Astrophysics and Space Science

, 364:218 | Cite as

Estimation of the lunar free libration modes based on the recent ephemerides

  • Yongzhang Yang
  • Qingbao He
  • Jinsong Ping
  • Jianguo YanEmail author
  • Wenzhao Zhang
Original Article


The accuracy of lunar laser ranging (LLR) experiment has already increased to about 1 cm in recent years. When the LLR data joined to the lunar dynamical model updated from DE430, three independent numerically integrated ephemerides called DE436, EPM2017 and INPOP17a were released in 2017. It prompts us to reanalysis the lunar rotation parameters of the ephemerides, and also, especially estimation of the lunar free libration modes. To this end, we performed a Fourier analysis and least-square fit with the data from these three ephemerides to estimate lunar free librations. The results show that the free librations for each ephemeris are consistent with each other and with the previously published DE421 and DE430 free libration values; minor differences in detail may result from differences in the physical parameters contained in each ephemeris. The Euler angles integrated in each ephemeris however, show a significant difference before 1800, relative to the difference after 1970. This indicates that special attention must be paid to backward integration when building ephemerides.


Moon Physical librations Numerical integration Frequencies analysis 



This research is supported by LIESMARS Special Research Funding, by the National Natural Science Foundation of China (U1831132, 41590851, 11373060) and by State Key Project for Science and Technology (2015CB857101), grant of Hubei Province Natural Science (2018CFA087), Open Project of Lunar and Planetary Science Laboratory, Macau University of Science and Technology (FDCT 119/2017/A3), and Open Funding of Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing (KF201813). Planetary ephemerides files can be downloaded from (DE436), (INPOP17a), and (EPM2017).

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Bretagnon, P.: Astron. Astrophys. 114, 278 (1982) ADSGoogle Scholar
  2. Calame, O.: Moon 15(3–4), 343 (1976) ADSCrossRefGoogle Scholar
  3. Chapront, J., Chapront-Touzé, M., Francou, G.: In: International Astronomical Union Colloquium, vol. 172, p. 317. Cambridge University Press, Cambridge (1999) Google Scholar
  4. Chapront-Touzé, M., Chapront, J.: Astron. Astrophys. 124, 50 (1983) ADSGoogle Scholar
  5. Eckhardt, D.H.: In: International Astronomical Union Colloquium, vol. 63, p. 193. Cambridge University Press, Cambridge (1981a) Google Scholar
  6. Eckhardt, D.H.: Moon Planets 25(1), 3 (1981b) ADSCrossRefGoogle Scholar
  7. Eckhardt, D.H.: In: Interactions Between Physics and Dynamics of Solar System Bodies p. 307. Springer, Berlin (1993) CrossRefGoogle Scholar
  8. Folkner, W., Park, R.: Tech. rep., Jet Propulsion Laboratory, California Institute of Technology (2016).
  9. Folkner, W.M., Williams, J.G., Boggs, D.H., Park, R.S., Kuchynka, P.: The Planetary and Lunar Ephemerides DE430 and DE431. The Interplanetary Network Progress Report, vol. 42-196. Jet Propulsion Laboratory, California Institute of Technology (2014).
  10. Hill, G.W.: Am. J. Math. 1(1), 5 (1878) CrossRefGoogle Scholar
  11. Jin, W., Li, J.: Earth Moon Planets 73(3), 259 (1996) ADSCrossRefGoogle Scholar
  12. Laskar, J.: Astron. Astrophys. 198, 341 (1988) ADSGoogle Scholar
  13. Murphy, T.: LLR Analysis Workshop, December 9–10 (2010).
  14. Newhall, X., Williams, J.G.: In: International Astronomical Union Colloquium, vol. 165, p. 21. Cambridge University Press, Cambridge (1997) Google Scholar
  15. Pavlov, D.A., Williams, J.G., Suvorkin, V.V.: Celest. Mech. Dyn. Astron. 126(1–3), 61 (2016) ADSCrossRefGoogle Scholar
  16. Petrova, N.: Earth Moon Planets 73(1), 71 (1996) ADSCrossRefGoogle Scholar
  17. Petrova, N., Zagidullin, A., Nefedyev, Y., Kosulin, V., Andreev, A.: Adv. Space Res. 60(10), 2303 (2017) ADSCrossRefGoogle Scholar
  18. Ping, J., Wang, M., Zhang, S., et al.: J. Deep Space Explor. 1(3), 192 (2014) Google Scholar
  19. Rambaux, N., Williams, J.: Celest. Mech. Dyn. Astron. 109(1), 85 (2011) ADSCrossRefGoogle Scholar
  20. Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J.: Astron. Astrophys. 282, 663 (1994) ADSGoogle Scholar
  21. SOFA, I.S.B.: (2019).
  22. Tang, W., Xu, P., Hu, S., Cao, J., Dong, P., Bu, Y., Chen, L., Han, S., Gong, X., Li, W., et al.: Adv. Space Res. 60(6), 1315 (2017) ADSCrossRefGoogle Scholar
  23. Viswanathan, V., Fienga, A., Minazzoli, O., Bernus, L., Laskar, J., Gastineau, M.: Mon. Not. R. Astron. Soc. 476(2), 1877 (2018) ADSCrossRefGoogle Scholar
  24. Yang, Y.-Z., Li, J.-L., Ping, J.-S., Hanada, H.: Res. Astron. Astrophys. 17, 127 (2017) ADSCrossRefGoogle Scholar
  25. Yoder, C.F.: Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 303(1477), 327 (1981) ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Information Engineering in Surveying, Mapping and Remote SensingWuhan UniversityWuhanChina
  2. 2.National Astronomical ObservatoriesChinese Academy of SciencesBeijingChina
  3. 3.Astronomy Department of Beijing Normal UniversityBeijingChina

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