Article on “Steady state obliquity…” published in CM&DA


In 2018 the space mission Bepi-Colombo will start its long-term journey to visit our innermost planet. It is Europe’s first mission to Mercury. The mission objectives include the investigation of the origin and evolution of planets very close to their parent star. I am interested in the orbital and rotational evolution of this planet. My article on the steady state obliquity of a rigid body has been published in CM&DA. In this work I investigated the influence of the gravitational field on the axial tilt and coupling between the rotation and orbit of planet Mercury. The axial tilt, or obliquity, of Mercury is close to but not exactly zero, and the orbital period of Mercury is about 88 days while the rotation period is very close to 2/3 of this value. Mercury therefore is currently situated in a 3:2 spin-orbit resonance with non-zero obliquity. The magnitude and stability of the axial tilt is determined by the kind of spin-orbit resonance and the gravitational field, i.e. the internal mass distributions of the planet. Mercury, very probably, was not always situated in the 3:2 spin-orbit resonance. Formation studies of the planets assume that planets form at much higher rotation rates. Over time dissipative effects like tidal friction slows down the rotation period to allow the coupling between the orbit and the rotation of the planet. In my study I investigate the influence of different spin-orbit resonances and an extended gravitational field on the magnitude of the axial tilt. I derive simple formulas that allow to predict this steady state obliquity for given spin-orbit resonance and gravitational field parameters. I also derive a simple mathematical model that allows to investigate the oscillations of Mercury around exact spin-orbit resonance. The amplitude and periods of these oscillations are the result of other physical effects, i.e. the internal composition of Mercury.



I was invited speaker in the 7th International Meeting on Celestial Mechanics. The meeting took place in San Martino al Cimino, a beautiful village situated near the city of  Viterbo, Italy. I would like to thank the organizers for the kind invitation, and the organization of the meeting. My presentation took place in the Monday afternoon session, where I summarized my recent scientific findings on Cassini state 1. It has also been made available on youtube thanks to the organizers.

Cassini states correspond to special orientations of the spin axis of a celestial body. If the orbit of a celestial body gets perturbed then the orientation of the rotational axis of the body tries to keep its alignment with the normal axis of the orbital plane. The steady state solution corresponds to the point in the figure.  The closed curves surrounding this point correspond to solutions of the problem where the rotational axis starts oscillating around the steady state solution.

The figure was created using Wolfram Mathematica and was first published in my article of CM&DA. See also my related work (I, II) as a co-author done with Marco Sansottera and Anne Lemaitre, and Benoit Noyelles (III).