Non-linear resonances in spin-orbit coupling problems with three degrees of freedom (Mercury)

A system of vector and matrix differential equations is developed which may be applied to the problem of determining the spin-orbit state of the planet Mercury. Unlike previous studies, this system of equations may be analyzed without requiring that the spin axis remain perpendicular to the plane of the orbit. A specific average is defined and a transformation from the original system of equations to the "averaged" equations is shown to exist. Theorems are proved which relate certain characteristics of solutions of the original system to a condition on the "averaged" equations. The theorems are applied to sample problems which illustrate the extreme conditions which may be faced and which point out differences in the conclusions deduced from these opposite extreme cases. Mathematical models are developed to describe the physical system composed of Mercury and the sun. These models are consistent with those presented in previous studies by other authors. The theorems are applied to the mathematical models to obtain results which are consistent with the present understanding of the problem.