Celestial Mechanics and Dynamical Astronomy

, Volume 108, Issue 3, pp 245–263 | Cite as

Celestial reference frames and the gauge freedom in the post-Newtonian mechanics of the Earth–Moon system

  • Sergei Kopeikin
  • Yi Xie
Original Article


We introduce the Jacobi coordinates adopted to the advanced theoretical analysis of the relativistic Celestial Mechanics of the Earth–Moon system. Theoretical derivation utilizes the relativistic resolutions on reference frames adopted by the International Astronomical Union (IAU) in 2000. The resolutions assume that the Solar System is isolated and space-time is asymptotically flat at infinity and the primary reference frame covers the entire space-time, has its origin at the Solar System barycenter (SSB) with spatial axes stretching up to infinity. The SSB frame is not rotating with respect to a set of distant quasars that are assumed to be at rest on the sky forming the International Celestial Reference Frame (ICRF). The second reference frame has its origin at the Earth–Moon barycenter (EMB). The EMB frame is locally inertial and is not rotating dynamically in the sense that equation of motion of a test particle moving with respect to the EMB frame, does not contain the Coriolis and centripetal forces. Two other local frames—geocentric and selenocentric—have their origins at the center of mass of Earth and Moon respectively and do not rotate dynamically. Each local frame is subject to the geodetic precession both with respect to other local frames and with respect to the ICRF because of their relative motion with respect to each other. Theoretical advantage of the dynamically non-rotating local frames is in a more simple mathematical description of the metric tensor and relative equations of motion of the Moon with respect to Earth. Each local frame can be converted to kinematically non-rotating one after alignment with the axes of ICRF by applying the matrix of the relativistic precession as recommended by the IAU resolutions. The set of one global and three local frames is introduced in order to decouple physical effects of gravity from the gauge-dependent effects in the equations of relative motion of the Moon with respect to Earth.


Gravitation Relativity Gauge invariance Post-Newtonian three-body problem Lunar ephemerides Lunar laser ranging Einstein–Infeld–Hoffmann (EIH) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alley C.O.: Laser ranging to retro-reflectors on the moon as a test of theories of gravity. In: Meystre, P., Scully, M.O. (eds) Quantum Optics, Experimental Gravitation, and Measurement Theory, NATO ASI Series: Physics, vol. 94, pp. 429–495. Plenum Press, New York (1983)Google Scholar
  2. Baierlein R.: Testing general relativity with laser ranging to the Moon. Phys. Rev. 162(5), 1275–1287 (1967). doi: 10.1103/PhysRev.162.1275 CrossRefADSGoogle Scholar
  3. Battat, J., Murphy, T., Adelberger, E., Hoyle, C.D., McMillan, R., Michelsen, E., Nordtvedt, K, Orin A, Stubbs C, Swanson H.E.: APOLLO: Testing Gravity with Millimeter-precision Lunar Laser Ranging. APS Meeting Abstracts p. 12.003 (2007)Google Scholar
  4. Bender P.L., Currie D.G., Dicke R.H., Eckhardt D.H., Faller J.E., Kaula W.M., Mulholland J.D., Plotkin H.H., Poultney S.K., Silverberg E.C., Wilkinson D.T., Williams J.G., Alley C.O.: The lunar laser ranging experiment. Science 182, 229–238 (1973). doi: 10.1126/science.182.4109.229 CrossRefADSGoogle Scholar
  5. Bender, P.L., Faller, J.E., Hall, J.L., Degnan, J.J., Dickey, J.O., Newhall, X.X., Williams, J.G., King RW, Macknik, L.O., O’Gara, D.: Microwave and optical lunar transponders. In: Mumma, M.J., Smith, H.J. (eds.) Astrophysics from the Moon, American Institute of Physics Conference Series, vol. 207, pp. 647–653 (1990). doi: 10.1063/1.39357
