Nice 2 model

Last updated July 17, 2024

The Nice 2 model is a model of the early evolution of the Solar System. The Nice 2 model resembles the original Nice model in that a late instability of the outer Solar System results in gravitational encounters between planets, the disruption of an outer planetesimal disk, and the migrations of the outer planets to new orbits. However, the Nice 2 model differs in its initial conditions and in the mechanism for triggering the late instability. These changes reflect the analysis of the orbital evolution of the outer Solar System during the gas disk phase and the inclusion of gravitational interactions between planetesimals in the outer disk into the model.

Description

The Nice 2 model begins with the outer planets in a stable quadruple resonance with each planet in resonance with its nearest neighbors.^[1] One example among several potential stable quadruple resonance configurations is Jupiter and Saturn in a 3:2 resonance, Saturn and Uranus in a 3:2 resonance, and Uranus and Neptune in a 4:3 resonance.^[2] Interactions with an outer planetesimal disk that is gravitationally stirred by Pluto-sized objects cause the planets to migrate inward while remaining in resonance.^[1] During this migration the eccentricity of the inner ice giant increases, leading to secular-resonance crossings.^[1] After several hundred million years, the resonant configuration is destabilized during one of these secular-resonance crossings.^[1] Gravitational encounters between the planets similar to those in the original Nice model begin shortly thereafter.

Development

The Nice 2 model addresses some weaknesses of the original Nice model. The first weakness is the artificial selection of the initial orbits of the outer planets to produce an instability that matches the timing of the Late Heavy Bombardment.^[2] The second weakness is the sensitivity of the timing of the instability to the location of the inner edge of the planetesimal disk.^[2] The Nice 2 model uses particular initial conditions, derived from the examination of the orbital evolution of giant planets orbiting in a gas disk,^[1] which may occur under appropriate circumstances.^[3] An instability trigger with no apparent correlation between the timing of the instability and the position of the inner edge of the planetesimal disk is the result of the incorporation of the interactions between planetesimals into the Nice 2 model.^[1]

Initial conditions

The initial orbits of the giant planets in the Nice 2 model correspond to a predicted orbital structure of the outer Solar System at the end of the gas disk phase.^[2] Models of giant planets orbiting in a gas disk predict that they would migrate toward the central star at a rate dependent on the mass of the planet and characteristics of the disk. In a system with multiple planets this migration can result in the convergence of the planet's orbits and their capture into mean-motion resonances.^[4]^[5] Investigations focusing on Jupiter and Saturn demonstrated that they can be captured in a 3:2 or 2:1 resonance depending on the characteristics of the protoplanetary disk.^[6]^[3]^[7] After capture into resonance, the gaps that Jupiter and Saturn formed in the disk's density distribution may overlap and their inward migration may be halted or reversed.^[4]^[5] When Uranus and Neptune are added in turn to the model they are captured into further resonances with the capture of the outer ice giant resulting in the inner ice giant having a higher eccentricity than the other planets.^[8] The result is a system in a quadruple resonance. A number of stable configurations have been identified with the particular final configuration depending on the starting locations of the planets.^[9]

Instability trigger

The inclusion of gravitational interactions between planetesimals in the outer disk revealed an alternative mechanism for triggering the late instability of the outer planets. During numerical simulations that included the gravitational interactions between planetesimals, a transfer of energy between the disk and the planets was observed. This energy transfer led to the migration of the planets toward the Sun and occurred even when there were no encounters between planetesimals and the planets. As the migration progressed the eccentricity of the inner ice giant increased. In some of the simulations the quadruple resonance was eventually destabilized resulting in gravitational encounters between planets. The instability was observed in 25% of the simulations with the timing varying between 300 million and 1 billion years. No correlation between the location of the inner edge of the planetesimal disk and the occurrence or the timing of the instability was apparent.^[1]

Closer investigation using a simpler model with one planet and a planetesimal disk indicated that the energy transfer was due to a coupling between the eccentricity of the planetesimals in the outer belt and the semi-major axis of the planet. As a result of this coupling an increase in the average eccentricity of the planetesimal belt driven via the gravitational stirring by Pluto-sized objects yields a decrease in the semi-major axis of the planet. The coupling was found to be proportional to the eccentricity of the planet and in a multiple planet system would have its greatest effect on the most eccentric planet.^[1]

