Spectral Analysis of Orbits via Discrete Fourier Transforms

We give some simple and direct algorithms for deriving the Fourier series which describe the quasi-periodic motion of regular orbits from numerical integrations of those orbits. The algorithms rely entirely on discrete Fourier transforms. We calibrate the algorithms by applying them to some orbits which were studied earlier using the NAFF method. The new algorithms reproduce the test orbits accurately, satisfy constraints which are consequences of Hamiltonian theory, and are faster. We discuss the rate at which the Fourier series converge, and practical limits on the degree of accuracy that can reasonably be achieved.