Statistics and universality in simplified models of planetary formation (original) (raw)
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Theoretical models of planetary system formation: mass vs. semi-major axis
Astronomy & Astrophysics, 2013
Context. Planet formation models have been developed during the past years to try to reproduce what has been observed of both the solar system and the extrasolar planets. Some of these models have partially succeeded, but they focus on massive planets and, for the sake of simplicity, exclude planets belonging to planetary systems. However, more and more planets are now found in planetary systems. This tendency, which is a result of radial velocity, transit, and direct imaging surveys, seems to be even more pronounced for low-mass planets. These new observations require improving planet formation models, including new physics, and considering the formation of systems. Aims. In a recent series of papers, we have presented some improvements in the physics of our models, focussing in particular on the internal structure of forming planets, and on the computation of the excitation state of planetesimals and their resulting accretion rate. In this paper, we focus on the concurrent effect of the formation of more than one planet in the same protoplanetary disc and show the effect, in terms of architecture and composition of this multiplicity. Methods. We used an N-body calculation including collision detection to compute the orbital evolution of a planetary system. Moreover, we describe the effect of competition for accretion of gas and solids, as well as the effect of gravitational interactions between planets. Results. We show that the masses and semi-major axes of planets are modified by both the effect of competition and gravitational interactions. We also present the effect of the assumed number of forming planets in the same system (a free parameter of the model), as well as the effect of the inclination and eccentricity damping. We find that the fraction of ejected planets increases from nearly 0 to 8% as we change the number of embryos we seed the system with from 2 to 20 planetary embryos. Moreover, our calculations show that, when considering planets more massive than ∼5 M ⊕ , simulations with 10 or 20 planetary embryos statistically give the same results in terms of mass function and period distribution.
Theory of planet formation and comparison with observation
EPJ Web of Conferences, 2011
The planetary mass-radius diagram is an observational result of central importance to understand planet formation. We present an updated version of our planet formation model based on the core accretion paradigm which allows us to calculate planetary radii and luminosities during the entire formation and evolution of the planets. We first study with it the formation of Jupiter, and compare with previous works. Then we conduct planetary population synthesis calculations to obtain a synthetic mass-radius diagram which we compare with the observed one. Except for bloated Hot Jupiters which can be explained only with additional mechanisms related to their proximity to the star, we find a good agreement of the general shape of the observed and the synthetic M − R diagram. This shape can be understood with basic concepts of the core accretion model.
Planet formation models: the interplay with the planetesimal disc
Astronomy & Astrophysics, 2012
Context. According to the sequential accretion model (or core-nucleated accretion model), giant planet formation is based first on the formation of a solid core which, when massive enough, can gravitationally bind gas from the nebula to form the envelope. The most critical part of the model is the formation time of the core: to trigger the accretion of gas, the core has to grow up to several Earth masses before the gas component of the protoplanetary disc dissipates. Aims. We calculate planetary formation models including a detailed description of the dynamics of the planetesimal disc, taking into account both gas drag and excitation of forming planets. Methods. We computed the formation of planets, considering the oligarchic regime for the growth of the solid core. Embryos growing in the disc stir their neighbour planetesimals, exciting their relative velocities, which makes accretion more difficult. Here we introduce a more realistic treatment for the evolution of planetesimals' relative velocities, which directly impact on the formation timescale. For this, we computed the excitation state of planetesimals, as a result of stirring by forming planets, and gas-solid interactions. Results. We find that the formation of giant planets is favoured by the accretion of small planetesimals, as their random velocities are more easily damped by the gas drag of the nebula. Moreover, the capture radius of a protoplanet with a (tiny) envelope is also larger for small planetesimals. However, planets migrate as a result of disc-planet angular momentum exchange, with important consequences for their survival: due to the slow growth of a protoplanet in the oligarchic regime, rapid inward type I migration has important implications on intermediate-mass planets that have not yet started their runaway accretion phase of gas. Most of these planets are lost in the central star. Surviving planets have masses either below 10 M ⊕ or above several Jupiter masses. Conclusions. To form giant planets before the dissipation of the disc, small planetesimals (∼0.1 km) have to be the major contributors of the solid accretion process. However, the combination of oligarchic growth and fast inward migration leads to the absence of intermediate-mass planets. Other processes must therefore be at work to explain the population of extrasolar planets that are presently known.
