Abstract
This is the first part of a study to develop a modern theory of physical libration of the Moon caused by a liquid core. We use a special approach to studying Moon’s rotation relying on Poincaré’s planetary model and special forms of equations of motion in Andoyer and Poincaré variables. We construct expansions of the force function of the problem (the second harmonic of the selenopotential) in Andoyer and Poincaré variables for a high-precision description of disturbed orbital motion of the Moon. We investigate the main regularities in lunar rotational motion taken as a body with a solid nonspherical mantle and an ellipsoidal liquid core. The motion of the ideal liquid of the core is simple according to Poincaré. The Cassini laws can be dinamically interpreted for the motion of a synchronous satellite with a liquid core. The Cassini angle (the inclination of the rotation axis relative to the normal to the ecliptic plane) determined by us is very consistent with its determinations from laser observations.
Similar content being viewed by others
References
Weber, R.C., Lin, P.-Y., Garnero, E.J., et al., Seismic detection of the lunar core, Science, 2011, vol. 331, no. 6015, pp. 309–312.
Poincaré, H., Sur la précession des corps déformables, Bull. Astron., 1910, vol. 27, pp. 321–356.
Barkin, Yu.V., Hanada, H., Matsumoto, K., et al., The effects of the physical librations of the Moon, caused by a liquid core and their possible detection from the longterm laser observations and in the Japanese lunar project ILO, in Book of Abstracts of the Third Moscow Solar System Symposium (SM-S3), 8–12 October 2012, Moscow, Russia, 2012, abstract 3MS3-MN-10, pp. 18–19. http://ms2012.cosmos.ru/3ms3-mn-10_h._hanada_2012_iki_moon_librations_121005.pdf.
Kudryavtsev, S.M., Long-term harmonic development of lunar ephemeris, Astron. Astrophys, 2007, vol. 471, no. 3, pp. 1069–1075. doi 10.1051/0004-6361:20077568
Barkin, Yu.V., Kudryavtsev, S.M., and Barkin, M.Yu., Perturbations of the first order of the moon rotation, in Proceedings of the International Conference “Astronomy and World Heritage: Across Time and Continents” (Kazan, 19–24 August 2009), Kazan: KSU, 2009, pp. 161–164.
Barkin, Yu.V. and Ferrándiz, J.M., New approach to development of Moon rotation theory, in The 35th Lunar and Planetary Science Conference, League City, Texas, 2004, abstract no. 1294. http://adsabs.harvard.edu/abs/2004LPI….35.1294B.
Rambaux, N. and Williams, J.G., The Moon’s physical librations and determination of their free modes, Celestial Mech. Dyn. Astron., 2011, vol. 109, no. 1, pp. 85–100. doi 10.1007/s10569-010-9314-2
Williams, J.G., Boggs, D.H., and Turyshev, S.G., LLR analysis—JPL model and data analysis, California Institute of Technology LLR Workshop, Harvard, Boston, MADecember 9–10, 2010.
Williams, J.G., Boggs, D.H., and Ratcliff, J.T., Lunar moment of inertia and Love number, in The 42nd Lunar and Planetary Science Conference, 2011.
Williams, J.G., Boggs, D.H., and Ratcliff, J.T., Lunar moment of inertia, Love number and core, in The 43rd Lunar and Planetary Science Conference, 2012.
Lamb, H., Hydrodynamics, New York: Dover Publications, 1945.
Araki, H, Tazawa, S., Noda, H., et al., Lunar global shape and polar topography derived from Kaguya-LALT laser altimetry, Science, 2009. V. 323, no. 5916, pp. 897–900. doi 10.1126/science.1164146
Williams, J.G., Boggs, D.H., Yoder, C.F., et al., Lunar rotational dissipation in solid body and molten core, J. Geophys. Res.: Planets, 2001, vol. 106, no. E11, pp. 27933–27968.
