Abstract
An analytical expansion of the disturbing function arising from direct planetary perturbations on the motion of satellites is derived. As a Fourier series, it allows the investigation of the secular effects of these direct perturbations, as well as of every argument present in the perturbation. In particular, we construct an analytical model describing the evection resonance between the longitude of pericenter of the satellite orbit and the longitude of a planet, and study briefly its dynamic. The expansion developed in this paper is valid in the case of planar and circular planetary orbits, but not limited in eccentricity or inclination of the satellite orbit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abu-El-Ata N., Chapront J.: Développements analytiques de l’inverse de la distance en mécanique céleste. Astron. Astrophys. 38, 57–66 (1975)
Bidart P.: MPP01, a new solution for planetary perturbations in the orbital motion of the Moon. Astron. Astrophys. 366, 351–358 (2001)
Breiter S.: The prograde C7 resonance for earth and mars satellite orbits. Celest. Mech. 77, 201–214 (2000)
Bretagnon P.: Termes à à longues périodes dans le système solaire. Astron. Astrophys. 30, 141–154 (1974)
Brown E.W.: An Introductory Treatise on the Lunar Theory. Cambridge University Press, Cambridge (1896)
Brown E.W.: The Inequalities in the Motion of the Moon due to the Direct Action of the Planets. Cambridge University Press, Cambridge (1908a)
Brown E.W.: On the Lunar inequalities due to planetary action. Mon. Not. R. Astron. Soc. 68, 148–170 (1908b)
Brumberg V.A., Ivanova T.: New approach to determining planetary perturbations in lunar theory. Celest. Mech. 26, 77–81 (1982)
Chapront J., Abu-El-Ata N.: Les perturbations planétaires de la Lune en variables elliptiques. I. Formulaire et séparation de Brown. Astron. Astrophys. 55, 83–94 (1977)
Chapront J., Francou G.: The lunar theory ELP revisited. Introduction of new planetary perturbations. Astron. Astrophys. 404, 735–742 (2003)
Chapront-Touzé M.: Construction itérative d’une solution du problème central de la Lune. Influence des petits diviseurs. Astron. Astrophys. 36, 5–16 (1974)
Chapront-Touzé M., Chapront J.: Les perturbations planétaires de la Lune. Astron. Astrophys. 91, 233–246 (1980)
Chapront-Touzé M., Chapront J.: The lunar ephemeris ELP 2000. Astron. Astrophys. 124, 50–62 (1983)
Chapront-Touzé M., Chapront J.: ELP 2000-85: a semi-analytical lunar ephemeris adequate for historical times. Astron. Astrophys. 190, 342–352 (1988)
Ćuk M., Burns J.A.: On the secular behavior of irregular satellites. Astron. J. 128, 2518–2541 (2004)
Ćuk M.: Excitation of lunar eccentricity by planetary resonances. Science 318, 244 (2007)
Delaunay C.: Note sur les mouvements du périgée et du noeud de la Lune. Compt. Rend. Hebdom. Acad. Sci 74, 17–21 (1872)
Deprit A., Henrard J., Rom A.: Analytical lunar ephemeris. I. Definition of the main problem. Astron. Astrophys. 10, 257–269 (1971a)
Deprit A., Henrard J., Rom A.: Analytical lunar ephemeris: the variational orbit. Astron. J. 76, 273–276 (1971b)
Ferraz-Mello S.: Canonical perturbations theories—degenerate systems and resonance. Astrophysics and Space Science Library, Springer (2007)
Ford E.B., Kozinsky B., Rasio F.A.: Secular evolution of hierarchical triple star systems. Astrophys. J. 535, 385–401 (2000)
Frouard J., Fouchard M., Vienne A.: About the dynamics of the evection resonance. Astron. Astrophys. 515, A54 (2010)
Frouard J., Vienne A., Fouchard M.: The long-term dynamics of the jovian irregular satellites. Astron. Astrophys. 532, A44 (2011)
Henrard J.: A new solution to the main problem of Lunar theory. Celest. Mech. 19, 337 (1979)
Hill G.W.: On the development of the perturbative function in periodic series. Analyst II, 161–180 (1875)
