The observed lifetimes of gaseous protoplanetary discs place strong constraints on gas and ice giant formation in the core accretion scenario. The approximately 10-Earth-mass solid core responsible ...for the attraction of the gaseous envelope has to form before gas dissipation in the protoplanetary disc is completed within 1–10 million years. Building up the core by collisions between km-sized planetesimals fails to meet this timescale constraint, especially at wide stellar separations. Nonetheless, gas-giant planets are detected by direct imaging at wide orbital distances. In this paper, we numerically study the growth of cores by the accretion of cm-sized pebbles loosely coupled to the gas. We measure the accretion rate onto seed masses ranging from a large planetesimal to a fully grown 10-Earth-mass core and test different particle sizes. The numerical results are in good agreement with our analytic expressions, indicating the existence of two accretion regimes, one set by the azimuthal and radial particle drift for the lower seed masses and the other, for higher masses, by the velocity at the edge of the Hill sphere. In the former, the optimally accreted particle size increases with core mass, while in the latter the optimal size is centimeters, independent of core mass. We discuss the implications for rapid core growth of gas-giant and ice-giant cores. We conclude that pebble accretion can resolve the long-standing core accretion timescale conflict. This requires a near-unity dust-to-gas ratio in the midplane, particle growth to mm and cm and the formation of massive planetesimals or low radial pressure support. The core growth timescale is shortened by a factor 30–1000 at 5 AU and by a factor 100–10 000 at 50 AU, compared to the gravitationally focused accretion of, respectively, low-scale-height planetesimal fragments or standard km-sized planetesimals.
The solid content of circumstellar disks is inherited from the interstellar medium: dust particles of at most a micrometer in size. Protoplanetary disks are the environment where these dust grains ...need to grow at least 13 orders of magnitude in size. Our understanding of this growth process is far from complete, with different physics seemingly posing obstacles to this growth at various stages. Yet, the ubiquity of planets in our galaxy suggests that planet formation is a robust mechanism. This chapter focuses on the earliest stages of planet formation, the growth of small dust grains towards the gravitationally bound “planetesimals”, the building blocks of planets. We will introduce some of the key physics involved in the growth processes and discuss how they are expected to shape the global behavior of the solid content of disks. We will consider possible pathways towards the formation of larger bodies and conclude by reviewing some of the recent observational advances in the field.
In the solar system giant planets come in two flavours: gas giants (Jupiter and Saturn) with massive gas envelopes, and ice giants (Uranus and Neptune) with much thinner envelopes around their cores. ...It is poorly understood how these two classes of planets formed. High solid accretion rates, necessary to form the cores of giant planets within the life-time of protoplanetary discs, heat the envelope and prevent rapid gas contraction onto the core, unless accretion is halted. We find that, in fact, accretion of pebbles (~cm sized particles) is self-limiting: when a core becomes massive enough it carves a gap in the pebble disc. This halt in pebble accretion subsequently triggers the rapid collapse of the super-critical gas envelope. Unlike gas giants, ice giants do not reach this threshold mass and can only bind low-mass envelopes that are highly enriched by water vapour from sublimated icy pebbles. This offers an explanation for the compositional difference between gas giants and ice giants in the solar system. Furthermore, unlike planetesimal-driven accretion scenarios, our model allows core formation and envelope attraction within disc life-times, provided that solids in protoplanetary discs are predominantly made up of pebbles. Our results imply that the outer regions of planetary systems, where the mass required to halt pebble accretion is large, are dominated by ice giants and that gas-giant exoplanets in wide orbits are enriched by more than 50 Earth masses of solids.
Laboratory experiments indicate that direct growth of silicate grains via mutual collisions can only produce particles up to roughly millimeters in size. On the other hand, recent simulations of the ...streaming instability have shown that mm/cm-sized particles require an excessively high metallicity for dense filaments to emerge. Using a numerical algorithm for stiff mutual drag force, we perform simulations of small particles with significantly higher resolutions and longer simulation times than in previous investigations. We find that particles of dimensionless stopping time τs = 10-2 and 10-3 – representing cm- and mm-sized particles interior of the water ice line – concentrate themselves via the streaming instability at a solid abundance of a few percent. We thus revise a previously published critical solid abundance curve for the regime of τs ≪ 1. The solid density in the concentrated regions reaches values higher than the Roche density, indicating that direct collapse of particles down to mm sizes into planetesimals is possible. Our results hence bridge the gap in particle size between direct dust growth limited by bouncing and the streaming instability.
The formation of planetary cores must proceed rapidly in order for the giant planets to accrete their gaseous envelopes before the dissipation of the protoplanetary gas disc (≲3 Myr). In orbits ...beyond 10 AU, direct accumulation of planetesimals by the cores is too slow. Fragments of planetesimals could be accreted faster, but planetesimals are likely too large for fragmentation to be efficient, and resonant trapping poses an additional hurdle. Here we instead investigate the accretion of small pebbles (mm-cm sizes) that are the natural outcome of an equilibrium between the growth and radial drift of particles. We construct a simplified analytical model of dust coagulation and pebble drift in the outer disc, between 5 AU and 100 AU, which gives the temporal evolution of the solid surface density and the dominant particle size. These two key quantities determine how core growth proceeds at various orbital distances. We find that pebble surface densities are sufficiently high to achieve the inside-out formation of planetary cores within the disc lifetime. The overall efficiency by which dust gets converted to planets can be high, close to 50% for planetary architectures similar to the solar system. Growth by pebble accretion in the outer disc is sufficiently fast to overcome catastrophic type I migration of the cores. These results require protoplanetary discs with large radial extent (≳100 AU) and assume a low number of initial seed embryos. Our findings imply that protoplanetary discs with low disc masses, as expected around low-mass stars (<1 M⊙), or with sub-solar dust-to-gas ratios, do not easily form gas-giant planets (M ≳ 100 ME), but preferentially form Neptune-mass planets or smaller (M ≲ 10 ME). This is consistent with exoplanet surveys which show that gas giants are relatively uncommon around stars of low mass or low metallicity.
