Skip to main content
SpringerLink
Log in

Effect of the Earth’s compression on the physical libration of the Moon

  • Published:
Solar System Research Aims and scope Submit manuscript

Abstract

The effect of the Earth’s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 106 than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon’s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon’s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth’s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth’s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L ☾ = 90° or L ☾ = 270°. The famous Cassini’s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth’s compression on the Moon’s libration in longitude, a harmonic with an amplitude of 0.03″ and period of T 8 = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth’s compression on the variation in spin angular velocity of the Moon proves to be negligible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abalakin, V.K., Aksenov, E.P., Grebennikov, E.A., et al., Spravochnoe rukovodstvo po nebesnoi mekhanike i astrodinamike (Handbook on Celestial Mechanics and Astrodynamics), Moscow: Nauka, 1976.

    Google Scholar 

  • Calame, O., Free Librations of the Moon Determined by an Analysis of Lazer Range Measurements, Moon, 1976, vol. 15, pp. 343–352.

    Article  ADS  Google Scholar 

  • Eckhardt, D.H., Theory of Libration of the Moon, Moon Planets, 1981, vol. 25, pp. 3–49.

    Article  ADS  MATH  Google Scholar 

  • Eckhardt, D.H., Planetary and Earth Figure Perturbations in the Libration of the Moon, in High Precision Earth Rotation and Earth-Moon Dynamics, Calame, O.D., Ed., Reidel Publ. Co., 1982, pp. 193–198.

  • Gutenberg, B., Physics of the Earth’s Interior, New York, 1959; Moscow: IL, 1963.

  • Henrard, J., Perturbations by the Oblateness of Earth and Moon, Celest. Mech., 1982, vol. 26, p. 95.

    Article  ADS  Google Scholar 

  • Khabibullin, Sh.T., Physical Libration of Luna 9, Izv. AOE, 1958, no. 31, pp. 1–182.

  • Kondratyev, B.P., On One Inaccuracy of Isaac Newton, Kvant, 2009, no. 5, pp. 38–41.

  • Kondratyev, B.P., The Vector Approach To the Problem of Physical Libration of the Moon. II. The Nonlinear Problem, Sol. Syst. Res., 2011, vol. 45, no. 5, pp. 447–456.

    Article  ADS  Google Scholar 

  • Korn, G.A. and Korn, T.M., Mathematical Handbook for Scientists and Engineers, New York: McGraw-Hill, 1961; Moscow: Nauka, 1970.

    MATH  Google Scholar 

  • Koziel, K., Constants of the Moon Free Libration on the Basis of Heliometric Observations from the Years 1841–1945, Earth, Moon Planets, 1989, vol. 45, pp. 153–159.

    Article  ADS  Google Scholar 

  • Kulikov, K.A. and Gurevich, V.B., Osnovy lunnoi astrometrii (Foundations of the Lunar Astrometry), Moscow: Nauka, 1972.

    Google Scholar 

  • Moutsalas, M.D., Librations of the Lunar Globe, in Physics and Astronomy of the Moon, Kopal, Z., Ed., New York: Acad. Press, 1971; Moscow: Mir, 1973.

    Google Scholar 

  • Sagitov, M.U., Lunnaya gravimetriya (Lunar Gravimetry), Moscow: Nauka, 1979.

    Google Scholar 

  • Wenjing, J. and Jinling, L., Determinations of Some Physical Parameters of the Moon with Lunar Lazer Ranging Date, Earth, Moon Planets, 1996, vol. 73, pp. 259–265.

    Article  Google Scholar 

  • Williams, J.G. and Dickey, J.O., Lunar Geophysics, Geodesy, and Dynamics, Proc. 13th Int. Workshop on Lazer Ranging, Washington, 2002, pp. 75–86.

Download references

Author information

Authors and Affiliations

  1. Udmurt State University, Izhevsk, Russia

    B. P. Kondratyev

  2. Pulkovo Astronomical Observatory, Russian Academy of Sciences, Pulkovskoe sh. 65, St. Petersburg, 196140, Russia

    B. P. Kondratyev

Authors
  1. B. P. Kondratyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © B.P. Kondratyev, 2012, published in Astronomicheskii Vestnik, 2012, Vol. 46, No. 5, pp. 369–379.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kondratyev, B.P. Effect of the Earth’s compression on the physical libration of the Moon. Sol Syst Res 46, 341–351 (2012). https://doi.org/10.1134/S003809461205005X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S003809461205005X

Keywords

Search

Navigation