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!

A periodic variation in the atmosphere of Venus

Hi there! Today’ post will be my first on Venus. More precisely, it deals with its atmosphere. As you may know, the planet Venus is known for its very thick atmosphere, which precludes optical observations of its surface. The study I present today is entitled “Discovery of a 150 day period in the Venus condensational clouds”, by Kevin McGouldrick and Constantine Tsang, who work in the city of Boulder, CO (I love this place). This study has been recently accepted for publication in Icarus.

Some Venus facts

Venus is the second innermost planet of the Solar System, which means that its orbit is interior to the one of the Earth. It is sometimes said to be a twin sister of the Earth because its diameter is 95% the diameter of the Earth. However, the meteorological conditions make it a very hostile place for life. The surface pressure is ~93 times the one of the Earth, the temperature is about 470˚C, and the atmosphere is essentially made of carbon dioxide.

Its rotation is very interesting, since it rotates very slowly, and in the retrograde direction. It has a rotation period of 245 days, while its orbital period around the Sun is only 225 days. This means that a Venusian day is longer than a Venusian year. This peculiar rotational state could result from the atmospheric tides, i.e. the way the dense atmosphere interacts with the gravitational forcing of the Sun, loses some energy, and also interacts with the surface. However, the atmosphere moves much faster, with a period of about 4.2 days.

The exploration of Venus

As a putative twin sister of the Earth and a nearby planet, Venus has been a priority target of the Space Race. This is why several American and Soviet probes reached it between 1962 and 1984, allowing major progress in our knowledge of the planet. Here are the probes:

  • 1962: Mariner 2 (USA). This probe was the first one to perform successfully a flyby of another planet than the Earth. It proved that the surface was hot, detected no magnetic field, and it improved our knowledge of the mass of the planet. Beside these results of Venus, it made measurements of the Solar wind and allowed many technological improvements in space navigation and telecommunication.
  • 1965: Venera 4 (USSR), Mariner 5 (USA). Venera 4 crashed on Venus after a fall in the atmosphere with a parachute, permitting the first in situ measurements of its chemical composition, and detection of a weak magnetic field, which Mariner 2 could not have detected. Mariner 5 made a flyby of Venus and analyzed its outer atmosphere.
  • 1969: Venera 5 & 6 (USSR) were technologically similar to Venera 4, but with specific improvements of the analysis of the atmosphere, based on the results of Venera 4.
  • 1970: Venera 7 (USSR) was the first probe to land on another planet than the Earth and to transmit data from the surface. It made the first accurate measurement of the temperature and the pressure at the surface.
  • 1972: Venera 8 (USSR) showed that the atmosphere of Venus was pretty clear below 50 km, meaning that the clouds had a higher altitude.
  • 1975: Venera 9 & 10 (USSR). These two probes were the first ones to send images of the surface of another planet than the Earth. Moreover, Venera 10 measured the velocity of the wind.
  • 1978: Venera 11 & 12 (USSR), Pioneer Venus Multiprobe (USA). Venera 11 & 12 made more accurate measurements of the composition of the atmosphere, and detected lightning and thunder. Pioneer Venus Multiprobe launched 4 probes to the surface of the planet, to analyse the atmosphere during their fall. One of these probes survived the impact, but did not have any imaging instrument. These probes identified 3 layers of clouds in the atmosphere.
  • 1978-1992: Pioneer Venus Orbiter (USA). This spacecraft was the companion of Pioneer Venus Multiprobe, and was inserted into orbit on Dec 4th 1978. Its orbit was very eccentric (0.8), and it contained 17 instruments, allowing to study the magnetic field of Venus, its gravity field, its atmosphere… It also monitored the water loss of the Halley’s comet in 1986.
  • 1981: Venera 13 & 14 (USSR) were landers, they made measurements of the atmosphere during the fall and took images of the surface.
  • 1983: Venera 15 & 16 (USSR). These probes were orbiters equipped with radars. They mapped ~25% of the surface.
  • 1984: Vega 1 & 2 (USSR + Europa). These two probes made a flyby of Venus to launch a lander devoted to make measurements of the atmosphere. After the flyby, the probes approached Halley’s comet and took ~1,500 images of it.
  • 1990: Flyby by Galileo (USA). Galileo was sent to Jupiter, but used the gravitational assistance of Venus on its way. This was the opportunity to study the composition of the clouds of Venus, in comparing the measurements at 1.74 and 2.30 μm, i.e. in the infrared. These two bandwidths correspond to minimal absorption by carbon dioxide and by water, so they can be used not only to detect a signal from the surface of Venus, i.e. the Solar light reflected by the surface, but also to estimate the temporal evolution and the composition of the clouds.
  • 1990-1994: Magellan (USA). This orbiter studied the gravity field of the planet, and also provided a detailed map. It particularly revealed the presence of many volcanoes, few impact craters and large lava plains, meaning that the surface is geologically young, and evidence of some tectonic activity, which is pretty different than the terrestrial one. It was revealed by low domical structures called coronae, produced by the upwelling and subsidence of magma from the mantle.
  • 1998-1999: 2 flybys by Cassini (USA), on its way to Saturn.
  • 2006-2015: Venus Express (Europa), see next paragraph.
  • Since 2015: Akatsuki (Japan). This spacecraft should have orbited Venus since 2010, but that maneuver failed. It then orbited the Sun during 5 years in safe mode before succeeding another orbital insertion in December 2015. This spacecraft essentially studies the dynamics of the atmosphere of Venus during a 2 year regular scientific mission, which has started in May 2016.

