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New Methods for Long-Time Numerical Integration of Planetary Orbits

Published online by Cambridge University Press:  07 August 2017

Hiroshi Kinoshita  and
Hiroshi Nakai
Hiroshi Kinoshita
Affiliation:
National Astronomical Observatory 2-21-1 Osawa, Mitaka, Tokyo 181, Japan E-Mail address:Kinoshita@c1.mtk.nao.ac.jp
Hiroshi Nakai
Affiliation:
National Astronomical Observatory 2-21-1 Osawa, Mitaka, Tokyo 181, Japan E-Mail address:Kinoshita@c1.mtk.nao.ac.jp

Abstract

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When planetary orbits are numerically integrated for a long time by conventional integrators, the most serious problem is secular errors in the energy and the angular momentum of the planetary system due to discretization (truncation) errors. The secular errors in the energy and the angular momentum mean that the semi-major axes, the eccentricities, and the inclinations of planetary orbits have a secular error which grows linearly with time. Recently symplectic integrators and linear symmetric multistep integrators are found not to produce the secular errors in the energy and the angular momentum due to the discretization errors. Here we describe briefly both methods and discuss favorable properties of these integrators for a long-term integration of planetary orbits.

Keywords

Numerical IntegrationSymplectic IntegratorsLinear Symmetric Multistep IntegratorsPlanetary Orbits
Type
Part VII - Dynamical Systems. Maps. Integrators
Information
Symposium - International Astronomical Union , Volume 152: Chaos, Resonance and Collective Dynamical Phenomena in the Solar System , 1992 , pp. 395 - 406
Copyright
Copyright © Kluwer 1992 

References

Applegate, J.H., Douglas, M.R., Gürsel, Y., Sussman, G.J. and Wisdom, J.: 1986, ‘The Outer Solar System for 200 Million Years’, Astron. J. 92, 176–194.Google Scholar
Butcher, J.C.: 1964, ‘On Runge-Kutta processes of higher order’, J. Austral. Math. Soc. 4, 179–194.Google Scholar
Cohen, C.J., Hubbard, E.C., and Oesterwinter, C.: 1973, ‘Elements of the Outer Planets for One Million Years’, Astron. Papers Amer. Ephemeris 92, 3–92.Google Scholar
Dahlquist, G.: 1976, ‘On One-Leg Multistep Method’, SIAM J. Numer. Anal. 20, 1130–1138.Google Scholar
Eirola, T. and Sanz-Serna, J.M.: 1990, ‘Conservation of Integrals and Symplectic Structure in the Integration of Differential Equations by Multistep Methods’, Applied Mathematics and Computation Reports, Universidad de Valladolid Report 1990/9, pp.9.Google Scholar
Gragg, W.: 1963, ‘Repeated Extrapolation to the Limit in the Numerical Solution of Ordinary Differential Equations’, Doctoral Dissertation, University of California, Los Angels, pp.103.Google Scholar
Henrici, P.: 1962, Discrete Variable Methods in Ordinary Differential Equations , Wiley,New York.407.Google Scholar
Kinoshita, H.: 1968, ‘Propagation of the Discretization Error in the Two-Body Problem’, Publ. Astron. Soc. Japan 20, 1–23.Google Scholar
Kinoshita, H. and Nakai, H.: 1984, ‘Motions of the Perihelions of Neptune and Pluto’, Celest. Mech. 34, 203–217.Google Scholar
Kinoshita, H.: 1989, ‘Numerical Integration Method in Dynamical Astronomy’, Celest. Mech. 45, 231–244.Google Scholar
Kinoshita, H.,Yoshida, H.,and Nakai, H.: 1991, ‘Symplectic Integrators and Applications to Dynamical Astronomy’, Celest. Mech. 50, 59–71.Google Scholar
Kinoshita, H. and Nakai, H.: 1991, ‘Error Analysis of Symmetric Integrators for Planetary Orbits’, in The Twenty-Fourth Symposium on Celestial Mechanics , ed. by Kinoshita, H. and Yoshida, H.,139–147.Google Scholar
Milani, A. and Nobili, A.: 1988, ‘Integration Error over Very Long Time Spans’, Celest. Mech. 43, 1–34.Google Scholar
Neri, F.: 1988, ‘Lie Algebras and Canonical Integration’, University of Maryland, Dept.of Physics , preprint.Google Scholar
Quinlan, D. and Tremaine, S.: 1990, ‘Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits’, Astron. J. 100, 1964–1700.CrossRefGoogle Scholar
Quinlan, G.D. and Toomre, A.T.: 1991, ‘Resonant Instabilities in Symmetric Multistep Methods’, private communication.Google Scholar
Quinlan, D.:1991, private communication.Google Scholar
Quinn, T. and Tremaine, S.: 1990, ‘Roundoff Error in Long-Term Planetary Orbit Integrals’, Astron. J. 99, 1016–1023.Google Scholar
Ruth, R.: 1983, ‘A Canonical Integration Technique’, IEEE Trans. Nucl. Sci. 30, (4), 2699–2671.Google Scholar
Sanz-Serna, J.M.: 1991, ‘Symplectic Runge-Kutta and Related Methods: Recent Results’, Applied Mathematics and Computation Reports, Universidad de Valladolid Report 1991/5, pp.11.Google Scholar
Sugimoto, D., Chikada, Y., Makino, J., Ito, T., Ebisuzaki, T., and Umemura, M.: 1990, ‘A Special-Purpose Computer for Gravitational Many-Body Problems’, Nature 345, 33–35.Google Scholar
Sussman, G.J. and Wisdom, J.: 1988, ‘Numerical Evidence That the Motion of Pluto is Chaotic’, Science 241, 433–437.Google Scholar
Yoshida, H.: 1990a, ‘Conserved Quantities of Symplectic Integrators for Hamiltonian Systems’, Physica D , in press.Google Scholar
Yoshida, H.: 1990b, ‘Construction of Higher Order Symplectic Integrators’, Phys. Lett. A. 150, 262–268.Google Scholar