Review of the works on the orbital evolution of solar system major planets / Kholshevnikov K.V., Kuznetsov E.D. // Solar System Research. - 2007. - V. 41, l. 4. - P. 265-300.

ISSN:
00380946
Type:
Article
Abstract:
The cognition history of the basic laws of motion of Solar system major planets is presented. Before Newton, the description of motion was purely kinematic, without relying on physics in view of its underdevelopment. From the standpoint of the modern mathematical theory of approximation, all of the models from Ptolemy's predecessors to Kepler inclusive differ only in details. The mathematical theory worked on an infinite time scale; the motion was represented by P. Bohl's quasi-periodic functions (a special case of H. Bohr's quasi-periodic functions). After Newton, the mathematical description of motion came to be based on physical principles and took the form of ordinary differential equations. The advent of General Relativity (GR) and other relativistic theories of gravitation in the 20th century changed little the mathematical situation in the field under consideration. Indeed, the GR effects in the Solar system are so small that the post-post-Newtonian approximation is sufficient. Therefore, the mathematical description using ordinary differential equations is retained. Moreover, the Lagrangian and Hamiltonian forms of the equations are retained. From the 18th century until the mid-20th century, all the theories of planetary motion needed for practice were constructed analytically by the small parameter method. In the early 20th century, Lyapunov and Poincaré established the convergence of the corresponding series for a sufficiently small time interval. Subsequently, K. Kholshevnikov estimated this interval to be on the order of several tens of thousands of years, which is in agreement with numerical experiments. The first works describing analytically (in the first approximation) the evolution on cosmogonic time scales appeared in the first half of the 19th century (Laplace, Lagrange, Gauss, Poisson). The averaging method was developed in the early 20th century based on these works. Powerful analytical and numerical methods that have allowed us to make significant progress in describing the orbital evolution of Solar system major planets appeared in the second half of the 20th century. This paper is devoted to their description. © 2007 Pleiades Publishing, Inc.
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DOI:
10.1134/S0038094607040016
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Affiliations Astronomical Institute, St. Petersburg State University, Universitetskii pr. 28, St. Petersburg 198504, Russian Federation; Ural State University, Yekaterinburg, Russian Federation
References Akim, E.L., Stepan'Yants, V.A., Numerical theory of the motion of the Earth and Venus based on data of radar and optical observations and tracking data for the Venera 9 and 10 Satellites (1977) Dokl. Akad. Nauk SSSR, 233, pp. 314-317. , [Sov. Phys. Dokl. (Engl. Transl.), vol. 25, pp. 135-137]; Akim, E.L., Brumberg, V.A., Kislik, M.D., Kovalevsky, J., Brumberg, V.A., A Relativistic Theory of Motion of Inner Planets (1986) Proc. IAU Symp. No. 114: Relativity in Celestial Mechanics and Astrometry, pp. 63-68. , Kluwer Dordrecht; Anolik, M.V., Krasinskii, G.A., Pius, L.Yu., Trigonometric Theory of Secular Perturbations of Major Planets (1969) Tr. Inst. Teor. Astron. Akad. Nauk SSSR, 14, pp. 3-47; Applegate, J.H., Douglas, M.R., Gürsel, Y., The Outer Solar System for 200 Million Years (1986) Astron. J., 92, pp. 176-194; Arnol'D, V.I., Little Denominators and the Problem of Stability in Classical and Celestial Mechanics (1963) Usp. Mat. Nauk, 18, pp. 91-192. , 6; Ash, M.E., Shapiro, I.I., Smith, W.B., Astronomical Constants and Planetary Ephemerides Deduced from Radar and Optical Observations (1967) Astron. J., 72, pp. 332-350; Bogolyubov, N.N., Mitropol'Skii, Yu.A., (2005) Asimptoticheskie Metody v Teorii Nelineinykh Kolebanii. Sobranie Nauchnykh Trudov, , Matematika i nelineinaya mekhanika Moscow Translated under the title Asymptotic Methods in the Theory of Nonlinear Oscillations, New York: Gordon and Breach, 1962; Bretagnon, P., Termes à Longues Périodes dans le Système Solaire (1974) Astron. Astrophys., 30, pp. 