ETD

• Celestial Mechanics
• Giant Planets
• Obliquity
• Planetary Migration
• Spin-Orbit Resonance

The obliquity of a planet is the angle between its spin axis and the normal to its orbital plane.
Giant planets are expected to form with rotation axis nearly perpendicular to their orbital planes, resulting in obliquities close to zero. However, the current tilts of giant planets in our solar system exhibit a wide range of values: Jupiter has an obliquity of approximately $3.1^\circ$, Saturn around $26.7^\circ$, Uranus around $97.8^\circ$ and Neptune around $28.3^\circ$. This suggests that some dynamical mechanisms must have influenced the planets' spin-axis orientations after their formation.

The evolution of the spin axis is driven by the stellar torque acting on the planet's rotational flattening. Up to quadrupole order in the gravitational potential, the equation of motion can be written as
\begin{equation}\label{spin_axis_equation}
\frac{d\mathbf{s}}{dt}
=
\alpha(\mathbf{s}\cdot\mathbf{n})(\mathbf{s}\times\mathbf{n}),
\end{equation}
where $\mathbf{s}$ is the unit spin-axis vector, $\textbf{n}$ is the unit vector perpendicular to the orbital plane and $\alpha$ is the precession constant which depends on the planet's shape and on its satellite system. Equation \eqref{spin_axis_equation} is not autonomous since $\textbf{n}(t)$ evolves with time and is strongly dependent on the perturbation from the other planets.
The evolution of the spin axis is governed by the interplay between the precession constant $\alpha$ and the secular evolution of the orbital normal $\mathbf{n}(t)$. In particular, the spin dynamics depends on the relative magnitude of $\alpha$ and the characteristic frequencies appearing in the quasi-periodic expansion of $\mathbf{n}(t)$. Secular spin--orbit resonances may arise when these quantities become comparable, allowing a substantial transfer of angular momentum from the orbital to the rotational dynamics.

In this work, we investigate whether the encounter with a spin-orbit resonance could provide a dynamical pathway from Neptune's initially small obliquity to its present value of $28.3^\circ$. To this end, we consider two complementary mechanisms. First, we increase the planetary precession constant by assuming the presence of a primordial satellite system, whose contribution enhances Neptune's precession constant $\alpha$. Second, we model the evolution of the orbital frequencies during planetary migration, which gradually modifies the secular architecture of the outer Solar System. The combined action of these effects can drive the system through a resonance crossing and potentially lead to capture of Neptune's spin axis into a secular spin--orbit resonance.

Once the conditions required for resonant tilting have been established, we investigate whether the primordial satellite system can survive the phase of planetary instability and, subsequently, how the system may evolve toward Neptune's present-day satellite architecture. In particular, we derive constraints on the encounter history of Neptune and explore mechanisms capable of terminating the spin--orbit resonance and removing the primordial satellites.

This work shows how resonant Hamiltonian dynamics and resonance capture can naturally produce large obliquity variations in planetary systems. In the specific case of Neptune, a primordial satellite system with total mass of order $10^{-4}M_N$ is sufficient to trigger resonant tilting during planetary migration. More generally, the results illustrate how secular resonances and bifurcation phenomena can shape the long-term rotational dynamics of planets.