Cassini's motions of the Moon and Mercury and possible excitations of free librations

On the basis of conditionally-periodic solutions of Hamiltonian systems at resonance of main frequencies Cassini's motions, their stability, Cassini's angle and periods of free librations of the Moon and Mercury have been recently studied and determined. The generalized formulations of Cassini's laws for the motion of the Moon and Mercury, that are considered as absolutely rigid non-spherical bodies, have been determined. The study of the second approximation equations of the desired quasi-periodic solutions in the case of the Moon allows us to determine the constant components of the first order for six Andoyer variables and the constant component of the second order for the angular velocity of the Moon. These effects are caused by the influence of the third harmonic of selenopotential. In this paper, these effects are described by analytical formulas, the dynamic and geometric interpretations are given, and a new interpretation of Mercury's motion under the generalized Cassini's laws has been proposed. Predictions of the existence of free librations of significant amplitude in the Mercury longitude, that are confirmed by the radar measurements data of the Mercury angular velocity, and in its pole motion in the body and in space have been made. The mechanism describing free librations of celestial bodies and their pole oscillations has been proposed due to the forced relative oscillations and wobble of the core-mantle system of celestial bodies (Moon, Mercury, Earth and other bodies in the solar system) under gravitational action of the external celestial bodies. The paper shows that the ascending node of equator of Mercury (and the intermediate plane orthogonal to the angular momentum) of epoch 2000.0 on the ecliptic does not coincide with the ascending node of orbital plane of Mercury on the same plane, and is ahead of it at an angle 23º4’. Angular momentum vector of the rotational motion of Mercury forms a constant angle ρG=4'1±1'1 with normal to the moveable plane of its orbit. The observed inclination of the angular velocityρω=2'1±0'1, can be considered as a possible evidence of a significant amplitude of the poles free motion of the Mercury rotation axis (c amplitude of about 2′- 4′).