Celestial Mechanics and Dynamical Astronomy

, Volume 126, Issue 1–3, pp 89–129 | Cite as

Secular tidal changes in lunar orbit and Earth rotation

Original Article
First Online:
Received:
Revised:
Accepted:

Abstract

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = −0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.

Keywords

Tides Lunar orbit Earth rotation Tidal acceleration Tidal dissipation Moon Lunar laser ranging (LLR) 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We thank the lunar laser ranging stations at McDonald Observatory, Texas, Observatoire de la Côte d’Azur, France, Haleakala Observatory, Hawaii, Apache Point Observatory, New Mexico, and Matera, Italy that provided the data sets that make LLR analyses possible. LLR data are available from the International Laser Ranging Service archive at http://ilrs.gsfc.nasa.gov/. We acknowledge extensive conversations with D. Pavlov about tidal modeling that benefited this paper. C. F. Yoder contributed to the early development of the solar perturbation scaling factors for LLR results. M. Efroimsky provided a valuable review. The research described in this paper was carried out at the Jet Propulsion Laboratory of the California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Government sponsorship acknowledged.

References

  1. Aleshkina, E.Y.: Lunar numerical theory and determination of parameters \(k_{2}\), \(\delta _{M}\) from analysis of LLR data. Astron. Astrophys. 394, 717–721 (2002). doi: 10.1051/0004-6361:20021149 ADSCrossRefGoogle Scholar
  2. Bills, B.G., Ray, R.D.: Lunar orbital evolution: a synthesis of recent results. Geophys. Res. Lett. 26, 3045–3048 (1999). doi: 10.1029/1999GL008348 ADSCrossRefGoogle Scholar
  3. Brown, E.W.: An introductory treatise on the lunar theory. Cambridge University Press, Cambridge (1896)MATHGoogle Scholar
  4. Brown, E.W.: Tables of the motion of the Moon. Yale University Press, New Haven (1919)Google Scholar
  5. Capitaine, N., Wallace, P.T., Chapront, J.: Expressions for IAU 2000 precession quantities. Astron. Astrohys. 412, 567–586 (2003). doi: 10.1051/0004-6361:20031539 ADSCrossRefGoogle Scholar
  6. Chapront, J., Chapront-Touzé, M.: Lunar motion: theory, and observations. Celesti. Mech. Dyn. Astron. 66, 31–38 (1996). doi: 10.1007/BF00048821 ADSCrossRefMATHGoogle Scholar
  7. Chapront-Touzé, M.: Perturbations due to the shape of the Moon in the lunar theory ELP 2000. Astron. Astrophys. 119, 256–260 (1983)ADSGoogle Scholar
  8. Chapront-Touzé, M., Chapront, J.: The lunar ephemeris ELP 2000. Astron. Astrophys. 124, 50–62 (1983)ADSMATHGoogle Scholar
  9. Chapront-Touzé, M., Chapront, J.: ELP 2000–85: a semi-analytical lunar ephemeris adequate for historical times. Astron. Astrophys. 190, 342–352 (1988)ADSGoogle Scholar
  10. Chapront-Touzé, M., Chapront, J.: Lunar Tables and Programs from 4000 B. C. to A. D. 8000. Willmann-Bell, Richmond (1991)MATHGoogle Scholar
  11. Chapront, J., Chapront-Touzé, M., Francou, G.: A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements. Astron. Astrophys. 387, 700–709 (2002)ADSCrossRefMATHGoogle Scholar
  12. Cheng, M., Tapley, B.D., Ries, J.C.: Deceleration in the Earth’s oblateness. J. Geophys. Res. 118, 740–747 (2013). doi: 10.1002/jgrb.50058 ADSCrossRefGoogle Scholar
  13. Christodoulidis, D.C., Smith, D.E., Williamson, R.G., Klosko, S.M.: Observed tidal breaking in the Earth/Moon/Sun system. J. Geophys. Res. 93, 6216–6236 (1988). doi: 10.1029/JB093iB06p06216 ADSCrossRefGoogle Scholar