The increase in the eccentricity of the inner ice giant was found to be due to the varying strengths of the coupling between the planetesimal disk and the planets. The inner ice giant, with its higher eccentricity due to its resonant capture of the outer ice giant, would normally migrate at a faster rate than the other planets. However, since the resonant configuration requires that the migration be synchronized, the inner ice giant must drag the other planets along. The increase in the inner ice giant's eccentricity is a result of this process.^[1]

Examination of the orbital evolution of the planets revealed that the destabilization of their orbits was due to secular resonance crossings. The increase of the eccentricity of the inner ice giant during the migration led to the slow variation of the precession frequencies of the planets. Secular resonances occurred when these frequencies became similar. The eccentricity of the inner ice giant fluctuated during these secular resonance crossings, sometimes dropping enough to cause the breaking of the quadruple resonance. Whether the quadruple resonance broke was determined by the strength of the secular resonance and the time spent in the secular resonance.^[1]

The nature of the instability mechanism is responsible for the lack of a correlation between the distance to the inner edge of the planetesimal belt and the timing of the instability. If the inner edge of the planetesimal disk is close the migration of the planets occurs at a faster rate. More secular resonance crossings occur but since less time is spent in each one only the strongest can break the quadruple resonance. The reverse is true for a more distant planetesimal belt. As a result of the conflict between these factors the timing and the occurrence of the instability is fairly independent of the distance to the inner edge of the planetesimal belt.^[1]

Potential issues and an alternative

A study using a numerical simulation that included gravitational interactions among all objects revealed that a dynamical instability occurred in less than 70 million years. Interactions between planetesimals dynamically heated the disk and lead to earlier interactions between the planetesimals and giant planets. This study used a limited number of planetesimals due to computational constraints so it is as yet unknown whether this result would apply to a more complete disk.^[10]

The combination of the late destabilization of a five planet resonant chain and an extended migration of Neptune is unlikely. Reproducing the orbital distribution of the Kuiper belt objects requires that Neptune undergo a migration of several AU, reaching 28 AU before the encounters between planets begin. This migration of Neptune is likely if the planetesimal disk began within 2 AU of Neptune's initial orbit. However, a late destabilization of the resonance chain requires a more distant disk, at least 4 AU beyond Neptune's orbit.^[11]

An early breaking of the resonance chain followed by a slow dust-driven migration may bridge this gap. The dust-driven is the result of collision among the planetesimals producing debris that is ground to dust in a collisional cascade. The dust then spirals toward the orbits of the planets due to Poynting–Robertson drag. Interactions with this dust disrupts the resonance chain and drive their migration toward the planetesimal disk over a several hundred million years period.^[11] The instability mechanism of the Nice 2 model becomes irrelevant if the dust generated by collisions among the planetesimals disrupts a resonant chain early.

Related Research Articles

The Kuiper belt is a circumstellar disc in the outer Solar System, extending from the orbit of Neptune at 30 astronomical units (AU) to approximately 50 AU from the Sun. It is similar to the asteroid belt, but is far larger—20 times as wide and 20–200 times as massive. Like the asteroid belt, it consists mainly of small bodies or remnants from when the Solar System formed. While many asteroids are composed primarily of rock and metal, most Kuiper belt objects are composed largely of frozen volatiles, such as methane, ammonia, and water. The Kuiper belt is home to most of the objects that astronomers generally accept as dwarf planets: Orcus, Pluto, Haumea, Quaoar, and Makemake. Some of the Solar System's moons, such as Neptune's Triton and Saturn's Phoebe, may have originated in the region.

In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationship is found between a pair of objects. The physical principle behind orbital resonance is similar in concept to pushing a child on a swing, whereby the orbit and the swing both have a natural frequency, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Neptune and Pluto. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance between bodies with similar orbital radii causes large planetary system bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.

A Kirkwood gap is a gap or dip in the distribution of the semi-major axes of the orbits of main-belt asteroids. They correspond to the locations of orbital resonances with Jupiter.

