LongTerm Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession
 Michael Efroimsky
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It was believed until very recently that a nearequatorial satellite would always keep up with the planet’s equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich [Astronomy & Astrophysics (2004) Vol. 415, pp. 1187–1199], this misconception originated from a wrong interpretation of a (mathematically correct) result obtained in terms of nonosculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet’s obliquity change. Some of these terms will be nontrivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet’s obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the firstorder terms responsible for the drift will be shortperiod and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the firstorder terms secular. For example, the planetary nutations will resonate with the satellite’s orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in a subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics.
 Arnold, V. I. (1989) Mathematical Methods of Classical Mechanics.. SpringerVerlag, New York
 Ashby, N., Allison, T. (1993) ‘Canonical planetary perturbation equations for velocity dependent forces and the lenseThirring precession’. Celest. Mech. Dyn. Astr 57: pp. 537585 CrossRef
 Brumberg, V. A., Evdokimova, L. S., Kochina, N. G. (1971) ‘Analytical methods for the orbits of artificial satellites of the Moon’. Celestial Mech. 3: pp. 197221 CrossRef
 Brumberg, V. A. (1992) Essential Relativistic Celestial Mechanics. Adam Hilger, Bristol
 Brouwer, D. (1959) ‘Solution of the problem of artificial satellite theory without drag’. The Astronomical J. 64: pp. 378397 CrossRef
 Chernoivan, V. A., Mamaev, I. S. (1999) ‘The restricted twobody problem and the Kepler problem in the constantcurvature spaces’. Reg. Chaot. Dynam 4: pp. 112124 CrossRef
 Defraigne, P., Rivoldini, A., Van Hoolst, T., Dehant, V. (2003) Mars nutation resonance due to free inner core nutation’. J. Geophys. Res. 108: pp. 5128 CrossRef
 Dehant, V., van Hoolst, T., Defraigne, P. (2000) ‘Comparison between the nutations of the planet Mars and the nutations of the Earth’. Surv. Geophys 21: pp. 89110 CrossRef
 Efroimsky, M.: 2002a, ‘Equations for the orbital elements. Hidden symmetry’, Preprint No 1844 of the Institute of Mathematics and its Applications, University of Minnesota http://www.ima.umn.edu/preprints/feb02/feb02.html.
 Efroimsky, M.: 2002b, ‘The implicit gauge symmetry emerging in the Nbody problem of celestial mechanics’, astroph/0212245.
 Efroimsky, M. and Goldreich, P.: 2003, ‘Gauge symmetry of the Nbody problem in the HamiltonJacobi approach’, J. Math. Phys. 44, 5958–5977 astroph/0305344.
 Efroimsky, M. and Goldreich, P.: 2004, ‘Gauge freedom in the Nbody problem of celestial mechanics’, Astron. Astrophys. 415, 1187–1199; astroph/0307130.
 Efroimsky, M.: 2004, ‘Longterm evolution of orbits about a precessing oblate planet, The case of uniform precession’, astroph/0408168
 Eubanks, T. M. (1993) ‘Variation in the orientation of the Earth’ In: Contributions of Space Geodesy to Geodynamics: Earth Dynamics. Amer. Geophys. Union, Washington
 Goldreich, P. (1965) ‘Inclination of satellite orbits about an oblate precessing planet’. Astron. J 70: pp. 59 CrossRef
 Kinoshita, T. (1993) ‘Motion of the orbital plane of a satellite due to a secular change of the obliquity of its mother planet’. Celest. Mech. Dyn. Astr. 57: pp. 359368 CrossRef
 Kozai, Y. (1960) ‘Effect of precession and nutation on the orbital elements of a close earth satellite’. The Astronomical J., 65: pp. 621623
 Laskar, J. (2004) ‘A comment on ‘Accurate Spin Axes and Solar System Dynamics: Climatic Variations for the Earth and Mars’. Astron. Astrophys 416: pp. 799800 CrossRef
 Laskar, J., Robutel, J. (1993) ‘The chaotic obliquity of the planets’. Nature 361: pp. 608612 CrossRef
 Laskar, J. (1990) ‘The chaotic motion of the solar system A numerical estimate of the size of the chaotic zones’. Icarus 88: pp. 266291 CrossRef
 Marsden, J., Ratiu, T (2003) Introduction to Mechanics and Symmetry.. Springer, NY
 Morbidelli, A. (2002) Modern Celestial Mechanics: Dynamics in the Solar System.. Taylor & Francis, London
 Murison, M. (1988) ‘Satellite Capture and the Restricted ThreeBody Problem’. Ph.D. thesis, University of Wisconsin, Madison
 Newman, W., Efroimsky, M. (2003) ‘The method of variation of constants and multiple time scales in orbital mechanics’. Chaos 13: pp. 476485 CrossRef
 Proskurin, V. F., Batrakov, Y. V. (1960) ‘Perturbations of the Motion of Artificial Satellites, caused by the Earth Oblateness’. The Bulletin of the Institute of Theoretical Astronomy 7: pp. 537548
 Richardson, D. L., Kelly, T. J. (1988) ‘Twobody motion in the postNewtonian approximation’. Celestial Mech 43: pp. 193210 CrossRef
 Touma, J., Wisdom, J. (1994) ‘LiePoisson integrators for rigid body dynamics in the solar system’. Astron J. 107: pp. 11891202 CrossRef
 Van Hoolst, T., Dehant, V., Defraigne, P. (2000) ‘Chandler wobble and free core nutation for Mars’. Planet. Space Sci 48: pp. 11451151 CrossRef
 Ward, W. (1973) ‘Largescale variations in the obliquity of Mars’. Science 181: pp. 260262
 Ward, W. (1974) ‘Climatic variations of Mars. Astronomical theory of insolation’. J. Geophys. Res 79: pp. 33753386 CrossRef
 Title
 LongTerm Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession
 Journal

Celestial Mechanics and Dynamical Astronomy
Volume 91, Issue 12 , pp 75108
 Cover Date
 20050101
 DOI
 10.1007/s105690042415z
 Print ISSN
 09232958
 Online ISSN
 15729478
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 nearequatorial satellites of oblate planets
 contact orbital elements
 Industry Sectors
 Authors

 Michael Efroimsky ^{(1)}
 Author Affiliations

 1. US Naval Observatory, 20392, Washington, DC, USA