  14. Deprit, A., Henrard, J., Rom, A.: Analytical lunar ephemeris: Delaunay’s theory. Astron. J. 76, 269–272 (1971)ADSCrossRefMATHGoogle Scholar
  15. Dickey, J.O., Bender, P.L., Faller, J.E., Newhall, X.X., Ricklefs, R.L., Ries, J.G., et al.: Lunar laser ranging: a continuing legacy of the Apollo program. Science 265, 482–490 (1994). doi: 10.1126/science.265.5171.482 ADSCrossRefGoogle Scholar
  16. Eckert, W.J., Jones, R., Clark, H.K.: Construction of the lunar ephemeris, in Improved Lunar Ephemeris 1952–1959. A Joint Supplement to the American Ephemeris and the (British) Nautical Almanac. U. S. Naval Observatory, U. S. Government Printing Office, pp. 283–363 (1954)Google Scholar
  17. Folkner, W.M., Williams, J.G., Boggs, D.H., Park, R.S., Kuchynka, P.: The planetary and lunar ephemerides DE 430 and DE431. The Interplanetary Network (IPN) Progress Report 42-196, Feb 15, 2014, Jet Propul. Lab., Pasadena, Calif. (2014). http://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf
  18. Hansen, K.S.: Secular effects of oceanic tidal dissipation on the Moon’s orbit and the Earth’s rotation. Rev. Geophys. Space Phys. 20, 457–480 (1982). doi: 10.1029/RG020i003p00457 ADSCrossRefGoogle Scholar
  19. Hartmann, T., Wenzel, H.-G.: The HW95 tidal potential catalogue. Geophys. Res. Lett. 22, 3553–3556 (1995). doi: 10.1029/95GL03324 ADSCrossRefGoogle Scholar
  20. Henning, W.G., O’Connell, R.J., Sasselov, D.D.: Tidally heated terrestrial exoplanets: viscoelastic response models. Astrophys. J. 707, 1000–1015 (2009). doi: 10.1088/0004-637X/707/2/1000 ADSCrossRefGoogle Scholar
  21. Henrard, J.: Analytic lunar ephemeris: a report. Publication of the department of mathematics. University of Namur, Belgium (1972)Google Scholar
  22. Hilton, J.L., Capitaine, N., Chapront, J., Ferrandiz, J.M., Fienga, A., Fukushima, T., et al.: Report of the international astronomical union division I working group on precession and the ecliptic. Celesti. Mech. Dyn. Astron. 94(3), 351–367 (2006). doi: 10.1007/s10569-006-0001-2 ADSCrossRefMATHGoogle Scholar
  23. Iorio, L.: On the anomalous secular increase in the eccentricity of the orbit of the Moon. Mon. Not. R. Astron. Soc. 415, 1266–1275 (2011). doi: 10.1111/j.1365-2966.2011.18777.x ADSCrossRefGoogle Scholar
  24. Iorio, L.: An empirical explanation of the anomalous increases in the astronomical unit and the lunar eccentricity. Astron. J. 142(68), 1–3 (2011b). doi: 10.1088/0004-6256/142/3/68 Google Scholar
  25. Iorio, L.: The lingering anomalous secular increase in the eccentricity of the orbit of the Moon: further attempts of explanation of cosmological origin. Galaxies 2(2), 259–262 (2014). doi: 10.3390/galaxies2020259 ADSCrossRefGoogle Scholar
  26. Kaula, W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. 2, 661–685 (1964). doi: 10.1029/RG002i004p00661 ADSCrossRefGoogle Scholar
  27. Kaula, W.M.: Theory of Satellite Geodesy. Dover Publications Inc, Mineola, New York, p. 124. (1966)Google Scholar
  28. Konopliv, A.S., Park, R.S., Yuan, D.-N., Asmar, S.W., Watkins, M.M., Williams, F.G., et al.: The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL primary mission. J. Geophys. Res. 118, 1415–1434 (2013). doi: 10.1002/jgre.20097 CrossRefGoogle Scholar
  29. Krasinsky, G.A.: Tidal effects in the Earth–Moon system and the Earth’s rotation. Celesti. Mech. Dyn. Astron. 75, 39–66 (1999). doi: 10.1023/A:1008381000993 ADSCrossRefMATHGoogle Scholar