Kaula W.M.: Development of the lunar and solar disturbing functions for a close satellite. Astroph. J. 67, 300 (1962)
Kaula, W.M., Yoder, C.F.: Lunar orbit evolution and tidal heating of the moon. Abstracts of the Lunar and Planetary Science Conference, 7, 440 (1976)
Kinoshita H., Nakai H.: Secular perturbations of fictitious satellites of Uranus. Celest. Mech. Dyn. Astron. 52, 293–303 (1991)
Kinoshita H., Nakai H.: Analytical solution of the kozai resonance and its application. Celest. Mech. Dyn. Astron. 75, 125–147 (1999)
Kinoshita H., Nakai H.: General solution of the Kozai mechanism. Celest. Mech. Dyn. Astron. 98, 67–74 (2007)
Kozai Y.: Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67, 591–598 (1962)
Laskar J.: Secular evolution of the solar system over 100 million years. Astron. Astrophys. 198, 341–362 (1988)
Laskar J., Boué G.: Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations. Astron. Astrophys. 522, A60 (2010)
Lee M.H., Peale S.J.: Secular evolution of hierarchical planetary systems. Astrophys. J. 592, 1201–1216 (2003)
Lidov, M.L., Analiz Evolucii Orbit Iskustvennich Sputnikov. Problemi Dvigenia Iskustvennich Nebesnich, Tel. Izd. Akad. Nauk SSSR. 119–134 (1961)
Lidov M.L.: The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Planet. Space Sci. 9, 719–759 (1962)
Morbidelli A.: Modern Celestial Mechanics: Aspects of Solar System Dynamics. Taylor & Francis, London (2002)
Murray C.D., Dermott S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)
Nesvorný D., Morbidelli A.: Three-body mean motion resonances and the chaotic structure of the asteroid belt. Astron. J. 116, 3029–3037 (1998)
Nesvorný D., Morbidelli A.: An analytic model of three-body mean motion resonances. Celest. Mech. Dyn. Astron. 71, 243–271 (1999)
Radau R.: Mémoires et observations. Remarques sur certaines inégalités à longue période du mouvement de la lune. Bull. Astronomique Serie I 9, 137–146 (1892)
Radau R.: Mémoires et observations. Remarques sur certaines inégalités à longue période du mouvement de la lune [Suite et fin]. Bull. Astronomique Serie I 9, 185–212 (1892)
Schmidt D.S.: The main problem of Lunar theory solved by the method of Brown. Moon Planets 23, 135–164 (1980)
Simon J.L., Bretagnon P.: First order perturbations of the four large planets. Astron. Astrophys. 42, 259–263 (1975)
Simon J.L., Bretagnon P.: Second order perturbations of the four large planets. Astron. Astrophys. 69, 369–372 (1978)
Standaert D.: Direct perturbations of the planets on the Moon’s motion. Celest. Mech. 22, 357–369 (1980)
Standaert D.: Comments about the direct perturbations of Venus and Mars on the Moon’s motion. Celest. Mech. 26, 113–119 (1982)
Standaert D.: Direct perturbations of the planets on the Moon’s motion: results and comparisons. Celest. Mech. 30, 21–29 (1983)
Tisserand F.: Traité de Mécanique Céleste, Tome I. Gauthier-Villars, Paris (1889)
Tisserand F.: Traité de Mécanique Céleste, Tome III. Gauthier-Villars, Paris (1894)
Touma J., Wisdom J.: Resonances in the early evolution of the earth-moon system. Astron. J. 115, 1653–1663 (1998)
Yokoyama, T., Vieira Neto, E., Winter, O.C., Sanchez, D.M., de Oliveira Brasil, P.I.: On the evection resonance and its connection to the stability of outer satellites. Mathematical Problems in Engineering 251978 (2008). doi:10.1155/2008/251978
Winter O.C., Boldrin L.A.G., Vieira Neto E., Vieira Martins R., Giuliatti Winter S.M., Gomes R.S., Marchis F., Descamps P.: On the stability of the satellites of asteroid 87 Sylvia. Mon. Not. R. Astron. Soc. 395, 218–227 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Frouard, J., Yokoyama, T. Analytical direct planetary perturbations on the orbital motion of satellites. Celest Mech Dyn Astr 115, 59–79 (2013). https://doi.org/10.1007/s10569-012-9447-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-012-9447-6