  6. Bertotti B., Ciufolini I., Bender P.L.: New test of general relativity—measurement of de sitter geodetic precession rate for lunar perigee. Phys. Rev. Lett. 58, 1062–1065 (1987). doi: 10.1103/PhysRevLett.58.1062 CrossRefMathSciNetADSGoogle Scholar
  7. Brumberg V.: On derivation of EIH (Einstein–Infeld–Hoffman) equations of motion from the linearized metric of general relativity theory. Cel. Mech. Dyn. Astron. 99, 245–252 (2007). doi: 10.1007/s10569-007-9094-5 MATHCrossRefMathSciNetADSGoogle Scholar
  8. Brumberg V.A.: Relativistic corrections in the theory of motion of the Moon. Bull. Inst. Theor. Astron. 6(10(83)), 733–756 (1958)Google Scholar
  9. Brumberg V.A.: Relativistic Celestial Mechanics. Nauka, Moscow (1972) (in Russian)MATHGoogle Scholar
  10. Brumberg, V.A.: Relativistic reduction of astronomical measurements and reference frames. In: Gaposchkin, E.M., Kolaczek, B. (eds.) ASSL vol. 86: IAU Colloq. 56: Reference Coordinate Systems for Earth Dynamics, pp. 283–294 (1981)Google Scholar
  11. Brumberg V.A.: Essential Relativistic Celestial Mechanics. Adam Hilger, New York (1991)MATHGoogle Scholar
  12. Brumberg, V.A., Kopejkin, S.M.: Relativistic theory of celestial reference frames. In: Kovalevsky, J., Mueller, I.I., Kolaczek, B. (eds.) ASSL vol. 154: Reference Frames, pp. 115–141 (1989a)Google Scholar
  13. Brumberg V.A., Kopejkin S.M.: Relativistic reference systems and motion of test bodies in the vicinity of the Earth. Nuovo. Cimento. B Ser. 103, 63–98 (1989)CrossRefADSGoogle Scholar
  14. Calame O., Mulholland J.D.: Lunar tidal acceleration determined from laser range measures. Science 166, 977–978 (1978)CrossRefADSGoogle Scholar
  15. Chapront J., Chapront-Touzé M., Francou G.: Determination of the lunar orbital and rotational parameters and of the ecliptic reference system orientation from LLR measurements and IERS data. Astron. Astrophys. 343, 624–633 (1999)ADSGoogle Scholar
  16. 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). doi: 10.1051/0004-6361:20020420 CrossRefADSGoogle Scholar
  17. Chapront-Touze M., Chapront J.: The lunar ephemeris ELP 2000. Astron. Astrophys. 124(1), 50–62 (1983)MathSciNetADSGoogle Scholar
  18. Chazy J.: La Théorie de la relativité et la mécanique céleste Tome I, 1928 et Tome II, 1930. Éditions Jacques Gabay, Paris (2005)Google Scholar
  19. Ciufolini, I.: Lunar laser ranging, gravitomagnetism and frame-dragging. ArXiv e-prints 0809.3219 (2008)Google Scholar
  20. Cook A.: The Motion of the Moon. Adam Hilger, Bristol (1988)MATHGoogle Scholar
  21. Currie D.G., Cantone C., Carrier W.D., Dell’Agnello S., Delle Monache G., Murphy T., Rubincam D., Vittori R.: A lunar laser ranging retro-reflector array for the 21st Century. LPI Contrib. 1415, 2145 (2008)ADSGoogle Scholar
  22. Damour T., Esposito-Farese G.: Tensor-multi-scalar theories of gravitation. Class. Quantum Gravity 9, 2093–2176 (1992)MATHCrossRefMathSciNetADSGoogle Scholar
  23. Damour T., Soffel M., Xu C.: General-relativistic celestial mechanics. I. Method and definition of reference systems. Phys. Rev. D 43, 3273–3307 (1991)CrossRefMathSciNetADSGoogle Scholar
  24. Damour T., Soffel M., Xu C.: General-relativistic celestial mechanics. IV. Theory of satellite motion. Phys. Rev. D 49, 618–635 (1994)CrossRefMathSciNetADSGoogle Scholar
  25. de Sitter W.: On Einstein’s theory of gravitation and its astronomical consequences second paper. Mon. Not. Roy. Astron. Soc. 77(2), 155–184 (1916)ADSGoogle Scholar
