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Free librations of the moon determined by an analysis of laser range measurements

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Among the lunar laser range measurements obtained during the past six years at McDonald Observatory, those available cover the period August 1969-November 1974, being 1377 normal observations made on the three Apollo reflectors and that of Lunokhod II. The fit of these data led to a rms residual of 55 cm. In this study, a large number of parameters have been resolved, including the geocentric coordinates of the telescope, the selenocentric coordinates of each of the reflectors, as well as orbital elements of the Moon. In addition, the interest has been directed more specially towards the study of the rotational motion of the Moon and particularly the problem of its free librations. The performed resolutions give the evidence of the three modes of free oscillations. The determined amplitudes arise to 1″.7 in longitude and 0″.5 and 8″.7 in latitude, with the respective periods of 2.9 years, 27.3 days and 75 years. In connection with these parameters, the fittings determined also the most of part of elements of lunar gravitational field: the moment of inertia parametersβ andγ, and a number of the third degree harmonics. These new results should now permit a research on the implications of these oscillations effects, concerning the impact history of the Moon and the properties of its internal structure. On the basis of the amplitudes determined here, one can already estimate an order of the magnitude for theQ dissipation coefficient comparable with that determined from seismic studies of the Moon.

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  1. Abbot, R. J., Shelus, P. J., Mulholland, J. D., and Silverberg, E. C.: 1973,Astron. J. 78, 784.

  2. Calame, O.: 1975, ‘Etude des mouvements libratoires lunaires et localisation des stations terrestres, à partir des mesures laser de distances’, University of Paris, Thèse.

  3. Eckhardt, D. H.: 1965,Astron. J. 70, 406.

  4. Eckhardt, D. J.: 1970,The Moon 1, 264.

  5. Habibullin, Sh. T.: 1968,Acta Astron. 18, 207.

  6. Koziel, K.: 1967,Icarus 7, 1.

  7. Latham, G. V.et al. (8 co-authors): 1971, in ‘Apollo 14 Preliminary Science Report” (NASA SP-272; pp. 133-161).

  8. Mulholland, J. D., Shelus, P. J., and Silverberg, E. C.: 1975, ‘Laser Observations of the Moon: Normal Points for 1973’,Astron. J. 80, 1087.

  9. Sekiguchi, N.: 1970,The Moon 1, 110.

  10. Shelus, P. J., Mulholland, J. D., and Silverberg, E. C.: 1975, Astron. J.80, 1087.

  11. Sjogren, W. L.: 1971,J. Geophys. Res. 76, 7021.

  12. Williams, J. G.: 1974 (private communication).

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Calame, O. Free librations of the moon determined by an analysis of laser range measurements. The Moon 15, 343–352 (1976). https://doi.org/10.1007/BF00562246

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  • Internal Structure
  • Gravitational Field
  • Rotational Motion
  • Orbital Element
  • Free Oscillation