Surface Density Of Planetesimals Research Articles

Planetesimal Drift in Eccentric Disks: Possible Outward Migration

Radial drift of solid particles in the protoplanetary disk is often invoked as a threat to planet formation, as it removes solid material from the disk before it can be assembled into planets. However, it may also concentrate solids at particular locations in the disk, thus accelerating the coagulation process. Planetesimals are thought to drift much faster in an eccentric disk, due to their higher velocities with respect to the gas, but their drift rate has only been calculated using approximate means. In this work, we show that in some cases previous estimates of the drift rate, based on a modification of the results for an axisymmetric disk, are highly inaccurate. In particular, we find that under some easily realized circumstances, planetesimals may drift outwards, rather than inwards. This results in the existence of radii in the disk that act as stable attractors of planetesimals. We show that this can lead to a local enhancement of more than an order of magnitude in the surface density of planetesimals, even when a wide dispersion of planetesimal size is considered.

Linking planetary embryo formation to planetesimal formation

Context. The growth-timescales of planetary embryos and their formation process are imperative for our understanding on how planetary systems form and develop. They determine the subsequent growth mechanisms during the life stages of a circumstellar disk. Aims. We quantify the timescales and spatial distribution of planetary embryos through collisional growth and fragmentation of dynamically forming 100 km sized planetesimals. In our study, the formation timescales of viscous disk evolution and planetesimal formation are linked to the formation of planetary embryos in the terrestrial planet zone. Methods. We connected a one-dimensional model for viscous gas evolution, dust and pebble dynamics, and pebble flux-regulated planetesimal formation to the N-body code LIPAD. Our framework enabled us to study the formation, growth, fragmentation, and evolution of planetesimals with an initial size of 100 km in diameter for the first million years of a viscous disk. Results. Our study shows the effect of the planetesimal surface density evolution on the preferential location and timescales of planetary embryo formation. Only the innermost embryos (<2 au) in our study form well within the lifetime of an active pebble flux for any disk studied. Higher planetesimal disk masses and steeper planetesimal surface density profiles result in more massive embryos within a larger area, rather than in a higher number of embryos. A one-dimensional analytically derived model for embryo formation based on the local planetesimal surface density evolution is presented. This model can reproduce the spatial distribution, formation rate, and total number of planetary embryos at a fraction of the computational cost of the N-body simulations. Conclusions. The formation of planetary embryos in the terrestrial planet zone occurs simultaneously with the formation of planetesimals. The local planetesimal surface density evolution and the orbital spacing of planetary embryos in the oligarchic regime are good constraints for modeling planetary embryo formation analytically. Our embryo formation model is a valuable asset in future studies of planet formation.

Planetesimal formation at the gas pressure bump following a migrating planet

Context. Many scenarios have been proposed to avoid known difficulties in planetesimal formation such as drift or fragmentation barriers. However, in these scenarios planetesimals in general only form at some specific locations in protoplanetary discs. On the other hand, it is generally assumed in planet formation models and population synthesis models that planetesimals are broadly distributed in the protoplanetary disc. Aims. We propose a new scenario in which planetesimals can form in broad areas of these discs. Planetesimals form at the gas pressure bump formed by a first-generation planet (e.g. formed by pebble accretion) and the formation region spreads inward in the disc as the planet migrates. Methods. We used a simple 1D Lagrangian particle model to calculate the radial distribution of pebbles in the gas disc perturbed by a migrating embedded planet. We consider that planetesimals form by streaming instability at the points where the pebble-to-gas density ratio on the mid-plane becomes larger than unity. In this work, we fixed the Stokes number of pebbles and the mass of the planet to study the basic characteristics of this new scenario. We also studied the effect of some key parameters, such as the gas disc model, the pebble mass flux, the migration speed of the planet, and the strength of turbulence. Results. We find that planetesimals form in wide areas of protoplanetary discs provided the flux of pebbles is typical and the turbulence is not too strong. The planetesimal surface density depends on the pebble mass flux and the migration speed of the planet. The total mass of the planetesimals and the orbital position of the formation area strongly depend on the pebble mass flux. We also find that the profile of the planetesimal surface density and its slope can be estimated by very simple equations. Conclusions. We show that our new scenario can explain the formation of planetesimals in broad areas. The simple estimates we provide for the planetesimal surface density profile can be used as initial conditions for population synthesis models.

