Unstable moons around spherical planets
In this work, we reverse the well-established notion that the orbital dynamics of close-in moons is controlled by the oblateness of their host planet. Instead of oblate planets, we place the moons around spherical planets, in a stable two-planet system mutually inclined by 10°. Some of the moons, surprisingly to the first sight, goes unstable by acquiring high eccentricities and inclinations, as shown on the right. The constantness in semi-major axis points it to a secular problem.
The explanation lies in how the orbital precession of the moon works in the planetary system. Normally, the orbital plane of a close-in moon (blue arrow in the plot below) precesses around the spin axis of the planet (orange arrow) due to the torque exerted by the planet's oblateness.
If we view the entire system in the frame with the moon hosting planet as the center, as in the above plot, the star orbits the host planet and is seen as a ring instead of a point mass by the moon, due to its much larger orbital period. Therefore, the star acts like an external oblateness and the moon's orbital plane also precess around the star's orbital plane.
Therefore, there're two torques acting on the moon - the planet oblateness and the star. For the close-in moons, because of their proximity to the planet, the precession is mainly driven by the planet's oblateness. For the Galilean moons, the precession periods range from a few to several hundered years.
The precession driven by the star is slower, since the stellar torque decreases towards the planet's center. For moons on Io-like orbits around a Jupiter, the stellar driven torque is of the order 10,000 years. The star's orbital plane also precesses around the system's average plane, since it's a non-coplanar system. The precession periods range from the order of 1000 to 10,000 years. Therefore, when the moon is placed around a spherical planet and is only slowly torqued around by the star, its orbital precession can hit resonances with the orbital precession of the star.
The left plot shows the Hamiltonian phase space plot of a close-in moon on an Io-like orbit around a Jupiter mass planets at 5 AU at different simulations corresponding to different orbital precession rates of the star. The moons at 3:2 and 1:1 resonances with the star exhibit circulating and librating motion, and they are stable, as well as moons outside the 3:2 resonance. Moons in between the 3:2 and 1:1 resonances, however,acquire high eccentricities and inclinations, and some go unstable.