141-154; Bretagnon, P., Théorie au Deuxième Ordre des Planétes Inférieures (1980) Astron. Astrophys., 84, pp. 329-341; Bretagnon, P., Construction d'une Théorie des Grosses Planétes par une Méthode Itérative (1981) Astron. Astrophys., 101, pp. 342-349; Bretagnon, P., Constantes d'Intégration et Éléments Moyens Pour l'Ensemble des Planétes (1982) Astron. Astrophys., 108, pp. 69-75; Bretagnon, P., Théorie du Mouvement de l'Ensemble des Planétes. Solution VSOP82 (1982) Astron. Astrophys., 114, pp. 278-288; Bretagnon, P., Théorie des Planètes Inférieures (1982) Celest. Mech., 26, pp. 161-167; Bretagnon, P., Amélioration des Théories Planètaries Analytiques (1984) Celest. Mech., 34, pp. 193-201; Bretagnon, P., Construction of a Planetary Solution with the Help of An N-Body Program, and Analytical Complements (1986) Celest. Mech., 38, pp. 181-190; Bretagnon, P., Francou, G., Planetary theories in rectangular and spherical variables. VSOP87 solutions (1988) Astron. Astrophys., 202, pp. 309-315; Bretagnon, P., Méthode Itérative de Construction d'une Théorie Générale Planètaire (1990) Astron. Astrophys., 231, pp. 561-570; Bretagnon, P., Simon, J.-L., Théorie Générale du Couple Jupiter-Saturne par une Méthode Itérative (1990) Astron. Astrophys., 239, pp. 387-398; Bretagnon, P., Francou, G., Ferraz-Mello, S., General Theory for the Outer Planets (1992) Proc. IAU Symp. No. 152: Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System, pp. 37-42. , Kluwer Dordrecht; Bretagnon, P., Ferraz-Mello, S., Morando, B., Arlot, J.E., Analytical Solution of the Motion of the Planets over Several Thousands of Years (1996) Proc. IAU Symp. No. 172: Dynamics, Ephemerides and Astrometry of the Solar System, pp. 17-28. , Kluwer Dordrecht; Brouwer, D., Van Woerkom, A.J.J., The Secular Variations of the Orbital Elements of the Principal Planets (1950) Astron. Pap. Am. Ephem., 13, pp. 81-107; Brumberg, V.A., A Numerical Development of a Generalized Planetary Theory (1967) Astron. Zh., 44, pp. 204-216. , 1. [Sov. Astron. (Engl. Transl.), vol. 11, no. 1, pp. 156-165]; Brumberg, V.A., Giacaglia, G.E.O., Application of Hill's Lunar Method in General Planetary Theory (1970) Periodic Orbits, Stability and Resonances, pp. 410-450. , Reidel Dordrecht; Brumberg, V.A., Egorova, A.V., Trigonometric linear theory of the second-order of secular perturbations and motion of major planets (1971) Nablyudeniya Iskusstv. Nebes. Tel, (62), pp. 42-72; Brumberg, V.A., Chapront, J., Construction of a General Planetary Theory of the First Order (1973) Celest. Meth, 8, pp. 335-356; Brumberg, V.A., An interative method of general planetary theory (1974) Proc. IAU Symp. No. 62: The Stability of the Solar System and of Small Stellar Systems, pp. 139-155. , Kozai, Y., Ed; Brumberg, V.A., Evodokimova, L.S., Skripnichenko, V.I., Quasiperiodic Intermediate Orbits of the Major Planets and Zero-Order Resonance (1975) Astron. Zh., 52, pp. 420-430. , 2. [Sov. Astron. (Engl. Transl.), vol. 19, no. 2, pp. 255-260]; Brumberg, V.A., Evdokimova, L.S., Skripnichenko, V.I., Secular Perturbations in General Planetary Theory (1975) Celest. Mech., 11, pp. 131-138; Brumberg, V.A., Perturbation Theory in Rectangular Coordinates (1978) Celest. Mech., 18, pp. 319-336; Brumberg, V.A., Evdokimova, L.S., Skripnichenko, V.I., Szebechely, V., Mathematical Results of the General Planetary Theory in Rectangular Coordinates (1978) Dynamics of Planets and Satellites and Theories of Their Motion, pp. 33-48. , Reidel Dordrecht; Brumberg, V.A., (1980) Analiticheskie Algoritmy Nebesnoi Mekhaniki (Analytical Algorithms of Celestial Mechanics), , Nauka Moscow; Brumberg, V.A., Kinoshita, H., Nakai, H., General Planetary Theory Revisited with the Aid of Elliptic Functions (1992) Proc. of the 25th Symp. Celest. Mech., p. 156. , Nat. Astron. Obs. Tokyo; Brumberg, V.A., General Planetary Theory in Elliptic Functions (1994) Celest. Mech. Dyn. Astron., 59, pp. 1-36; Brumberg, V.A., (1995) Analytical Techniques of Celestial Mechanics, , Springer Heidelberg; Brumberg, V.A., Ferraz-Mello, S., Morando, B., Arlot, J.E., Theory Compression with Elliptic Functions (1996) Proc IAU Symp. No. 172: Dynamics, Ephemerides and Astrometry of the Solar System, pp. 89-100. , Kluwer Dordrecht; Brumberg, V.A., Klioner, S.A., Ferraz-Mello, S., Morando, B., Arlot, J.E., Numerical