Effect of pebble flux-regulated planetesimal formation on giant planet formation

Context. The formation of gas giant planets by the accretion of 100 km diameter planetesimals is often thought to be inefficient. A diameter of this size is typical for planetesimals and results from self-gravity. Many models therefore use small kilometer-sized planetesimals, or invoke the accretion of pebbles. Furthermore, models based on planetesimal accretion often use the ad hoc assumption of planetesimals that are distributed radially in a minimum-mass solar-nebula way. Aims. We use a dynamical model for planetesimal formation to investigate the effect of various initial radial density distributions on the resulting planet population. In doing so, we highlight the directive role of the early stages of dust evolution into pebbles and planetesimals in the circumstellar disk on the subsequent planet formation. Methods. We implemented a two-population model for solid evolution and a pebble flux-regulated model for planetesimal formation in our global model for planet population synthesis. This framework was used to study the global effect of planetesimal formation on planet formation. As reference, we compared our dynamically formed planetesimal surface densities with ad hoc set distributions of different radial density slopes of planetesimals. Results. Even though required, it is not the total planetesimal disk mass alone, but the planetesimal surface density slope and subsequently the formation mechanism of planetesimals that enables planetary growth through planetesimal accretion. Highly condensed regions of only 100 km sized planetesimals in the inner regions of circumstellar disks can lead to gas giant growth. Conclusions. Pebble flux-regulated planetesimal formation strongly boosts planet formation even when the planetesimals to be accreted are 100 km in size because it is a highly effective mechanism for creating a steep planetesimal density profile. We find that this leads to the formation of giant planets inside 1 au already by pure 100 km planetesimal accretion. Eventually, adding pebble accretion regulated by pebble flux and planetesimal-based embryo formation as well will further complement this picture.

Jupiter’s heavy-element enrichment expected from formation models

Aims.The goal of this work is to investigate Jupiter’s growth by focusing on the amount of heavy elements accreted by the planet, and to compare this with recent structure models of Jupiter.Methods.Our model assumes an initial core growth dominated by pebble accretion, and a second growth phase that is characterised by a moderate accretion of both planetesimals and gas. The third phase is dominated by runaway gas accretion during which the planet becomes detached from the disc. The second and third phases were computed in detail, considering two different prescriptions for the planetesimal accretion and fits from hydrodynamical studies to compute the gas accretion in the detached phase.Results.In order for Jupiter to consist of ~20–40M⊕of heavy elements as suggested by structure models, we find that Jupiter’s formation location is preferably at an orbital distance of 1 ≲a≲ 10 au once the accretion of planetesimals dominates. We find that Jupiter could accrete between ~1 and ~15M⊕of heavy elements during runaway gas accretion, depending on the assumed initial surface density of planetesimals and the prescription used to estimate the heavy-element accretion during the final stage of the planetary formation. This would yield an envelope metallicity of ~0.5 to ~3 times solar. By computing the solid (heavy-element) accretion during the detached phase, we infer a planetary mass-metallicity (MP–MZ) relation ofMZ~MP2/5, when a gap in the planetesimal disc is created, and ofMZ~MP1/6without a planetesimal gap.Conclusions.Our hybrid pebble-planetesimal model can account for Jupiter’s bulk and atmospheric enrichment. The high bulk metallicity inferred for many giant exoplanets is difficult to explain from standard formation models. This might suggest a migration history for such highly enriched giant exoplanets and/or giant impacts after the disc’s dispersal.

