The Nice model of Solar System

When I see the previous video (via Keplero) I immediatly thought to the Nice model, a simulation model about our Solar System developed by Rodney Gomes, Harold F. Levison, Alessandro Morbidelli, Kleomenis Tsiganis in three papers published on Nature vol.235
First of all a little resume about Solar System formation thoery. Following Kant-Laplace model, our Solar System was born from massive and dense clouds of molecular hydrogen—giant molecular clouds. In this nebula occurs planets formation. In particular theory supposed that giant planets formed on circular and coplanar orbits^{(3, 5)}. In this picture, all planets aresubstantially formed in the same position of the actual System. The Nice model suggests that all Solar System objects are formed in a different position and a perturbation in orbits forced the actuall more stable orbits.
The original model's core was developed by Gomes, Morbidelli and Levison in 2004⁽⁷⁾

We study planetary migration in a gas-free disk of planetesimals. In the case of our Solar System we show that Neptune could have had either a damped migration, limited to a few AUs, or a forced migration up to the disk’s edge, depending on the disk's mass density. We also study the possibility of runaway migration of isolated planets in very massive disk, which might be relevant for extra-solar systems. We investigate the problem of the mass depletion of the Kuiper belt in the light of planetary migration and conclude that the belt lost its pristine mass well before that Neptune reached its current position. Therefore, Neptune effectively hit the outer edge of the proto-planetary disk. We also investigate the dynamics of massive planetary embryos embedded in the planetesimal disk. We conclude that the elimination of Earth-mass or Mars-mass embryos originally placed outside the initial location of Neptune also requires the existence of a disk edge near 30AU.

In this first paper there's an analytic toy model for migration process. First of all they calculate the variation in time of the semi-major axis $a_P$ of the planet:
\[\frac{\text{d} a_P}{\text{d} t} = \frac{k}{2 \pi} \frac{M(t)}{M_P} \frac{1}{\sqrt{a_P}}\]
where $M(t)$ is the amount of material in orbits that cross the orbit of the planet, $M_P$ the mass of the planet, $k$ a parameter of the distribution of those orbits.
The evolution of $M(t)$ is described by the following equation:
\[\dot M (t) = -\frac{M(t)}{\tau} + 2 \pi a_P |\dot a_P| \sigma (a_P)\]
where $\tau$ is decay time of planetesimals, $\sigma$ the surface density of not yet scattered planetesimals, $\dot a_P$ the planetary migration rate, $\dot M (t)$ the decay of the planetesimal population due to the planetesimal's finite dynamical lifetime.
Sobstituting the first equation in the second, it can obtain:
\[\dot M (t) = \left ( \frac{1}{\tau} + |k| \sqrt{a_P} \frac{\sigma(a_P)}{M_P} \right ) M(t)\]
And the solution of this equation is given by
\[M(t) = M(0) \text{e}^{\alpha t}\]
where
\[\alpha = \frac{1}{\tau} + |k| \sqrt{a_P} \frac{\sigma(a_P)}{M_P}\]
that is time independent.
We can have two different situations: $\alpha$ negative, $\alpha$ positive.
In the first case the migration speed is too low to compensate the loss of planetesimals: the migration mode is called damped migration.
In the second case $M(t)$ grows exponentially, and migration, called forced migration, is self-sustained.

After the giant planets were formed and the circumsolar gaseous nebula was dissipated, the Solar System was composed of the Sun, the planets and a debris disk of small planetesimals.

Planets' migration so is caused by the change of angular momentum during the scattering with planetesimals.

Numerical simulations⁽⁴⁾ show that Jupiter was forced to move inward, while Saturn, Uranus and Neptune drifted outward.

An example of the output produced by Nice simulations is the following plot:

where MMR means mean motion resonance, $a$ the semimajor axis, and $q$ is the minimum and $Q$ the maximum of heliocentric distances. The plot is taken from a
N-body simulation with 35$M_E$ 'hot' disk composed of 3500 particles and truncated at 30 AU. Planets are Uranus (U), Neotune (N), Saturn (S), Jupiter (J).

The secular perturbations that Jupiter and Saturn exert onUranus andNeptune force the eccentricities of the ice giants to increase by an amount that depends on the masses and semimajor axes of all planets⁽⁶⁾.

So, following researchers,amorecompact system is more chaotic and unstable, and ice giants scattered outward.

