Planetary dynamics

In a planetary system, planets mutually interact gravitationally, which modifies their orbits around the central star. These orbital variations can be:

Fast, of the order of the orbital period. For instance, this is the case in an unstable system from which a planet is ejected. The semi-major axis of the planet suffers from a random walk until ejection.
Resonant, or semi-fast. This is the case if there is a commensurability between the orbital periods of two planets. For instance, if the orbital period of a planet is equal to two times the orbital period of another planet in the system. The semi-major axis and the eccentricity of both planets then oscillate regularly.
Secular, of the order of several hundred of orbital periods. This is the case if the system is short-term stable and far from resonances. The semi-major axes of planets is then constant. Orbits are deformed and rotates in space: this is what we call the "orbital precession".

These types of evolution are not mutually exclusive. For instance, a resonance can produce a strong secular evolution of the eccentricity of a planet, eventually leading to close encounters between planets, and therefore to fast orbital chaos.

In today's Solar System, the orbits of the planets mostly vary in a secular way. The secular dynamics of the terrestrial planets is yet chaotic. This secular chaos (or "slow chaos") may lead to colisions between planets on a timescale of several billions of years.

Refinement of orbital elements

Observed exoplanetary systems generally have ages ranging between several hundred million years and several billions of years. Therefore, it is very unlikely to observe them in a state of fast chaos able to shortly lead to planetary ejections or collisions.

This fact can be used to refine our knowledge of the orbits of these exoplanets: for this, one just needs to exclude orbital parameters leading to fast chaos.

As part of a study led by Sophia Sulis, and together with Charlie Lebarbé, we have used this method to constrain the mass and orbital inclination of the exoplanet HD 73344 c. However, the constraint set by the system's stability is not very stringent.

© Sulis et al. (2024), A low-mass sub-Neptune planet transiting the bright active star HD 73344, A&A 688, A14 — **Figure:** Stability map of the exoplanet HD 73344 b as a function of the orbital elements of the exoplanet HD 73344 c. Red regions show a strong short-term chaos. Black regions are very stable. The observed location is shown by a cross.

Yet, because of secular mutual interactions between the two planets in this system, the orbit of the interior planet, HD 73344 b, may strongly change over time. As a result, this exoplanet may only transit in front of its host star (as observed today) during a fraction of its secular evolution.

This fact can be used to strongly constrain the inclination and mass of exoplanet HD 73344 c: for this, one just needs to exclude orbital parameters for which the transit probability of planet b is very low [xxv].

Long-term evolution

Pulsars are remnants of massive stars at the end of their life. They spin very fast and emit a periodic radiation that can be observed from the Earth. The precise timing of this signal can be used to detect minute gravitational perturbations in the neighbourhood of pulsars. This method led to the discovery of the first exoplanets.

I am involved in the project of Guillaume Voisin in his study of pulsar PSR J0337+1715, which is orbited by two white dwarfs. A particular kind of signal in the timing of this pulsar led him to propose the existence of a small planet in orbit around this triple system, whose mass would be about half the mass of our Moon.

© Voisin et al. (2024), Explanation of the exceptionally strong timing noise of PSR J0337+1715 by a circum-ternary planet and consequences for gravity tests, A&A in press — **Figure:** Long-term evolution of the orbital elements of the planet proposed to orbit the triple system PSR J0337+1715.

Because of its hierarchical architecture, this system essentially interacts in a secular way. We have shown that the orbit proposed for the small planet is long-term stable, thanks to its peculiar inclination lying in the neighbourhood of an equilibrium point [xxvii]. Hence, this planet may be the only survivor of a population of small planets that had less stable orbits.

Last update: 2024-11.