Astrophysics and Space Science

, 363:186 | Cite as

On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon

  • B. P. KondratyevEmail author
Original Article


From the observations of the gravitational field and the figure of the Moon, it is known that its center of mass (briefly COM) does not coincide with the center of figure (COF), and the line “COF/COM” is not directed to the center of the Earth, but deviates from it to the South–East. Here we study the deviation of the lunar COM to the East from the mean direction to Earth.

At first, we consider the optical libration of a satellite with synchronous rotation around the planet for an observer at a point on second (empty) orbit focus. It is found that the main axis of inertia of the satellite has asymmetric nonlinear oscillations with amplitude proportional to the square of the orbit eccentricity. Given this effect, a mechanism of tidal secular evolution of the Moon’s orbit is offered that explains up to \(20\%\) of the known displacement of the lunar COM to the East. It is concluded that from the alternative—evolution of the Moon’s orbit with a decrease or increase in eccentricity—only the scenario of evolution with a monotonous increase in orbit eccentricity agrees with the displacement of lunar COM to the East. The precise calculations available confirm that now the eccentricity of the lunar orbit is actually increasing and therefore in the past it was less than its modern value, \(e = 0.0549\).

To fully explain the displacement of the Moon’s COM to the East was deduced a second mechanism, which is based on the reliable effect of tidal changes in the shape of the Moon. For this purpose the differential equation which governs the process of displacement of the Moon’s COM to the East with inevitable rounding off its form in the tidal increase process of the distance between the Earth and the Moon is derived. The second mechanism not only explains the Moon’s COM displacement to the East, but it also predicts that the elongation of the lunar figure in the early epoch was significant and could reach the value \(\varepsilon\approx0.31\). Applying the theory of tidal equilibrium figures, we can estimate how close to the Earth the Moon could have formed.


