The Rotation of the Nonrigid Earth at the Second Order. II. The Poincaré Model: Nonsingular Complex Canonical Variables and Poisson Terms

Abstract

We develop a Hamiltonian analytical theory for the rotation of a Poincaré Earth model (rigid mantle and liquid core) at the second order with respect to the lunisolar potential and moving ecliptic term. Since the Andoyer variables considered in the first-order solution present virtual singularities, i.e., vanishing divisors, we introduce a set of nonsingular complex canonical variables. This choice allows for applying the Hori canonical perturbation method in a standard way. We derive analytical expressions for the first- and second-order solutions of the precession and nutation of the angular momentum axis (Poisson terms). Contrary to first-order theories, there is a part of the Poisson terms that does depend on the Earth's structure. The resulting numerical amplitudes, not incorporated in the International Astronomical Union nutation standard, are not negligible considering current accuracies. They are at the microarcsecond level for a few terms, with a very significant contribution in obliquity of about 40 μas for the nutation argument with period −6798.38 days. The structure-dependent amplitudes present a large amplification with respect to the rigid model due to the fluid core resonance. The features of such resonance, however, are different from those found in first-order solutions. The most prominent is that it does not depend directly on the second-order nutation argument but rather on the combination of first-order arguments generating it. It entails that some first-order approaches, like those based on the transfer function, cannot be applied to obtain the second-order contributions.