We show that condensation is an efficient particle growth mechanism that leads to growth beyond decimetre-sized pebbles close to an ice line in protoplanetary discs. As coagulation of dust particles ...is frustrated by bouncing and fragmentation, condensation could be a complementary, or even dominant, growth mode in the early stages of planet formation. Ice particles diffuse across the ice line and sublimate, and vapour diffusing inwards across the ice line recondenses onto already existing particles, causing them to grow. We develop a numerical model of the dynamical behaviour of ice particles close to the water ice line, approximately 3 AU from the host star. Particles move with the turbulent gas, modelled as a random walk. They also sediment towards the midplane and drift radially towards the central star. Condensation and sublimation are calculated using a Monte Carlo approach. Our results indicate that, with a turbulent α-value of 0.01, growth from millimetre to at least decimetre-sized pebbles is possible on a time scale of 1000 years. We find that particle growth is dominated by ice and vapour transport across the radial ice line, with negligible growth caused by transport across the atmospheric ice line. Ice particles mix outwards by turbulent diffusion, leading to net growth across the entire cold region. The resulting particles are large enough to be sensitive to concentration by streaming instabilities, pressure bumps and vortices, which can cause further growth into planetesimals. In our model, particles are considered to be homogeneous ice particles. Considering the more realistic composition of ice condensed onto rocky ice nuclei might affect the growth time scales, by release of refractory ice nuclei after sublimation. We also ignore sticking and fragmentation in particle collisions. These effects will be the subject of future investigations.
We present simulations of the nonlinear 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 timescale) trigger an upward cascade to large particle clumps with peak overdensitles 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 overdensitles 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.
Recent numerical simulations have shown long-lived axisymmetric sub- and super-Keplerian flows in protoplanetary disks. These zonal flows are found in local as well as global simulations of disks ...unstable to the magnetorotational instability. This paper covers our study of the strength and lifetime of zonal flows and the resulting long-lived gas over- and underdensities as functions of the azimuthal and radial size of the local shearing box. We further investigate dust particle concentrations without feedback on the gas and without self-gravity. The strength and lifetime of zonal flows increase with the radial extent of the simulation box, but decrease with the azimuthal box size. Our simulations support earlier results that zonal flows have a natural radial length scale of 5-7 gas pressure scale heights. This is the first study that combines three-dimensional MHD simulations of zonal flows and dust particles feeling the gas pressure. The pressure bumps trap particles with St = 1 very efficiently. We show that St = 0.1 particles (of some centimeters in size if at 5 AU in a minimum mass solar nebula) reach a hundred-fold higher density than initially. This opens the path for particles of St = 0.1 and dust-to-gas ratio of 0.01 or for particles of St > or =, slanted 0.5 and dust-to-gas ratio 10 super(-4) to still reach densities that potentially trigger the streaming instability and thus gravoturbulent formation of planetesimals.
We present local simulations that verify the linear streaming instability that arises from aerodynamic coupling between solids and gas in protoplanetary disks. This robust instability creates ...enhancements in the particle density in order to tap the free energy of the relative drift between solids and gas, generated by the radial pressure gradient of the disk. We confirm the analytic growth rates found by Youdin and Goodman using grid hydrodynamics to simulate the gas and, alternatively, particle and grid representations of the solids. Since the analytic derivation approximates particles as a fluid, this work corroborates the streaming instability when solids are treated as particles. The idealized physical conditions-axisymmetry, uniform particle size, and the neglect of vertical stratification and collisions-provide a rigorous, well-defined test of any numerical algorithm for coupled partiole-gas dynamics in protoplanetary disks. We describe a numerical particle-mesh implementation of the drag force, which is crucial for resolving the coupled oscillations. Finally, we comment on the balance of energy and angular momentum in two-component disks with frictlonal coupling. A companion paper details the nonlinear evolution of the streaming instability into saturated turbulence with dense particle clumps.
The full spatial structure and temporal evolution of the accretion flow into the envelopes of growing gas giants in their nascent discs is only accessible in simulations. Such simulations are ...constrained in their approach of computing the formation of gas giants by dimensionality, resolution, consideration of self-gravity, energy treatment and the adopted opacity law. Our study explores how a number of these parameters affect the measured accretion rate of a Saturn-mass planet. We present a global 3D radiative hydrodynamics framework using the FARGOCA-code. The planet is represented by a gravitational potential with a smoothing length at the location of the planet. No mass or energy sink is used; instead luminosity and gas accretion rates are self-consistently computed. We find that the gravitational smoothing length must be resolved by at least ten grid cells to obtain converged measurements of the gas accretion rates. Secondly, we find gas accretion rates into planetary envelopes that are compatible with previous studies, and continue to explain those via the structure of our planetary envelopes and their luminosities. Our measured gas accretion rates are formally in the stage of Kelvin–Helmholtz contraction due to the modest entropy loss that can be obtained over the simulation timescale, but our accretion rates are compatible with those expected during late run-away accretion. Our detailed simulations of the gas flow into the envelope of a Saturn-mass planet provide a framework for understanding the general problem of gas accretion during planet formation and highlight circulation features that develop inside the planetary envelopes. Those circulation features feedback into the envelope energetics and can have further implications for transporting dust into the inner regions of the envelope.
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