 20 AU range from 0.075 to 0.125 M ⊙ , roughly consistent with the high-end of the mass distribution inferred for disks around T Tauri stars.Minimum outer temperatures range from 30 to 100 K, as expected from studies of the early protosolar nebula and suggested by the modeling of protoplanetary disks spectra. The mass of the central star is also varied although it is usually assumed equal to that of the Sun. Overall the initial disks span minimum Q parameters between 0.8 and 2, with most models being around ∼ 1.4. The disks are evolved assuming either a locally isothermal equation of state or an adiabatic equation of state with varying γ. Heating by (artificial) viscosity and shocks is included when the adiabatic equation of state is used. When overdensities above a specific threshold appear as a result of gravitational instability in a locally isothermal calculation, the equation of state is switched to adiabatic to account for the increased optical depth. We show that when a disk has a minimum Q parameter less than 1.4 strong trailing spiral instabilities, typically three or four armed modes, form and grow until fragmentation occurs along the arms after about 5 mean disk orbital times. The resulting clumps contract quickly to densities several orders of magnitude higher than the initial disk density, and the densest of them survive even under adiabatic conditions. These clumps are stable to tidal disruption and merge quickly, leaving 2-3 protoplanets on fairly eccentric orbits (the mean eccentricity being around
Gravoturbulent Formation of Planetesimals
The Astrophysical Journal, 2006
We explore the effect of magnetorotational turbulence on the dynamics and concentrations of boulders in local box simulations of a sub-Keplerian protoplanetary disc. The solids are treated as particles each with an independent space coordinate and velocity. We find that the turbulence has two effects on the solids. 1) Meter and decameter bodies are strongly concentrated, locally up to a factor 100 times the average dust density, whereas decimeter bodies only experience a moderate density increase. The concentrations are located in large scale radial gas density enhancements that arise from a combination of turbulence and shear. 2) For meter-sized boulders, the concentrations cause the average radial drift speed to be reduced by 40%. We find that the densest clumps of solids are gravitationally unstable under physically reasonable values for the gas column density and for the dust-to-gas ratio due to sedimentation. We speculate that planetesimals can form in a dust layer that is not in itself dense enough to undergo gravitational fragmentation, and that fragmentation happens in turbulent density fluctuations in this sublayer. Subject headings: instabilities -MHD -planetary systems: formation -planetary systems: protoplanetary disks -turbulence 1 The code is available at
Turbulent Clustering of Protoplanetary Dust and Planetesimal Formation
The Astrophysical Journal, 2011
We study the clustering of inertial particles in turbulent flows and discuss its applications to dust particles in protoplanetary disks. Using numerical simulations, we compute the radial distribution function (RDF), which measures the probability of finding particle pairs at given distances, and the probability density function of the particle concentration. The clustering statistics depend on the Stokes number, St, defined as the ratio of the particle friction timescale, τ p , to the Kolmogorov timescale in the flow. In agreement with previous studies, we find that, in the dissipation range, the clustering intensity strongly peaks at St 1, and the RDF for St ∼ 1 shows a fast power-law increase toward small scales, suggesting that turbulent clustering may considerably enhance the particle collision rate. Clustering at inertial-range scales is of particular interest to the problem of planetesimal formation. At these large scales, the strongest clustering is from particles with τ p in the inertial range. Clustering of these particles occurs primarily around a scale where the eddy turnover time is ∼τ p. We find that particles of different sizes tend to cluster at different locations, leading to flat RDFs between different particles at small scales. In the presence of multiple particle sizes, the overall clustering strength decreases as the particle size distribution broadens. We discuss particle clustering in two recent models for planetesimal formation. We argue that, in the model based on turbulent clustering of chondrule-size particles, the probability of finding strong clusters that can seed planetesimals may have been significantly overestimated. We discuss various clustering mechanisms in simulations of planetesimal formation by gravitational collapse of dense clumps of meter-size particles, in particular the contribution from turbulent clustering due to the limited numerical resolution.
Toward Planetesimals: Dense Chondrule Clumps in the Protoplanetary Nebula
The Astrophysical Journal, 2008
We outline a scenario which traces a direct path from freely-floating nebula particles to the first 10-100km-sized bodies in the terrestrial planet region, producing planetesimals which have properties matching those of primitive meteorite parent bodies. We call this primary accretion. The scenario draws on elements of previous work, and introduces a new critical threshold for planetesimal formation. We presume the nebula to be weakly turbulent, which leads to dense concentrations of aerodynamically size-sorted particles having properties like those observed in chondrites. The fractional volume of the nebula occupied by these dense zones or clumps obeys a probability distribution as a function of their density, and the densest concentrations have particle mass density 100 times that of the gas. However, even these densest clumps are prevented by gas pressure from undergoing gravitational instability in the traditional sense (on a dynamical timescale). While in this state of arrested development, they are susceptible to disruption by the ram pressure of the differentially orbiting nebula gas. However, self-gravity can preserve sufficiently large and dense clumps from ram pressure disruption, allowing their entrained particles to sediment gently but inexorably towards their centers, producing 10-100 km "sandpile" planetesimals. Localized radial pressure fluctuations in the nebula, and interactions between differentially moving dense clumps, will also play a role that must be allowed for in future studies. The scenario is readily extended from meteorite parent bodies to primary accretion throughout the solar system.