The nebular hypothesis is the most widely accepted model in the field of cosmogony to explain the formation and evolution of the Solar System. It suggests the Solar System is formed from gas and dust orbiting the Sun which clumped up together to form the planets. The theory was developed by Immanuel Kant and published in his Universal Natural History and Theory of the Heavens (1755) and then modified in 1796 by Pierre Laplace. Originally applied to the Solar System, the process of planetary system formation is now thought to be at work throughout the universe. The widely accepted modern variant of the nebular theory is the solar nebular disk model (SNDM) or solar nebular model. It offered explanations for a variety of properties of the Solar System, including the nearly circular and coplanar orbits of the planets, and their motion in the same direction as the Sun's rotation. Some elements of the original nebular theory are echoed in modern theories of planetary formation, but most elements have been superseded.

Hot Jupiters are a class of gas giant exoplanets that are inferred to be physically similar to Jupiter but that have very short orbital periods. The close proximity to their stars and high surface-atmosphere temperatures resulted in their informal name "hot Jupiters".

Planetary migration occurs when a planet or other body in orbit around a star interacts with a disk of gas or planetesimals, resulting in the alteration of its orbital parameters, especially its semi-major axis. Planetary migration is the most likely explanation for hot Jupiters. The generally accepted theory of planet formation from a protoplanetary disk predicts that such planets cannot form so close to their stars, as there is insufficient mass at such small radii and the temperature is too high to allow the formation of rocky or icy planetesimals.

An ice giant is a giant planet composed mainly of elements heavier than hydrogen and helium, such as oxygen, carbon, nitrogen, and sulfur. There are two ice giants in the Solar System: Uranus and Neptune.

The scattered disc (or scattered disk) is a distant circumstellar disc in the Solar System that is sparsely populated by icy small Solar System bodies, which are a subset of the broader family of trans-Neptunian objects. The scattered-disc objects (SDOs) have orbital eccentricities ranging as high as 0.8, inclinations as high as 40°, and perihelia greater than 30 astronomical units (4.5×10⁹ km; 2.8×10⁹ mi). These extreme orbits are thought to be the result of gravitational "scattering" by the gas giants, and the objects continue to be subject to perturbation by the planet Neptune.

There is evidence that the formation of the Solar System began about 4.6 billion years ago with the gravitational collapse of a small part of a giant molecular cloud. Most of the collapsing mass collected in the center, forming the Sun, while the rest flattened into a protoplanetary disk out of which the planets, moons, asteroids, and other small Solar System bodies formed.

This page describes exoplanet orbital and physical parameters.

Detached objects are a dynamical class of minor planets in the outer reaches of the Solar System and belong to the broader family of trans-Neptunian objects (TNOs). These objects have orbits whose points of closest approach to the Sun (perihelion) are sufficiently distant from the gravitational influence of Neptune that they are only moderately affected by Neptune and the other known planets: This makes them appear to be "detached" from the rest of the Solar System, except for their attraction to the Sun.

<span class="mw-page-title-main">Nice model</span> Scenario for the dynamical evolution of the Solar System

The Nicemodel is a scenario for the dynamical evolution of the Solar System. It is named for the location of the Côte d'Azur Observatory—where it was initially developed in 2005—in Nice, France. It proposes the migration of the giant planets from an initial compact configuration into their present positions, long after the dissipation of the initial protoplanetary disk. In this way, it differs from earlier models of the Solar System's formation. This planetary migration is used in dynamical simulations of the Solar System to explain historical events including the Late Heavy Bombardment of the inner Solar System, the formation of the Oort cloud, and the existence of populations of small Solar System bodies such as the Kuiper belt, the Neptune and Jupiter trojans, and the numerous resonant trans-Neptunian objects dominated by Neptune.

The Late Heavy Bombardment (LHB), or lunar cataclysm, is a hypothesized astronomical event thought to have occurred approximately 4.1 to 3.8 billion years (Ga) ago, at a time corresponding to the Neohadean and Eoarchean eras on Earth. According to the hypothesis, during this interval, a disproportionately large number of asteroids and comets collided into the terrestrial planets and their natural satellites in the inner Solar System, including Mercury, Venus, Earth and Mars. These came from both post-accretion and planetary instability-driven populations of impactors. Although it gained widespread credence, definitive evidence remained elusive. However, recent re-appraisal of the cosmo-chemical constraints suggest there was no late spike in the bombardment rate.

