Abstract
In this paper we develop a new numerical method to integrate the equations of motion of a celestial body. The idea is to replace the differential equation for the fast moving component by an equation for the energy per unit mass. We use a simple first-order explicit method for the approximation of the new system. It is shown that the radial error is much smaller than that of some numerical schemes. It will be of interest to have a more extensive comparison with state-of-the-art methods currently in use for long-term trajectory propagation. The evaluation of energy is also more accurate than in other known schemes. This method also conserves the energy per unit mass in the case of perturbation-free flight. The idea can be extended to higher-order methods and implicit schemes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ANANTHAKRISHNAIAH, U. “P-Stable Obrechkoff Methods with Minimal Phase-Lag for Periodic Initial-Value Problems,” Mathematics of Computation, Vol. 49, 1987, pp. 553–559.
ASH, J. H. “Analysis of Multistp Methods for Second-Order Ordinary Differential Equations,” Ph.D. Thesis, University of Toronto, 1969.
BATE, R. R., MUELLER, D. D., WHITE, J. E. Fundamentals of Astrodynamics, Dover Publications, New York, 1971.
BRUSA, L. and NIGRO, L. “A One-Step Method for Direct Integration of Structural Dynamic Equations,” International Journal for Numerical Methods in Engineering, Vol. 15, 1980, pp. 685–699.
BETTIS, D. G. “Numerical Integration of Products of Fourier and Ordinary Polynomials,” Numerische Mathematik, Vol. 14, 1970, pp. 421–434.
BURRAGE, K. Parallel and Sequential Methods for Ordinary Differential Equations, Oxford University Press, New York, 1995.
BUTCHER, J. C. The Numerical Analysis of Ordinary Differential Equations, Wiley & Sons, New York, 1987.
CRANK, J. and NICOLSON, P. “A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type,” Proceedings of the Cambridge Philosophical Society, Vol. 43, 1947, pp. 50–67.
DER, G. J. “Trajectory Propagation Using High-Order Numerical Integrators,” Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, Monterey, CA, February 9–11, 1998, Paper Number AAS 98-210.
FUKUSHIMA, T. “Symmetric Multistep Methods Revisited,” Proceedings of the 30th Symposium on Celestial Mechanics, 4–6 March 1998, Hayama, Kanagawa, Japan, T. Fukushima, T. Ito, T. Fuse, and H. Umerhara (eds), pp. 229–247.
GAUTSCHI, W. “Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials,” Numerische Mathematik, Vol. 3, 1961, pp. 381–397.
LAMBERT, J. D. Computational Methods in Ordinary Differential Equations, John Wiley & Sons, London, 1973.
LAMBERT, J. D. Numerical Methods for Ordinary Differential Systems, The Initial Value Problem, John Wiley & Sons, London, 1991.
LAMBERT, J. D. and WATSON, I. A. “Symmetric Multistep Methods for Periodic Initial Value Problems,” Journal of the Institute of Mathematics and its Applications, Vol. 18, 1976, pp. 189–202.
LYCHE, T. “Chebyshevian Multistep Methods for Ordinary Differential Equations,” Numerische Mathematik, Vol. 19, 1972, pp. 65–72.
NETA, B. and FORD, C. H. “Families of Methods for Ordinary Differential Equations Based on Trigonometric Polynomials,” Journal of Computational and Applied Mathematics, Vol. 10, 1984, pp. 33–38.
NETA, B. “Trajectory Propagation Using Information on Periodicity,” Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Boston, MA, August 10–12, 1998, Paper Number AIAA 98-4577.
NETA, B. and FUKUSHIMA, T. “Obrechkoff Versus Super-Implicit Methods for the Integration of Keplerian Orbits,” presented as paper AIAA 2000-4029 at the AIAA/AAS Astrodynamics Specialist Conference, Denver, CO, August 14–17, 2000.
NETA, B. and FUKUSHIMA, T. “Obrechkoff Versus Super-Implicit Methods for the Solution of First and Second Order Initial Value Problems,” Computers and Mathematics with Applications, special issue on Numerical Methods in Physics, Chemistry and Engineering, T. E. Simos and G. Abdelas (guest editors), accepted for publication.
OBRECHKOFF, N. “On Mechanical Quadrature” (Bulgarina, French summary), Spisanie Bulgarie Akademie Nauk, Vol. 65, 1942, 191–289.
STIEFEL, E. and BETTIS, D. G. “Stabilization of Cowell’s Method,” Numerische Mathematik, Vol. 13, 1969, pp. 154–175.
VALLADO, D. Fundamentals of Astrodynamics and Applications, McGraw Hill, New York, 1997.
VAN DOOREN, R. “Stabilization of Cowell’s Classical Finite Difference Method for Numerical Integration,” Journal of Computational Physics, Vol. 16, 1974, pp. 186–192.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Neta, B., Lipowski, Y. A New Scheme for Trajectory Propagation. J of Astronaut Sci 50, 255–268 (2002). https://doi.org/10.1007/BF03546251
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03546251