Evolution of the Planetary Obliquity under the Competition of Eccentric Kozai-Lidov Resonance and the Equilibrium Tide

The planetary obliquity is a significant factor in determining the physical properties of planetary surfaces and the climate. Since direct detection is limited by the observation accuracy, dynamical theories are helpful to predict the evolution of the planetary obliquity. Based on the conservation of angular momentum, the competition between the Eccentric Kozai-Lidov (EKL) resonance and the equilibrium tide is utilized in this work to investigate the diverse secular evolution paths of planetary obliquity. For close-in S-type terrestrial planets in binary star systems, when the initial timescale ratio of the secular resonance and the tidal dissipation t_tide/t_kl > 1, the planetary obliquity will first undergo the excitation and the planetary rotation axis will be triggered to flip, then the obliquity enters the quasi-equilibrium state between 40° and 60°. The maximum obliquity can reach 130° when t_tide/t_kl > 10⁴. The qualitative simulation results indicate the maximum obliquity increases with the semi-major axis ratio a₁/a₂ and the perturbing body’s eccentricity e₂. The equilibrium timescale t_eq and the timescale in relation to the obliquity going down to zero, are both positively correlated with a₁. When t_tide is comparable to t_kl, an abrupt rise of t_eq will occur at a₁~0.5 au and a₂=20 au. For several observed hot-Jupiters in binary star systems, we present the equilibrium obliquity map to predict their possible obliquity. For terrestrial planets around M-dwarfs, we further estimate the initial orbital conditions to maintain the stable obliquity in the habitable zone.