  26. Dickey J.O., Newhall X.X., Williams J.G.: Investigating relativity using lunar laser ranging—geodetic precession and the nordtvedt effect. Adv. Space Res. 9, 75–78 (1989)CrossRefADSGoogle Scholar
  27. Eddington A.S.: The Mathematical Theory of Relativity. Chelsea, New York (1975)Google Scholar
  28. Eichhorn H.: Inertial systems—definitions and realizations. Cel. Mech. 34, 11–18 (1984). doi: 10.1007/BF01235787 CrossRefADSGoogle Scholar
  29. Einstein A.: Die grundlage der allgemeinen relativitätstheorie. Ann. Phys. 354(7), 769–822 (1916). doi: 10.1002/andp.19163540702 CrossRefGoogle Scholar
  30. Einstein A., Infeld L., Hoffmsann B.: The gravitational equations and the problem of motion. Ann. Math. 39(1), 65–100 (1938)CrossRefGoogle Scholar
  31. Eling, C., Jacobson, T., Mattingly, D.: Einstein-Æther theory. In: Liu, J.T., Duff, M.J., Stelle, K.S., Woodward, R.P. (eds.) Deserfest: A Celebration of the Life and Works of Stanley Deser, pp. 163–179 (2006)Google Scholar
  32. Estabrook F.B.: Post-newtonian N-body equations of the brans-dicke theory. Astrophys. J. 158, 81–83 (1969). doi: 10.1086/150172 CrossRefMathSciNetADSGoogle Scholar
  33. Fienga A., Manche H., Laskar J., Gastineau M.: INPOP06: a new numerical planetary ephemeris. Astron. Astrophys. 477, 315–327 (2008). doi: 10.1051/0004-6361:20066607 CrossRefADSGoogle Scholar
  34. Fock V.A.: The Theory of Space, Time and Gravitation. Pergamon Press, New York (1959)MATHGoogle Scholar
  35. Gullstrand A.: Allgemeine lösung des statischen einkörproblems in der einsteinschen gravitationstheorie. Arkiv. Mat. Astron. Fys. 16(8), 1–15 (1922)Google Scholar
  36. Gutzwiller M.C.: Moon–Earth–Sun: the oldest three-body problem. Rev. Mod. Phys. 70(2), 589–639 (1998)CrossRefADSGoogle Scholar
  37. Hakim R.: An elementary introduction to relativistic gravitation. Cel. Mech. Dyn. Astron. 72, 1–36 (1998). doi: 10.1023/A:1008372412952 CrossRefMathSciNetADSGoogle Scholar
  38. Huang C., Jin W., Xu H.: The terrestrial and lunar reference frame in lunar laser ranging. J. Geod. 73, 125–129 (1999). doi: 10.1007/s001900050227 CrossRefADSGoogle Scholar
  39. Huang, C.L., Jin, W.J., Xu, H.G.: The terrestrial and lunar reference frame in LLR. Shanghai Obs. Ann. pp. 169–175 (1996)Google Scholar
  40. Infeld L., Plebański J.: Motion and Relativity. Oxford University Press, Oxford (1960)MATHGoogle Scholar
  41. Iorio, L.: Will it be possible to measure intrinsic gravitomagnetism with lunar laser ranging? ArXiv e-prints 0809.4014 (2008)Google Scholar
  42. Klioner S.A., Voinov A.V.: Relativistic theory of astronomical reference systems in closed form. Phys. Rev. D 48, 1451–1461 (1993)CrossRefADSGoogle Scholar
  43. Kopeikin, S.: The gravitomagnetic influence on earth-orbiting spacecrafts and on the lunar orbit. ArXiv e-prints 0809.3392 (2008)Google Scholar
  44. Kopeikin, S., Vlasov, I.: Parametrized post-newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem. Phys. Rep. 400, 209–318 (2004). doi: 10.1016/j.physrep.2004.08.004, gr-qc/0403068
  45. Kopeikin, S., Xie, Y.: Post-newtonian reference frames for advanced theory of the lunar motion and a new generation of lunar laser ranging. ArXiv e-prints 0902.2416 (2009)Google Scholar
  46. Kopeikin, S.M.: Comment on “Gravitomagnetic Influence on Gyroscopes and on the Lunar Orbit”. Phys. Rev. Lett. 98(22), 229,001–+ (2007a). doi: 10.1103/PhysRevLett.98.229001, arXiv:gr-qc/0702120
  47. Kopeikin, S.M.: Relativistic reference frames for astrometry and navigation in the Solar System. In: Belbruno, E. (ed.) New Trends in Astrodynamics and Applications III, American Institute of Physics Conference Series, vol. 886, pp. 268–283 (2007b). doi: 10.1063/1.2710062
