Dale H. Boggs

Progress in lunar laser ranging tests of relativistic gravity

Analyses of laser ranges to the Moon provide increasingly stringent limits on any violation of the equivalence principle (EP); they also enable several very accurate tests of relativistic gravity. These analyses give an EP test of Delta(MG/MI)EP=(-1.0+/-1.4) x 10(-13). This result yields a strong equivalence principle (SEP) test of Delta(MG/MI)SEP=(-2.0+/-2.0) x 10(-13). Also, the corresponding SEP violation parameter eta is (4.4+/-4.5) x 10(-4), where eta=4beta-gamma-3 and both beta and gamma are post-Newtonian parameters. Using the Cassini gamma, the eta result yields beta-1=(1.2+/-1.1) x 10(-4). The geodetic precession test, expressed as a relative deviation from general relativity, is Kgp=-0.0019+/-0.0064. The search for a time variation in the gravitational constant results in G /G=(4+/-9) x 10(-13) yr(-1); consequently there is no evidence for local (approximately 1 AU) scale expansion of the solar system.

Lunar rotational dissipation in solid body and molten core

Analyses of Lunar Laser ranges show a displacement in direction of the Moons pole of rotation which indicates that strong dissipation is acting on the rotation. Two possible sources of dissipation are monthly solid-body tides raised by the Earth (and Sun) and a fluid core with a rotation distinct from the solid body. Both effects have been introduced into a numerical integration of the lunar rotation. Theoretical consequences of tides and core on rotation and orbit are also calculated analytically. These computations indicate that the tide and core dissipation signatures are separable. They also allow unrestricted laws for tidal specific dissipation Q versus frequency to be applied. Fits of Lunar Laser ranges detect three small dissipation terms in addition to the dominant pole-displacement term. Tidal dissipation alone does not give a good match to all four amplitudes. Dissipation from tides plus fluid core accounts for them. The best match indicates a tidal Q which increases slowly with period plus a small fluid core. The core size depends on imperfectly known properties of the fluid and core-mantle interface. The radius of a core could be as much as 352 km if iron and 374 km for the Fe-FeS eutectic composition. If tidal Q versus frequency is assumed to be represented by a power law, then the exponent is −0.19±0.13. The monthly tidal Q is 37 (−4,+6), and the annual Q is 60 (−15,+30). The power presently dissipated by solid body and core is small, but it may have been dramatic for the early Moon. The outwardly evolving Moon passed through a change of spin state which caused a burst of dissipated power in the mantle and at the core-mantle boundary. The energy deposited at the boundary plausibly drove convection in the core and temporarily powered a dynamo. The remanent magnetism in lunar rocks may result from these events, and the peak field may mark the passage of the Moon through the spin transition.

LUNAR LASER RANGING TESTS OF THE EQUIVALENCE PRINCIPLE WITH THE EARTH AND MOON

A primary objective of the lunar laser ranging (LLR) experiment is to provide precise observations of the lunar orbit that contribute to a wide range of science investigations. In particular, time series of the highly accurate measurements of the distance between the Earth and the Moon provide unique information used to determine whether, in accordance with the equivalence principle (EP), these two celestial bodies are falling toward the Sun at the same rate, despite their different masses, compositions, and gravitational self-energies. Thirty-five years since their initiation, analyses of precision laser ranges to the Moon continue to provide increasingly stringent limits on any violation of the EP. Current LLR solutions give (-1.0 ± 1.4) × 10-13 for any possible inequality in the ratios of the gravitational and inertial masses for the Earth and Moon, Δ(MG/MI). This result, in combination with laboratory experiments on the weak equivalence principle, yields a strong equivalence principle (SEP) test of Δ(...

Lunar interior properties from the GRAIL mission

The Gravity Recovery and Interior Laboratory (GRAIL) mission has sampled lunar gravity with unprecedented accuracy and resolution. The lunar GM, the product of the gravitational constant G and the mass M, is very well determined. However, uncertainties in the mass and mean density, 3345.56 ± 0.40 kg/m3, are limited by the accuracy of G. Values of the spherical harmonic degree-2 gravity coefficients J2 and C22, as well as the Love number k2 describing lunar degree-2 elastic response to tidal forces, come from two independent analyses of the 3 month GRAIL Primary Mission data at the Jet Propulsion Laboratory and the Goddard Space Flight Center. The two k2 determinations, with uncertainties of ~1%, differ by 1%; the average value is 0.02416 ± 0.00022 at a 1 month period with reference radius R = 1738 km. Lunar laser ranging (LLR) data analysis determines (C − A)/B and (B − A)/C, where A < B < C are the principal moments of inertia; the flattening of the fluid outer core; the dissipation at its solid boundaries; and the monthly tidal dissipation Q = 37.5 ± 4. The moment of inertia computation combines the GRAIL-determined J2 and C22 with LLR-derived (C − A)/B and (B − A)/C. The normalized mean moment of inertia of the solid Moon is Is/MR2 = 0.392728 ± 0.000012. Matching the density, moment, and Love number, calculated models have a fluid outer core with radius of 200–380 km, a solid inner core with radius of 0–280 km and mass fraction of 0–1%, and a deep mantle zone of low seismic shear velocity. The mass fraction of the combined inner and outer core is ≤1.5%.

