Numerical Fourier expansions of the planetary disturbing function (Q5938340)

The paper describes the experience with numerical Fourier expansion of planetary disturbing functions. Two point masses \(M\) of \(M'\) move in elliptic orbits around a common primary point mass. Using the numerical Fourier analysis, the disturbing functions \(R\) and \(R'\) can be computed for any given set of 12 elements of the two elliptic orbits. Any kind of anomaly (mean anomaly, true anomaly, eccentricity anomaly etc.) can be used as an independent trigonometric variable. The series considered as Fourier series converge for all values of parameters for which the two bodies do not collide or their orbits do not intersect. The author uses a fast Fourier transform package in which the round-off errors can be estimated and always controlled. More compact form of the expansion of disturbing functions can be obtained when one of the orbits is chosen as a reference plane. In this case one has Fourier seires with four variables. The theoretical and practical convergence of Fourier series is illustrated on examples, and some of the results are summarized in tables.