Efficiency of the Elliptic Functions Expansions of the First-Order Intermediare for General Planetary Theory (1996) Proc. IAU Symp. No. 172: Dynamics, Ephemerides and Astrometry of the Solar System, pp. 101-104. , Kluwer Dordrecht; Budnikova, N.A., Determination of perturbations using the Laplace-Newcomb method on high-performing computers (1971) Nablyudeniya Iskusstv. Nebes. Tel, (62), pp. 73-90; Carpino, M., Milani, A., Nobili, A.M., Long-Term Numerical Integrations and Synthetic Theories for the Motion of the Outer Planets (1987) Astron. Astrophys., 181, pp. 182-194; Chambers, J.E., A Hybrid Symplectic Integrator that Permits Close Encounters between Massive Bodies (1999) Monthly Notices of the Royal Astronomical Society, 304, pp. 793-799; Chapront, J., Construction d'une Théorie Littérale Planètaire jusqu'au Second Ordre des Masses (1970) Astron. Astrophys, 7, pp. 175-203; Chapront, J., Simon, J.L., Variations Séeculaires au Premier Ordre des Éléments des Quatre Grosses Planétes. Comparaison avec le Verrier et Galliot (1972) Astron. Astrophys., 19, pp. 231-234; Chapront, J., Bretagnon, P., Mehl, M., Un Formulaire Pour le Calcul des Perturbations d'Ordres Éléeves dans les Problémes Planètaires (1975) Celest. Mech., 11, pp. 379-399; Chapront, J., Simon, J.L., Planetary Theories with the Aid of the Expansions of Elliptical Functions (1996) Celest. Mech. Dyn. Astron., 63, pp. 171-188; Cohen, C.J., Hubbard, E.C., Libration of the Close Approaches of Pluto to Neptune (1965) Astron. J., 70, pp. 10-13; Cohen, C.J., Hubbard, E.C., Oesterwinter, C., Elements of the Outer Planets for One Million Years (1973) Astron. Pap. Am. Ephem., 22, pp. 3-82; Cohen, C.J., Hubbard, E.C., Oesterwinter, C., Planetary Elements for 10000000 Years (1973) Celest. Mech., 7, pp. 438-448; Danilov, V.M., Dorogavtseva, L.V., Estimates of Relaxation Times in Numerical Dynamical Models of Open Star Clusters (2003) Astron. Zh., 80, pp. 526-534. , 6. [Astron. Rep. (Engl. Transl.), vol. 47, no. 6, pp. 483-491]; Davydov, V.L., Molchanov, A.M., Numerical experiments in the problem of evolution of two-planetary system (1971) Preprint Inst. Appl. Math. Acad. Sci. SSSR, (16). , Moscow; Duncan, M.J., Lissauer, J.J., The Effects of Post-Main-Sequence Solar Mass Loss on the Stability of Our Planetary System (1998) Icarus, 134, pp. 303-310; Duriez, L., Téorie Générale Planètarie en Variables Elliptiques. I. Développement des Equations (1977) Astron. Astrophys., 54, pp. 93-112; Duriez, L., Théoréme de Poisson en Variables Héliocentriques. Conditions d'Application de ce Théoréme Relatif à l'Invariabilité des Grands Axes des Orbites Planètaries a l'Ordre deux des Masses (1978) Astron. Astrophys., 68, pp. 199-216; Eckert, W.J., Brouwer, D., Clemence, G.M., Coordinates of five outer planets, 1653-2060 (1951) Astron. Pap. Am. Ephem., 12; Eroshkin, G.I., Glebova, N.I., Fursenko, M.A., (1992) Dopolneniya 27-28A K "astronomicheskomu Ezhegodniku", pp. 1-8. , ITA RAN St. Petersburg. (Additions 27-28A to "Astronomical Year-Book"); Everhart, E., Implicit Single Methods for Integrating Orbits (1974) Celest. Mech., 10, pp. 35-55; Gerasimov, I.A., Chazov, V.V., Rykhlova, L.V., Tagaeva, D.A., Construction of the Theory of Motion for Solar-System Bodies Based on a Universal Method for Perturbative Function Calculation (2000) Astron. Vestn., 34, pp. 559-566. , 6. [Sol. Syst. Res. (Engl. Transl.), vol. 34, no. 6, pp. 509-516]; Glebova, N.I., Refinement of Planets' Ephemerides Based on Processing Optical and Radar Observations in 1960-1980 (1984) Byull. Inst. Teor. Astron. Akad. Nauk SSSR, 15, pp. 241-250; Grebenikov, E.A., Ryabov, Yu.A., (1971) Novye Kachestvennye Metody v Nebesnoi Mekhanike (New Qualitative Methods in Celestial Mechanics), , Nauka Moscow; Guzzo, M., The Web of Three-Planet Resonances in the Outer Solar System (2005) Icarus, 174, pp. 273-284; Guzzo, M., The Web of Three-Planet Resonances in the Outer Solar System. II. A Source of Orbital Instability for Uranus and Neptune (2006) Icarus, 181, pp. 475-485; Hamid, S.E., First-order planetary theory (1968) Smiths. Astrophys. Obs. Spec. Rep., p. 235; Hayes, W.B., Chaos in the Outer Solar System May Be Indeterminate (2005) Bull. Am. Astron. Soc., 37, p. 1414; Hayes, W.B., (2007) Is the Outer Solar System