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.

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

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 16th International Workshop on Laser Ranging, SLR – the Next Generation, ed. Stanislaw Schillak, 101-120, 2009, http://www.astro.amu.edu.pl/ILRS_Workshop_2008/index.php .

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.

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.

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.

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, 42nd 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, 114th Meeting of the Geodetic Society of Japan, Uji City, Kyoto Prefecture, Japan, November 8-10, 2010.

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.

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.

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

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.

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.

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

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.

Normal Points• 17474 normal points, March 1970 – October 2010• Stations Ap 11 Ap 14 Ap 15 Lk1 Lk2 Total

– McDonald 468 495 2356 132 3451– MLRS – Saddle 10 26 236 3 275– MLRS – Mt 226 236 2398 2 12 2874– OCA 876 775 7285 285 9221– Haleakala 20 23 633 18 694– APO 176 180 506 29 51 942– Matera 1 16 17– Total 1776 1736 13430 31 501 17474

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.

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.

Weighted RMS Residual by Year

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.

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

Periodogram of Residuals 1970-2010 Low Frequency End

High Frequency RMS

Periodogram of Residuals 1970-2010 Monthly Region

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.

Apache Point Position

• Station coordinate rates are determined with uncertainties of 6-13 mm/yr.

• Overall bias is 0.19±0.20 nsec.

Station Motion Solution Model

mm/yr mm/yrMcD East –12.9±2.0 –12.7 North –4.3±3.6 –6.2 Up 1.0±2.1 1.OCA East 19.6±0.8 20.7 North 16.4±2.6 15.9 Up 2.9±2.1 1.APO East –14.4±5.9 –13.3

North 0.8±12.7 –8.2Up 8.8±7.9 1.

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.

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 J2 0.46 & 0.45 km

–Moon J2 & C22 0.2 m

–Earth C22 0.5 mm

–Solar radiation pressure 4 mm

Relativistic Effects on Orbit

Cause Amplitude–Lorentz contraction 0.95 m–Solar potential 6 cm–Time transformation 5 & 5 cm–Other relativity 5 cm

Sources: Chapront-Touzé and Chapront; Vokrouhlicky; Williams and Dickey

Causes of Perigee and Node Precessions

Cause perigee rate rate "/yr

"/yr

Sun 146,425.38 –69,671.67Planets 2.47 –1.44Earth J2 6.33 –5.93Moon J2 & C22 –0.0176 –0.1705Relativity 0.0180 0.0190

Orbit — Tidal Dissipation for DE421

• Semimajor axis +38.14 mm/yr

• Tidal acceleration –25.85 “/cent2

• Both Earth and Moon have tidal dissipation. Dissipation from the Earth gives –26.1 “/cent2 while the Moon gives +0.2 “/cent2.

• Artificial satellite tide results predict tidal acceleration –26.25 “/cent2 from Earth.

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.

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.

Retroreflector Locations

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

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

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.