The five-planet Nice model is a numerical model of the early Solar System that is a revised variation of the Nice model. It begins with five giant planets, the four that exist today plus an additional ice giant between Saturn and Uranus in a chain of mean-motion resonances.

The E-belt asteroids were the population of a hypothetical extension of the primordial asteroid belt proposed as the source of most of the basin-forming lunar impacts during the Late Heavy Bombardment.

The jumping-Jupiter scenario specifies an evolution of giant-planet migration described by the Nice model, in which an ice giant is scattered inward by Saturn and outward by Jupiter, causing their semi-major axes to jump, and thereby quickly separating their orbits. The jumping-Jupiter scenario was proposed by Ramon Brasser, Alessandro Morbidelli, Rodney Gomes, Kleomenis Tsiganis, and Harold Levison after their studies revealed that the smooth divergent migration of Jupiter and Saturn resulted in an inner Solar System significantly different from the current Solar System. During this migration secular resonances swept through the inner Solar System exciting the orbits of the terrestrial planets and the asteroids, leaving the planets' orbits too eccentric, and the asteroid belt with too many high-inclination objects. The jumps in the semi-major axes of Jupiter and Saturn described in the jumping-Jupiter scenario can allow these resonances to quickly cross the inner Solar System without altering orbits excessively, although the terrestrial planets remain sensitive to its passage.

In planetary astronomy, the grand tack hypothesis proposes that Jupiter formed at a distance of 3.5 AU from the Sun, then migrated inward to 1.5 AU, before reversing course due to capturing Saturn in an orbital resonance, eventually halting near its current orbit at 5.2 AU. The reversal of Jupiter's planetary migration is likened to the path of a sailboat changing directions (tacking) as it travels against the wind.

Pebble accretion is the accumulation of particles, ranging from centimeters up to meters in diameter, into planetesimals in a protoplanetary disk that is enhanced by aerodynamic drag from the gas present in the disk. This drag reduces the relative velocity of pebbles as they pass by larger bodies, preventing some from escaping the body's gravity. These pebbles are then accreted by the body after spiraling or settling toward its surface. This process increases the cross section over which the large bodies can accrete material, accelerating their growth. The rapid growth of the planetesimals via pebble accretion allows for the formation of giant planet cores in the outer Solar System before the dispersal of the gas disk. A reduction in the size of pebbles as they lose water ice after crossing the ice line and a declining density of gas with distance from the sun slow the rates of pebble accretion in the inner Solar System resulting in smaller terrestrial planets, a small mass of Mars and a low mass asteroid belt.

In planetary science a streaming instability is a hypothetical mechanism for the formation of planetesimals in which the drag felt by solid particles orbiting in a gas disk leads to their spontaneous concentration into clumps which can gravitationally collapse. Small initial clumps increase the orbital velocity of the gas, slowing radial drift locally, leading to their growth as they are joined by faster drifting isolated particles. Massive filaments form that reach densities sufficient for the gravitational collapse into planetesimals the size of large asteroids, bypassing a number of barriers to the traditional formation mechanisms. The formation of streaming instabilities requires solids that are moderately coupled to the gas and a local solid to gas ratio of one or greater. The growth of solids large enough to become moderately coupled to the gas is more likely outside the ice line and in regions with limited turbulence. An initial concentration of solids with respect to the gas is necessary to suppress turbulence sufficiently to allow the solid to gas ratio to reach greater than one at the mid-plane. A wide variety of mechanisms to selectively remove gas or to concentrate solids have been proposed. In the inner Solar System the formation of streaming instabilities requires a greater initial concentration of solids or the growth of solid beyond the size of chondrules.

Planet Nine is a hypothetical ninth planet in the outer region of the Solar System. Its gravitational effects could explain the peculiar clustering of orbits for a group of extreme trans-Neptunian objects (ETNOs), bodies beyond Neptune that orbit the Sun at distances averaging more than 250 times that of the Earth. These ETNOs tend to make their closest approaches to the Sun in one sector, and their orbits are similarly tilted. These alignments suggest that an undiscovered planet may be shepherding the orbits of the most distant known Solar System objects. Nonetheless, some astronomers question this conclusion and instead assert that the clustering of the ETNOs' orbits is due to observational biases, resulting from the difficulty of discovering and tracking these objects during much of the year.

