ApJ to be submitted: Beyond Cassini States: Analytical Framework for Non-Trivial Obliquity Steady States in Multi-Planetary Super-Earth Systems

Abstract: Secular forcing in compact multiplanet systems can maintain large planetary obliquities even when stellar tides act to align a planet’s spin with its orbit. We develop an analytical framework for the nontrivial obliquity steady states previously found numerically for close-in super-Earths subject to two orbital-precession modes. In a frame co-precessing with the inner orbit, the dominant mode defines an integrable Colombo’s top, while the weaker mode provides periodic forcing and tides provide weak dissipation. Numerical integrations identify these states as stable periodic attractors, equivalent to dissipative limit cycles in the extended phase space. We associate the attractors with nonlinear $m:n$ commensurabilities between the intrinsic libration or circulation frequency and the difference between the orbital precession frequencies. Their phase-space locations follow from the resonance condition, while their existence further requires the cycle-averaged Hamiltonian changes produced by forcing and tides to balance. This balance locks the phase difference between the spin response and orbital forcing. We develop and test this formulation for the $1:1$ libration and circulation branches. For higher-order branches, the perturbative harmonic structure and scaling arguments provide a qualitative interpretation and suggest that progressively weaker dissipation is required. The nonlinear resonant attractors therefore provide a route to long-lived high-obliquity behavior beyond the classical Cassini states in compact planetary systems.

Paper Link: (to be submitted)