Forced obliquity variations of Mercury

The spin pole of Mercury is very nearly, but not quite, aligned with its orbit pole. Tidal dissipation has driven the free obliquity to very small values, and the high rate of spin pole precession allows the forced obliquity variations to remain small despite significant variations in orbital inclination and eccentricity. We present calculations of the obliquity for a 10 million year time span, centered on the present. The obliquity remains small, with typical values of 2-4 minutes of arc. The dominant period of obliquity oscillations is 895 kyr, which is also the main period at which the orbital inclination varies. If the orbit pole precession rate were uniform, dissipation would have driven Mercury into a Cassini state, in which the spin pole and orbit pole remain coplanar with the invariable pole, as the spin pole precesses about the moving orbit pole. However, due to the nonuniform orbit precession rate, this simple coplanar configuration is not maintained, except on a mode-by-mode basis. That is, when the orbit pole motion is represented as a sum of normal modes of the coupled oscillations of the planetary system, the spin pole coprecesses with the orbit pole at each modal frequency. This is a generalization of Cassini's second and third laws of lunar rotation to the case of nonuniform orbit precession. We compare results of a linearized obliquity model with a numerical integration of the equations of motion. The two solutions agree at the level of a few seconds of arc.