  30. Lemoine, F.G., Goossens, S., Sabaka, T.J., Nicholas, J.B., Mazarico, E., Rowlands, D.D., et al.: High-degree gravity models from GRAIL primary mission data. J. Geophys. Res. Planets 118, 1676–1698 (2013). doi: 10.1002/jgre.20118 ADSCrossRefGoogle Scholar
  31. Lyard, F., Lefevre, F., Letellier, T., Francis, O.: Modeling the global ocean tides: modern insights from FES2004. Ocean Dyn. 56, 394–415 (2006). doi: 10.1007/s10236-006-0086-x ADSCrossRefGoogle Scholar
  32. Mathews, P.M., Herring, T.A., Buffet, B.A.: Modeling of nutation and precession: New nutation series for non-rigid Earth and insights into the Earth’s interior. J. Geophys. Res. 107 (B4), ETG 3-1–ETG 3-26 (2002). doi: 10.1029/2001JB000390
  33. Meyer, J., Elkins-Tanton, L., Wisdom, J.: Coupled thermal-orbital evolution of the early Moon. Icarus 208, 1–10 (2010). doi: 10.1016/j.icarus.2010.01.029 Corrigendum to Coupledthermal-orbital evolution of the early Moon. doi: 10.1016/j.icarus.2010.12.008
  34. Mignard, F.: The lunar orbit revisited, III. Moon Pl. 24, 189–207 (1981). doi: 10.1007//BF00910608 ADSCrossRefMATHGoogle Scholar
  35. Murphy Jr., T.W., Adelberger, E.G., Battat, J.B.R., Carey, L.N., Hoyle, C.D., LeBlanc, P., et al.: APOLLO: The Apache point observatory lunar laser-ranging operation: instrument description and first detections. Publ. Astron. Soc. Pacific 120, 20–37 (2008). doi: 10.1086/526428, arXiv:0710.0890 [astro-ph]
  36. Murphy Jr., T.W., Adelberger, E.G., Battat, J.B.R., Hoyle, C.D., Johnson, N.H., McMillan, R.J., et al.: APOLLO: millimeter lunar laser ranging. Class. Quantum Grav. 29, 184005 (2012). doi: 10.1088/0264-9381/29/18/184005 ADSCrossRefGoogle Scholar
  37. Murphy, T.W.: Lunar laser ranging: the millimeter challenge. Rep. Prog. Phys. 76, 076901 (2013). doi: 10.1088/0034-4885/76/7/076901 ADSCrossRefGoogle Scholar
  38. Newhall, X.X., Standish, E.M., Williams, J.G.: DE 102, a numerically integrated ephemeris of the Moon and planets spanning forty-four centuries. Astron. Astrophys. 125, 150–167 (1983)ADSMATHGoogle Scholar
  39. Pavlov, D.A., Williams, J.G., Suvorkin, V.V.: Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models. Submitted to Celest. Mech. Dyn. Astron. (2016) (in press)Google Scholar
  40. Petit, G., Luzum, B.: IERS Conventions (2010). IERS Tech. Note 36, pp. 179, Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main (2010). http://www.iers.org/TN36/
  41. Poliakow, E.: Numerical modeling of the paleotidal evolution of the Earth-Moon system. In: Proceedings of International Astronomical Union Colloquium 197. Dynamics of Populations of Planetary Systems. August–September 2004, Belgrade\(.\) Edited by Knezevic, Z., Milani, A., Cambridge Univ. Press, pp. 445–452 (2005). doi: 10.1017/S174392130400897X
  42. Rambaux, N., Williams, J.G.: The Moon’s physical librations and determination of their free modes. Celesti. Mech. Dyn. Astron. 109, 85–100 (2011). doi: 10.1007/s10569-010-9314-2 ADSCrossRefMATHGoogle Scholar
  43. Ray, R.D., Eanes, R.J., Lemoine, F.G.: Constraints on energy dissipation in the Earth’s body tide from satellite tracking and altimetry. Geophys. J. Int. 144, 471–480 (2001). doi: 10.1046/j.1365-246x.2001.00356.x ADSCrossRefGoogle Scholar
  44. Ray, R.D., Erofeeva, S.Y.: Long-period tidal variations in the length of day. J. Geophys. Res. Solid Earth 119, 1498–1509 (2014). doi: 10.1002/2013JB010830 ADSCrossRefGoogle Scholar
  45. Rubincam, D.P.: Tidal friction in the Earth–Moon system and Laplace planes: Darwin redux. Icarus 266, 24–43 (2016). doi: 10.1016/j.icarus.2015.10.024 ADSCrossRefGoogle Scholar