  48. Kopeikin, S.M., Fomalont, E.B.: Gravimagnetism, causality, and aberration of gravity in the gravitational light-ray deflection experiments. Gen. Relat. Gravi. 39, 1583–1624 (2007). doi: 10.1007/s10714-007-0483-6, arXiv:gr-qc/0510077
  49. Kopeikin S.M., Pavlis E., Pavlis D., Brumberg V.A., Escapa A., Getino J., Gusev A., Müller J., Ni W.T., Petrova N.: Prospects in the orbital and rotational dynamics of the Moon with the advent of sub-centimeter lunar laser ranging. Adv. Space Res. 42, 1378–1390 (2008). doi: 10.1016/j.asr.2008.02.014,0710.1450 CrossRefADSGoogle Scholar
  50. Kopejkin S.M.: Celestial coordinate reference systems in curved space-time. Cel. Mech. 44, 87–115 (1988). doi: 10.1007/BF01230709 MATHCrossRefMathSciNetADSGoogle Scholar
  51. Kostelecky, A.: Theory and status of lorentz violation. APS Meeting Abstracts p. I3.00001 (2008)Google Scholar
  52. Kudryavtsev S.M.: Accurate harmonic development of lunar ephemeris LE-405/406. Highlights Astron. 14, 472 (2007). doi: 10.1017/S1743921307011477 ADSGoogle Scholar
  53. Landau L.D., Lifshitz E.M.: The classical Theory of Fields. Pergamon Press, Oxford (1975)Google Scholar
  54. Lestrade J.F., Bretagnon P.: Relativistic perturbations for all the planets. Astron. Astrophys. 105(1), 42–52 (1982)MATHADSGoogle Scholar
  55. Li, G.Y., Zhao, H.B., Xia, Y., Zeng, F., Luo, Y.J.: PMOE planetary/lunar ephemeris framework†. In: IAU Symposium, IAU Symposium, vol. 248, pp. 560–562 (2008). doi: 10.1017/S1743921308020140
  56. Lorentz, H.A., Droste, J.: The collected papers of H.A. Lorentz.: Chap The Motion of a System of Bodies Under the Influence of their Mutual Attraction, According to Einstein’s theory, pp. 330–355. Nijhoff, The Hague (1937)Google Scholar
  57. Lorimer D.R., Kramer M.: Handbook of Pulsar Astronomy. Cambridge University Press, Cambridge (2004)Google Scholar
  58. Mashhoon B., Theiss D.S.: Relativistic effects in the motion of the Moon. In: Lämmerzahl, C., Everitt, C.W.F., Hehl, F.W. (eds) Gyros,Clocks, Interferometers: Testing Relativistic Gravity in Space, pp. 310–316. Springer, Berlin (2001)CrossRefGoogle Scholar
  59. Meyer, F., Seitz, F., Mueller, J.: Algorithm for reliable normal point calculation of noisy LLR measurements. In: Schreiber, U., Werner, C., Kamerman, G.W., Singh, U.N. (eds.) Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 4546, pp. 154–159 (2002)Google Scholar
  60. Moyer T.D.: Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation. Wiley, Hoboken, New Jersey (2003)CrossRefGoogle Scholar
  61. Mueller J., Schneider M., Soffel M., Ruder H.: Testing einstein’s theory of gravity by analyzing lunar laser ranging data. Astrophys. J. Lett. 382, L101–L103 (1991). doi: 10.1086/186222 CrossRefADSGoogle Scholar
  62. Mukhanov V.: Physical Foundations of Cosmology. Cambridge University Press, Cambridge, UK (2005)MATHCrossRefGoogle Scholar
  63. Müller J., Nordtvedt K.: Lunar laser ranging and the equivalence principle signal. Phys. Rev. D 58(6), 062,001 (1998). doi: 10.1103/PhysRevD.58.062001 CrossRefGoogle Scholar
  64. Müller J., Nordtvedt K., Vokrouhlický D.: Improved constraint on the α1 PPN parameter from lunar motion. Phys. Rev. D 54, 5927–5930 (1996). doi: 10.1103/PhysRevD.54.R5927 CrossRefADSGoogle Scholar
  65. Müller, J., Williams, J.G., Turyshev, S.G.: Lunar laser ranging contributions to relativity and geodesy. In: Dittus, H., Lämmerzahl, C., Turyshev, S.G. (eds.) Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space, Astrophysics and Space Science Library, vol. 349, pp. 457–472 (2008a)Google Scholar
