Category Archives: Asteroids: Centaurs

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.

To know more

Feel interested? Please leave a comment!

Surviving as a Trojan of Neptune

Hi! Today I will tell you about a study recently accepted for publication in Astronomy & Astrophysics, by Rodney Gomes and David Nesvorný, on the survival of the asteroids which precede and follow Neptune on its orbit.

The coorbital resonance

In the Solar System, the mean motion resonances are ubiquitous. When the orbital frequencies of two bodies are commensurate, interesting phenomena might happen: they could have a more stable orbit, or they could experience a permanent forcing which raise their eccentricity and / or inclination, and in some cases could result in an ejection. A resonance has particularly strong effects on a small body which orbit resonates with the one of a large planet. This is for instance how the giant planets shaped the asteroid belt.

Here, we deal with the coorbital resonance, which is a very specific and interesting case. This happens between two bodies which have on average the same orbital frequencies, and the perturbations associated result in some zones of stability. In particular, there are five equilibrium positions for the coorbital restricted 3-body problem, i.e. if we consider the Sun, a planet, here Neptune, and a small body. These equilibriums are known as Lagrangian points, and the most remarkable of them are denoted L4 and L5. They precede and follow the planet at an angular distance of 60°, and are stable equilibriums. As a consequence, they are likely to accumulate several small bodies, and this is verified by the observations, which have detected asteroids which coorbit with Jupiter, Uranus and Neptune.

At this time, 6,288 of these objects have been detected for Jupiter, 1 for Uranus, and 17 for Neptune.

The planetary migration

Since 2004 and the first version of the Nice model, the giant planets are assumed to have formed closer to the Sun than they are now, and have migrated to their current orbit. The reason for this migration is that they were form in a large proto-planetary disk, full of planetesimals which drove migration. The asteroids are some of these planetesimals. This raises the following question: could the coorbital (or Lagrangian) asteroids survive this migration?

Long-term numerical integrations

Addressing this problem requires long-term and intensive numerical simulations. The issue is this: you need to simulate the evolution of the Solar System over 4.5 Gyr. For that, you write down the gravity equations ruling the motion of the planets and the planetesimals (these are many objects… the authors considered 60,000 of them), and you propagate them numerically.

To propagate them, you start from a given position and velocity of each of your bodies (initial conditions), and the equations give you the time-derivative at this point. You then use it to extrapolate the trajectory in the time, and you reiterate…

Of course, this algorithm does not give exact results. To lower the error, you should take a small time-step, but a too small time-step requires more iterations, and at each iteration you add an error due to the internal accuracy of the computer. To make your life easier, numerical integrators have been developed to improve the accuracy for a given time-step. In this study, the authors use two very well-known tools, SWIFT and MERCURY, dedicated to the integration of the motion of the planets and asteroids.

In this paper

The authors show it is difficult to get Trojans of Neptune that survive over the lifetime of the Solar System. In a first numerical integration, they do get captures, but none of them survive. Then they consider planetesimals which are very close to the observed Trojan, and they get some captures.

Something interesting is that they show that the orbital inclination of these Trojans can be excited during the migration process. For that, the migration should be slow enough, i.e. over 150 Myr, while previous studies, which assumed a migration ten times faster, did not excite the inclinations up to observed values.

Some perspectives

Even if it is now accepted that the planets have migrated, several competing scenarios exist (Nice, Nice 2, Grand Tack,…) and some are probably to come, just because there are many ranges of initial conditions which are possible, many possible assumptions on the initial state of the proto-planetary nebula… and these scenarios should of course impact the capture of Trojans of the giant planets.