Evaluation of the Second, Fourth and Sixth Harmonics in the Earth's Gravitational Potential

Abstract

IN an article¹ in Nature in 1958, it was explained how the rate of rotation of the orbital plane of an Earth satellite (the rate of regression of the nodes) could be used to evaluate the even harmonics in the Earth's gravitational potential U. At an exterior point distant r from the Earth's centre and having co-latitude θ, U may be expressed in terms of Legendre polynomials P _n as : where G is the gravitational constant, M the mass of the Earth, R its equatorial radius, and the J _n are constants to be determined. J ₁ is zero if the equator is chosen to pass through the Earth's centre of mass. The motion of a satellite in the gravitational field specified by equation (1) has been investigated theoretically^2–4, and the rate of rotation of the orbital plane, Ω̇, may in principle be expressed in the form : where the F _n and F _mn are functions of the orbital elements. Since J ₂ is of order 10⁻³, the J _n are of order 10⁻⁶ when n > 2, and the F _n are of the same order as the F _mn, only the J ₂ ² term in the second series in equation (2) need be considered, and even in that term an approximate value of J ₂ is adequate. Thus each observed value of Ω̇ provides one linear relation between the J _n, and observed values from k satellites provide k simultaneous equations between the J _n. These equations are not ill-conditioned if the k orbits differ widely enough. It happens that the odd-numbered J _n have little effect on Ω̇, and the available values⁵ for J ₃ and J ₅, namely : can be substituted into equation (2) without introducing significant error. Thus k satellites with widely different orbits—and, in particular, with widely differing orbital inclinations—will yield values of J ₂, J ₄, … J _2k if we are prepared to assume that the contributions of J ₇, J ₉ … and of J _{2k + 2}, J _{2k + 4}, … are negligible.