  66. Müller J., Soffel M., Klioner S.A.: Geodesy and Relativity. J. Geod. 82, 133–145 (2008). doi: 10.1007/s00190-007-0168-7 MATHCrossRefADSGoogle Scholar
  67. Murphy T.W., Michelson E.L., Orin A.E., Adelberger E.G., Hoyle C.D., Swanson H.E., Stubbs C.W., Battat J.B.: Apollo: a new push in lunar laser ranging. Int. J. Mod. Phys. D 16, 2127–2135 (2007). doi: 10.1142/S0218271807011589 CrossRefADSGoogle Scholar
  68. Murphy, T.W., Adelberger, E.G., Battat, J.B.R., Carey, L.N., Hoyle, C.D., Leblanc, P., Michelsen, E.L., Nordtvedt, K., Orin, A.E., Strasburg, J.D., Stubbs, C.W., Swanson, H.E., Williams, E.: The apache point observatory lunar laser-ranging operation: instrument description and first detections. Publ. Astron. Soc. Pac. 120, 20–37 (2008). doi: 10.1086/526428, 0710.0890Google Scholar
  69. Murphy, T.W. Jr., Nordtvedt, K., Turyshev, S.G.: Gravitomagnetic influence on gyroscopes and on the lunar orbit. Phys. Rev. Lett. 98(7):071,102 (2007a). doi: 10.1103/PhysRevLett.98.071102, gr-qc/0702028
  70. Murphy, T.W. Jr., Nordtvedt, K., Turyshev, S.G.: Murphy, nordtvedt, and turyshev reply:. Phys. Rev. Lett. 98(22), 229,002 (2007b). doi: 10.1103/PhysRevLett.98.229002, 0705.0513
  71. Newhall, X.X., Standish, E.M. Jr., Williams, J.G.: Planetary and lunar ephemerides, lunar laser ranging and lunar physical librations. (Lecture). In: Ferraz-Mello, S., Morando, B., Arlot, J.E. (eds.) Dynamics, Ephemerides, and Astrometry of the Solar System, IAU Symposium, vol. 172, pp. 37–44 (1996)Google Scholar
  72. Nordtvedt K.: Existence of the gravitomagnetic interaction. Int. J. Theor. Phys. 27, 1395–1404 (1988). doi: 1007/BF00671317 MATHCrossRefGoogle Scholar
  73. Nordtvedt, K.: Lunar laser ranging—a comprehensive probe of the post-newtonian long range interaction. In: Lämmerzahl, C., Everitt, C.W.F., Hehl, F.W. (eds.) Gyros, Clocks, Interferometers ...: Testing Relativistic Gravity in Space, Lecture Notes in Physics, vol. 562, pp. 317–329. Springer Verlag, Berlin (2001)CrossRefGoogle Scholar
  74. Painlevé P.: La mécanique classique et la théorie de la relativité. C. R. Acad. Sci. (Paris) 173, 677–680 (1921)Google Scholar
  75. Pearlman M., Degnan J., Bosworth J.: The international laser ranging service. Adv. Space Res. 30(2), 135–143 (2002). doi: 10.1016/S0273-1177(02)00277-6 CrossRefADSGoogle Scholar
  76. Petrova N.M.: On equations of motion and tensor of matter for a system of finite masses in general theory of relativity. Zh. Exp. Theor. Phys. 19, 989–999 (1949)MathSciNetGoogle Scholar
  77. Pitjeva E.V.: High-precision ephemerides of planetsEPM and determination of some astronomical constants. Sol. Sys. Res. 39(3), 176–186 (2005)CrossRefADSGoogle Scholar
  78. Rambaux, N., Williams, J.G., Boggs, D.H.: A dynamically active Moon—lunar free librations and excitation mechanisms. In: Lunar and Planetary Institute Conference Abstracts, Lunar and Planetary Inst. Technical Report, vol. 39, p. 1769 (2008)Google Scholar
  79. Roy A.E.: Orbital Motion. Institute of Physics Publishing, Bristol (2005)Google Scholar
  80. Soffel M., Ruder H., Schneider M.: The dominant relativistic terms in the lunar theory. Astron. Astrophys. 157(2), 357–364 (1986)MATHADSGoogle Scholar
  81. Soffel, M., Klioner, S.A., Petit, G., Wolf, P., Kopeikin, S.M., Bretagnon, P., Brumberg, V.A., Capitaine, N., Damour, T., Fukushima, T., Guinot, B., Huang, T.Y., Lindegren, L., Ma, C., Nordtvedt, K., Ries, J.C., Seidelmann, P.K., Vokrouhlický, D., Will, C.M., Xu, C.: The IAU 2000 resolutions for astrometry, celestial mechanics, and metrology in the relativistic framework: explanatory supplement. Astron. J. (USA) 126, 2687–2706 (2003). doi: 10.1086/378162, eprint astro-ph/0303376Google Scholar
