On the dynamics of small bodies beyond Neptune

Hi there! Today I will present you a study on the possible dynamics of some Trans-Neptunian Objects (TNOs). This study, Study and application of the resonant secular dynamics beyond Neptune by M. Saillenfest, M. Fouchard, G. Tommei and G.B. Valsecchi, has recently been accepted for publication in Celestial Mechanics and Dynamical Astronomy.
This is a theoretical study, which presents some features of the dynamics that could one day be observed. This manuscript follows another one by the same authors, in which a theory of the “resonant secular dynamics” is presented. Here it is applied to small bodies, which are thought to be in mean-motion resonances with Neptune. This study results from a French-Italian collaboration.

The Kozai-Lidov mechanism

The dynamics that is presented here uses the so-called Kozai-Lidov mechanism. This is a mechanism which has been simultaneously and independently discovered in Russia (by Lidov) and in Japan (by Kozai), and which considers the following configuration: a massive central body, another massive one called the perturber, and a test-particle, i.e. a massless body, which orbits the central one. This problem is called the Restricted 3-body problem. Originally, the central body was the Earth, the perturber the Moon, and the test-particle an artificial satellite of the Earth. In such a case, the orbit of the test-particle is an ellipse, which is perturbed by the perturber; this results in variation of the elliptical elements, i.e. eccentricity, inclination… moreover, the orientation of the ellipse is moving…

To describe the problem, I need to introduce the following orbital elements:

  • The semimajor axis a, which is half the long axis of the orbit,
  • the mean anomaly M, which locates the satellite on the ellipse,
  • the eccentricity e, which is positive and smaller than 1. It tells us how eccentric the orbit is (e=0 means that the orbit is circular),
  • the pericentre ω, which is the point of the orbit which is the closest to the central body (undefined if the orbit is circular),
  • the inclination I, which is the angle between the orbital plane and the reference plane,
  • the ascending node Ω, which locates the intersection between the orbital plane and the reference plane.

The Kozai-Lidov mechanism allows a confinement of the pericentre with respect to the ascending node, and it can be shown that it results in a raise of the eccentricity of the inclination. Exploiting such a mechanism gives frozen orbits, i.e. configurations for which the orbit of an artificial orbiter, even inclined and eccentric, will keep the same spatial orientation.

These recent years, this mechanism has been extended for designing space missions around other objects than the Earth, but also to explain the dynamics of some exoplanetary systems, of small distant satellites of the giant planets, and of Trans-Neptunian Objects, as it is the case here. In this last problem, the central body is the Sun, the perturber is a giant planet (more specifically here, it is Neptune), and the test-particle is a TNO, with the hope to explain the inclined and eccentric orbit of some of them. A notable difference with the original Kozai-Lidov problem is that here, the test-particle orbits exterior to the perturber. Another difference is that its dynamics is also resonant.

Resonant and secular dynamics

The authors do not speak of resonant secular dynamics, but of dynamics that is both resonant and secular. The difference is that the involved resonance is not a secular one. Let me explain.

The authors consider that the TNO is in a mean-motion resonance with Neptune. This implies an integer commensurability between its orbital period around the Sun and the one of Neptune, with results in large variations of its semi-major axis. If we look at the orbital elements, this affects the mean anomaly M, while, when a resonance is secular, M is not affected.

So, these objects are in a mean-motion resonance with Neptune. Moreover, they have an interested secular dynamics. By secular, I mean that the mean anomaly is not affected, but something interesting involves the node and/or the pericentre. And this is where comes Kozai-Lidov. The paper studies the objects which are trapped into a mean-motion resonance with Neptune, and which are likely to present a confinement of the pericentre ω, which could explain a significant eccentricity and a high inclination.

For that, they make an analytical study, which theory had been developed in the first paper, and which is applied here.

Why an analytical study?

The modern computing facilities allow to simulate the motion of millions of test-particles over the age of the Solar System, in considering the gravitational interaction of the planets, the galactic tide, a star passing by… and this results in clusters of populations of fictitious TNOs. Very well. But when you do that, you do not know why this particular object behaves like that. However, an analytical study will give you zones of stability for the orbits, which are preferred final states. It will tell you: there will probably be some objects in this state, BECAUSE… and in the case of this study, the because has something to do with the Kozai-Lidov mechanism. Moreover, the because also gives you some confidence in your results, since you have an explanation why you get what you get.

To make things short, a numerical study shows you many things, while an analytical one proves you a few things. A comprehensive study of the problem requires combining the two approaches.

This paper

This paper specifically deals with fictitious objects, which are in mean-motion with Neptune, and are likely to be affected by the Kozai-Lidov mechanism. After many calculations presented in the first paper, the authors show that the problem can be reduced to one degree of freedom, in a Hamiltonian formalism.

The Hamiltonian formalism is a common and widely used way to treat problems of celestial mechanics. It consists in expressing the total energy of the problem, i.e. kinetic + potential energy, and transform it so that trajectories can be described. These trajectories conserve the total energy, which may seem weird for a physical problem. Actually there is some dissipation in the dynamics of TNOs, but so small that it can be neglected in many problems. The most recent numerical studies in this topic consider the migration of the planets, which is not a conservative process. In the paper I present you today, this migration is not considered. This is one of the approximations required by the analytical study.

The remaining degree of freedom is the one relevant to the Kozai-Lidov mechanism. The one associated with the mean-motion resonance is considered to be constant. For that it involves the area enshrouded by the libration of the resonant argument, which is constant (hypothesis of the adiabatic invariant). So, the authors get a one degree-of-freedom Hamiltonian, for which they draw phase spaces, showing the trajectory in the plane q vs. ω, q=a(1-e) being the distance between the Sun and the pericentre of the TNO, i.e. its closest distance to the Sun. These phase portraits depend on other parameters, like the mean-motion resonance with Neptune that is considered, and a parameter η, which combines the inclination and the eccentricity.

The results are a catalog of possible trajectories, some of them presenting a confinement of the pericentre ω. For a large cloud of objects, this would result in an accumulation of pericentres in a constrained zone. The authors try to find confirmation of their results with existing objects, but their limited number and the inaccuracy on their location make this comparison inconclusive. They also point out that the orbits of Sedna and 2012VP113 cannot be explained by this mechanism.


The future observations of TNOs will give us access to more objects and more accurate trajectories, and it is to be hoped that some of them will fit into the trajectories found by the authors. That would be a great success for that, and that would be deserved regarding the effort necessary to achieve such an analytical study.

As I said, such a problem needs analytical and numerical studies, but some of the authors (Marc Fouchard and Giovanni Valsecchi) are also involved in such a numerical exploration, which starts from a fictitious Oort cloud and simulates the excitation of the eccentricity and inclination of some of the objects.

For the two studies to meet, it should also be investigated how the planetary migration, which results from models of formation and evolution of the Solar System, affects the zones of stability due to the Kozai-Lidov mechanism.

Finally, we should not forget the quest for the Planet Nine. As the authors honestly point out, an additional planet could break down some of the conclusions.

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