Thus, the flux of small bodies towards Saturn and Jupiter, and hence their rate of migration, increases abruptly.

Consequently to this fast migration period, planetary eccenticities and inclinations became more stable. And...

The final orbits of the planets depend on the evolution of the system immediately after the resonance crossing event.

So simulation of Nice model explain the actually characteristics of the orbits of Saturn, Uranus, Jupitar, Neptune. They also conclude that

• Saturn suffered an encounter with one or both ice giants;
• The final orbital separation of Jupiter and Saturn depends on the amount of mass that they process during the evolution of the system;
• The eccentricities of Jupiter and Saturn are probably the result of the fact that these planets crossed the 1:2 MMR;
• The actual disk may indeed have been by the presence of a large number of Pluto-sized objects^{(1, 14)}

Another important success of Nice model is the capturing of trojans by Jupiter and Neptune: we define trojans a lot of asteroids captured by the two planets and sharing their orbits. In this case there is some interesting conclusions:

• Trojans origins are probably in the progenitor of Oort cloud and Kuipert belt.
• Trojans are apparently poor in water and organics. Simulations explain also this fact: before capturing, asteroids have a very eccentricity phase, so they relatively close to the Sun.
• Trojans are captured immediatly after Jupiter and Saturn crossed.

In conclusion, researchers

believe that the
Trojans represent observational evidence for this resonance crossing, which has also been shown elsewhere⁽⁸⁾ to produce the correct planetary orbits.

After the study of the external planets, Gomes and collegues focused their attention to the internal planets, in particular to study the origin of the late heavy bombardment period⁽¹⁰⁾: during this period (approximately 4.1 to 3.8 billion years ago) a large number of asteroids' impacts shooted moon surface and, for inference, also Earth, Mars, Venus, Mercury. Following results of their simulations, Gomes and collegues inferred that the LHB was caused by giant planets migration.
The results are showed using the following images:

Left the Solar System before giants' migration; center the beginning of the migration; right the Solar System after migration.
If you prefer, here there is the official video simulation:

Nice scheme for the LHB

is the result of a generic migration-delaying mechanism, followed by an instability, which is itself induced by a deterministic mechanism of orbital excitation of the planets⁽⁸⁾.This revised planetary migration scheme naturally accounts for the currently observed planetary orbits⁽⁸⁾, the LHB, the present orbital distribution of the main-belt asteroids and the origin of Jupiter's Trojans⁽⁹⁾.

In 2008 the same group with the add of Christa VanLaerhoven, try to explain the formation of Kuipert belt⁽¹²⁾.

The Kuiper belt is the relic of the primordial planetesimal disk, shaped by various dynamical and collisional processes that occurred when the Solar System was evolving towards its present structure.

In this case researchers performed a lot of simulations, becuse:

in all the simulations that we performed of the Nice model, the number of objects left in the classical Kuiper belt at the end of the simulations was too small to be analyzed. We believe that there are two reasons for this; both related to the fact that we used a relatively small number of disk particles. First, we do not expect the capture efficiency to be very large and thus we should not have captured many objects. Perhaps more importantly, Neptune's orbital evolution had a stochastic component that was too prominent, due to the fact that the disk was represented by unphysically massive objects.

They updated simulation using information extracted from other papers^{(2, 11)} and using the following expression for gravitational force ${\vec F}_{UN}$ between Uranus and Neptune:
\[\vec F_{UN} = G \frac{m_U m_N}{(\vec x_{UN}^2 + \vec \epsilon^2)^{\frac{3}{2}}}\vec x_{UN}\]
where $\vec x_{UN}$ is the vector of relative position between Uranus and Neptune, $\vec \epsilon$ is a constant vector with modulo 0.8 AU.
They performed in particular three runs in which they changed the distribution of the objects. For example we can see the results of the trans-neptunian region:

In conclusion: the Nice model, named by the location of the Observatoire de la Côte d'Azu, in which model was developed (the original group of researchers are there in the same moment!), explain the evolution of Solar System starting from a more compact position of planets. After a distortion, that changed an unstable situation of orbits, the giants migrated in actual position, generated the capture of trojans, the Late Heavy Bombardment period. In a further analisys, the group try to explain observation about Kuipert belt in order to obtain a more strong evidence for their model. The results don't perform an ultimately confirmation of the model, but seem to indicate that the way is right.
I personally think that the following quotation⁽¹³⁾ is a precise indication about the index of model in astronomy community:

In the frame of this model, the orbital and size distribution of the Trojan and Hildas asteroids can be explained. The structure of the Kuiper Belt is also well reproduced. In
addition, the irregular satellites of Saturn, Uranus and Neptune can be captured during the global instability. For these reasons, the Nice model is one of the most impressive results of the last decade on Solar System formation.