  46. Samain, E., Mangin, J.F., Veillet, C., Torre, J.-M., Fridelance, P., Chabaudie, J.E., et al.: Millimetric lunar laser ranging at OCA (Observatoire de la Côte d’Azur). Astron. Astrophys. Suppl. Ser. 130, 235–244 (1998). doi: 10.1051/aas:1998227 ADSCrossRefGoogle Scholar
  47. Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touzé, M., Francou, G., Laskar, J.: Numerical expressions for precession formulae and mean elements for the Moon and planets. Astron. Astrophys. 282, 663–683 (1994)ADSGoogle Scholar
  48. Standish, E.M., Williams, J.G.: Orbital ephemerides of the Sun, Moon, and planets. Chapter 8. In: Urban, S., Seidelmann, P.K. (eds.) Explanatory Supplement to the Astronomical Almanac, 3rd edition, pp. 305–345. University Science Books, Mill Valley, CA (2013). http://iau-comm4.jpl.nasa.gov/XSChap8.pdf
  49. Stephenson, F.R., Morrison, L.V.: Long-term fluctuations in the Earth’s rotation: 700 BC to AD 1990. Philos. Trans. R. Soc. 351, 165–202 (1995). doi: 10.1098/rsta.1995.0028 ADSCrossRefGoogle Scholar
  50. Webb, D.J.: Tides and the evolution of the Earth–Moon system. Geophys. J. R. Astron. Soc. 70, 261–271 (1982). doi: 10.1111/j1365-246X.1982tb06404.x ADSCrossRefGoogle Scholar
  51. Williams, J.G., Sinclair, W.S., Yoder, C.F.: Tidal acceleration of the Moon. Geophys. Res. Lett. 5, 943–946 (1978). doi: 10.1029/GL005i011p00943 ADSCrossRefGoogle Scholar
  52. Williams, J.G.: Contributions to the Earth’s obliquity rate, precession, and nutation. Astron. J. 108, 711–724 (1994). doi: 10.1086/117108 ADSCrossRefGoogle Scholar
  53. Williams, J.G., Boggs, D.H., Yoder, C.F., Ratcliff, J.T., Dickey, J.O.: Lunar rotational dissipation in solid body and molten core. J. Geophys. Res. 106, 27933–27968 (2001). doi: 10.1029/2000JE001396 ADSCrossRefGoogle Scholar
  54. Williams, J.G., Boggs, D.H.: Lunar core and mantle. What does LLR see? In: Proceedings of 16th International Workshop on Laser Ranging, SLR—the Next Generation, October 2008, Poznan, Poland, ed. Stanislaw Schillak, pp. 101–120 (2009). http://www.astro.amu.edu.pl/ILRS_Workshop_2008/index.php
  55. Williams, J.G, Boggs, D.H., Folkner, W.M.: DE430 Lunar Orbit, Physical Librations, and Surface Coordinates. IOM 335-JW,DB,WF-20130722-016, July 22, 2013, Jet Propul. Lab., Pasadena, Calif. (2013). http://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/de430_moon_coord.pdf
  56. Williams, J.G., Turyshev, S.G., Boggs, D.H.: The past and present Earth-Moon system: the speed of light stays steady as tides evolve. Planet. Sci. 3, 2 (2014a). doi: 10.1186/s13535-014-0002-5. http://www.planetary-science.com/content/3/1/2
  57. Williams, J.G., Konopliv, A.S., Boggs, D.H., Park, R.S., Yuan, D.-N., Lemoine, F.G., et al.: Lunar interior properties from the GRAIL mission. J. Geophys. Res. Planets 119, 1546–1578 (2014b). doi: 10.1002/2013JE004559 ADSCrossRefGoogle Scholar
  58. Williams, J.G., Boggs, D.H.: Tides on the Moon: theory and determination of dissipation. J. Geophys. Res. Planets 120(4), 689–724 (2015). doi: 10.1002/2014JE004755 ADSCrossRefGoogle Scholar
  59. Yoder, C.F., Williams, J.G., Sinclair, W.S., Parke, M.E.: Tidal variations of Earth rotation. J. Geophys. Res. 86, 881–891 (1981). doi: 10.1029/JB086iB02p00881 ADSCrossRefGoogle Scholar
  60. Yoder, C.F., Williams, J.G., Dickey, J.O., Schutz, B.E., Eanes, R.J., Tapley, B.D.: Secular variation of Earth’s gravitational harmonic J2 coefficient from Lageos and the nontidal acceleration of Earth rotation. Nature 303, 757–762 (1983). doi: 10.1038/303757a0 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations