1. LLR Analysis – JPL Model and Data Analysis James G. Williams, Dale H. Boggs and Slava G. Turyshev Jet Propulsion Laboratory, California Institute of Technology LLR Workshop, Harvard, Boston, MA December 9-10, 2010 Copyright 2010. All rights reserved.

2. Introduction Presentation outlines JPL LLR activities & analysis –Publications, abstracts & meetings –Ephemerides –Activities: model, computation, data analysis & other –Simulations –Residual analysis – annual rms & spectra –Model comments

3. LLR Publications 2009 J. G. Williams, S. G. Turyshev, and D. H. Boggs, Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon, International Journal of Modern Physics D, 18 (7), 1129-1175, doi:10.1142/S021827180901500X, 2009. J. G. Williams and D. H. Boggs, Lunar Core and Mantle. What Does LLR See?, in Proceedings of 16 th International Workshop on Laser Ranging, SLR – the Next Generation, ed. Stanislaw Schillak, 101-120, 2009, http://www.astro.amu.edu.pl/ILRS_Workshop_2008/i ndex.php.

4. LLR Publications 2010 Rambaux, N., and J. G. Williams, The Moon’s physical librations and determination of their free modes, Celestial Mechanics and Dynamical Astronomy, doi:10.1007/s10569-010-9314-2, (Oct 26) 2010. Fok, H. S., C. K. Shum, Y. Yi, H. Araki, J. Ping, J. G. Williams, G. Fotopoulos, H. Noda, S. Goossens, Q. Huang, Y. Ishihara, K. Matsumoto, J. Oberst, and S. Sasaki, Accuracy assessment of lunar topography models, Earth, Planets and Space, special issue New results of lunar science with KAGUYA (SELENE), in press, 2010.

5. Abstracts & Meetings 2009 J. G. Williams, D. H. Boggs, and J. T. Ratcliff, A Larger Lunar Core?, abs. #1452 of the Lunar and Planetary Science Conference XXXX, March 23-27, 2009. J. G. Williams, Lunar Laser Ranging: Relativistic Model and Tests of Gravitational Physics, IAU Symposium 261, Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and Data Analysis, April 27-May 1, 2009, Virginia Beach, VA, 2009.

6. Abstracts & Meetings 2009-2010 N. Rambaux and J. G. Williams, A new determination of the normal modes of the Moon’s libration, Division on Dynamical Astronomy meeting, Virginia Beach, VA, May 2-5, 2009, Bull. Amer. Astron. Soc., 41, #2, 907, 2009. Rambaux, N., J. Williams, and J. Laskar, Dynamically active Moon, European Planetary Science Congress, Potsdam, Germany, September 13-18, 2009. Williams, J. G., S. G. Turyshev, and W. M. Folkner, Lunar Geophysics and Lunar Laser Ranging, Ground-Based Geophysics on the Moon, Tempe, AZ, January 21-22, 2010.

7. Abstracts & Meetings 2010 J. G. Williams, D. H. Boggs, and J. T. Ratcliff, Lunar Fluid Core Moment, abstract #2336 of the Lunar and Planetary Science Conference XXXXI, The Woodlands, TX, March 1-5, 2010. Williams, J. G., D. H. Boggs, and H. Noda, Exploring the Lunar Interior with Tides, Gravity and Orientation, abs. 21.10, 42 nd DPS meeting, Pasadena, CA, Oct 3- 8, 2010, Bull. Amer. Astron. Soc., 42, #4, 987, 2010 Noda, H., J. G. Williams, and D. H. Boggs, Simulation Study of Lunar Laser Ranging by JPL Software, 114 th Meeting of the Geodetic Society of Japan, Uji City, Kyoto Prefecture, Japan, November 8-10, 2010.

8. Publicly Available Ephemerides Joint lunar and planetary fits lead to DE418 in 2007 and DE421 in 2008. DE421 is recommended for the lunar orbit and physical librations. Memos are available for both. DE421 is being used by LRO and will be used by GRAIL.

