GENERAL THEORY OF THE ROTATION OF THE NON-RIGID EARTH AT THE SECOND ORDER. I. THE RIGID MODEL IN ANDOYER VARIABLES

Abstract

This paper is the first part of an investigation where we will present an analytical general theory of the rotation of the non-rigid Earth at the second order, which considers the effects of the interaction of the rotation of the Earth with itself, also named as the spin-spin coupling. Here, and as a necessary step in the development of that theory, we derive complete, explicit, analytical formulae of the rigid Earth rotation that account for the second-order rotation-rotation interaction. These expressions are not provided in this form by any current rigid Earth model. Working within the Hamiltonian framework established by Kinoshita, we study the second-order effects arising from the interaction of the main term in the Earth geopotential expansion with itself, and with the complementary term arising when referring the rotational motion to the moving ecliptic. To this aim, we apply a canonical perturbation method to solve analytically the canonical equations at the second order, determining the expressions that provide the nutation-precession, the polar motion, and the length of day. In the case of the motion of the equatorial plane, nutation-precession, we compare our general approach with the particular study for this motion developed by Souchay et al., showing the existence of new terms whose numerical values are within the truncation level of 0.1 μas adopted by those authors. These terms emerge as a consequence of not assuming in this work the same restrictive simplifications taken by Souchay et al. The importance of these additional contributions is that, as the analytical formulae show, they depend on the Earth model considered, in such a way that the fluid core resonance could amplify them significatively when extending this theory to the non-rigid Earth models.