Celestial Mechanics and Dynamical Astronomy

, Volume 116, Issue 2, pp 141–174 | Cite as

High precision symplectic integrators for the Solar System

  • Ariadna Farrés
  • Jacques LaskarEmail author
  • Sergio Blanes
  • Fernando Casas
  • Joseba Makazaga
  • Ander Murua
Original Article


Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term studies in the Solar System. These comparisons are made in Jacobi and heliocentric coordinates and the implementation of the algorithms is fully detailed for practical use. We conclude that high order integrators should be privileged, with a preference for the new \((10,6,4)\) method of Blanes et al. (2013).


Symplectic integrators Hamiltonian systems Planetary motion Jacobi coordinates Heliocentric coordinates Splitting sympletic methods 



This work was supported by GTSNext project. The work of SB, FC, JM and AM has been partially supported by Ministerio de Ciencia e Innovación (Spain) under project MTM2010-18246-C03 (co-financed by FEDER Funds of the European Union).


  1. Blanes, S., Casas, F., Farrés, A., Laskar, J., Makazaga, J., Murua, A.: New families of symplectic splitting methods for numerical integration in dynamical astronomy. Appl. Number. Math. (2013). doi: 10.1016/j.apnum.2013.01.003
  2. Candy, J., Rozmus, W.: A symplectic integration algorithm for separable hamiltonian functions. J. Comput. Phys. 92(1), 230–256 (1991)MathSciNetADSCrossRefzbMATHGoogle Scholar
  3. Chambers, J.E.: A hybrid symplectic integrator that permits close encounters between massive bodies. Mon. Notices R. Astron. Soc. 304, 793–799 (1999)ADSCrossRefGoogle Scholar
  4. Chambers, J.E., Murison, M.A.: Pseudo-high-order symplectic integrators. Astron. J. 119(1), 425 (2000)ADSCrossRefGoogle Scholar
  5. Danby, J.M.A.: Fundamentals of Celestial Mechanics. 2nd Edition, revised and enlarged. XII, p. 484. Willmann-Bell, London (1992)Google Scholar
  6. Duncan, M.J., Levison, H.F., Lee, M.H.: A multiple time step symplectic algorithm for integrating close encounters. Astrono. J. 116, 2067–2077 (1998)ADSCrossRefGoogle Scholar
  7. Fienga, A., Laskar, J., Kuchynka, P., Manche, H., Desvignes, G., Gastineau, M., Cognard, I., Theureau, G.: The INPOP10a planetary ephemeris and its applications in fundamental physics. Celest. Mech. Dyn. Astron. 111, 363–385 (2011)ADSCrossRefGoogle Scholar
  8. Gladman, B., Duncan, M.: On the fates of minor bodies in the outer solar system. Astron. J. 100, 1680–1693 (1990)ADSCrossRefGoogle Scholar
  9. Goldman, D., Kaper, T.: \(N\)th-order operator splitting schemes and nonreversible systems. SIAM J. Numer. Anal. 33, 349–367 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  11. Kahan, W.: Pracniques: further remarks on reducing truncation errors. Commun. ACM 8, 40 (1965)CrossRefGoogle Scholar
  12. Kinoshita, H., Yoshida, H., Nakai, H.: Symplectic integrators and their application to dynamical astronomy. Celest. Mech. Dyn. Astron. 50, 59–71 (1991)ADSCrossRefzbMATHGoogle Scholar
  13. Koseleff, P.V.: Calcul formel pour les méthodes de lie en mécanique hamiltonienne. PhD thesis, Ecole Polytechnique (1993a)Google Scholar
  14. Koseleff, P.V.: Relations among lie formal series and construction of symplectic integrators. In: Cohen, G. D., Mora, T., Moreno, O. (eds.) Applied Algebra, Algebraic Algorithms and Error Correcting Codes. 10th International symposium, (AAECC-10), San Juan de Puerto Rico, Puerto Rico, May 10–14, 1993, proceedings. Lect. Not. Comp. Sci, vol 673, pp. 213–230. Springer, New York (1993b)Google Scholar
  15. Koseleff, P.V.: Exhaustive search of symplectic integrators using computer algebra. Integration algorithms and classical mechanics, Fields Inst. Commun. 10, 103–120 (1996)Google Scholar
  16. Laskar, J.: A numerical experiment on the chaotic behaviour of the solar system. Nature 338, 237 (1989)ADSCrossRefGoogle Scholar
  17. Laskar, J.: The chaotic motion of the solar system—a numerical estimate of the size of the chaotic zones. Icarus 88, 266–291 (1990a)ADSCrossRefGoogle Scholar
  18. Laskar, J.: Les Méthodes Modernes de la Mecánique Céleste (Goutelas, France, 1989), Editions Frontières, chap Systèmes de Variables et Eléments, pp. 63–87 (1990b)Google Scholar
  19. Laskar, J.: Analytical framework in poincaré variables for the motion of the solar system. In: Roy, A. (ed.) Predictability, Stability, and Chaos in N-Body Dynamical Systems, pp. 93–114. NATO, Plenum Press, ASI (1991)Google Scholar