The Astrophysical Journal, 2008
Remnant planetesimals might have played an important role in reducing the orbital eccentricities of the terrestrial planets after their formation via giant impacts. However, the population and the size distribution of remnant planetesimals during and after the giant impact stage are unknown, because simulations of planetary accretion in the runaway growth and giant impact stages have been conducted independently. Here we report results of direct N-body simulations of the formation of terrestrial planets beginning with a compact planetesimal disk. The initial planetesimal disk has a total mass and angular momentum as observed for the terrestrial planets, and we vary the width (0.3 and 0.5AU) and the number of planetesimals (1000-5000). This initial configuration generally gives rise to three final planets of similar size, and sometimes a fourth small planet forms near the location of Mars. Since a sufficient number of planetesimals remains, even after the giant impact phase, the final orbital eccentricities are as small as those of the Earth and Venus.
2011
The planetary mass-radius diagram is an observational result of central importance to understand planet formation. We present an updated version of our planet formation model based on the core accretion paradigm which allows us to calculate planetary radii and luminosities during the entire formation and evolution of the planets. We first study with it the formation of Jupiter, and compare with previous works. Then we conduct planetary population synthesis calculations to obtain a synthetic mass-radius diagram which we compare with the observed one. Except for bloated Hot Jupiters which can be explained only with additional mechanisms related to their proximity to the star, we find a good agreement of the general shape of the observed and the synthetic M - R diagram. This shape can be understood with basic concepts of the core accretion model.
Dynamical models of terrestrial planet formation
Advanced Science …, 2011
We review the problem of the formation of terrestrial planets, with particular emphasis on the interaction of dynamical and geochemical models. The lifetime of gas around stars in the process of formation is limited to a few million years based on astronomical observations, while isotopic dating of meteorites and the Earth-Moon system suggest that perhaps 50-100 million years were required for the assembly of the Earth. Therefore, much of the growth of the terrestrial planets in our own system is presumed to have taken place under largely gas-free conditions, and the physics of terrestrial planet formation is dominated by gravitational interactions and collisions. The earliest phase of terrestrial-planet formation involve the growth of km-sized or larger planetesimals from dust grains, followed by the accumulations of these planetesimals into ∼100 lunar-to Marsmass bodies that are initially gravitationally isolated from one-another in a swarm of smaller planetesimals, but eventually grow to the point of significantly perturbing one-another. The mutual perturbations between the embryos, combined with gravitational stirring by Jupiter, lead to orbital crossings and collisions that drive the growth to Earth-sized planets on a timescale of 10 7 − 10 8 years. Numerical treatment of this process has focussed on the use of symplectic integrators which can rapidy integrate the thousands of gravitationally-interacting bodies necessary to accurately model planetary growth. While the general nature of the terrestrial planets-their sizes and orbital parameters-seem to be broadly reproduced by the models, there are still some outstanding dynamical issues. One of these is the presence of an embryo-sized body, Mars, in our system in place of the more massive objects that simulations tend to yield. Another is the effect such impacts have on the geochemistry of the growing planets; re-equilibration of isotopic ratios of major elements during giant impacts (for example) must be considered in comparing the predicted compositions of the terrestrial planets with the geochemical data. As the dynamical models become successful in reproducing the essential aspects of our own terrestrial planet system, their utility in predicting the distribution of terrestrial planet systems around other stars, and interpreting observations of such systems, will increase.
