In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the "constants" with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the "constants". The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and "constants" stays unaltered under disturbance. "Constants" satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of "constants" is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating.In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function (Efroimsky & Goldreich
We focus on the updating of a specific contribution to the precession of the equator in longitude, usually named as "second order." It stems from the crossing of certain terms of the lunisolar gravitational potential. The IAU2006 precession theory assigns it the value of −46.8 mas/cy that was derived for a rigid Earth model. Instead of that model, we consider a two-layer Earth composed of an elastic mantle and a liquid core, working out the problem within the Hamiltonian framework developed by Getino and Ferrándiz. The targeted effect is obtained without further simplifying assumptions through Hori's canonical perturbation method applied up to the second order of perturbation. On account of using a more realistic Earth model, the revised value of the second-order contribution is significantly changed and reaches −55.29 mas/cy. That variation of the second-order contribution is larger than other contributions included in IAU2006. It must be compensated with an increase of −8.51 mas/cy in the value of the lunisolar first-order component ¢ p A of the precession of the equator rate, which is derived from the total rate by subtracting the remaining contributions accounted for in IAU2006 precession. The updating of the second-order contribution implies that the ¢ p A parameter has to be changed, from 5040684.593 to 5040693.104 mas/cy in absence of potential revisions of other contributions. It entails a proportional variation of Earth's dynamical ellipticity H d , for which the estimation associated with IAU2006, 0.00327379448, should be updated to 0.00327380001, about 1.7 ppm larger.
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