Image extracted from When giants roamed by Joe Hahn

Observation In (12) there is the first time in which authors call Nice model with a more formal name: the model of the dynamical evolutionof the giant planets.

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Other links about Nice model:
The Nice Way To Build A Solar System by Steve Nerlich
The Nice way to make a solar system by John G. Cramer
Uranus, Neptune Swapped Spots, New Model Says by Anne Minard

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(1) S. Stern On the number of planets in the outer solar system - Evidence of a substantial population of 1000-km bodies. Icarus 90, 271-281 (1991)
(2) R. Malhotra, The origin of Pluto's orbit: Implications for the Solar System beyond Neptune, Astron. J. 110, 420–429 (1995)
(3) Patrick J.B. et al., Formation of the giant planets by concurrent accretion of solids and gas. Icarus 164, 62-85 (1996)
(4) J.M. Hans, R. Malhotra Orbital evolution of planets embedded in a planetesimal disk. Astron. J. 117, 3041-3053 (1999)
(5) Lubow S.H., Seibert M., Artymowicz P. Disck accretion onto high-mass planets. Astrophys. J. 526, 1001-1012 (1999)
(6) C. Murray, S.F. Dermott Solar System Dynamics, Cambridge University Press, Cambridge, UK (1999)
(7) Rodney Gomes, Alessandro Morbidelli, Harold Levison Planetary migration in a planetesimal disk: why did Neptune stop at 30 AU? Icarus 470, 492-507 (2004)
(8) K. Tsiganis, R. Gomes, A. Morbidelli, H.F. Levison, Origin of the orbital architectureof the giant planets of the Solar System. Nature 435 (459-461) (2005)
(9) Chaotic capture ofJ upiter's Trojan asteroids in the early Solar System, Nature 435, 462-465 (2005)
(10) R. Gomes, H.F. Levison, K. Tsiganis, A. Morbidelli, Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets, Nature 435, 466-469 (2005)
(11) J. Kominami, H. Tanaka, S. Ida, Orbital evolution and accretion of protoplanets tidally interacting with a gas disk, Icarus 178, 540–552 (2005)
(12) H.F. Levison, A. Morbidelli, C. VanLaerhoven, R. Gomes, K. Tsiganis, Origin of the structure of the Kuiper belt during a dynamical instability in the orbits of Uranus and Neptune, Icarus 196, 258–273 (2008)
(13) A. Crida, Solar System formation (2009 - arXiv)
(14) For space vagabond, it could be interesting T. Sumi et al. Unbound or distant planetary mass population detected by gravitational microlensing. Nature, 473 (7347), 349-352 (2011)

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Gomes, Rodney, Morbidelli, Alessandro, & Levison, Harold (2004). Planetary migration in a planetesimal disk: why did Neptune stop at 30 AU? Icarus, 170 (2), 492-507 DOI: 10.1016/j.icarus.2004.03.011

Tsiganis, K., Gomes, R., Morbidelli, A., & Levison, H. (2005). Origin of the orbital architecture of the giant planets of the Solar System Nature, 435 (7041), 459-461 DOI: 10.1038/nature03539

Morbidelli, A., Levison, H., Tsiganis, K., & Gomes, R. (2005). Chaotic capture of Jupiter's Trojan asteroids in the early Solar System Nature, 435 (7041), 462-465 DOI: 10.1038/nature03540

Gomes, R., Levison, H., Tsiganis, K., & Morbidelli, A. (2005). Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets Nature, 435 (7041), 466-469 DOI: 10.1038/nature03676

Levison, H., Morbidelli, A., VanLaerhoven, C., Gomes, R., & Tsiganis, K. (2008). Origin of the structure of the Kuiper belt during a dynamical instability in the orbits of Uranus and Neptune Icarus, 196 (1), 258-273 DOI: 10.1016/j.icarus.2007.11.035