Chaotic?, , astroph/0702179; Hayes, W.B., Surfing on the edge: Chaos vs. Near-integrability in the system of Jovian planets (2007) Nature, , in press; Ito, T., Tanikawa, K., Very long-term numerical integrations of solar system planetary orbits (2002) Proc. 8th IAU Asian-Pacific, Regional Meeting, V.II. Astron. Soc. Jpn., pp. 45-46; Ito, T., Tanikawa, K., Long-term integrations and stability of planetary orbits in our solar system (2002) Mon. Not. R. Astron. Soc., 336, pp. 483-500; Ivanova, T.V., Poisson series processor PSP (1997) Preprint Inst. Theor. Astron., Russ. Acad. Sci., (64). , St. Petersburg; Ivanova, T., A New Echeloned Poisson Series Processor (EPSP) (2001) Celest. Mech. Dyn. Astron., 80, pp. 167-176; Jorba, A., Zou, M., A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods (2005) Exp. Math., 14, pp. 99-117; Kholshevnikov, K.V., (1985) Asimptoticheskie Metody Nebesnoi Mekhaniki (Asymptotic Methods in Celestial Mechanics), , Leningr. Gos. Univ. Leningrad; Kholshevnikov, K.V., Preservation of the Form of Area Integrals in Averaging Transformations (1991) Astron. Zh., 68, pp. 660-663. , 3. [Sov. Astron. (Engl. Transl.), vol. 35, no. 3, p. 325]; Kholshevnikov, K.V., Precision of Epicyclic Theory (1994) Istoriko-astronomicheskie Issledovaniya. Vyp. 24 (Historical and Astronomical Investigations), pp. 181-191. , Yanus Moscow; Kholshevnikov, K.V., D'Alembertian Functions in Celestial Mechanics (1997) Astron. Zh., 74, pp. 146-153. , 1. [Astron. Rep. (Engl. Transl.), vol. 41, no. 1, pp. 135-141]; Kholshevnikov, K.V., Tublina, O.K., Coordinates in Keplerian Motion as D'Alembertian Functions (1998) Astron. Zh., 75, pp. 476-480. , 3. [Astron. Rep. (Engl. Transl.), vol. 42, no. 3, pp. 420-424]; Kholshevnikov, K.V., The Hamiltonian in the Planetary or Satellite Problem as a D'Alembertian Function (2001) Astron. Zh., 78, pp. 669-672. , 7. [Astron. Rep. (Engl. Transl.), vol. 45, no. 7, pp. 577-579]; Kholshevnikov, K.V., Greb, A.V., Noncanonical Parametrization of Poisson Brackets in Celestial Mechanics (2001) Astron. Vestn., 35, pp. 457-462. , 5. [Sol. Syst. Res. (Engl. Transl.), vol. 35, no. 5, pp. 415-419]; Kholshevnikov, K.V., Greb, A.V., Kuznetsov, E.D., The Expansion of the Hamiltonian of the Planetary Problem into the Poisson Series in All Keplerian Elements (Theory) (2001) Astron. Vestn., 35, pp. 267-272. , 3. [Sol. Syst. Res. (Engl. Transl.), vol. 35, no. 3, pp. 243-248]; Kholshevnikov, K.V., Greb, A.V., Kuznetsov, E.D., The Expansion of the Hamiltonian of the Two-Planetary Problem into the Poisson Series in All Elements: Estimation and Direct Calculation of Coefficient (2002) Astron. Vestn., 36, pp. 75-87. , 1. [Sol. Syst. Res. (Engl. Transl.), vol. 36, no. 1, pp. 68-79]; Kholshevnikov, K.V., Kuznetsov, E.D., Finkelstein, A., Capitaine, N., Evolution of a Two-Planetary Regular System on a Cosmogonic Time Scale (2004) Journees-2003. Astrometry, Geodynamics and Solar System Dynamics: From Milliarcseconds to Microarcseconds, pp. 286-287. , IAA RAS St. Petersburg; Kholshevnikov, K.V., Kuznetsov, E.D., Byrd, G.G., Kholshevnikov, K.V., Myllri, A.A., Behaviour of a Weakly Perturbed Two-Planetary System on a Cosmogonic Time-Scale, Order and Chaos in Stellar and Planetary Systems (2004) Astron. Soc. Pac. Conf. Ser., pp. 99-105. , Astron. Soc. Pac. San Francisco; Kholshevnikov, K.V., Kuznetsov, E.D., Knežević, Z., Milani, A., Behavior of a Two-Planetary System on a Cosmogonic Time-Scale (2005) Proc. IAU Coll. No. 197: Dynamics of Populations of Planetary Systems, pp. 107-112. , Cambridge Univ. Press Cambridge; Kinoshita, H., Nakai, H., Motions of the Perihelios of Neptune and Pluto (1984) Celest. Mech., 34, pp. 203-217; Kinoshita, H., Nakai, H., Ferraz-Mello, S., New Methods for Long-Time Numerical Integrations of the Planetary Orbits (1992) Proc. IAU Symp. No. 152: Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System, pp. 395-406. , Kluwer Dordrecht; Kinoshita, H., Nakai, H., Long-Term Behavior of the Motion of Pluto over 5.5 Billion Years (1995) Earth, Moon Planets, 71, pp. 165-173. , 3; Kinoshita, H., Nakai, H., Ferraz-Mello, S., Morando, B., Arlot, J.E., The Motion of Pluto over the Age of the Solar System (1996) Proc. IAU Symp. No. 172: Dynamics, Ephemerides and Astrometry of the Solar System, pp. 61-70. , Kluwer