Abstract
A celestial-mechanical model for the motion of two viscoelastic spheres in the gravitational field of a massive point is considered, treating them as a double planet. The spheres move along quasi-circular orbits in a single plane, with their rotational axes perpendicular to this plane. The deformation of the spheres is described using the classical theory of small deformations. A Kelvin-Voigt model is adopted for the viscous forces. A system of evoutionary equations is obtained and applied to analyze the joint translational-rotational tidal evolution of the Earth and Moon in the gravitational field of the Sun. This system has been numerically integrated several billion years into the past and into the future. The results are compared with the predictions of other theories, paleontological data, and astronomical observations.
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References
G. H. Darwin, The Tides and Kindred Phenomena in the Solar System (CreateSpace Independent Publishing Platform, 2013; Nauka, Moscow, 1965).
G. J. F. MacDonald, Rev. Geophys. 2, 467 (1964).
P. Goldreich, Rev. Geophys. 4, 411 (1966).
V. V. Beletskii, Preprint Inst. Prikl. Matem. ANSSSR No. 43 (IPM AN SSSR, Moscow, 1978).
D. J. Webb, Geophys. J. R. Astron. Soc. 70, 261 (1982).
G. A. Krasinsky, Celest. Mech. Dyn. Astron. 84, 27 (2002).
J. Touma and J. Wisdom, Astron. J. 108, 1943 (1994).
M. Efroimsky and V. Lainey, J. Geophys. Res. — Planets 112, E12003 (2007).
F. Mignard, Moon Planets 20, 301 (1979).
F. Mignard,Moon Planets 23, 185 (1980).
S. Ferraz-Mello, A. Rodriguez, and H. Hussmann, Celest. Mech. Dyn. Astron. 101, 171 (2008).
M. Efroimsky and J. G. Williams, Celest. Mech. Dyn. Astron. 104, 257 (2009).
M. Efroimsky and V. V. Makarov, Astrophys. J. 764, id. 26 (2013).
V. A. Churkin. Preprint Inst. Prikl. Astron. RAN No. 121 (IPA RAN, St. -Petersburg, 1998).
V. A. Churkin, Tr. Inst. Prikl. Astron. RAN, No. 4, 187 (1999).
V. A. Churkin, Tr. Inst. Prikl. Astron. RAN, No. 5, 225 (2000).
M. Efroimsky, Celest. Mech. Dynam. Astron. 112, 283 (2012).
V. G. Vil’ke, Analytical Mechanics of Systems with an Infinite Number of Degrees of Freedom (Mekhmat MGU, Moscow, 1997) [in Russian].
V. G. Vil’ke, Prikl. Mat. Mekh. 44, 395 (1980).
V. G. Vil’ke, S. A. Kopylov, and Yu. G. Markov, Prikl. Mat. Mekh. 49, 25 (1985).
Yu. G. Markov and I. S. Minyaev, Astron. Vestn. 28, 59 (1994).
V. G. Vil’ke and A. V. Shatina, Kosmich. Issled. 39, 316 (2001).
A. A. Zlenko, The Equations of Motion of Two Viscoelastic Spheres in the Central Force Field in the Double-Planet Problem (Mosk. Avtodorozhn. Inst. (Gos. Tekh. Univ.), Moscow, 2009) [in Russian]; Available from VINITI RAN No. 581-V2009 (2009).
A. A. Zlenko, Kosmich. Issled. 49, 569 (2011).
A. A. Zlenko, Kosmich. Issled. 50, 490 (2012).
B. Luzum, N. Capitaine, A. Fienda, W. Folkner, T. Fukushima, J. Hilton, C. Hohenkerk,G. Krasinsky, G. Petit, E. Pitjeva, M. Soffel, and P. Wallace, Celest. Mech. Dyn. Astron. 110, 293 (2011).
Astronomical Year-Book 2012 (Nauka, St. Petersburg, 2011) [in Russian].
J. G. Williams and D. L. Boggs, in Proceedings of the 16th International Workshop on Laser Ranging, Ed. by S. Schillak (Space Res. Centre, Polish Acad. Sci., Warsaw, 2009), p. 101.
F. R. Stefenson and L. V. Morrison, Phil. Trans. R. Soc. A 351, 165 (1995).
G. E. Williams, Geophys. Res. Lett. 29, 421 (1997).
C. D. Murray and S. F. Dermott, Solar System Dynamics (Cambridge Univ. Press, Cambridge, 2000; Fizmatlit, Moscow, 2010).
G. A. Krasinsky, Soobshch. Inst. Prikl. Astron. RAN 148 (2002).
M. R. Walter, Science 170, 1331 (1970).
W. M. Kaula, Rev. Geophys. Space 9, 217 (1971).
G. H. Darvin, Phil. Trans. R. Soc. London 171, 713 (1880).
W. K. Hartman and D. R. Davis, Icarus 24, 504 (1975).
E. V. Pitjeva and N. P. Pitjev, Solar Syst. Res. 46, 78 (2012).
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Original Russian Text © A.A. Zlenko, 2015, published in Astronomicheskii Zhurnal, 2015, Vol. 92, No. 1, pp. 80–96.
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Zlenko, A.A. A celestial-mechanical model for the tidal evolution of the Earth-Moon system treated as a double planet. Astron. Rep. 59, 72–87 (2015). https://doi.org/10.1134/S1063772915010096
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DOI: https://doi.org/10.1134/S1063772915010096