9. Recent Activities – Model & Computation Completed move off of old computer. Found a problem with integration error growth for some integrated partial derivatives. Changed the computation approach. Added Lunokhod 1 to retroreflector arrays. Extended model for Earth tide perturbations on the lunar orbit. Made the diurnal and semidiurnal time delays frequency dependent. There are now three Love number and five time delay parameters.

10. Activity – Data Analysis Attempts to iterate fluid core moment solutions fail to converge. Too nonlinear. Fit new data from OCA (7 nights) & Matera (1 night) Lunokhod 1 data analysis Try out new Earth tide model. The anomalous de/dt decreases ~20% to 1.1x10 -11 /yr. Add an OCA bias in April-May 1992 Reweight last decade of OCA data

11. Simulations Simulation software was prepared by Dale. Simulations were done at JPL by H. Noda using the new software. Simulated future data (300 /yr) and (1 or 2) new LLR sites on Moon. The simulations for new LLR sites were less impressive than expected. We had to rush to get first results before Noda left and did not have a chance to simulate a wider variety of cases.

12. Free Physical Librations The free physical librations in longitude (1.3”) and wobble (3.3”x8.2”) show that the Moon’s rotation is not quite in a fully damped state (Rambaux and Williams, 2010). The longitude free librations may be caused by geologically recent resonance passage (Eckhardt, 1993). The Moon appears to be showing some geophysical activity, possibly from the fluid core (Yoder, 1981), that is stimulating the wobble.

13. Other Related Activities SGT, JGW and WMF involved in new corner cube design SGT, JGW and WMF involved in Lunette Discovery proposal for geophysical landers JGW is on two working groups: Lunar Geodesy & Cartography, & ILN Site Selection JGW is a Co-I on GRAIL mission: two 2012 lunar orbiters for high accuracy gravity DHB works on Earth orientation software

14. Global and Post-Fit Analysis Global least-squares fit for many dynamical and geometrical solution parameters. Based on accurate numerical integration, accurate model, rotation matrices for orientation of Earth and Moon, etc. Post-fit analysis looks for what has not been fit by global solution.

15. Normal Points 17474 normal points, March 1970 – October 2010 StationsAp 11Ap 14Ap 15 Lk1 Lk2Total –McDonald 468 4952356 132 3451 –MLRS – Saddle 10 26 236 3 275 –MLRS – Mt 226 2362398 2 122874 –OCA 876 7757285 2859221 –Haleakala 20 23 633 18 694 –APO 176 180 506 29 51 942 –Matera 1 16 17 –Total1776173613430 31 501 17474

16. Residual Analysis Residuals show physical libration signatures – retroreflector residuals separate ~0.1 nsec. Spectra of post-fit residuals show long period power in range residuals and also physical librations. All range spectral background ~1 mm.

17. Residual Analysis Apache Point (APO) ranges have rms residuals of ~0.11 nsec or ~1.7 cm For both APO and OCA the rms scatter varies from year to year. Perhaps this is due to some signature.

18. Weighted RMS Residual by Year

19. Distinguishing Dynamics from Noise Over a long span of time missing dynamical effects in orbit radius or lunar orientation and rotation will give one or more lines in spectra of residuals. Random effects will show as a broad background in spectra.

20. Smoothed Periodogram of Residuals 1970-2010 is Highest at Low Frequency

21. Periodogram of Residuals 1970-2010 Low Frequency End  High Frequency RMS

22. Periodogram of Residuals 1970-2010 Monthly Region

23. Earth Model The model for the Earth has become more complex through the years. Major complexities come from the oceans, atmosphere and ground water. These variations cause small effects on tides, loading, nutation, UT1 and polar motion.

24. Apache Point Position Station coordinate rates are determined with uncertainties of 6-13 mm/yr. Overall bias is 0.19±0.20 nsec.

25. Station Motion SolutionModel mm/yrmm/yr McDEast–12.9±2.0–12.7 North –4.3±3.6 –6.2 Up 1.0±2.1 1. OCAEast 19.6±0.8 20.7 North 16.4±2.6 15.9 Up 2.9±2.1 1. APOEast–14.4±5.9–13.3 North 0.8±12.7 –8.2 Up 8.8±7.9 1.

