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/i
ndex.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 38, 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
McD East
North
Up
OCA East
North
Up
APO East
North
Up
Solution
mm/yr
–12.9±2.0
–4.3±3.6
1.0±2.1
19.6±0.8
16.4±2.6
2.9±2.1
–14.4±5.9
0.8±12.7
8.8±7.9
Model
mm/yr
–12.7
–6.2
1.
20.7
15.9
1.
–13.3
–8.2
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
–Ellipticity
–Solar perturbations
–Jupiter perturbation
–Venus perturbations
–Earth J2
–Moon J2 & C22
–Earth C22
–Solar radiation pressure
Amplitude
20905 & 570 km
3699 & 2956 km
1.06 km
0.73, 0.68 & 0.60 km
0.46 & 0.45 km
0.2 m
0.5 mm
4 mm
Relativistic Effects on Orbit
Cause
– Lorentz contraction
– Solar potential
– Time transformation
– Other relativity
Amplitude
0.95 m
6 cm
5 & 5 cm
5 cm
Sources: Chapront-Touzé and Chapront;
Vokrouhlicky; Williams and Dickey
Causes of Perigee and Node
Precessions
Cause
perigee rate
"/yr
node rate
"/yr
Sun
146,425.38 –69,671.67
Planets
2.47
–1.44
Earth J2
6.33
–5.93
Moon J2 & C22 –0.0176
–0.1705
Relativity
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
1985 1119
1986 272
1987 261
1988 487
1989 404
1990 684
1991 465
1992 419
1993 648
1994 785
1995 1018
1996 1015
1997 819
1998 695
1999 844
2000 895
2001 375
2002 242
2003 245
2004 487
2005 331
2006 166
2007 209
2008 369
2009 239
2010 263
Total 17473
nsec
0.636
0.392
0.268
0.272
0.300
0.297
0.206
0.233
0.228
0.180
0.143
0.112
0.107
0.104
0.095
0.112
0.148
0.157
0.129
0.102
0.116
0.112
0.101
0.134
0.147
0.099
0.177
cm
9.55
5.87
4.03
4.09
4.50
4.46
3.09
3.49
3.42
2.70
2.14
1.68
1.60
1.56
1.43
1.68
2.22
2.35
1.94
1.54
1.73
1.67
1.52
2.01
2.20
1.49
2.66
Norm
0.982
0.966
1.159
1.073
1.298
1.209
1.019
1.121
1.051
1.002
0.906
0.925
0.916
0.805
0.749
0.760
0.898
0.944
0.798
0.761
0.794
0.804
0.863
1.174
1.228
0.918
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.