References

1 2 3 4 5 6 7 8 9 10 11 Levison, Harold F.; Morbidelli, Alessandro; Tsiganis, Kleomenis; Nesvorný, David; Gomes, Rodney (2011). "Late Orbital Instabilities in the Outer Planets Induced by Interaction with a Self-gravitating Planetesimal Disk". The Astronomical Journal. 142 (5): 152. Bibcode:2011AJ....142..152L. doi: 10.1088/0004-6256/142/5/152 .
1 2 3 4 Morbidelli, Alessandro (2010). "A coherent and comprehensive model of the evolution of the outer Solar System". Comptes Rendus Physique. 11 (9–10): 651–659. arXiv: 1010.6221 . Bibcode:2010CRPhy..11..651M. doi:10.1016/j.crhy.2010.11.001.
1 2 Zhang, H.; Zhou, J.-L. (2010). "On the Orbital Evolution of a Giant Planet Pair Embedded in a Gaseous Disk. I. Jupiter-Saturn Configuration". The Astrophysical Journal. 714 (1): 532–548. arXiv: 1002.2201 . Bibcode:2010ApJ...714..532Z. doi:10.1088/0004-637X/714/1/532.
1 2 Masset, F.; Snellgrove, M. (2001). "Reversing type II migration: resonance trapping of a lighter giant protoplanet". Monthly Notices of the Royal Astronomical Society. 320 (4): L55–L59. arXiv: astro-ph/0003421 . Bibcode:2001MNRAS.320L..55M. doi: 10.1046/j.1365-8711.2001.04159.x .
1 2 Morbidelli, Alessandro; Crida, Aurélien (2007). "The dynamics of Jupiter and Saturn in the gaseous proto-planetary disk". Icarus. 191 (1): 158–171. arXiv: 0704.1210 . Bibcode:2007Icar..191..158M. doi:10.1016/j.icarus.2007.04.001.
↑ Pierens, A.; Nelson, R. P. (2008). "Constraints on resonant–trapping for two planets embedded in a protoplanetary disc". Astronomy and Astrophysics. 482 (1): 333–340. arXiv: 0802.2033 . Bibcode:2008A&A...482..333P. doi:10.1051/0004-6361:20079062.
↑ D'Angelo, G.; Marzari, F. (2012). "Outward Migration of Jupiter and Saturn in Evolved Gaseous Disks". The Astrophysical Journal. 757 (1): 50 (23 pp.). arXiv: 1207.2737 . Bibcode:2012ApJ...757...50D. doi:10.1088/0004-637X/757/1/50.
↑ Morbidelli, Alessandro; Tsiganis, Kleomenis; Crida, Aurélien; Levison, Harold F.; Gomes, Rodney (2007). "Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and Their Relationship to the Current Orbital Architecture". The Astronomical Journal. 134 (5): 1790–1798. arXiv: 0706.1713 . Bibcode:2007AJ....134.1790M. doi:10.1086/521705.
↑ Batygin, Konstantin; Brown, Michael E. (2010). "Early Dynamical Evolution of the Solar System: Pinning Down the Initial Conditions of the Nice Model". The Astrophysical Journal. 76 (2): 1323–1331. arXiv: 1004.5414 . Bibcode:2010ApJ...716.1323B. doi:10.1088/0004-637X/716/2/1323.
↑ Reyes-Ruiz, M.; Aceves, H.; Chavez, C. E. (2014). "Stability of the Outer Planets in Multiresonant Configurations with a Self-gravitating Planetesimal Disk". The Astrophysical Journal. 804 (2): 91. arXiv: 1406.2341 . Bibcode:2015ApJ...804...91R. doi:10.1088/0004-637X/804/2/91.
1 2 Deienno, Rogerio; Morbidelli, Alessandro; Gomes, Rodney S.; Nesvorny, David (2017). "Constraining the giant planets' initial configuration from their evolution: implications for the timing of the planetary instability". The Astronomical Journal. 153 (4): 153. arXiv: 1702.02094 . Bibcode:2017AJ....153..153D. doi: 10.3847/1538-3881/aa5eaa .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Levison_2011-1] 1 2 3 4 5 6 7 8 9 10 11 Levison, Harold F.; Morbidelli, Alessandro; Tsiganis, Kleomenis; Nesvorný, David; Gomes, Rodney (2011). "Late Orbital Instabilities in the Outer Planets Induced by Interaction with a Self-gravitating Planetesimal Disk". The Astronomical Journal. 142 (5): 152. Bibcode:2011AJ....142..152L. doi: 10.1088/0004-6256/142/5/152 .

