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 $t_{\mathrm{eq}}$ 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 $t_{\mathrm{eq}}$ and $r_{t}$ with the maximum likelihood method. The spin-axis orientation of S-type terrestrials orbiting M-dwarfs reverses over $90^\circ$ when $r_{t} > 100$, then enter the quasi-equilibrium state between $40^\circ$ and $60^\circ$, while the maximum obliquity can reach $130^\circ$ 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 $\tau$ Boot Ab are explored to have a great possibility to be head-down over the secular evolution of spin.