Skip to main content
SpringerLink
Log in

Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

A Correction to this article was published on 02 December 2019

Abstract

The aim of this work is to combine the model of orbital and rotational motion of the Moon developed for DE430 with up-to-date astronomical, geodynamical, and geo- and selenophysical models. The parameters of the orbit and physical libration are determined in this work from lunar laser ranging (LLR) observations made at different observatories in 1970–2013. Parameters of other models are taken from solutions that were obtained independently from LLR. A new implementation of the DE430 lunar model, including the liquid core equations, was done within the EPM ephemeris. The postfit residuals of LLR observations make evident that the terrestrial models and solutions recommended by the IERS Conventions are compatible with the lunar theory. That includes: EGM2008 gravitational potential with conventional corrections and variations from solid and ocean tides; displacement of stations due to solid and ocean loading tides; and precession-nutation model. Usage of these models in the solution for LLR observations has allowed us to reduce the number of parameters to be fit. The fixed model of tidal variations of the geopotential has resulted in a lesser value of Moon’s extra eccentricity rate, as compared to the original DE430 model with two fit parameters. A mixed model of lunar gravitational potential was used, with some coefficients determined from LLR observations, and other taken from the GL660b solution obtained from the GRAIL spacecraft mission. Solutions obtain accurate positions for the ranging stations and the five retroreflectors. Station motion is derived for sites with long data spans. Dissipation is detected at the lunar fluid core-solid mantle boundary demonstrating that a fluid core is present. Tidal dissipation is strong at both Earth and Moon. Consequently, the lunar semimajor axis is expanding by 38.20 mm/yr, the tidal acceleration in mean longitude is \(-25.90 {{}^{\prime \prime }}/\mathrm{cy}^2\), and the eccentricity is increasing by \(1.48\times 10^{-11}\) each year.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Avdyushev, V.: Gauss–Everhart integrator. Comput. Technol. (Vychislitelnye Tekhnologii) 15, 31–46 (2010). (in Russian)

    MATH  Google Scholar 

  • Bizouard, C., Gambis, D.: The combined solution C04 for earth orientation parameters consistent with international terrestrial reference frame 2005. In: Drewes, H. (ed.) Geodetic Reference Frames, IAG Symposium Munich, Germany, 9–14 October 2006, pp. 265–270. Springer, Berlin (2009)

  • Bizouard, C., Gambis, D.: The combined solution C04 for earth orientation parameters consistent with international terrestrial reference frame 2008. IERS notice (2011). http://hpiers.obspm.fr/iers/eop/eopc04/C04.guide.pdf

  • Chapront-Touzé, M., Chapront, J.: ELP 2000–85—a semi-analytical lunar ephemeris adequate for historical times. A&A 190(1–2), 342–352 (1998)

    ADS  Google Scholar 

  • Cunningham, L.E.: On the computation of the spherical harmonic terms needed during the numerical integration of the orbital motion of an artificial satellite. Celest. Mech. 2(2), 207–216 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Ferland, R., Piraszewski, M.: The IGS-combined station coordinates, earth rotation parameters and apparent geocenter. J. Geod. 83(3), 385–392 (2009)

    Article  ADS  Google Scholar 

  • Fienga, A., Laskar, J., Gastineau, M., Verma, A.: INPOP new release: INPOP13c. Technical report Observatoire de Paris (2013). http://www.imcce.fr/fr/presentation/equipes/ASD/inpop/inpop13c.pdf

  • Findler, R., Clements, J., Flanagan, C., Flatt, M., Krishnamurthi, S., Steckler, P., et al.: DrScheme: a programming environment for scheme. J. Funct. Program. 12(2), 159–182 (2002)

    Article  MATH  Google Scholar 

  • Finkelstein, A.M., Ipatov, A.V., Skurikhina, E.A., Surkis, I.F., Smolentsev, S.G., Fedotov, L.V.: Geodynamic observations on the quasar VLBI network in 2009–2011. Astron. Lett. 38(6), 394–398 (2012)

    Article  ADS  Google Scholar 

  • Flatt, M.: PLT: reference: racket. Technical Report PLT-TR-2010-1, PLT Design Inc. (2010). http://racket-lang.org/tr1/

  • Folkner, W., Williams, J., Boggs, D., Park, R., Kuchynka, P.: The planetary and Lunar ephemerides DE430 and DE431. IPN Progress Report 42-196, NASA JPL (2014)