[Morbidelli_2010-2] 1 2 3 4 Morbidelli, Alessandro (2010). "A coherent and comprehensive model of the evolution of the outer Solar System". Comptes Rendus Physique. 11 (9–10): 651–659. arXiv: 1010.6221 . Bibcode:2010CRPhy..11..651M. doi:10.1016/j.crhy.2010.11.001.

[zhang_2010-3] 1 2 Zhang, H.; Zhou, J.-L. (2010). "On the Orbital Evolution of a Giant Planet Pair Embedded in a Gaseous Disk. I. Jupiter-Saturn Configuration". The Astrophysical Journal. 714 (1): 532–548. arXiv: 1002.2201 . Bibcode:2010ApJ...714..532Z. doi:10.1088/0004-637X/714/1/532.

[masset_snellgrove_2001-4] 1 2 Masset, F.; Snellgrove, M. (2001). "Reversing type II migration: resonance trapping of a lighter giant protoplanet". Monthly Notices of the Royal Astronomical Society. 320 (4): L55–L59. arXiv: astro-ph/0003421 . Bibcode:2001MNRAS.320L..55M. doi: 10.1046/j.1365-8711.2001.04159.x .

[Morbidelli_Crida_2007-5] 1 2 Morbidelli, Alessandro; Crida, Aurélien (2007). "The dynamics of Jupiter and Saturn in the gaseous proto-planetary disk". Icarus. 191 (1): 158–171. arXiv: 0704.1210 . Bibcode:2007Icar..191..158M. doi:10.1016/j.icarus.2007.04.001.

[Pierens_Nelson_2008-6] Pierens, A.; Nelson, R. P. (2008). "Constraints on resonant–trapping for two planets embedded in a protoplanetary disc". Astronomy and Astrophysics. 482 (1): 333–340. arXiv: 0802.2033 . Bibcode:2008A&A...482..333P. doi:10.1051/0004-6361:20079062.

[dangelo_marzari_2012-7] D'Angelo, G.; Marzari, F. (2012). "Outward Migration of Jupiter and Saturn in Evolved Gaseous Disks". The Astrophysical Journal. 757 (1): 50 (23 pp.). arXiv: 1207.2737 . Bibcode:2012ApJ...757...50D. doi:10.1088/0004-637X/757/1/50.

[Morbidelli_et_al_2007-8] Morbidelli, Alessandro; Tsiganis, Kleomenis; Crida, Aurélien; Levison, Harold F.; Gomes, Rodney (2007). "Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and Their Relationship to the Current Orbital Architecture". The Astronomical Journal. 134 (5): 1790–1798. arXiv: 0706.1713 . Bibcode:2007AJ....134.1790M. doi:10.1086/521705.

[Batygin_2010-9] Batygin, Konstantin; Brown, Michael E. (2010). "Early Dynamical Evolution of the Solar System: Pinning Down the Initial Conditions of the Nice Model". The Astrophysical Journal. 76 (2): 1323–1331. arXiv: 1004.5414 . Bibcode:2010ApJ...716.1323B. doi:10.1088/0004-637X/716/2/1323.

[Reyes-Ruiz_etal_2014-10] Reyes-Ruiz, M.; Aceves, H.; Chavez, C. E. (2014). "Stability of the Outer Planets in Multiresonant Configurations with a Self-gravitating Planetesimal Disk". The Astrophysical Journal. 804 (2): 91. arXiv: 1406.2341 . Bibcode:2015ApJ...804...91R. doi:10.1088/0004-637X/804/2/91.

[Deienno_etal_2017-11] 1 2 Deienno, Rogerio; Morbidelli, Alessandro; Gomes, Rodney S.; Nesvorny, David (2017). "Constraining the giant planets' initial configuration from their evolution: implications for the timing of the planetary instability". The Astronomical Journal. 153 (4): 153. arXiv: 1702.02094 . Bibcode:2017AJ....153..153D. doi: 10.3847/1538-3881/aa5eaa .

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]