Correspondence between theory and observations of polar motion

SUMMARY The Earth's orientation in space changes in response to the action of a variety of torques, generated both externally and internally. External torques arising from the gravitational forces of the Sun and Moon act upon the Earth, causing it to undergo periodic motions known as the lunisolar precession and nutations. Internal dynamical processes that change the deformable Earth's inertia tensor, or that generate relative angular momentum, also cause the Earth's orientation in space to change, the resulting motions being known as wobble (or polar motion) and changes to the length‐of‐day. Historically, it was thought that observations determined the location of the rotation pole within some rotating reference frame fixed to the solid Earth, and theoretical expressions for the nutations and wobble were developed and given in terms of the Earth's rotation axis. However, it has been (relatively recently) argued that observatories, being located on the surface of the Earth, are more nearly moving with the Earth's surface, and hence observations more nearly reflect the motion of the Earth's figure axis, rather than its rotation or angular momentum axes. This argument was accepted by the International Astronomical Union, and the current 1980 Theory of Nutation refers to an axis, the celestial ephemeris pole, that is more closely associated with the Earth's figure axis than it is with the Earth's rotation axis. Polar motion values, as reported by modern Earth rotation services, give the location of the celestial ephemeris pole within some rotating, body‐fixed reference frame. The celestial ephemeris pole does not correspond to either the Earth's instantaneous figure axis, its instantaneous rotation axis, or its instantaneous angular momentum axis, but rather corresponds to an axis that exhibits no nearly diurnal motions in either the terrestrial, body‐fixed reference frame or the celestial, space‐fixed frame. The focus of this paper is not on the nutational motions of the Earth generated by external torques, but rather on the wobble. The goal of this paper is to write in terms of reported values the standard theoretical equation describing the Earth's wobble, namely, the linearized conservation of angular momentum equation known as the Liouville equation. In terms of the location m ( t ) = m 1 ( t ) + im 2 ( t ) of the rotation pole, the Liouville equation is usually written aswhere the complex‐valued x‐functions are functions of perturbations to the Earth's inertia tensor and relative angular momentum. In order to rewrite this equation in terms of the location p ( t ) = p 1 ( t ) + ip 2 ( t ) of the reported pole, a relation between the locations of the rotation pole and the celestial ephemeris pole must be obtained. This is accomplished in the time domain by considering the properties of the time‐dependent transformation matrix that relates components of a position vector in the rotating, body‐fixed frame to its components in the celestial, space‐fixed frame, the resulting relation beingUsing this expression relating the location of the rotation pole to that of the celestial ephemeris pole, the linearized conservation of angular momentum equation becomesThus, there is no x‐term in the Liouville equation describing long‐period polar motions when it is written in terms of reported valves.