Abstract
The main limit to the time span of a numerical integration of the planetary orbits is no longer set by the availability of computer resources, but rather by the accumulation of the integration error. By the latter we mean the difference between the computed orbit and the dynamical behaviour of the real physical system, whatever the causes. The analysis of these causes requires an interdisciplinary effort: there are physical model and parameters errors, algorithm and discretisation errors, rounding off errors and reliability problems in the computer hardware and system software, as well as instabilities in the dynamical system. We list all the sources of integration error we are aware of and discuss their relevance in determining the present limit to the time span of a meaningful integration of the orbit of the planets. At present this limit is of the order of 108 years for the outer planets. We discuss in more detail the truncation error of multistep algorithms (when applied to eccentric orbits), the coefficient error, the method of Encke and the associated coordinate change error, the procedures used to test the numerical integration software and their limitations. Many problems remain open, including the one of a realistic statistical model of the rounding off error; at present, the latter can only be described by a semiempirical model based upon the simpleN 2 ∈ formula (N=number of steps, ∈=machine accuracy), with an unknown numerical coefficient which is determined only a posteriori.
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References
Applegate, J.H., Douglas, M.R., Gursel, Y., Hunter, P., Seitz, C. and Sussman, G.J.: 1985,IEEE Transactions on Computers c 34, 822–831.
Applegate, J.H., Douglas, M.R., Gursel, Y., Sussman, G.J. and Wisdom, J.: 1986,Astron. J. 92, 176–194.
Barlier, F., Carpino, M., Farinella, P., Mignard, F., Milani, A. and Nobili, A.M.: 1986,Annales Geophysicae 4 A, 193–210.
Broucke, R.A. and Cefola, P.J.: 1972,Celest. Mech. 5, 303–310.
Brouwer, D.: 1937,Astron. J. 46, 149–153.
Cappellari, J.O., Velez, C.E. and Fuchs, A.J.: 1976,Mathematical Theory of the Goddard Trajectory Determination System, Goddard Space Flight Center, NASA Rep. X-582-76-77.
Carpino, M., Milani, A. and Nobili, A.M.: 1987,Astron. Astrophys. 181, 182–194.
Cohen, C.J., Hubbard, E.C. and Oesterwinter, C.: 1973,Astr. Pap. American Ephemeris 22, Part I.
Danby, J.M.A. and Burkardt, T.M.: 1983,Celest. Mech. 31, 95–107.
Dormand, J.R. and Prince, P.J.: 1978,Celest. Mech. 18, 223–232.
Fabri, E. and Penco, U.: 1984, private communication.
Flammia, A.: 1988, inTable ronde de planetologie dynamique, Aussois, C. Froesclé and D. Benest eds., in press.
Fox, K.: 1984,Celest. Mech. 33, 127–142.
Kinoshita, H.: 1968,pubbl. Astron. Soc. Japan 20, 1–23.
Kinoshita, H. and Nakai, H.: 1984,Celest. Mech. 34, 203–217.
Knuth, D.: 1981,The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading Mass..
Kovalevsky, J.: 1963,Introduction à la mécanique céleste, A. Colin, Paris.
Lighthill, J.: 1986,Proc. Royal Soc London A,407, 35–50.
Marsden, B.: 1984, private communication.
Milani, A. and Nobili, A.M.: 1983,Celest. Mech. 31, 241–291.
Milani, A., Nobili, A.M., Fox, K. and Carpino, M.: 1986,Nature 319, 386–388.
Milani, A., Nobili, A.M. and Carpino, M.: 1987a,Astron. Astrophys. 172, 265–279.
Milani, A., Nobili, A.M. and Farinella, P.: 1987b,Non-gravitational perturbations and satellite geodesy, A. Hilger, Bristol.
Nobili, A.M., Milani, A. and Carpino, M.: 1988, ‘Fundamental frequencies and small divisors int he orbits of the outer planets’,Astron. Astrophys., in press.
Roy, A.E.: 1982,Orbital motion, Adam Hilger Ltd., Bristol.
Sussman, G.J.: 1985, private communication.
Walker, I.W., Emslie, G.A. and Roy, A.E.: 1980,Celest. Mech. 22, 371–402.
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Milani, A., Nobili, A.M. Integration error over very long time spans. Celestial Mechanics 43, 1–34 (1987). https://doi.org/10.1007/BF01234550
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DOI: https://doi.org/10.1007/BF01234550