26. LLR Geophysics & Geodesy The main limitation to LLR results for the Earth is the small number of stations operating during the last two decades. All are in the northern hemisphere. For those sites there are station positions & motions, UT0 and variation of latitude results. Can also determine Earth orientation precession & obliquity rates, annual & 18.6 yr nutations, orientation in space, and diurnal & semidiurnal UT1 variations.

27. Largest Radial Amplitudes by Cause Cause Amplitude –Ellipticity 20905 & 570 km –Solar perturbations 3699 & 2956 km –Jupiter perturbation 1.06 km –Venus perturbations 0.73, 0.68 & 0.60 km –Earth J 2 0.46 & 0.45 km –Moon J 2 & C 22 0.2 m –Earth C 22 0.5 mm –Solar radiation pressure 4 mm

28. Relativistic Effects on Orbit Cause Amplitude –Lorentz contraction0.95 m –Solar potential6 cm –Time transformation5 & 5 cm –Other relativity5 cm Sources: Chapront-Touzé and Chapront; Vokrouhlicky; Williams and Dickey

29. Causes of Perigee and Node Precessions Cause perigee rate  rate "/yr "/yr Sun 146,425.38 –69,671.67 Planets 2.47–1.44 Earth J 2 6.33–5.93 Moon J 2 & C 22 –0.0176–0.1705 Relativity 0.0180 0.0190

30. Orbit — Tidal Dissipation for DE421 Semimajor axis +38.14 mm/yr Tidal acceleration –25.85 “/cent 2 Both Earth and Moon have tidal dissipation. Dissipation from the Earth gives –26.1 “/cent 2 while the Moon gives +0.2 “/cent 2. Artificial satellite tide results predict tidal acceleration –26.25 “/cent 2 from Earth.

31. Orbit — Eccentricity Rate Tides on Earth 1.6x10 –11 /yr Tides on Moon–0.4x10 –11 /yr Anomalous rate(1.2±0.3)x10 –11 /yr Total 2.4x10 –11 /yr After adding frequency-dependent tidal time delays, the anomalous eccentricity rate contributes –4 mm/yr to perigee & +4 mm/yr to apogee distance rates. Cause is unknown.

32. Libration Model The libration model is missing an inner core. An inner core would add three free modes. None of these frequencies are known. The outer (fluid) core is modeled, but the single free mode is very long period.

33. Retroreflector Locations

34. Comparison of Apollo 11 & 14 Periodograms. Do Any Features Match? Why is Apollo 14 background higher than Apollo 11? Both spectra are high adjacent to the longitude libration resonance at 1056 d

35. APO RSS vs Full Weight year N nsec cm Norm 1985 1119 0.636 9.55 0.982 1986 272 0.392 5.87 0.966 1987 261 0.268 4.03 1.159 1988 487 0.272 4.09 1.073 1989 404 0.300 4.50 1.298 1990 684 0.297 4.46 1.209 1991 465 0.206 3.09 1.019 1992 419 0.233 3.49 1.121 1993 648 0.228 3.42 1.051 1994 785 0.180 2.70 1.002 1995 1018 0.143 2.14 0.906 1996 1015 0.112 1.68 0.925 1997 819 0.107 1.60 0.916 1998 695 0.104 1.56 0.805 1999 844 0.095 1.43 0.749 2000 895 0.112 1.68 0.760 2001 375 0.148 2.22 0.898 2002 242 0.157 2.35 0.944 2003 245 0.129 1.94 0.798 2004 487 0.102 1.54 0.761 2005 331 0.116 1.73 0.794 2006 166 0.112 1.67 0.804 2007 209 0.101 1.52 0.863 2008 369 0.134 2.01 1.174 2009 239 0.147 2.20 1.228 2010 263 0.099 1.49 0.918 Total 17473 0.177 2.66 1.013

36. Data Needs Accurate data with a good distribution vs fundamental arguments that are well distributed over 5 retroreflectors. Lack of a southern hemisphere station makes post-fit analysis more difficult. When OCA gathers more data, we can compare OCA and APO.