Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

Abstract

We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes de référence spatio-temporels. l'Observatoire de Paris, pp 74-81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251-283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117-180, 1995), but considering the contribution of the convective terms. Specifically, we 123 100 A. Escapa derive analytical formulas that determine the elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron. 105:261-274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded the motion of the tide rai...

The kinematical mechanism for the perturbation of the rotational axis in the rotation of the elastic Earth

“…Also, in the process of this investigation, is shown the condition for the approximate conclusion in Kubo (2009) mentioned above holding strictly, the condition being in agreement with the one derived by quite different methods in Escapa (2011).…”

“…If this supposition is not valid, the conclusion may be erroneous. Escapa (2011) found this error by calculating exactly the convective and direct perturbations by extending the result of Getino and Ferrándiz (1995). Also, Kubo (2011) investigated the Oppolzer-like terms from a kinematical point of view and obtained an alternative rigorous expression for the direct perturbation of the figure axis, which must be equivalent to the corresponding equations in Escapa (2011).…”

“…Escapa (2011) found this error by calculating exactly the convective and direct perturbations by extending the result of Getino and Ferrándiz (1995). Also, Kubo (2011) investigated the Oppolzer-like terms from a kinematical point of view and obtained an alternative rigorous expression for the direct perturbation of the figure axis, which must be equivalent to the corresponding equations in Escapa (2011). Actually Kubo (2011) showed the equivalence of these two expressions, at least to the first order with respect to the ratioΘ ν /ω, Θ ν being a function mentioned above and ω the rotational speed of the Earth.…”

Dynamical adjustments in IAU 2000A nutation series arising from IAU 2006 precession

“…In doing that, we emphasize the dependence of the nutations on the precession quantities and incorporate some contributions that are usually discarded but are needed to ensure the consistency of the model. Further explanations can be found, for example, in Kinoshita (1977); Souchay et al (1999); Efroimsky & Escapa (2007); and Getino et al (2010) for rigid models and in ; Escapa et al (2001);and Escapa (2011) for non-rigid ones. The rotational motion of the Earth around its center of mass O is characterized by relating a quasi-inertial reference system A92, page 2 of 13 OXYZ with a reference system Oxyz attached to the Earth in some prescribed way; for example by Tisserand mean axes, Oz = e z being the symmetry axis, also named the Earth figure axis.…”

“…On the other one, the Hamiltonian formalism can be extended to consider other non-rigid Earth models (e.g., Getino & Ferrándiz 1995;Escapa et al 2001, Escapa 2011. Hence, the developments made here can be directly incorporated to those theories, providing an integrated framework to study the Earth's rotational motion.…”