The role of the mutual inclination in the dynamics of a two eccentric giant-planet system

We investigate a fundamental planetary system comprising a Sun-like star, a Jupiter-like planet, and a Neptune-like planet across a wide range of orbital configurations, not limited to the well-studied hierarchical case. We present resonance atlases illustrating the domains of thousands of mutual mean-motion resonances (MMRs) and their connection to both chaotic and regular dynamics. Our analysis reveals that an eccentric Jupiter generates a significant chaotic region in its vicinity due to close encounters and overlapping MMRs, particularly extended in semi-major axis for inclinations below 30 degrees. Notably, some strong resonances, such as the 1:1, persist within these chaotic zones.

Employing a semi-analytical method to study secular dynamics, we identify two regimes for equilibrium configurations. At low mutual inclinations, equilibrium arises from oscillations of mutual pericenters around 0 or 180 degrees. Conversely, at high mutual inclinations equilibrium is characterized by specific values of the arguments of pericenters that are integer multiples of 90 degrees.

Through numerical integration of the full equations of motion, we calculate the three fundamental frequencies of the system across diverse configurations and analyze their dependence on orbital elements. Based on this frequency analysis, we find two distinct dynamical regimes separated by a critical mutual inclination between 30 and 40 degrees, where a secular resonance occurs due to the coincidence of the two fundamental pericenter frequencies. Below this critical inclination, the dynamics resembles the classical secular model for low-inclination orbits, featuring three well-defined fundamental frequencies and free and forced modes, with mutual inclination remaining quasi-constant. Above this threshold, the dynamics shifts dramatically, exhibiting increased variations in mutual inclination and the emergence of combined fundamental frequencies, potentially dominated by the secular resonance or the von Zeipel-Lidov-Kozai (vZLK) mechanism. The observed change in dynamical regimes correlates with the transition in the equilibrium configurations of the pericenters.

While the planetary mass ratio has a limited qualitative impact on our findings, it directly influences the individual fundamental frequencies. Furthermore, the relative placement of the Neptune-mass planet (interior or exterior to Jupiter) leads to different dynamical behaviors at increasing mutual inclinations. An interior Neptune exhibits gradual secular changes in eccentricity and inclination, whereas an exterior Neptune shows a rapid increase in these elements following the onset of the vZLK mechanism.