  82. Soffel M., Klioner S., Müller J., Biskupek L.: Gravitomagnetism and lunar laser ranging. Phys. Rev. D 78(2), 024,033 (2008). doi: 10.1103/PhysRevD.78.024033 CrossRefGoogle Scholar
  83. Soffel M.H.: Relativity in Astrometry, Celestial Mechanics and Geodesy. Springer, Berlin (1989)Google Scholar
  84. Standish, E.M.: JPL Planetary and Lunar Ephemerides, DE405/LE405. JPL IOM 312.F-98-048. Jet Propulsion Laboratory (1998)Google Scholar
  85. Standish, E.M. : Planetary and lunar ephemerides: testing alternate gravitational theories. In: Macias, A., Lämmerzahl, C., Camacho, A. (eds.) Recent Developments in Gravitation and Cosmology, American Institute of Physics Conference Series, vol. 977, pp. 254–263 (2008). doi: 10.1063/1.2902789
  86. Standish, E.M., Williams, G.: Dynamical reference frames in the planetary and Earth–Moon systems. In: Lieske, J.H., Abalakin, V.K. (eds.) Inertial Coordinate System on the Sky, IAU Symposium, vol. 141, pp. 173–180 (1990)Google Scholar
  87. Synge J.L.: Relativity: The General Theory. Series in Physics. North-Holland Publication Co., Amsterdam (1964) c1964Google Scholar
  88. Tao J.H., Huang T.Y., Han C.H.: Coordinate transformations and gauges in the relativistic astronomical reference systems. Astron. Astrophys. 363, 335–342 (2000)ADSGoogle Scholar
  89. Walter H.G., Sovers O.J.: Astrometry of Fundamental Catalogues: the Evolution from Optical to Radio Reference Frames. Springer-Verlag, Berlin, Heidelberg (2000)Google Scholar
  90. Will C.M.: Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  91. Williams J.G.: A scheme for lunar inner core detection. Geophys. Res. Lett. 34, 3202 (2007). doi: 10.1029/2006GL028185 CrossRefGoogle Scholar
  92. Williams J.G., Dicke R.H., Bender P.L., Alley C.O., Currie D.G., Carter W.E., Eckhardt D.H., Faller J.E., Kaula W.M., Mulholland J.D.: New test of the equivalence principle from lunar laser ranging. Phys. Rev. Lett. 36, 551–554 (1976). doi: 10.1103/PhysRevLett.36.551 CrossRefADSGoogle Scholar
  93. Williams, J.G., Newhall, X.X., Yoder, C.F., Dickey, J.O.: Lunar free libration. In: Lunar and Planetary Institute Conference Abstracts, Lunar and Planetary Inst. Technical Report, vol. 27, p. 1439 (1996)Google Scholar
  94. Williams, J.G., Boggs D.H., Ratcliff, J.T., Yoder, C.F., Dickey, J.O.: Influence of a fluid lunar core on the moon’s orientation. In: Lunar and Planetary Institute Conference Abstracts, Lunar and Planetary Inst. Technical Report, vol. 32, p. 2028 (2001a)Google Scholar
  95. 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 CrossRefADSGoogle Scholar
  96. Williams, J.G., Boggs D.H., Ratcliff J.T., Dickey, J.O. : Lunar rotation and the lunar interior. In: Mackwell, S., Stansbery, E. (eds.) Lunar and Planetary Institute Conference Abstracts, Lunar and Planetary Inst. Technical Report, vol. 34, p. 1161 (2003)Google Scholar
  97. Williams, J.G., Turyshev, S.G., Murphy, T.W.: Improving LLR tests of gravitational theory. Int. J. Mod. Phys. D 13, 567–582 (2004). doi: 10.1142/S0218271804004682, arXiv:gr-qc/0311021
  98. Williams, J.G., Boggs, D.H., Ratcliff, J.T.: Lunar tides, fluid core and core/mantle boundary. In: Lunar and Planetary Institute Conference Abstracts, Lunar and Planetary Inst. Technical Report, vol. 39, p. 1484 (2008)Google Scholar
  99. Xu H., Jin W.: Tidal acceleration of the moon. Shanghai Obs. Ann. 15, 129–133 (1994)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Physics & AstronomyUniversity of Missouri-ColumbiaColumbiaUSA
  2. 2.Astronomy DepartmentNanjing UniversityNanjingChina

Personalised recommendations