Evolution of the Planetary Obliquity: The Eccentric Kozai-Lidov Mechanism Coupled with Tide

The planetary obliquity plays a significant role in determining physical properties of planetary surfaces and climate. As direct detection is constrained due to the present observation accuracy, kinetic theories are helpful to predict the evolution of the planetary obliquity. Here the coupling effect between the eccentric Kozai-Lidov (EKL) effect and the equilibrium tide is extensively investigated, the planetary obliquity performs to follow two kinds of secular evolution paths, based on the conservation of total angular momentum. The equilibrium timescale of the planetary obliquity t_eq varies along with r_t, which is defined as the initial timescale ratio of the tidal dissipation and secular perturbation. We numerically derive the linear relationship between t_eq and r_t with the maximum likelihood method. The spin-axis orientation of S-type terrestrials orbiting M-dwarfs reverses over 90^∘ when r_t > 100, then enter the quasi-equilibrium state between 40^∘ and 60^∘, while the maximum obliquity can reach 130^∘ when r_t > 10^4. Numerical simulations show that the maximum obliquity increases with the semi-major axis ratio a_1/a_2, but is not so sensitive to the eccentricity e_2. The likelihood of obliquity flip for S-type terrestrials in general systems with a_2 < 45 AU is closely related to m_1. The observed potential oblique S-type planets HD 42936 b, GJ 86 Ab and τ Boot Ab are explored to have a great possibility to be head-down over the secular evolution of spin.