  • Hohenkerk, C.: SOFA and the algorithms for transformations between time scales and between reference systems. In: Schuh, H., Bhm, S., Nilsson, T., Capitaine, N. (eds.) Proceedings of the Journées 2011 “Systèmes de référence spatio-temporels”, pp. 21–24. Vienna University of Technology (2012)

  • Konopliv, A.S., Park, R.S., Yuan, D.N., Asmar, S.W., Watkins, M.M., Williams, J.G., et al.: The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL primary mission. J. Geophys. Res. Planets 118(7), 1415–1434 (2013)

    Article  ADS  Google Scholar 

  • Kopeikin, S.M.: Theory of relativity in observational radio astronomy. Sov. Astron. 34(1), 5–9 (1990)

    ADS  Google Scholar 

  • Krasinsky, G., Prokhorenko, S., Yagudina, E.: New version of EPM-ERA lunar theory. In: Capitaine, N. (ed.) Proceedings of the Journées 2010 “Systèmes de référence spatio-temporels, pp. 61–64. Observatoire de Paris (2011)

  • Krasinsky, G., Vasilyev, M.: ERA-7. Knowledge base and programming system for dynamical astronomy: manual. Institute of Applied Astronomy RAS (2006)

  • Krasinsky, G.A.: Selenodynamical parameters from analysis of LLR observations of 1970–2001. Commun. IAA RAS 148, 1–27 (2002)

    Google Scholar 

  • Krasinsky, G.A., Novikov, F.A., Scripnichenko, V.I.: Problem oriented language for ephemeris astronomy and its realisation in the system ERA. Celest. Mech. 45(1), 219–229 (1988)

    Article  ADS  Google Scholar 

  • Krasinsky, G.A., Vasilyev, M.V.: Era: knowledge base for ephemeris and dynamical astronomy. In: Wytrzyszczak, I.M., Lieske, J.H., Feldman, R.A. (eds.) Dynamics and Astrometry of Natural and Artificial Celestial Bodies: Proceedings of IAU Colloquium 165 Poznań, Poland July 1–5, 1996, pp. 239–244. Springer, Dordrecht (1997)

    Chapter  Google Scholar 

  • Manche, H., Bouquillon, S., Fienga, A., Laskar, J., Francou, G.: Towards INPOP07, adjustments to LLR data. In: Capitaine, N. (ed.) Proceedings of the Journées 2007 “Systèmes de référence spatio-temporels”, pp. 70–73. Observatoire de Paris (2008)

  • Manche, H., Fienga, A., Laskar, J., Bouquillon, S., Francou, G., Gastineau, M.: LLR residuals of INPOP10a and constraints on post-newtonian parameters. In: Schuh, H., Bhm, S., Nilsson, T., Capitaine, N. (eds.) Proceedings of the Journées 2011 “Systèmes de référence spatio-temporels”, pp. 65–68. Vienna University of Technology (2012)

  • Mathews, P.M., Dehant, V., Gipson, J.M.: Tidal station displacements. J. Geophys. Res. Solid Earth 102(B9), 20469–20477 (1997)

    Article  Google Scholar 

  • Mendes, V.B., Pavlis, E.C.: High-accuracy zenith delay prediction at optical wavelengths. Geophys. Res. Lett. 31(14), L14602 (2004)

  • Mendes, V.B., Prates, G., Pavlis, E.C., Pavlis, D.E., Langley, R.B.: Improved mapping functions for atmospheric refraction correction in SLR. Geophys. Res. Lett. 29(10), 53–1–53–4 (2002)

    Article  Google Scholar 

  • Murphy, T.: Lunar laser ranging: the millimeter challenge. Rep. Prog. Phys. 76, 076,901 (2013)

    Article  Google Scholar 

  • Murphy, T., Adelberger, E., Battat, J., Hoyle, C., Johnson, N., McMillan, R., et al.: APOLLO: millimeter lunar laser ranging. Class. Quantum Grav. 29, 184,005 (2012)

    Article  Google Scholar 

  • Newhall, X., Williams, J.G., Dickey, J.O.: Earth rotation (UTO–UTC) from lunar laser ranging. IERS Tech. Note No. 5, 41–45 (1990)