Constraints on terrestrial planet formation timescales and equilibration processes in the Grand Tack scenario from Hf-W isotopic evolution

We examine 141 N-body simulations of terrestrial planet late-stage accretion that use the Grand Tack scenario, coupling the collisional results with a hafnium-tungsten (Hf-W) isotopic evolution model. Accretion in the Grand Tack scenario results in faster planet formation than classical accretion models because of higher planetesimal surface density induced by a migrating Jupiter. Planetary embryos that grow rapidly experience radiogenic ingrowth of mantle 182W that is inconsistent with the measured terrestrial composition, unless much of the tungsten is removed by an impactor core that mixes thoroughly with the target mantle. For physically Earth-like surviving planets, we find that the fraction of equilibrating impactor core kcore≥ 0.6 is required to produce results agreeing with observed terrestrial tungsten anomalies (assuming equilibration with relatively large volumes of target mantle material; smaller equilibrating mantle volumes would require even larger kcore). This requirement of substantial core re-equilibration may be difficult to reconcile with fluid dynamical predictions and hydrocode simulations of mixing during large impacts, and hence this result does not favor the rapid planet building that results from Grand Tack accretion.

Formation of Solar system analogues – II. Post-gas-phase growthand water accretion in extended discs via N-body simulations

This work is the second part of a project that attempts to analyze the formation of Solar system analogues (SSAs) from the gaseous to the post-gas phase, in a self-consistently way. In the first paper (PI) we presented our model of planet formation during the gaseous phase which provided us with embryo distributions, planetesimal surface density, eccentricity and inclination profiles of SSAs, considering different planetesimal sizes and type I migration rates at the time the gas dissipates. In this second work we focus on the late accretion stage of SSAs using the results obtained in PI as initial conditions to carry out N-body simulations. One of our interests is to analyze the formation of rocky planets and their final water contents within the habitable zone. Our results show that the formation of potentially habitable planets (PHPs) seems to be a common process in this kind of scenarios. However, the efficiency in forming PHPs is directly related to the size of the planetesimals. The smaller the planetesimals, the greater the efficiency in forming PHPs. We also analyze the sensitivity of our results to scenarios with type I migration rates and gap-opening giants, finding that both phenomena act in a similar way. These effects seem to favor the formation of PHPs for small planetesimal scenarios and to be detrimental for scenarios formed from big planetesimals. Finally, another interesting result is that the formation of water-rich PHPs seems to be more common than the formation of dry PHPs.

Close-in planetesimal formation by pile-up of drifting pebbles

The consistency of planet formation models suffers from the disconnection between the regime of small and large bodies. This is primarily caused by so-called growth barriers: the direct growth of larger bodies is halted at centimetre-sized objects and particular conditions are required for the formation of larger, gravitationally bound planetesimals. We aim to connect models of dust evolution and planetesimal formation to identify regions of protoplanetary discs that are favourable for the formation of kilometre-sized bodies and the first planetary embryos. We combine semi-analytical models of viscous protoplanetary disc evolution, dust growth and drift including backreaction of the dust particles on the gas, and planetesimal formation via the streaming instability into one numerical code. We investigate how planetesimal formation is affected by the mass of the protoplanetary disc, its initial dust content, and the stickiness of dust aggregates. We find that the dust growth and drift leads to a global redistribution of solids. The pile-up of pebbles in the inner disc provides local conditions where the streaming instability is effective. Planetesimals form in an annulus with its inner edge lying between 0.3 AU and 1 AU and its width ranging from 0.3 AU to 3 AU. The resulting surface density of planetesimals follows a radial profile that is much steeper than the initial disc profile. These results support formation of terrestrial planets in the solar system from a narrow annulus of planetesimals, which reproduces their peculiar mass ratios.

