Numerical Fourier Expansions of the Planetary Disturbing Function

Abstract

Various Fourier expansions of the planetary disturbing function can be computed numerically with the use of numerical Fourier analysis. The task to compute the most general five-dimensional Fourier expansion of disturbing function has become feasible with typical server-class computers quite recently. In such an expansion two anomalies, two arguments of perihelions and two longitudes of the node are independent angular variables, while two semi-major axes, two eccentricities and two inclinations are fixed numerically. The semianalytical expansion of the disturbing function resulting from numerical Fourier analysis theoretically converges for any values of the parameters except for those sets of parameters which allow the bodies to collide. Various aspects of the numerical computation of the Fourier expansion are discussed. Theoretical and practical convergence of the Fourier series is discussed and illustrated.