    Google Scholar 

  • Pavlov, D., Skripnichenko, V.: Rework of the ERA software system: ERA-8. In: Malkin, Z., Capitaine, N. (eds.) Proceedings of the Journées 2014 “Systèmes de référence spatio-temporels”, pp. 243–246. Pulkovo Observatory (2015)

  • Petit, G., Luzum, B.: IERS Conventions 2010 (IERS Technical Note 36). Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main (2010)

  • Pitjeva, E.: Updated IAA RAS planetary ephemerides-EPM2011 and their use in scientific research. Sol. Syst. Res. 47(5), 386–402 (2013)

    Article  ADS  Google Scholar 

  • Pitjeva, E., Pitjev, N.: Development of planetary ephemerides EPM and their applications. Celest. Mech. Dyn. Astron. 119(3–4), 237–256 (2014)

    Article  ADS  Google Scholar 

  • Ratcliff, J., Gross, R.: Combinations of earth orientation measurements: SPACE2014, COMB2014, and POLE2014. JPL publication 15-8, NASA (2015)

  • Samain, E., Mangin, J., Veillet, C., Torre, J.M., Fridelance, P., Chabaudie, J., et al.: Millimetric lunar laser ranging at OCA (Observatoire de la Côte d’Azur). Astron. Astrophys. Suppl. Ser. 130, 235–244 (1998)

    Article  ADS  Google Scholar 

  • Shelus, P.J.: MLRS: a lunar/artificial satellite laser ranging facility at the McDonald Observatory. IEEE Trans. Geosci. Rem. Sens. GE–234, 385–390 (1985)

    Article  ADS  Google Scholar 

  • Standish, E., Newhall, X., Williams, J., Yeomans, D.: Orbital ephemerides of the Sun, Moon, and Planets. In: Seidelmann, P.K. (ed.) Explanatory Supplement to the Astronomial Almanac. University Science Books (1992)

  • Vasilyev, M., Yagudina, E.: Russian lunar ephemeris EPM-ERA 2012. Sol. Syst. Res. 48(2), 158–165 (2014)

    Article  ADS  Google Scholar 

  • Williams, J., Boggs, D.: Tides on the Moon: theory and determination of dissipation. J. Geophys. Res. 120, 689–724 (2015)

    Article  Google Scholar 

  • Williams, J., Boggs, D.: Secular tidal changes in lunar orbit and Earth rotation. Celest. Mech. Dyn, Astron 126 this issue (2016) . doi:10.1007/s10569-016-9702-3

  • Williams, J.G., Boggs, D.H., Folkner, W.M.: DE430 Lunar Orbit, Physical Librations, and Surface Coordinates. Jet Propulsion Laboratory Interoffice Memorandum 335-JW,DB,WF-20130722-016, California Institute of Technology (2013)

  • 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. Planets 106(E11), 27933–27968 (2001)

    Article  ADS  Google Scholar 

  • 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(7), 1546–1578 (2014)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

D. Pavlov would like to thank Elena Pitjeva, Eleonora Yagudina, Sergey Kurdubov, Vladimir Skripnichenko, and numerous other colleagues from the IAA RAS for helpful comments and advice throughout this work; and Matthew Flatt from the University of Utah for his help in programming on the Racket platform. This work would not have been possible without the effort of personnel at observatories doing lunar laser ranging: Apache Point (Murphy et al. 2012; Murphy 2013), McDonald Laser Ranging Station (Shelus 1985), Observatoire de la Côte d’Azur (Samain et al. 1998), Giuseppe Bianco at Matera Laser Ranging Observatory, and Lunar Ranging Experiment (LURE) at the Haleakala observatory in the past. The POLAC website was of great help, where Christophe Barache, Sébastien Bouquillon, Teddy Carlucci, and Gerard Francou carefully collected LLR observations from different sources. An anonymous reviewer provided a lot of comments and suggestions that allowed to improve the article substantially. A portion of 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.

Author information

Authors and Affiliations

  1. Institute of Applied Astronomy RAS, Kutuzov embankment 10, St. Petersburg, 191187, Russia

    Dmitry A. Pavlov & Vladimir V. Suvorkin

  2. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, 91109, USA

    James G. Williams

Authors
  1. Dmitry A. Pavlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James G. Williams
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vladimir V. Suvorkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry A. Pavlov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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. Celest Mech Dyn Astr 126, 61–88 (2016). https://doi.org/10.1007/s10569-016-9712-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-016-9712-1

Keywords

Search

Navigation