Ice lines, planetesimal composition and solid surface density in the solar nebula

To date, there is no core accretion simulation that can successfully account for the formation of Uranus or Neptune within the observed 2–3 Myr lifetimes of protoplanetary disks. Since solid accretion rate is directly proportional to the available planetesimal surface density, one way to speed up planet formation is to take a full accounting of all the planetesimal-forming solids present in the solar nebula. By combining a viscously evolving protostellar disk with a kinetic model of ice formation, which includes not just water but methane, ammonia, CO and 54 minor ices, we calculate the solid surface density of a possible giant planet-forming solar nebula as a function of heliocentric distance and time. Our results can be used to provide the starting planetesimal surface density and evolving solar nebula conditions for core accretion simulations, or to predict the composition of planetesimals as a function of radius. We find three effects that favor giant planet formation by the core accretion mechanism: (1) a decretion flow that brings mass from the inner solar nebula to the giant planet-forming region, (2) the fact that the ammonia and water ice lines should coincide, according to recent lab results from Collings et al. [Collings, M.P., Anderson, M.A., Chen, R., Dever, J.W., Viti, S., Williams, D.A., McCoustra, M.R.S., 2004. Mon. Not. R. Astron. Soc. 354, 1133–1140], and (3) the presence of a substantial amount of methane ice in the trans-saturnian region. Our results show higher solid surface densities than assumed in the core accretion models of Pollack et al. [Pollack, J.B., Hubickyj, O., Bodenheimer, P., Lissauer, J.J., Podolak, M., Greenzweig, Y., 1996. Icarus 124, 62–85] by a factor of 3–4 throughout the trans-saturnian region. We also discuss the location of ice lines and their movement through the solar nebula, and provide new constraints on the possible initial disk configurations from gravitational stability arguments.

Accretion of the gaseous envelope of Jupiter around a 5–10 Earth-mass core

New numerical simulations of the formation and evolution of Jupiter are presented. The formation model assumes that first a solid core of several M ⊕ accretes from the planetesimals in the protoplanetary disk, and then the core captures a massive gaseous envelope from the protoplanetary disk. Earlier studies of the core accretion–gas capture model [Pollack, J.B., Hubickyj, O., Bodenheimer, P., Lissauer, J.J., Podolak, M., Greenzweig, Y., 1996. Icarus 124, 62–85] demonstrated that it was possible for Jupiter to accrete with a solid core of 10–30 M ⊕ in a total formation time comparable to the observed lifetime of protoplanetary disks. Recent interior models of Jupiter and Saturn that agree with all observational constraints suggest that Jupiter's core mass is 0–11 M ⊕ and Saturn's is 9–22 M ⊕ [Saumon, G., Guillot, T., 2004. Astrophys. J. 609, 1170–1180]. We have computed simulations of the growth of Jupiter using various values for the opacity produced by grains in the protoplanet's atmosphere and for the initial planetesimal surface density, σ init , Z , in the protoplanetary disk. We also explore the implications of halting the solid accretion at selected core mass values during the protoplanet's growth. Halting planetesimal accretion at low core mass simulates the presence of a competing embryo, and decreasing the atmospheric opacity due to grains emulates the settling and coagulation of grains within the protoplanet's atmosphere. We examine the effects of adjusting these parameters to determine whether or not gas runaway can occur for small mass cores on a reasonable timescale. We compute four series of simulations with the latest version of our code, which contains updated equation of state and opacity tables as well as other improvements. Each series consists of a run without a cutoff in planetesimal accretion, plus up to three runs with a cutoff at a particular core mass. The first series of runs is computed with an atmospheric opacity due to grains (hereafter referred to as ‘grain opacity’) that is 2% of the interstellar value and σ init , Z = 10 g / cm 2 . Cutoff runs are computed for core masses of 10, 5, and 3 M ⊕ . The second series of Jupiter models is computed with the grain opacity at the full interstellar value and σ init , Z = 10 g / cm 2 . Cutoff runs are computed for core masses of 10 and 5 M ⊕ . The third series of runs is computed with the grain opacity at 2% of the interstellar value and σ init , Z = 6 g / cm 2 . One cutoff run is computed with a core mass of 5 M ⊕ . The final series consists of one run, without a cutoff, which is computed with a temperature dependent grain opacity (i.e., 2% of the interstellar value for T < 350 K ramping up to the full interstellar value for T > 500 K ) and σ init , Z = 10 g / cm 2 . Our results demonstrate that reducing grain opacities results in formation times less than half of those for models computed with full interstellar grain opacity values. The reduction of opacity due to grains in the upper portion of the envelope with T ⩽ 500 K has the largest effect on the lowering of the formation time. If the accretion of planetesimals is not cut off prior to the accretion of gas, then decreasing the surface density of planetesimals lowers the final core mass of the protoplanet, but increases the formation timescale considerably. Finally, a core mass cutoff results in a reduction of the time needed for a protoplanet to evolve to the stage of runaway gas accretion, provided the cutoff mass is sufficiently large. The overall results indicate that, with reasonable parameters, it is possible that Jupiter formed at 5 AU via the core accretion process in 1 Myr with a core of 10 M ⊕ or in 5 Myr with a core of 5 M ⊕ .