Dordrecht; Kislik, M.D., Kolyuka, Yu.F., Kotel'Nikov, V.A., Unified Relativistic Theory of the Motion of the Inner Planets of the Solar System (1980) Dokl. Akad. Nauk SSSR, 255, pp. 545-547. , [Sov. Phys. Dokl. (Engl. Transl.), vol. 25, pp. 867-869]; Klioner, S.A., Kinoshita, H., Nakai, H., Some Typical Algorithms of the Perturbation Theory within Mathematica and Their Analysis (1992) Proc. 25th Symposium on Celestial Mechanics, p. 172. , Nat. Astron. Obs. Tokyo; Klioner, S.A., On the Expansions of Intermediate Orbit for General Planetary Theory (1997) Celest. Mech. Dyn. Astron., 66, pp. 345-363; Krasinsky, G.A., Pius, L.Yu., Secular perturbations of major planets (1971) Nablyudeniya Iskusstv. Nebes. Tel, (62), pp. 93-112; Krasinsky, G.A., Pitjeva, E.V., Sveshnikov, M.A., Sveshnikova, E.S., The AT-1 Analytical Theory of Motion of Inner Planets and its Application to Solving Problems of Ephemeris Astronomy (1978) Tr. Inst. Teor. Astron. Akad. Nauk SSSR, 17, pp. 46-53; Krasinsky, G.A., Pitjeva, E.V., Sveshnikov, M.A., Sveshnikova, E.S., Some Results on the Processing of Radar, Laser, and Optical Observations of the Inner Planets and the Moon (1981) Dokl. Akad. Nauk SSSR, 261, pp. 1320-1324. , [Sov. Phys. Dokl. (Engl. Transl.), vol. 26, pp. 1103-1105]; Krasinsky, G.A., Pitjeva, E.V., Sveshnikov, M.A., Sveshnikova, E.S., Improvement of the Ephemerides of the Inner Planets and the Moon Using Radar, Laser, and Meridian Measurements during 1961-1980 (1982) Byull. Inst. Teor. Astron., Akad. Nauk SSSR, 15, pp. 169-175; Krasinsky, G.A., Aleshkina, E.Yu., Pitjeva, E.V., Sveshnikov, M.L., Kovalevsky, J., Brumberg, V.A., Relativistic Effects from Planetary and Lunar Observations of the XVIII-XX Centuries (1986) Proc. IAU Symp. No. 114: Relativity in Celestial Mechanics and Astrometry, pp. 315-328. , Kluwer Dordrecht; Krasinsky, G.A., Pitjeva, E.V., Sveshnikov, M.L., Chunajeva, L.I., (1989) Dvizhenie Vneshnikh Planet Na Stoletnem Intervale Vremeni (Motion of Outer Planets during the Centenary Interval of Time), (4816), pp. V89. , Available from VINITI, Leningrad; Krasinsky, G.A., Pitjeva, E.V., Sveshnikov, M.L., Chunajeva, L.I., The Motion of Major Planets from Observations 1769-1988 and Some Astronomical Constants (1993) Celest. Mech. Dyn. Astron., 55, pp. 1-23; Kuznetsov, E.D., Kholshevnikov, K.V., Expansion of the Hamiltonian of the Two-Planetary Problem into the Poisson Series in All Elements: Application of the Poisson Series Processor (2004) Astron. Vestn., 38, pp. 171-179. , 2. [Sol. Syst. Res. (Engl. Transl.), vol. 38, no. 2, pp. 147-154]; Kuznetsov, E.D., Kholshevnikov, K.V., Dynamical Evolution of Weakly Disturbed Two-Planetary System on Cosmogonic Time-Scales: The Sun-Jupiter-Saturn System (2006) Astron. Vestn., 40, pp. 263-275. , 3. [Sol. Syst. Res. (Engl. Transl.), vol. 40, no. 3, pp. 239-250]; Laskar, J., Progress in General Planetary Theory (1984) Celest. Mech., 34, pp. 219-221; Laskar, J., Accurate Methods in General Planetary Theory (1985) Astron. Astrophys., 144, pp. 133-146; Laskar, J., Secular Terms of Classical Planetary Theory Using the Results of General Theory (1986) Astron. Astrophys., 157, pp. 59-70; Laskar, J., Secular Evolution of the Solar System over 10 Million Years (1988) Astron. Astrophys., 198, pp. 341-362; Laskar, J., A Numerical Experiment on the Chaotic Behaviour of the Solar System (1989) Nature, 338, pp. 237-238; Laskar, J., The Chaotic Motion of the Solar System. A Numerical Estimate of the Size of the Chaotic Zones (1990) Icarus, 88, pp. 266-291; Laskar, J., Ferraz-Mello, S., A Few Points on the Stability of the Solar System (1992) Proc. IAU Symp. No. 152: Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System, pp. 1-16. , Kluwer Dordrecht; Laskar, J., Frequency Analysis of Dynamical System (1993) Celest. Mech. Dyn. Astron., 56, pp. 191-196; Laskar, J., Large Scale Chaos in the Solar System (1994) Astron. Astrophys., 287, pp. 9-L12; Laskar, J., Large Scale Chaos and Marginal Stability in the Solar System (1996) Celest. Mech. Dyn. Astron., 64, pp. 115-162; Laskar, J., Ferraz-Mello, S., Morando, B., Arlot, J.E., Marginal Stability and Chaos in the Solar System (1996) Proc. IAU Symp. No. 172: Dynamics, Ephemerides and Astrometry of the Solar System, pp. 75-88. , Kluwer Dordrecht; Laskar, J., Robutel, P., Stability of the Planetary Three-Body Problem. I. Expansion of the Planetary Hamiltonian (1995) Celest. Mech. Dyn. Astron., 62, pp. 193-217; Laskar, J., Robutel, P., High Order Symplectic Integrators for Perturbed Hamiltonian Systems (2001) Celest. Mech. Dyn. Astron., 80, pp. 39-62; Lecar, M., Franklin, F.A., Holman, M.J., Chaos in the Solar System (2001) Ann. Rev. Astron. Astrophys., 39, pp. 581-631; Lestrade, J.