Venus Express (VEX)

This ESA spacecraft has been launched in November 2005, and was inserted in orbit in April 2006, originally for a 2-year mission… which was completed 9 years later! The main objective of that mission was to understand the dynamics of the atmosphere of Venus, with the hope of a better understanding of the atmospheric evolution in general. It contained 7 instruments, 3 of them being devoted to spectrometry (VIRTIS, SPICAV and PFS), one to radioscience (VeRa, for Venus Radioscience), one was a magnometer (MAG), one for imaging (VMC, for Venus Monitoring Camera), and the last one, ASPERA-4, investigated the interaction between the Solar wind and the Venusian atmosphere. We are today particularly interested by VIRTIS, for Visible and Infrared Thermal Imaging Spectrometer, which measured the emitted radiance in 1.74 μm and 2.30 μm of the night-side of Venus.
Venus Express had a polar and highly eccentric orbit. Its high eccentricity resulted in a large variation of the distance between the probe and the planet, i.e. from 460 to 63,000 km, with a period of 24 hours. As a consequence, the field of view and resolution of the measurements experienced large variations.
An interesting thing to notice is the fact that Venus Express reused some technologies designed for Mars Express and Rosetta.

This paper

The authors analyzed the emitted radiance in the infrared at different latitudes, for the two wavelengths 1.74 μm and 2.30 μm. Unfortunately, they do not have measurements later than 2008 October 27, because of the failure of the instrument’s cooling system (keep in mind that infrared is very sensitive to the temperature). Moreover, they used only data taken at a distance larger than 10,000 km. The variation of this radiance characterizes the dynamics of the lower region of the clouds, at an altitude between 50 and 55 km. Observing at these two wavelengths permits to draw conclusions on the size of the particles constituting the clouds. Actually, 4 sizes of particles are expected in the clouds of Venus, and in this specific region:

  • Mode 1 particles: they have an average diameter of 0.6 μm, and are expected in the upper region,
  • Mode 2 particles: they have an average diameter of 2 μm, and are expected in the upper region as well,
  • Mode 2′ particles: they have an average diameter of 3 μm, and are expected in the lower and middle regions,
  • Mode 3 particles, with a diameter of 7 μm, are expectd in the lower region.

So, for our lower clouds, we expect only Mode 2′ and Mode 3 particles.

The authors used VIRTIS data, and after denoising they averaged the measurements over 7 days, since they are interested only in the long-term dynamics. Since the atmosphere is rotating, the authors could thus only detect variations in time and in latitude, but not in longitude.

And the results are these: the radiance steadily increases at mid-latitudes, while it decreases near the poles, which could reveal a circulation of clouds over a very-long term. This long-term variation should be a periodic effect, which future measurements by Akatsuki should help to understand.
Moreover, the authors noticed a 150-day periodic variation in the cloud coverage, especially in the 1.74 μm radiance data, at mid-latitude. This is an unexpected result, which had already been hinted by the same authors 4 years before, with less data. The cause of this periodicity still needs to be elucidated. The authors notice that this period is almost two thirds of the rotation period of Venus, but this may be by chance. This could be the manifestation of a Hadley-like circulation, i.e. a kind of circular motion of the atmosphere driven by variations of its temperature, itself controlled by the latitude and the altitude.

Some links

That’s it for today! As usual, I am interested in your feedback. Let me know what you think about this article, what kind of articles you are interested in, if you have specific questions on the science behind…