Quantifying Orbital Migration from Exoplanet Statistics and Host Metallicities

We investigate how the statistical distribution of extrasolar planets may be combined with knowledge of the host stars' metallicity to yield constraints on the migration histories of gas giant planets. At any radius, planets that barely manage to form around the lowest metallicity stars accrete their envelopes just as the gas disk is being dissipated, so the lower envelope of planets in a plot of metallicity versus semimajor axis defines a sample of nonmigratory planets that will have suffered less than average migration subsequent to gap opening. Under the assumption that metallicity largely controls the initial surface density of planetesimals, we use simplified core accretion models to calculate how the minimum metallicity needed for planet formation varies as a function of semimajor axis. Models that do not include core migration prior to gap opening (type I migration) predict that the critical metallicity is largely flat between the snow line and a ≈ 6 AU, with a weak dependence on the initial surface density profile of planetesimals. When slow type I migration is included, the critical metallicity is found to increase steadily from 1 to 10 AU. Large planet samples that include planets at modestly greater orbital radii than present surveys therefore have the potential to quantify the extent of migration in both type I and type II regimes.

Orbital migration and the period distribution of exoplanets

We use the model for the migration of planets introduced in Del Popolo et al. (2003, MNRAS, 339, 556) to calculate the observed mass and semimajor axis distribution of extra-solar planets. The assumption that the surface density in planetesimals is proportional to that of gas is relaxed, and in order to describe disc evolution we use a method which, using a series of simplifying assumptions, is able to simultaneously follow the evolution of gas and solid particles for up to $10^7 ~{\rm yr}$. The distribution of planetesimals obtained after $10^7 ~{\rm yr}$ is used to study the migration rate of a giant planet through the model described in the present paper. The disk and migration models are used to calculate the distribution of planets as function of mass and semimajor axis. The results show that the model can give a reasonable prediction of planets' semi-major axes and mass distribution. In particular there is a pile-up of planets at $a \simeq 0.05$ AU, a minimum near 0.3 AU, indicating a paucity of planets at that distance, and a rise for semi-major axes larger than 0.3 AU, out to 3 AU. The semi-major axis distribution shows that the more massive planets (typically, masses larger than $4~ M_{\rm J}$) form preferentially in the outer regions and do not migrate much. Intermediate-mass objects migrate more easily whatever the distance at which they form, and that the lighter planets (masses from sub-Saturnian to Jovian) migrate easily.