Accretion of terrestrial planets from oligarchs in a turbulent disk
Icarus, 2007
We have investigated the final accretion stage of terrestrial planets from Mars-mass protoplanets that formed through oligarchic growth in a disk comparable to the minimum mass solar nebula (MMSN), through N-body simulation including random torques exerted by disk turbulence due to Magneto-Rotational-Instability. For the torques, we used the semi-analytical formula developed by Laughlin et al. (2004). The damping of orbital eccentricities (in all runs) and type-I migration (in some runs) due to the tidal interactions with disk gas are also included. Without any effect of disk gas, Earth-mass planets are formed in terrestrial planet regions in a disk comparable to MMSN but with too large orbital eccentricities to be consistent with the present eccentricities of Earth and Venus in our Solar system. With the eccentricity damping caused by the tidal interaction with a remnant gas disk, Earth-mass planets with eccentricities consistent with those of Earth and Venus are formed in a limited range of disk gas surface density (∼ 10 −4 times MMSN). However, in this case, on average, too many (6) planets remain in terrestrial planet regions, because the damping leads to isolation between the planets. We have carried out a series of N-body simulations including the random torques with different disk surface density and strength of turbulence. We found that the orbital eccentricities pumped up by the turbulent torques and associated random walks in semimajor axes tend to delay isolation of planets, resulting in more coagulation of planets. The eccentricities are still damped after planets become isolated. As a result, the number of final planets decreases with increase in strength of the turbulence, while Earth-mass planets with small eccentricities are still formed. In the case of relatively strong turbulence, the number of final planets are 4-5 at 0.5-2AU, which is more consistent with Solar system, for relatively wide range of disk surface density (∼ 10 −4-10 −2 times MMSN).
The Astrophysical Journal
While numerical simulations have been playing a key role in the studies of planet-disk interaction, testing numerical results against observations has been limited so far. With the two directly imaged protoplanets embedded in its circumstellar disk, PDS70 offers an ideal testbed for planet-disk interaction studies. Using twodimensional hydrodynamic simulations we show that the observed features can be well explained with the two planets in formation, providing strong evidence that previously proposed theories of planet-disk interaction are in action, including resonant migration, particle trapping, size segregation, and filtration. Our simulations suggest that the two planets are likely in 2:1 mean motion resonance and can remain dynamically stable over million-year timescales. The growth of the planets at 10 −8-10 −7 M Jup yr −1 , rates comparable to the estimates from Hα observations, does not destabilize the resonant configuration. Large grains are filtered at the gap edge and only small, (sub-)μm grains can flow to the circumplanetary disks (CPDs) and the inner circumstellar disk. With the submillimeter continuum ring observed outward of the two directly imaged planets, PDS70 provides the first observational evidence of particle filtration by gap-opening planets. The observed submillimeter continuum emission at the vicinity of the planets can be reproduced when (sub-)μm grains survive over multiple CPD gas viscous timescales and accumulate therein. One such possibility is if (sub-)μm grains grow in size and remain trapped in pressure bumps, similar to what we find happening in circumstellar disks. We discuss potential implications to planet formation in the solar system and mature extrasolar planetary systems.
Accretion in Protoplanetary Disks by Collisional Fusion
The Astrophysical Journal, 2010
The formation of a solar system such as ours is believed to have followed a multi-stage process around a protostar and its associated accretion disk. Whipple first noted that planetesimal growth by particle agglomeration is strongly influenced by gas drag, and Cuzzi and colleagues have shown that when midplane particle mass densities approach or exceed those of the gas, solid-solid interactions dominate the drag effect. The size dependence of the drag creates a "bottleneck" at the meter scale with such bodies rapidly spiraling into the central star, whereas much smaller or larger particles do not. Independent of whether the origin of the drag is angular momentum exchange with gas or solids in the disk, successful planetary accretion requires rapid planetesimal growth to km scales. A commonly accepted picture is that for collisional velocities V c above a certain threshold value, V th ∼ 0.1-10 cm s −1 , particle agglomeration is not possible; elastic rebound overcomes attractive surface and intermolecular forces. However, if perfect sticking is assumed for all ranges of interparticle collision speeds the bottleneck can be overcome by rapid planetesimal growth. While previous work has dealt with the influences of collisional pressures and the possibility of particle fracture or penetration, the basic role of the phase behavior of matter-phase diagrams, amorphs and polymorphs-has been neglected. Here it is demonstrated for compact bodies that novel aspects of surface phase transitions provide a physical basis for efficient sticking through collisional melting/amphorphization/polymorphization and subsequent fusion/annealing to extend the collisional velocity range of primary accretion to ∆V c ∼ 1-100 m s −1 V th , which encompasses both typical turbulent RMS speeds and the velocity differences between boulder sized and small grains ∼ 1-50 m s −1. Therefore, as inspiraling meter sized bodies collide with smaller particles in this high velocity collisional fusion regime they grow sufficiently rapidly to ∼ 0.1-1 km scale and settle into stable Keplerian orbits in ∼ 10 5 years before photoevaporative wind clears the disk of source material. The basic theory applies to low and high melting temperature materials and thus to the inner and outer regions of a nebula.