Lunar Laser Ranging Tests of the Equivalence Principle

The lunar laser ranging (LLR) experiment provides precise observations of the lunar orbit that contribute to a wide range of science investigations. In particular, time series of highly accurate measurements of the distance between the Earth and Moon provide unique information that determine whether, in accordance with the equivalence principle (EP), both of these celestial bodies are accelerating toward the Sun at the same rate, despite their different masses, compositions, and gravitational self-energies. Analyses of precise laser ranges to the Moon continue to provide increasingly stringent limits on any violation of the EP. Current LLR solutions give ( − 0.8 ± 1.3) × 10−13 for any possible inequality in the ratios of the gravitational and inertial masses for the Earth and Moon, (mG/mI)E − (mG/mI)M. Such an accurate result allows other tests of gravitational theories. Focusing on the tests of the EP, we discuss the existing data and data analysis techniques. The robustness of the LLR solutions is demonstrated with several different approaches to solutions. Additional high accuracy ranges and improvements in the LLR data analysis model will further advance the research of relativistic gravity in the solar system, and will continue to provide highly accurate tests of the EP.

Lunar Laser Ranging Science: Gravitational Physics and Lunar Interior and Geodesy

Laser pulses fired at retroreflectors on the Moon provide very accurate ranges. Analysis yields information on Earth, Moon, and orbit. The highly accurate retroreflector positions have uncertainties less than a meter. Tides on the Moon show strong dissipation, with Q = 33 ± 4 at a month and a weak dependence on period. Lunar rotation depends on interior properties; a fluid core is indicated with radius ∼20% that of the Moon. Tests of relativistic gravity verify the equivalence principle to ±1.4 × 10−13, limit deviations from Einstein’s general relativity, and show no rate for the gravitational constant G˙/G with uncertainty 9 × 10−13/year.

Tides on the Moon: Theory and determination of dissipation

Solid body tides on the Moon vary by about ±0.1 m each month. In addition to changes in shape, the Moons gravity field and orientation in space are affected by tides. The tidal expressions for an elastic sphere are compact, but dissipation introduces modifications that depend on the forcing period. Consequently, a Fourier representation of the tide-raising potential is needed. A mathematical model for the distortion-caused tidal potential may be used for the analysis of precise spacecraft tracking data. Since tides affect gravitational torques on the Moon from the Earths attraction, the lunar orientation is also affected. Expressions for five periodic perturbations of orientation are presented. The rheological properties of lunar materials determine how the Moon responds to different tidal periods. New lunar laser ranging solutions for the tidal orientation terms are presented. The quality factor Q is 38 ± 4 at 1 month, 41 ± 9 at 1 year, ≥74 at 3 years, and ≥58 at 6 years. The ranging results can be matched with absorption band models that peak at ~120 days and single relaxation time models that peak at ~100 days. Combined models are possibilities. Dissipation can modify laser ranging solutions; previously reported core flattening is too uncertain to be useful. Strong lunar tidal dissipation, modeled to arise in the deep hot mantle, appears to be from a region with radius ≥535 km. Classical Maxwell-type dissipation is too weak to detect at 3 and 6 year periods.

Angular momentum exchange among the solid Earth, atmosphere, and oceans: A case study of the 1982–1983 El Niño event

The 1982-1983 El Nino/Southern Oscillation (ENSO) event was accompanied by the largest interannual variation in the Earths rotation rate on record. In this study we demonstrate that atmospheric forcing was the dominant cause for this rotational anomaly, with atmospheric angular momentum (AAM) integrated from 1000 to 1 mbar (troposphere plus stratosphere) accounting for up to 92% of the interannual variance in the length of day (LOD). Winds between 100 and 1 mbar contributed nearly 20% of the variance explained, indicating that the stratosphere can play a significant role in the Earths angular momentum budget on interannual time scales. Examination of LOD, AAM, and Southern Oscillation Index (SOI) data for a 15-year span surrounding the 1982-1983 event suggests that the strong rotational response resulted from constructive interference between the low-frequency (approximately 4-6 year) and quasi-biennial (approximately 2-3 year) components of the ENSO phenomenon, as well as the stratospheric Quasi-Biennial Oscillation (QBO). Sources of the remaining LOD discrepancy (approximately 55 and 64 microseconds rms residual for the European Centre for Medium-Range Forecasting (EC) and U.S. National Meteorological Center (NMC) analyses) are explored; noise and systematic errors in the AAM data are estimated to contribute 18 and 33 microseconds, respectively, leaving a residual (rms) of 40 (52) microseconds unaccounted for by the EC (NMC) analysis. Oceanic angular momentum contributions (both moment of inertia changes associated with baroclinic waves and motion terms) are shown to be candidates in closing the interannual axial angular momentum budget.

Williams, Turyshev, and Boggs Reply:

Analyses of laser ranges to the Moon provide increasingly stringent limits on any violation of the equivalence principle (EP); they also enable several very accurate tests of relativistic gravity. These analyses give an EP test of Delta(MG/MI)EP=(-1.0+/-1.4) x 10(-13). This result yields a strong equivalence principle (SEP) test of Delta(MG/MI)SEP=(-2.0+/-2.0) x 10(-13). Also, the corresponding SEP violation parameter eta is (4.4+/-4.5) x 10(-4), where eta=4beta-gamma-3 and both beta and gamma are post-Newtonian parameters. Using the Cassini gamma, the eta result yields beta-1=(1.2+/-1.1) x 10(-4). The geodetic precession test, expressed as a relative deviation from general relativity, is Kgp=-0.0019+/-0.0064. The search for a time variation in the gravitational constant results in G /G=(4+/-9) x 10(-13) yr(-1); consequently there is no evidence for local (approximately 1 AU) scale expansion of the solar system.