Moon: interiors and formation Gravitation 


  1. Archinal, B.A., Rosiek, M.R., Kirk, R.L., Redding, B.L.: US geological survey, Virginia (2006) Google Scholar
  2. Barker, M.K., Mazarico, E., Neumann, G.A., Zuber, M.T., Kharuyama, J., Smith, D.E.: Icarus 273, 346 (2016) ADSCrossRefGoogle Scholar
  3. Bohme, S.: Astron. Nachr. 256, 356 (1953) Google Scholar
  4. Calame, O.: Moon 15, 343 (1976) ADSCrossRefGoogle Scholar
  5. Chandrasekhar, S.: Ellipsoidal Equilibrium Figures. Yale University Press, New Haven (1969) zbMATHGoogle Scholar
  6. Chapront, J., Chapront-Touzé, M., Francou, G.: Astron. Astrophys. 387, 700 (2002) ADSCrossRefGoogle Scholar
  7. Darwin, G.H.: Tidal Friction in Cosmogony, Scientific Papers 2. Cambridge University Press, Cambridge (1908) Google Scholar
  8. Deprit, A.: In: Kopal, Z. (ed.) Physics and Astronomy of the Moon. Academic Press, New York (1971) Google Scholar
  9. Dickey, J.O., Bender, P.L., Faller, J.E., Newhall, X.X., Ricklefs, R.L., Ries, J.G., Shelus, P.J., Veillet, C., Whipple, A.L., Wiant, J.R., Williams, J.G., Yoder, C.F.: Science 265(5171), 482 (1994) ADSCrossRefGoogle Scholar
  10. Eckhardt, D.H.: Moon 1(2), 264 (1970) ADSCrossRefGoogle Scholar
  11. Eckhardt, D.H.: Moon Planets 25, 3 (1981) ADSCrossRefGoogle Scholar
  12. Folkner, W.M., Williams, J.G., Boggs, D.H., Park, R.S., Kuchynka, P.: The Interplanetary Network Progress Report, 42–196, 1–81 (2014) Google Scholar
  13. Garrick-Bethell, I., Nimmo, F., Mark, A., Wieczorek, M.A.: Science 330(6006), 949 (2010). ADSCrossRefGoogle Scholar
  14. Goldreich, P.: Rev. Geophys. Space Phys. 4, 411 (1966) ADSCrossRefGoogle Scholar
  15. Iz, H., Ding, X.L., Dai, C.L., Shum, C.K.: J. Geod. 1(4), 348 (2011) Google Scholar
  16. Khabibullin, Sh.T.: Trudi AOE (31), 1 (1958) Google Scholar
  17. Khabibullin, Sh.T.: Trudi KMAO (34), 3 (1966) Google Scholar
  18. Kondratyev, B.P.: Dinamika ellipsoidal’nykh gravitiruiushchikh figure. Nauka, Moscow (1989) Google Scholar
  19. Kondratyev, B.P.: Kvant (5), 38 (2009) Google Scholar
  20. Kondratyev, B.P.: Sol. Syst. Res. 45, 38 (2011a) Google Scholar
  21. Kondratyev, B.P.: Sol. Syst. Res. 45, 458 (2011b) Google Scholar
  22. Kondratyev, B.P.: Astron. Rep. 62(8), 542 (2018). ADSCrossRefGoogle Scholar
  23. Koziel, K.: Earth Moon Planets 45, 153 (1989) ADSCrossRefGoogle Scholar
  24. Laskar, J.: Celest. Mech. Dyn. Astron. 56, 191 (1993) ADSCrossRefGoogle Scholar
  25. Laskar, J., Fienga, A., Gastineau, M., Manche, H.: Astron. Astrophys. 532, A89, 15 pages (2011) ADSCrossRefGoogle Scholar
  26. Lemoine, F.G., Goossens, S., Sabaka, T.J., Nicholas, J.B., Mazarico, E., Rowlands, D.D., Loomis, B.D., Chinn, D.S., Neumann, G.A., Smith, D.E., Zuber, M.T.: Geophys. Res. Lett. 41, 3382 (2014). ADSCrossRefGoogle Scholar
  27. Lipsky, Yu.N., Nikonov, V.A.: Astron. Ž. 48, 445 (1971) ADSGoogle Scholar
  28. Macdonald, G.J.F.: Rev. Geophys. 2, 467 (1964). ADSCrossRefGoogle Scholar
  29. Migus, A.: Moon Planets 23, 391 (1980) ADSCrossRefGoogle Scholar
  30. Moons, M.: Moon Planets 27, 257 (1982) ADSCrossRefGoogle Scholar
  31. Moutsoulas, M.D.: Physics and astronomy of the moon, 29 (1971) Google Scholar
  32. Murray, K., Dermott, S.: Dynamics of the Solar System. Cambridge University Press, Cambridge (2009) zbMATHGoogle Scholar
  33. Rambaux, N., Williams, J.: Celest. Mech. Dyn. Astron. 109, 85 (2011) ADSCrossRefGoogle Scholar
  34. Sagitov, M.U.: Lunar Gravimetry. Nauka, Moscow (1979) Google Scholar
  35. Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J.: Astron. Astrophys. 282, 663 (1994) ADSGoogle Scholar
  36. Simon, S.J., Stewart, S.T., Petaev, M.I., Leinhardt, Z.M., Mace, M.T., Jacobsen, S.B.: J. Geophys. Res. 123(4), 910–951 (2018). CrossRefGoogle Scholar
  37. Tian, Z., Wisdom, J., Elkins-Tanton, L.: Icarus 281, 90 (2017) ADSCrossRefGoogle Scholar
  38. Urey, H.C.: Mon. Not. R. Astron. Soc. 131, 212 (1966) ADSCrossRefGoogle Scholar
  39. Weber, R., Lin, P., Garnero, E.J., Williams, Q., Lognonne, P.: Science 331, 309 (2011) ADSCrossRefGoogle Scholar
  40. Wenjing, J., Jinling, L.: Earth Moon Planets 73, 259 (1996) CrossRefGoogle Scholar
  41. Wieczorek, M.A., Neumann, G.A., Nimmo, F., Kiefer, W.S., Taylor, G.J., Melosh, H.J., Phillips, R.J., Solomon, S.C., Andrews-Hanna, J.C., Asmar, S.W., Konopliv, A.S., Lemoine, F.G., Smith, D.E., Watkins, M.M., Williams, J.G., Zuber, M.T.: Science 339(6120), 671 (2013). ADSCrossRefGoogle Scholar
  42. Williams, J.: J. Geophys. Res. 106, 933 (2001) Google Scholar
  43. Williams, J.G., Dickey, J.O.: In: 13th International Workshop on Laser Ranging, Washington (2002) Google Scholar
  44. Williams, J., Boggs, D., Ratcliff, J.: In: Lunar Pl. Sci. Conf., p. 1579 (2014a) Google Scholar
  45. Williams, J., Konopliv, A., Boggs, D.: J. Geophys. Res., Planets 119, 1546 (2014b) ADSCrossRefGoogle Scholar
  46. Zhong, S.: Proc. Int. Astron. Union 9(S298), 457 (2014). CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Sternberg Astronomical InstituteM.V. Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Faculty of Physics of the M.V. Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Central Astronomical Observatory at PulkovoSaint-PetersburgRussia

Personalised recommendations