Abstract
Context. As a result of Titan’s migration and Saturn’s probable capture in secular spin–orbit resonance, recent works show that Saturn’s obliquity could be steadily increasing today and may reach large values in the next billions of years. Satellites around high-obliquity planets are known to be unstable in the vicinity of their Laplace radius, but the approximations used so far for Saturn’s spin axis are invalidated in this regime. Aims. We aim to investigate the behaviour of a planet and its satellite when the satellite crosses its Laplace radius while the planet is locked in secular spin–orbit resonance. Methods. We expand on previous works and revisit the concept of Laplace surface. We use it to build an averaged analytical model that couples the planetary spin-axis and satellite dynamics. Results. We show that the dynamics is organised around a critical point, S1, at which the phase-space structure is singular, located at 90° obliquity and near the Laplace radius. If the spin-axis precession rate of the planet is maintained fixed by a resonance while the satellite migrates outwards or inwards, then S1 acts as an attractor towards which the system is forced to evolve. When it reaches the vicinity of S1, the entire system breaks down, either because the planet is expelled from the secular spin–orbit resonance or because the satellite is ejected or collides into the planet. Conclusions. Provided that Titan’s migration is not halted in the future, Titan and Saturn may reach instability between a few gigayears and several tens of gigayears from now, depending on Titan’s migration rate. The evolution would destabilise Titan and drive Saturn towards an obliquity of 90°. Our findings may have important consequences for Uranus. They also provide a straightforward mechanism for producing transiting exoplanets with a face-on massive ring, a configuration that is often put forward to explain some super-puff exoplanets.
Highlights
We aim to investigate the behaviour of a planet and its satellite when the satellite crosses its Laplace radius while the planet is locked in secular spin–orbit resonance
We show that the dynamics is organised around a critical point, S1, at which the phase-space structure is singular, located at 90◦ obliquity and near the Laplace radius
A secular spin–orbit resonance occurs when the spin-axis precession rate of a planet becomes commensurate with a frequency that appears in its orbital precession
Summary
A secular spin–orbit resonance occurs when the spin-axis precession rate of a planet becomes commensurate with a frequency (or a combination of frequencies) that appears in its orbital precession. The obliquity of a planet cannot increase forever, and there must exist some kind of dynamical barrier, either on the planet’s or on the satellite’s side, that would halt the tilting at some point Even though this final outcome may not be directly relevant for Saturn and Titan because of the large timescales involved (see below), its generic nature makes it important even from the point of view of pure celestial mechanics as other planets and exoplanets may have been affected. Their study confirms the secular instabilities reported by Tremaine et al (2009) and Tamayo et al (2013), and their fully unaveraged model allows for other kinds of instability to appear, driven by the evection and ‘ivection’ resonances2 These previous results show that studying Saturn’s tilting mechanism in a general way requires one to keep an eye on both the satellite’s and planet’s dynamics.
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