Solar System Research, v. 53, N 5, p. 332-361, 2019
Migration of planetesimals from the feeding zone of the terrestrial planets, which was divided into seven regions depending on the distance to the Sun, was simulated. The influence of gravity of all planets was taken into account. In some cases, the embryos of the terrestrial planets rather than the planets themselves were considered; their masses were assumed to be 0.1 or 0.3 of the current masses of the planets. The arrays of orbital elements of migrated planetesimals were used to calculate the probabilities of their collisions with the planets, the Moon, or their embryos. As distinct from the earlier modeling of the evolution of disks of the bodies coagulating in collisions, this approach makes it possible to calculate more accurately the probabilities of collisions of planetesimals with planetary embryos of different masses for some evolution stages. When studying the composition of planetary embryos formed from planetesimals, which initially were at different distances from the Sun, we considered the narrower zones, from which planetesimals came, as compared to those examined earlier, and analyzed the temporal changes in the composition of planetary embryos rather than only the final composition of planets. Based on our calculations, we drew conclusions on the process of accumulation of the terrestrial planets. The embryos of the terrestrial planets, the masses of which did not exceed a tenth of the current planetary masses, accumulated planetesimals mainly from the vicinity of their orbits. When planetesimals fell onto the embryos of the terrestrial planets from the feeding zone of Jupiter and Saturn, these embryos had not yet acquired the current masses of the planets, and the material of this zone (including water and volatiles) could be accumulated in the inner layers of the terrestrial planets and the Moon. For planetesimals which initially were at a distance of 0.7–0.9 AU from the Sun, the probabilities of their infall onto the embryos of the Earth and Venus, the mass of which is 0.3 of the present masses of the planets, differed less than twofold for these embryos. The total mass of planetesimals, which initially were in each part of the region between 0.7 and 1.5 AU from the Sun and collided with the almost-formed Earth and Venus, apparently differed by less than two times for these planets. The inner layers of each of the terrestrial planets were mainly formed from the material located in the vicinity of the orbit of a certain planet. The outer layers of the Earth and Venus could accumulate the same material for these two planets from different parts of the feeding zone of the terrestrial planets. The Earth and Venus could acquire more than half of their masses in 5 Myr. The material ejection that occurred in impacts of bodies with the planets, which was not taken into account in the model, may enlarge the accumulation time for the planets. A relatively rapid growth of the bulk of the Martian mass can be explained by the formation of Mars’ embryo (the mass of which is several times less than that of Mars) due to contraction of a rarified condensation. For the mass ratio of the Earth’s and lunar embryos equal to 81 (the same as that for the masses of the Earth and the Moon), the ratio of the probabilities for infalls of planetesimals onto the Earth’s and lunar embryos did not exceed 54 for the considered variants of calculations; and it was highest for the embryos’ masses approximately three times less than the present masses of these celestial bodies. Special features in the formation of the terrestrial planets can be explained even under a relatively gentle decrease of the semi-major axis of Jupiter’s orbit due to ejection of planetesimals by Jupiter into hyperbolic orbits. In this modeling, it is not necessary to consider the migration of Jupiter to the orbit of Mars and back, as in the Grand Tack model, and sharp changes in the orbits of the giant planets falling into a resonance, as in the Nice model.
Arxiv preprint arXiv: …, 2010
We review the current theoretical understanding how growth from micrometer sized dust to massive giant planets occurs in disks around young stars. After introducing a number of observational constraints from the solar system, from observed protoplanetary disks, and from the extrasolar planets, we simplify the problem by dividing it into a number of discrete stages which are assumed to occur in a sequential way. In the first stage -the growth from dust to kilometer sized planetesimals -the aerodynamics of the bodies are of central importance. We discuss both a purely coagulative growth mode, as well as a gravoturbulent mode involving a gravitational instability of the dust. In the next stage, planetesimals grow to protoplanets of roughly 1000 km in size. Gravity is now the dominant force. The mass accretion can be strongly non-linear, leading to the detachment of a few big bodies from the remaining planetesimals. In the outer planetary system (outside a few AU), some of these bodies can become so massive that they eventually accrete a large gaseous envelope. This is the stage of giant planet formation, as understood within the core accretiongas capture paradigm. We also discuss the direct gravitational collapse model where giant planets are thought to form directly via a gravitational fragmentation of the gas disk. In the inner system, protoplanets collide in the last stage -probably after the dispersal of the gaseous disk -in giant impacts until the separations between the remaining terrestrial planets become large enough to allow long term stability. We finish the review with some selected questions.
