Variation rate of the planet’s obliquity $\left(\dot{\beta }\right)$ as a function of the planet’s spin rotation period (horizontal axis) and obliquity (vertical axis) for Earth-sized dry (left panels) and global-ocean (right panels) planets. Top: full calculation of the planet’s tidal response (FULL). Middle: calculation using, for all tidal forcing terms, the degree-2 Love number computed in the equatorial plane assuming the coplanar-circular configuration (APPROX). Bottom: relative difference between the FULL and APPROX cases, defined by Eq. (93). The variation rate of the planet’s obliquity was computed from the three-dimensional tidal torque using Eq. (77). The black dot at (P_rot,β) = (1 day, 0°) designates the actual Earth-Moon system in the coplanar-circular approximation.