On the determination of the long period tidal perturbations in the elements of artificial earth satellites

In the present article we develop the theory of the long period tidal effects in the motion of artificial satellites assuming the variability of elastic parameters of the Earth (Love numbers) across the parallels. The dependence of Love numbers on the longitude produces perturbations of the period of one day or less and hence is neglected in the present theory. In this respect we follow in the footsteps of Kaula (1969). If the deviations ofk ₂ andk ₃ from pure constants are not taken into consideration, then the perturbations caused by the variability ofk ₂ andk ₃ across the parallels will be misinterpreted as the perturbations caused byk ₄...-terms, and the spurious values ofk ₄... will be deduced. It is extremely doubtful, however, that the real effects caused byk ₄,k ₅,..., are significant enough to be detected. The short period effects with the period of the revolution of the satellite, or less, were removed from the differential equations for the variation of elements of the satellite by the averaging over the orbit of the satellite. These differential equations are in the form convenient for numerical integration over a long interval of time and also suitable for developing the tidal effects into trigonometric series with the arguments ω, Ω of the satellite andl, l′, F, D, Γ of the Moon. The numerical integration can be performed using some simple quadrature formula, without resorting to a predictor-corrector system.