1. LLR Analysis – Relativistic Model and Tests of Gravitational Physics James G. Williams Dale H. Boggs Slava G. Turyshev Jet Propulsion Laboratory California Institute of Technology, USA LLR Workshop, Boston, MA Dec 9-10, 2010 Copyright 2010. All rights reserved.

2. Introduction Lunar Laser Ranging (LLR) from 1970-2010, from 4 Earth stations (McDonald, Haleakala, OCA, Apache Point) to 5 retroreflector arrays, probes gravitational physics. The lunar orbit is sensitive to: –Equivalence principle. –Geodetic precession. –Change in the gravitational constant G. –Change in scale. Gravitational physics results depend on accurate models for data analysis.

3. Lunar Laser Ranges Laser pulses sent from stations on the Earth toward the Moon bounce off of retroreflector arrays and return to the Earth. Ranging started in 1969 & continues to 2010.

4. Ranging from Earth Station to Moon Retroreflector

5. Retroreflector Arrays

6. Fit and Model LLR range data is fit with a weighted least- squares approach. Observed time of flight from transmit to receive time (range) is compared with a computed time of flight. Computed range depends on orbits and rotations of Earth & Moon, station & reflector positions, general relativity, tides on Earth & Moon, atmosphere, station motion, + other model effects.

7. Frames, Clocks, Computation–1 Observed time-of-flight “ranges” use clocks at stations to measure interval from transmit to receive times. Clocks are compared with broadcast UTC. Numerically integrate Moon and planet orbits in solar system barycentric (SSB) frame. Also integrate dynamical partial derivatives for orbit & other parameters.

8. Frames, Clocks, Computation–2 Calculate time-of-flight: iterate up-leg (Earth station to Moon reflector) and down-leg (Moon to Earth) computation. Calculated time-of-flight starts with transmit time, converts to TDB, retrieves Earth and Moon positions and velocities from integrated ephemeris.

9. Frames, Clocks, Computation–3 Lorentz contractions and scale changes (from potential) are applied to geocentric station locations and Moon centered reflector positions. Relativistic (Shapiro) delay and atmospheric delay are added to the geometrical ranges.

10. Frames, Clocks, Computation–4 The calculated receive time – transmit time interval in TDB is converted to the station time interval. Partial derivatives of range with respect to solution parameters are computed after the light-time iterations converge. The least-squares solution fits the range model to observed range.

11. Sensitivities Most of the sensitivity to gravitational physics parameters comes through the orbit dynamics, but PPN ϒ also depends on the gravitational (Shapiro) time delay. We can solve for equivalence principle (M g /M i ), PPN β & ϒ, geodetic precession, (dG/dt)/G, & d(scale)/dt.

12. Data Weighting, Residuals During the fits, the data are weighted. Weighting depends on combined instrumental error (from station) and model error (~0.1 nsec = 1.5 cm). The quality of the fits can be monitored by examining postfit residuals by time, station, & retroreflector. We can examine spectra of residuals.

13. 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

14. 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

15. 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

16. Normal Points 17481 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 – peak 226 2362398 2 122874 –OCA 876 7757285 2859221 –Haleakala 20 23 633 18 694 –APO 176 182 509 30 52 949 –Matera 1 16 17 –Total1776173813433 32 502 17481

17. Equivalence Principle The equivalence principle (EP) test is sensitive to the gravitational/inertial mass ratio difference between the Earth and Moon. M g /M i = (1.4±1.2)x10 –13 Equivalent to a range amplitude of 4.1±3.5 mm at 29.53 days period.

18. Strong Equivalence Principle and PPN β Convert the LLR EP result to strong EP by subtracting Adelberger’s (2001) weak EP result of (1.0±1.4)x10 –13. That gives (M g /M i ) SEP = (0.4±1.9)x10 –13 η = 0.00009±0.00042 Using the Cassini result for ϒ leads to PPN β–1 = (3±11)x10 –5

19. EP Warning The preceding EP solution fixed the 4 annual nutation coefficients to IERS values. Also solving for the 4 imperfectly known nutation coefficients gives EP uncertainties ~2.5 times larger! Those LLR-determined nutation uncertainties are 0.2-0.5 mas. VLBI should give better values, but coefficients vary ~0.1 mas due to geophysical effects. Seasonal & diurnal geophysical effects affect station positions, geocenter motion, and nutations. These will corrupt the LLR EP test.

20. EP Warning Continued A caution – projecting future LLR EP uncertainties by scaling in proportion to much smaller range errors is not valid. Geophysical effects will become limiting error sources at some level.

21. Geodetic Precession The correction to the nominal geodetic or de Sitter-Fokker precession is modeled as a dimensionless factor K gp = 0.0035±0.0026 That geodetic precession rate change is 0.07±0.05 mas/yr compared to 19.2 mas/yr. The accuracy of the geodetic precession depends on time span and will improve for longer spans.

22. dG/dt and Scale Change The rate of change of the gravitational constant is = (8±5)x10 –13 /yr Scale rate is geometrical, no dynamics d(scale)/dt = (-5±5)x10 –12 /yr The solar system does not share the cosmic expansion of 1/13.7x10 9 yr = 7.3x10 –11 /yr by either comparison.

23. Future Future LLR data is important. Both high accuracy ranges, such as Apache Point provides, and improved accuracy analysis software benefit all results. A longer data span benefits geodetic precession and change in G. New retroreflectors could be placed on the Moon by future missions.

24. Conclusions Lunar Laser Ranging continues to test gravitational physics. A high accuracy model is necessary for LLR data analysis. LLR results for equivalence principle, geodetic precession, dG/dt and scale rate are consistent with general relativity

25. Looking Forward Laser ranging to the Moon can continue to provide and improve gravitational physics results.