  20. Laskar, J., Robutel, P.: High order symplectic integrators for perturbed hamiltonian systems. Celest. Mech. Dyn. Astron. 80, 39–62 (2001)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. Laskar, J., Quinn, T., Tremaine, S.: Confirmation of resonant structure in the solar system. Icarus 95, 148–152 (1992)ADSCrossRefGoogle Scholar
  22. Laskar, J., Robutel, P., Joutel, F., Gastineau, M., Correia, A.C.M., Levrard, B.: A long-term numerical solution for the insolation quantities of the earth. Astron. Astrophys. 428, 261–285 (2004)ADSCrossRefGoogle Scholar
  23. Laskar, J., Fienga, A., Gastineau, M., Manche, H.: La2010: a new orbital solution for the long-term motion of the earth. Astron. Astrophys. 532, 89 (2011a)ADSCrossRefGoogle Scholar
  24. Laskar, J., Gastineau, M., Delisle, J.B., Farrés, A., Fienga, A.: Strong chaos induced by close encounters with ceres and vesta. Astron. Astrophys. 532, L4 (2011b)ADSCrossRefGoogle Scholar
  25. Lourens, L. J., Hilgen, F. J., Shackleton, N. J., Laskar, J., Wilson, D.: The neogene period. In: Gradstein, F., Ogg, J., Smith, A. (eds.) A Geologic. Time scale, pp 409–440. Cambridge University Press, UK (2004)Google Scholar
  26. McLachlan, R.I.: Composition methods in the presence of small parameters. BIT Numer. Math. 35, 258–268 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  27. McLachlan, R.I.: Families of high-order composition methods. Numer. Algorithms 31, 233–246 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  28. McLachlan, R., Quispel, R.: Splitting methods. Acta Numerica 11, 341–434 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Milankovitch, M.: Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem. Spec. Acad. R, Serbe, Belgrade (1941)Google Scholar
  30. Morbidelli, A.: Modern integrations of solar system dynamics. Annu. Rev. Earth Planet. Sci. 30, 89–112 (2002)ADSCrossRefGoogle Scholar
  31. Murua, A., Sanz-Serna, J.: Order conditions for numerical integrators obtained by composing simpler integrators. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 357(1754), 1079–1100 (1999)MathSciNetADSCrossRefzbMATHGoogle Scholar
  32. Quinn, T.R., Tremaine, S., Duncan, M.: A three million year integration of the earth’s orbit. Astron. J. 101, 2287–2305 (1991)ADSCrossRefGoogle Scholar
  33. Saha, P., Tremaine, S.: Long-term planetary integration with individual time steps. Astron. J. 108, 1962–1969 (1994)ADSCrossRefGoogle Scholar
  34. Sheng, Q.: Solving linear partial differential equations by exponential splitting. IMA J. Numer. Anal. 9, 199–212 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sussman, G.J., Wisdom, J.: Chaotic evolution of the solar system. Science 257, 56–62 (1992)MathSciNetADSCrossRefzbMATHGoogle Scholar
  36. Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and monte carlo simulations. Phys. Lett. A 146(6), 319–323 (1990)MathSciNetADSCrossRefGoogle Scholar
  37. Suzuki, M.: General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32(2), 400–407 (1991)MathSciNetADSCrossRefzbMATHGoogle Scholar
  38. Touma, J., Wisdom, J.: Lie-poisson integrators for rigid body dynamics in the solar system. Astron. J. 107, 1189–1202 (1994)ADSCrossRefGoogle Scholar
  39. Viswanath, D.: How many timesteps for a cycle? Analysis of the Wisdom–Holman algorithm. BIT Numer. Math. 42, 194–205 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Wisdom, J.: Symplectic correctors for canonical heliocentric n-body maps. Astron. J. 131(4), 2294 (2006)Google Scholar
  41. Wisdom, J., Holman, M.: Symplectic maps for the n-body problem. Astron. J. 102, 1528–1538 (1991)ADSCrossRefGoogle Scholar
  42. Wisdom, J., Holman, M., Touma, J.: Symplectic correctors. Fields Inst. Commun. 10, 217 (1996)MathSciNetGoogle Scholar
  43. Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150(5–7), 262–268 (1990)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Ariadna Farrés
    • 1
  • Jacques Laskar
    • 1
    Email author
  • Sergio Blanes
    • 2
  • Fernando Casas
    • 3
  • Joseba Makazaga
    • 4
  • Ander Murua
    • 4
  1. 1.Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMCParisFrance
  2. 2.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain
  3. 3.Departament de Matemàtiques, Institut de Matemàtiques i Aplicacions de CastellóCastellónSpain
  4. 4.Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, UPV/EHUDonostia/San SebastiánSpain

Personalised recommendations