On the Formation Timescale and Core Masses of Gas Giant Planets

Numerical simulations show that the migration of growing planetary cores may be dominated by turbulent fluctuations in the protoplanetary disk, rather than by any mean property of the flow. We quantify the impact of this stochastic core migration on the formation timescale and core mass of giant planets at the onset of runaway gas accretion. For standard solar nebula conditions, the formation of Jupiter can be accelerated by almost an order of magnitude if the growing core executes a random walk with an amplitude of a few tenths of an AU. A modestly reduced surface density of planetesimals allows Jupiter to form within 10 Myr, with an initial core mass below 10 , in better agreement with observational constraints. For extrasolar planetary systems, the M results suggest that core accretion could form massive planets in disks with lower metallicities, and shorter lifetimes, than the solar nebula. Subject headings: accretion, accretion disks — planetary systems: formation — planets and satellites: formation — planets and satellites: individual (Jupiter) — solar system: formation

PLANETARY MIGRATION IN EVOLVING PLANETESIMAL DISKS

In the current paper, we further improved the model for the migration of planets introduced and extended to time-dependent planetesimal accretion disks by Del Popolo. In the current study, the assumption of Del Popolo, that the surface density in planetesimals is proportional to that of gas, is relaxed. In order to obtain the evolution of planetesimal density, we use a method developed by Stepinski and Valageas which is able to simultaneously follow the evolution of gas and solid particles for up to 107 years. Then, the disk model is coupled to migration model introduced by Del Popolo in order to obtain the migration rate of the planet in the planetesimal. We find that the properties of solids known to exist in protoplanetary systems, together with reasonable density profiles for the disk, lead to a characteristic radius in the range 0.03–0.2 AU for the final semi-major axis of the giant planet.Hence our model can explain the properties of discovered extrasolar giant planets.

Evolution of planetesimal discs and planetary migration

In this paper, we further develop the model for the migration of planets introduced by Del Popolo, Gambera & Ercan and extended to time-dependent planetesimal accretion discs by Del Popolo & Eksi. More precisely, the assumption of Del Popolo & Eksi that the surface density in planetesimals is proportional to that of the gas was released. Indeed, the evolution of the radial distribution of solids is governed by many processes: gas-solid coupling, co- agulation, sedimentation, evaporation/condensation, so that the distribution of planetesimals emerging from a turbulent disc does not necessarily reflect that of the gas. In order to describe this evolution we use a method developed by Stepinski & Valageas, which, using a series of simplifying assumptions, is able to simultaneously follow the evolution of gas and solid particles for up to 10 7 yr. This model is based on the premise that the transformation of solids from dust to planetesimals occurs through hierarchical coagulation. Then, the distribution of planetesimals obtained after 10 7 yr is used to study the migration rate of a giant planet through the migration model introduced by Del Popolo, Gambera & Ercan. This allows us to investigate the dependence of the migration rate on the disc mass, on its time evolution and on the value of the dimensionless viscosity parameter α. We find that in the case of discs having a total mass of 10 −3 -10 −1 M� , and 10 −4 10 −3 and only in the case where α< 10 −3 do the planets move to a minimum value of orbital radius of �2 au. Moreover, the observed distribution of planets in the period range 0-20 d can be easily obtained from our model. Therefore, dynamical friction between planets and the plan- etesimal disc provides a good mechanism to explain the properties of observed extrasolar giant planets.

A Reduced Efficiency of Terrestrial Planet Formation following Giant Planet Migration

Substantial orbital migration of massive planets may occur in most extrasolar planetary systems. Since migration is likely to occur after a significant fraction of the dust has been locked up into planetesimals, ubiquitous migration could reduce the probability of forming terrestrial planets at radii of the order of 1 AU. Using a simple time dependent model for the evolution of gas and solids in the disk, I show that replenishment of solid material in the inner disk, following the inward passage of a giant planet, is generally inefficient. Unless the timescale for diffusion of dust is much shorter than the viscous timescale, or planetesimal formation is surprisingly slow, the surface density of planetesimals at 1 AU will typically be depleted by 1-2 orders of magnitude following giant planet migration. Conceivably, terrestrial planets may exist only in a modest fraction of systems where a single generation of massive planets formed and did not migrate significantly.