-F., Bretagnon, P., Perturbations Relativistes pour l'Ensemble des Planétes (1982) Astron. Astrophys., 105, pp. 42-52; Levitan, B.M., (1953) Pochti-periodicheskie Funktsii, , Gostekhizdat Moscow (Almost-Periodic Functions); Lissauer, J.J., Chaotic Motion in the Solar System (1999) Rev. Mod. Phys., 71, pp. 835-845. , 3; Locatelli, U., Giorgilli, A., Invariant Tori in the Secular Motions of the Three-Body Planetary Systems (2000) Celest. Mech. Dyn. Astron., 78, pp. 47-74; Malkin, I.G., (1966) Teoriya Ustoichivosti Dvizheniya, , Nauka Moscow (The Theory of Motion Stability); Message, P.J., Asymptotic Series for Planetary Motion in Periodic Terms in Three Dimensions (1982) Celest. Mech., 26, pp. 25-39; Milani, A., Nobili, A.M., Fox, K., Carpino, M., Long-Term Changes in the Semimajor Axes of the Outer Planets (1986) Nature, 319, pp. 386-388; Milani, A., Nobili, A.M., Carpino, M., Secular Variations of the Semimajor Axes: Theory and Experiments (1987) Astron. Astrophys., 172, pp. 265-279; Milani, A., Nobili, A.M., Intergation Error over Very Long Time Spans (1988) Celest. Mech., 43, pp. 1-34; Moisson, X., Solar System Planetary Motion to Third Order of Masses (1999) Astron. Astrophys., 341, pp. 318-327; Moisson, X., Bretagnon, P., Analytical Planetary Solution VSOP2000 (2000) Celest. Mech. Dyn. Astron., 80, pp. 205-213; Murray, N., Holman, M., The Origin of Chaos in the Outer Solar System (1999) Science, 283, pp. 1877-1881; Nacozy, P.E., On the Stability of the Solar System (1976) Astron. J., 81, pp. 787-791; Nacozy, P.E., A Discussion of Long-Term Numerical Solutions of the Jupiter-Saturn-Sun System (1977) Celest. Mech., 16, pp. 77-86; Nacozy, P.E., Numerical studies on the stability of the solar system (1979) Proc. IAU Symp. No. 81: Dynamics of the Solar System, 81, pp. 17-21. , Duncombe, R.L., Ed; Nekhoroshev, N.N., Exponential Estimation of the Time of Stability of Close to Integrable Hamiltonian Systems (1977) Usp. Mat. Nauk, 32, pp. 5-66. , 6; Nekhoroshev, N.N., Exponential Estimation of the Time of Stability of Close to Integrable Hamiltonian Systems, 2 (1979) Tr. Seminara I.G. Petrovskogo, 5, pp. 5-50. , (Proc. of the I.G. Petrovskii Seminar); Newhall, X.X., Standish, E.M., Williams, J.G., DE102: A Numerical Integrated Ephemerides of the Moon and Planets Spanning Forty-Four Centuries (1983) Astron. Astrophys., 125, pp. 150-167; Newman, W.I., Varadi, F., Lee, A.Y., Numerical Integration, Lyapunov Exponents and the Outer Solar System (2000) Bull. Am. Astron. Soc., 32, p. 859; Newman, W.I., Lee, A.Y., Symplectic Integration Methods and Chaos: Timestep Selection and Lyapunov Time (2005) Bull. Am. Astron. Soc., 37, p. 531; Nobili, A.M., Milani, A., Carpino, M., Fundamental Frequencies and Small Divisors in the Orbits of the Outer Planets (1989) Astron. Astrophys., 210, pp. 313-336; Oesterwinter, C., Cohen, Ch.J., New Orbital Elements for Moon and Planets (1972) Celest. Mech., 5, pp. 317-395; Pitjeva, E.V., Refinement of Ephemerides of Major Planets and Estimation of the Value of Secular Change of Gravitational Constant on the Basis of Radar Observations of Spacecraft and Planets in 1961-1995 (1997) Tr. Inst. Prikt. Astron. Ross. Akad. Nauk, No. 1: Astrometr. Geodin., 1, pp. 249-261; Pitjeva, E.V., A new numerical theory of motion of major planets EPM98 and its comparison with the DE403 ephemeris of the jet propulsion laboratory, USA (1998) Tr. Inst. Prikt. Astron., Ross. Akad. Nauk, No. 3: Astrometr. Geodin., pp. 5-23; Pitjeva, E.V., Wytrzyszczak, I.M., Lieske, J.H., Feldman, R.A., The Ephemerides of the Inner Planets from Spacecraft Range Data and Radar Observations 1961-1995 (1997) Proc. IAU Coll. No. 165: Dynamics and Astrometry of Natural and Artificial Celestial Bodies, pp. 251-256. , Kluwer Dordrecht; Pitjeva, E.V., Modern Numerical Ephemerides of Planets and the Importance of Ranging Observations for their Creation (2001) Celest. Mech. Dyn. Astron., 80, pp. 249-271; Pitjeva, E.V., EPM2002 and EPM2002C - Two versions of high accuracy numerical planetary ephemerides constructed for TDB and TCB time scales (2004) Tr. Inst. Prikt. Aastron., Ross. Akad. Nauk, (11), pp. 91-106; Pitjeva, E.V., Finkelstein, A., Capitaine, N., Numerical Ephemerides of Planets and the Moon - EPM and Improvement of Some Astronomical Constants (2004) Journees-2003. Astrometry, Geodynamics and Solar System Dynamics: From Milliarcseconds to Microarcseconds, pp. 243-250. , IAA RAS St. Petersburg; Pitjeva, E.V., Kurtz, D.W., Precise Determination of the Motion of Planets and Some Astronomical Constants from Modern Observations (2004) Proc. IAU Coll. No. 196: Transits of Venus: New Views of the Solar System and Galaxy, , Cambridge Univ. Press Cambridge; Pitjeva, E.V., Modern numerical theories of motion of the sun, moon and major planets (2004) Tr. Inst. Prikl. Astron., Ross. Akad. Nauk, No. 10: Ephemeris Astron., 10, pp. 112-134; Pitjeva, E.V., High-Precision Ephemerides of Planets EPM and Determination of Some Astronomical Constants (2005) Astron. Vestn., 39, pp. 202-213. , 3. [Sol. Syst. Res. (Engl. Transl.), vol. 39, no. 3, pp. 176-136]; Poincaré, A., (1905) Leçons de Mecanique Céleste, , V.I, Paris; Quinn, T.R., Tremaine, S., Duncan, M., A Three Million Year Integration of the Earth's Orbit (1991) Astron. J., 101, pp. 2287-2305; Richardson, D.L., A Third Order Intermediate Orbit for Planetary Theory (1982) Celest. Mech., 26, pp. 187-195; Richardson, D.L., Walker, C.F., Numerical Simulation of the Nine-Body Planetary System Spanning Two Million Years (1989) J. Astronaut. Sci., 37, pp. 159-182; Robutel, P., An Application of KAM Theory to the Planetary Three Body Problem (1993) Celest. Mech. Dyn. Astron., 56, pp. 197-199; Robutel, P., The Stability of the Planetary Three-Body Problem: Influence of the Secular Resonances (1993) Celest. Mech. Dyn. Astron., 57, pp. 97-98; Robutel, P., Stability of the Planetary Three-Body Problem. II. KAM Theory and Existence of Quasiperiodic Motions (1995) Celest. Mech. Dyn. Astron., 62, pp. 219-261; Roy, A.E., Walker, I.W., MacDonald, A.J., Project LONGSTOP (1988) Vistas Astron., 32, pp. 95-116. , 2; Sharaf, Sh.G., Budnikova, N.A., On Secular Variations of the Earth's Orbital Parameters that Influenced Climate in Geological Past (1967) Byull. Inst. Teor. Astron., Akad. Nauk SSSR, 11, pp. 231-261. , 4; Simon, J.L., Théorie du Mouvement des Quatre Grosses Planétes. Solution TOP82 (1983) Astron. Astrophys., 120, pp. 197-202; Simon, J.L., Chapront, J., Perturbations du Second Ordre des Planétes Jupiter et Saturne. Comparaison avec le Verrier (1974) Astron. Astrophys., 32, pp. 51-64; Simon, J.L., Bretagnon, P., Perturbations du Premier Ordre des Quatre Grosses Planétes. Variations Littérales (1975) Astron. Astrophys., 42, pp. 259-263; Simon, J.L., Bretagnon, P., Résultats des Perturbations du Premier Ordre des Quatre Grosses Planétes. Variations Littérales (1975) Astron. Astrophys., Suppl. Ser., 22, pp. 107-160; Simon, J.L., Bretagnon, P., Perturbations du Deuxième Ordre des Quatre Grosses Planétes. Variations Sécularies du Demi-Grand Axe (1978) Astron. Astrophys., 69, pp. 369-372; Simon, J.L., Bretagnon, P., Résultats des Perturbations du Deuxième Ordre des Quatre Grosses Planétes (1978) Astron. Astrophys., Suppl. Ser., 34, pp. 183-194; Simon, J.L., Francou, G., Théorie au Troisième Ordre des Masses des Quatre Grosses Planétes (1981) Astron. Astrophys., 103, pp. 223-243; Simon, J.L., Francou, G., Amélioration des Théories de Jupiter et Saturne par Analyse Harmonique (1982) Astron. Astrophys., 114, pp. 125-130; Simon, J.L., Bretagnon, P., Théorie du Mouvement de Jupiter et Saturne sur un Intervalle de Temps de 6000 Ans. Solution JASON84 (1984) Astron. Astrophys., 138, pp. 169-178; Simon, J.L., Joutel, F., Calcul de Perturbations Mutuelles de Jupiter et Saturne en Fonction d'une Seule Variable Angulaire (1988) Astron. Astrophys., 205, pp. 328-334; Simon, J.L., Joutel, F., Bretagnon, P., Calcul de Perturbations Mutuelles des Quatre Grosses Planétes en Fonction d'une Seule Variable Angulaire (1992) Astron. Astrophys., 265, pp. 308-323; Simon, J.L., Bretagnon, P., Chapront, J., Numerical Expressions for Precession Formulae and Mean Elements for the Moon