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A method of symplectic integrations with adaptive time-steps for individual Hamiltonians in the planetary N-body problem

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Abstract

A new algorithm is developed for long-term integrations of the N-body problem. The method uses symplectic integrations of the Hamiltonian equations of motion for each body. This allows one to employ individual adaptive time-steps in computations. The efficiency of this technique is demonstrated by several tests performed for typical problems of Solar System dynamics.

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References

  • Aarseth S. (2003). Gravitational N-Body Simulations. Tools and Algorithms. Cambridge monographs on Mathematical Physics. Cambridge University Press, Cambridge

    Google Scholar 

  • Brouwer D. and Woerkom A.J.J. (1950). The secular variations of the orbital elements of the principal planets. Astron. Pap., Washington 13: 85–107

    Google Scholar 

  • Chambers J.E. (1999). A hybrid symplectic integrator that permits close encounters between massive bodies. Mon. Not. Roy. Astron. Soc. 304: 793–799

    Article  ADS  Google Scholar 

  • Duncan M.J., Levison H.F. and Lee M.H. (1998). A multiple time step symplectic algorithm for integrating close encounters. Astron. J. 116: 2067–2077

    Article  ADS  Google Scholar 

  • Emel’yanenko V.V. (2002). An explicit symplectic integrator for cometary orbits. Celest. Mech. Dyn. Astron. 84: 331–341

    Article  MATH  ADS  Google Scholar 

  • Fukushima T. (2003). Efficient orbit integration by scaling for Kepler energy consistency. Astron. J. 126: 1097–1111

    Article  ADS  Google Scholar 

  • Gladman B., Duncan M. and Candy J. (1991). Symplectic integrators for long-term integrations in Celestial Mechanics. Celest. Mech. Dyn. Astron. 52: 221–240

    Article  ADS  Google Scholar 

  • Hairer E. and Söderlind G. (2005). Explicit, time reversible, adaptive step size control. SIAM J. Sci. Comput. 26: 1838–1851

    Article  MATH  Google Scholar 

  • Lee M.H., Duncan M.J. and Levison H.F. (1997). Variable timestep integrators for long-term orbital integrations. In: Clarke, D.A. and West, M.J. (eds) Proc. 12th Kingston Meeting, Computational Astrophysics., pp 32–38. Astronomical Society of the Pacific, San Francisco

    Google Scholar 

  • Mikkola S. (1997). Practical symplectic methods with time transformation for the few-body problem. Celest. Mech. Dyn. Astron. 67: 145–165

    Article  MATH  ADS  Google Scholar 

  • Mikkola S. and Aarseth S. (2002). A time-transformed leapfrog scheme. Celest. Mech. Dyn. Astron. 84: 343–354

    Article  MATH  ADS  Google Scholar 

  • Mikkola S. and Innanen K. (2002). Individual accuracy checks for massive bodies and particles in symplectic integration. Astron. J. 124: 3445–3448

    Article  ADS  Google Scholar 

  • Mikkola S. and Tanikawa K. (1999). Explicit symplectic algorithms for time-transformed Hamiltonians. Celest. Mech. Dyn. Astron. 74: 287–295

    Article  MATH  ADS  Google Scholar 

  • Mikkola S. and Wiegert P. (2002). Regularizing time transformations in symplectic and composite integration. Celest. Mech. Dyn. Astron. 82: 375–390

    Article  MATH  ADS  Google Scholar 

  • Preto M. and Tremaine S. (1999). A class of symplectic integrators with adaptive timestep for separable Hamiltonian systems. Astron. J. 118: 2532–2541

    Article  ADS  Google Scholar 

  • Rauch K.P. and Holman M. (1999). Dynamical chaos in the Wisdom–Holman integrator: origins and solutions. Astron. J. 117: 1087–1102

    Article  ADS  Google Scholar 

  • Saha P. and Tremaine S. (1992). Symplectic integrators for Solar System dynamics. Astron. J. 104: 1633–1640

    Article  ADS  Google Scholar 

  • Saha P. and Tremaine S. (1994). Long-term planetary integration with individual time steps. Astron. J. 108: 1962–1969

    Article  ADS  Google Scholar 

  • Sanz-Serna J.M. and Calvo M.P. (1994). Numerical Hamiltonian Problems. Chapman & Hall, London

    MATH  Google Scholar 

  • Sharaf Sh.G. and Budnikova N.A. (1967). On the secular changes of the orbital elements of the Earth influencing a climate of the geological past. Bull. Inst. Theor. Astron. Akad. Nauk SSSR 11: 231–261

    Google Scholar 

  • Stiefel E.L. and Scheifele G. (1971). Linear and Regular Celestial Mechanics. Springer-Verlag, Berlin, Heidelberg, New York

    MATH  Google Scholar 

  • Szebehely V. (1967). Theory of Orbits. Academic Press, New York

    Google Scholar 

  • Wisdom J. and Holman M. (1991). Symplectic maps for the N-body problem. Astron. J. 102: 1528–1538

    Article  ADS  Google Scholar 

  • Wisdom J., Holman M. and Touma J. (1996). Symplectic correctors. Fields Inst. Commun. 10: 217–244

    Google Scholar 

  • Yoshida H. (1990). Construction of higher order symplectic integrators. Phys. Lett. A 150: 262–268

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Computational and Celestial Mechanics, South Ural University, Lenina 76, Chelyabinsk, 454080, Russia

    Vacheslav Vasilievitch Emel’yanenko

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  1. Vacheslav Vasilievitch Emel’yanenko
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Correspondence to Vacheslav Vasilievitch Emel’yanenko.

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Emel’yanenko, V.V. A method of symplectic integrations with adaptive time-steps for individual Hamiltonians in the planetary N-body problem. Celestial Mech Dyn Astr 98, 191–202 (2007). https://doi.org/10.1007/s10569-007-9077-6

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  • DOI: https://doi.org/10.1007/s10569-007-9077-6

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