Matsumoto, K., Goossens, S., Ishihara, Y., et al., An improved lunar gravity field model from SELENE and historical tracking data: Revealing the farside gravity features, J. Geophys. Res., 2010, vol. 115, E06007. doi 10.1029/2009JE003499
Sevilla, M. and Romero, P., Polar motion for an elastic Earth model with a homogeneous liquid core using a canonical theory, Bull. Géod., 1987, vol. 61, no. 1, pp. 1–20.
Ferrándiz, J.M. and Barkin, Yu.V., Model of the Earth with the eccentric and the moveable liquid core, Journées Systèmes de Référence Spatio-Temporels and IXLohrmann-Kolloquium, Dresden, Germany, 13–15 September 1999, Motion of Celestial Bodies, Astrometry and Astronomical Reference Frames, Soffel, M. and Capitaine, N., Eds., Dresden, Germany: Senckenberg naturhistorische Sammlungen Dresden, 2000, p. 192.
Ferrándiz, J.M. and Barkin, Yu.V., On integrable cases of the Poincaré problem, Astron. Astrophys. Trans., 2001, vol. 19, pp. 769–780.
Barkin, Yu.V., An analytical theory of the lunar rotational motion, in Proceedings of the International Symposium Figure and Dynamics of the Earth, Moon and Planets (September 15–20, 1986, Prague, Czechoslovakia), Holota, P., Ed., Prague, 1987, pp. 657–677.
Barkin, Yu.V., Dynamics of the system of nonspherical celestial bodies and theory of a rotating Moon, Doctoral (Phys.–Math.) Dissertation, Moscow: Sternberg Astronomical Institute, Moscow State University, 1989.
Kinoshita, H., Theory of rotation of the rigid Earth, Celestial Mech., 1977, vol. 15, pp. 277–326.
Sasao, T., Okubo, S., and Saito, M., A simple theory on the dynamical effects of a stratified fluid core upon nutational motion of the Earth, in nutation and the Earth rotation, Proc. IAU Symp., 1980, no. 78, pp. 165–183.
Beletskii, V.V., On Cassini’s laws, Preprint 79 of Inst. Appl. Math. USSR Acad. Sci., 1971.
Beletskii, V.V., Dvizhenie iskusstvennogo sputnika otnositel’no tsentra mass (Artificial Satellite Motion Relative to the Center of Mass), Moscow: Nauka, 1965.
Beletskii, V.V., Dvizhenie sputnika otnositel’no tsentra mass v gravitatsionnom pole (Satellite Motion Relative to the Center of Mass in the Gravitational Field), Moscow: Mosk. Gos. Univ., 1975.
Simon, J.L., Bretagnon, P., Chapront, J., et al., Numerical expressions for precession formulae and mean elements for the Moon and planets, Astron. Astrophys., 1994, no. 282, pp. 663–683.
Getino, J. and Ferrándiz, J.M., A Hamiltonian theory for an elastic Earth: Secular rotational acceleration, Celestial Mech., 1991, vol. 52, pp. 381–396.
Navarro, J.F., Teoría analítica de la rotación de la tierra rígida mediante manipulación simbólica especifica, Doctoral Dissertation, Alicante, Department of Applied Mathematics, University of Alicante, 2006.
Calame, O., Free librations of the Moon determined by an analysis of laser range measurements, The Moon, 1976, vol. 15, nos. 3–4, pp. 343–352.
Cassini, J.D., Traité de l’origine et du progrès de l’astronomie, Paris. 1693.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.V. Barkin, 2016, published in Kosmicheskie Issledovaniya, 2016, Vol. 54, No. 4, pp. 347–365.
Rights and permissions
About this article
Cite this article
Barkin, Y.V. Theory of physical libration of the Moon caused by a liquid core: Cassini’s motion. Cosmic Res 54, 325–342 (2016). https://doi.org/10.1134/S0010952516030023
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0010952516030023