and the Planets (1994) Astron. Astrophys., 282, pp. 663-683; Sokolov, L.L., Kholshevnikov, K.V., On the Representability of Solutions of Three-Body Problems by Conditionally Periodic Functions - Part One. Part Two (1980) Astron. Zh., 57, pp. 168-177. , 1. [Sov. Astron. (Engl. Transl.), vol. 24, no. 1, pp. 99; no. 2, pp. 223]; Standish Jr., E.M., The JPL Planetary Ephemerides (1982) Celest. Mech., 26, pp. 181-186; Standish Jr., E.M., Orientation of the JPL Ephemerides, DE200/LE200, to the Dynamical Equinox of J2000 (1982) Astron. Astrophys., 114, pp. 297-302; Standish Jr., E.M., The Observational Basis for JPL's DE200, Planetary Ephemerides of the Astronomical Almanac (1990) Astron. Astrophys., 233, pp. 252-271; Standish Jr., E.M., An Approximation to the Inter Planet Ephemeris Errors in JPL's DE200 (1990) Astron. Astrophys., 233, pp. 272-274; Standish, E.M., Newhall, X.X., Williams, J.G., Folkner, W.M., JPL planetary and lunar ephemerides, DE403/LE403 (1995) Interoffice Memorandum, pp. 31410-127. , JLP; Standish, E.M., Newhall, X.X., Ferraz-Mello, S., Morando, B., Arlot, J.E., New Accuracy Levels for Solar System Ephemerides (1996) Proc. IAU Symp. No. 172: Dynamics, Ephemerides and Astrometry of the Solar System, pp. 29-36. , Kluwer Dordrecht; Standish, E.M., JPL planetary and lunar ephemerides, DE405/LE405 (1998) Interoffice Memorandum, , JLP; Standish, E.M., JPL planetary ephemerides, DE410 (2003) Interoffice Memorandum, , JPL; Subbotin, M.F., (1968) Vvedenie v Teoreticheskuyu Astronomiyu, , Nauka Moscow (Introduction to Theoretical Astronomy); Sukhotin, A.A., Algorithm of the Gauss-alphen-goryachev method in Lagrangian variables and its computer implementation (1981) Astron. Geodeziya, (9), pp. 67-73; Sukhotin, A.A., Evolution of orbital elements of outer planets over a time period of 800 thousand years (1984) Astron. Geodeziya, (12), pp. 80-91; Sukhotin, A.A., Kholshevnikov, K.V., Evolution of orbital elements of outer planets over 200 thousand years, calculated by alphen-Goryachev method (1986) Astron. Geodeziya, (14), pp. 5-21; Sussman, G.J., Wisdom, J., Numerical Evidence that the Motion of Pluto is Chaotic (1988) Science, 241, pp. 433-437; Sussman, G.J., Wisdom, J., Chaotic Evolution of the Solar System (1992) Science, 257, pp. 56-62; Varadi, F., Ghil, M., Kaula, W.M., Jupiter, Saturn, and Edge of Chaos (1999) Icarus, 139, pp. 286-294; Varadi, F., Runnegar, B., Ghil, M., Successive Refinements in Long-Term Integrations of Planetary Orbits (2003) Astrophys. J., 592, pp. 620-630; Vashkov'Yak, M.A., Quantitative characteristics of evolution of orbits in the restricted circular twice-averaged three-body problem (1979) Preprint of Keldysh Inst. Appl. Math., (157). , Acad. Sci. USSR, Moscow; Vashkov'Yak, M.A., Evolution of Orbits in the Restricted Circular Twice-Averaged Three-Body Problem. 1. Qualitative Investigations (1981) Kosm. Issled., 19, pp. 5-18. , 1. [Cosmic Res. (Engl. Transl.), vol. 19, no. 1, pp. 1-10]; Vashkov'Yak, M.A., Evolution of Orbits in the Restricted Circular Twice-Averaged Three-Body Problem. 2. Quantative Characteristics (1981) Kosm. Issled., 19, pp. 165-177. , 2. [Cosmic Res. (Engl. Transl.), vol. 19, no. 2, pp. 99-109]; Williams, C.A., Van Flandern, N., Wright, E.A., First Order Planetary Perturbations with Elliptic Functions (1987) Celest. Mech., 40, pp. 367-391; Williams, C.A., Ferraz-Mello, S., A Planetary Theory with Elliptic Functions and Elliptic Integrals Exhibiting no Small Divisors (1992) Proc. IAU Symp. No. 152: Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System, pp. 43-48. , Kluwer Dordrecht; Wisdom, J., Holman, M., Symplectic Maps for the N-Body Problem (1991) Astron. J., 102, pp. 1528-1538; Wisdom, J., Ferraz-Mello, S., Long-Term Evolution of the Solar System (1992) Proc. IAU Symp. No. 152: Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System, pp. 17-24. , Kluwer Dordrecht; Zhang, J.-X., A Study of the Planetary Secular Perturbations (1982) Chin. Astron. Astrophys., 6, pp. 137-144
Correspondence Address Kholshevnikov, K.V.; Astronomical Institute, St. Petersburg State University, Universitetskii pr. 28, St. Petersburg 198504, Russian Federation
Language of Original Document English
Abbreviated Source Title Sol. Syst. Res.
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