Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Binney, J. and Spergel, D.: 1982, 'Spectral stellar dynamics', Astrophys. J. 252, 308.
Binney, J. and Spergel, D.: 1984, 'Spectral stellar dynamics-II. The action integrals', Monthly Notices Royal Astron. Soc. 206, 159.
Binney, J. and Tremaine, S.: 1987, Galactic Dynamics. Princeton Univ. Press, Princeton, NJ, USA.
Briggs, W. L. and Henson, V. E.: 1995, 'The DFT: an owner's manual of the discrete Fourier transform', Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
Carpintero, D. D. and Aguilar, L. A.: 1998, 'Orbit classification in arbitrary 2D and 3D potentials', Monthly Notices Royal Astron. Soc. 298, 1.
Copin, Y., Zhao, H. S. and de Zeeuw, P. T.: 2000, 'Probing a regular orbit with spectral dynamics', Monthly Notices Royal Astron. Soc. 318, 781.
Davis, P. J.: 1975, 'Interpolation and Approximation', Dover, New York, NY, USA.
Hairer, E., Norsett, S. P. and Wanner, G.: 1991, 'Solving Ordinary Differential Equations I', Springer Verlag, New York, NY, USA.
Lanczos, C.: 1956, 'Applied Analysis', Prentice Hall, Englewood Cliffs, NJ, USA.
Laskar, J.: 1990, 'The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones', Icarus 88, 266.
Laskar, J.: 1993, 'Frequency analysis for multi-dimensional systems. Global dynamics and diffusion', Physica D 67, 257.
Laskar, J.: 1999, Introduction to frequency map analysis', in Siwo, C. (ed.), Hamiltonian Systems with Three or more Degrees of Freedom. Kluwer, Dordrecht, Netherlands.
Merritt, D. and Valluri, M.: 1999, 'Resonant orbits in triaxial galaxies', Astron. J. 118, 1177.
Papaphilippou, Y. and Laskar, J.: 1996, 'Frequency map analysis and global dynamics in a galactic potential with two degrees of freedom', Astron. Astrophys. 307, 427.
Papaphilippou, Y. and Laskar, J.: 1998, 'Global dynamics of triaxial galactic models through frequency map analysis', Astron. Astrophys. 329, 451.
Ratcliff, S. J., Chang, K. M. and Schwarzschild, M.: 1984, 'Stellar orbits in angle variables', Astrophys. J. 279, 610.
Schwarzschild, M.: 1979, 'A numerical model for a triaxial stellar system in dynamical equilibrium', Astrophys. J. 232, 236.
Valluri, M. and Merritt, D.: 1998, 'Regular and chaotic dynamics of triaxial stellar systems', Astrophys. J. 506, 686.
Wachlin, F. C. and Ferraz-Mello, S.:1998, 'Frequency map analysis of the orbital structure in elliptical galaxies', Monthly Notices Royal Astron. Soc. 298, 22.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hunter, C. Spectral Analysis of Orbits via Discrete Fourier Transforms. Space Science Reviews 102, 83–99 (2002). https://doi.org/10.1023/A:1021360731798
Issue Date:
DOI: https://doi.org/10.1023/A:1021360731798