N-body simulations of planet formation via pebble accretion
Astronomy and Astrophysics, 2021
Aims. The connection between initial disc conditions and final orbital and physical properties of planets is not well-understood. In this paper, we numerically study the formation of planetary systems via pebble accretion and investigate the effects of disc properties such as masses, dissipation timescales, and metallicities on planet formation outcomes. Methods. We improved the N-body code SyMBA that was modified for our Paper I by taking account of new planet-disc interaction models and type II migration. We adopted the 'two-α' disc model to mimic the effects of both the standard disc turbulence and the mass accretion driven by the magnetic disc wind. Results. We successfully reproduced the overall distribution trends of semi-major axes, eccentricities, and planetary masses of extrasolar giant planets. There are two types of giant planet formation trends, depending on whether or not the disc's dissipation timescales are comparable to the planet formation timescales. When planet formation happens fast enough, giant planets are fully grown (Jupiter mass or higher) and are distributed widely across the disc. On the other hand, when planet formation is limited by the disc's dissipation, discs generally form low-mass cold Jupiters. Our simulations also naturally explain why hot Jupiters (HJs) tend to be alone and how the observed eccentricity-metallicity trends arise. The low-metallicity discs tend to form nearly circular and coplanar HJs in situ, because planet formation is slower than high-metallicity discs, and thus protoplanetary cores migrate significantly before gas accretion. The high-metallicity discs, on the other hand, generate HJs in situ or via tidal circularisation of eccentric orbits. Both pathways usually involve dynamical instabilities, and thus HJs tend to have broader eccentricity and inclination distributions. When giant planets with very wide orbits ("super-cold Jupiters") are formed via pebble accretion followed by scattering, we predict that they belong to metal-rich stars, have eccentric orbits, and tend to have (∼80%) companions interior to their orbits.
N-body simulations of planet formation via pebble accretion
Astronomy & Astrophysics, 2017
Context. Planet formation with pebbles has been proposed to solve a couple of long-standing issues in the classical formation model. Some sophisticated simulations have been performed to confirm the efficiency of pebble accretion. However, there has not been any global N-body simulations that compare the outcomes of planet formation via pebble accretion to observed extrasolar planetary systems. Aims. In this paper, we study the effects of a range of initial parameters of planet formation via pebble accretion, and present the first results of our simulations. Methods. We incorporate a published pebble-accretion model into the N-body code SyMBA, along with the effects of gas accretion, eccentricity and inclination damping, and planet migration in the disc. Results. We confirm that pebble accretion leads to a variety of planetary systems, but have difficulty in reproducing observed properties of exoplanetary systems, such as planetary mass, semimajor axis, and eccentricity distributions. The main reason behind this is an overly efficient type-I migration, which closely depends on the disc model. However, our simulations also lead to a few interesting predictions. First, we find that formation efficiencies of planets depend on the stellar metallicities, not only for giant planets, but also for Earths (Es) and Super-Earths (SEs). The dependency for Es/SEs is subtle. Although higher metallicity environments lead to faster formation of a larger number of Es/SEs, they also tend to be lost later via dynamical instability. Second, our results indicate that a wide range of bulk densities observed for Es and SEs is a natural consequence of dynamical evolution of planetary systems. Third, the ejection trend of our simulations suggest that one free-floating E/SE may be expected for two smaller-mass planets. Key words. planets and satellites: formation-planets and satellites: dynamical evolution and stability-planetary systemsplanets and satellites: general-protoplanetary disks 1 Recent studies of protostellar discs suggest that the MRI turbulence may not be efficient in the planet-forming region (∼1-10 AU), and that the angular momentum transfer may be largely done by magnetocentrifugal disc winds (Turner et al. 2014). If this were the case, the pressure bumps would need to be created by other mechanisms, or pebbles would need to be trapped by other means such as vortices (e.g. Barge & Sommeria 1995; Raettig et al. 2015).