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Tides and Earth-Moon system evolution are coupled over geological time. Tidal energy dissipation on Earth slows Earth's rotation rate, increases obliquity, lunar orbit semi-major axis and eccentricity, and decreases lunar inclination. Tidal and core-mantle boundary dissipation within the Moon decrease inclination, eccentricity and semi-major axis. Here we integrate the Earth-Moon system backwards for 4.5Ga with orbital dynamics and explicit ocean tide models that are "high-level" (i.e., not idealized). To account for uncertain plate tectonic histories, we employ Monte Carlo simulations, with tidal energy dissipation rates (normalized relative to astronomical forcing parameters) randomly selected from ocean tide simulations with modern ocean basin geometry and with 55, 116, and 252Ma reconstructed basin paleogeometries. The normalized dissipation rates depend upon basin geometry and Earth's rotation rate. Faster Earth rotation generally yields lower normalized dissipation rates. The Monte Carlo results provide a spread of possible early values for the Earth-Moon system parameters. Of consequence for ocean circulation and climate, absolute (un-normalized) ocean tidal energy dissipation rates on the early Earth may have exceeded today's rate due to a closer Moon. Prior to ∼3Ga , evolution of inclination and eccentricity is dominated by tidal and core-mantle boundary dissipation within the Moon, which yield high lunar orbit inclinations in the early Earth-Moon system. A drawback for our results is that the semi-major axis does not collapse to near-zero values at 4.5Ga, as indicated by most lunar formation models. Additional processes, missing from our current efforts, are discussed as topics for future investigation.

Only a few sites on Earth are technically equipped to carry out Lunar Laser Ranging (LLR) to retroreflector arrays on the surface of the Moon. Despite the weak signal, they have successfully provided LLR range data for about 49 years, generating about 26,000 normal points. Recent system upgrades and new observatories have made millimeter-level range accuracy achievable. Based on appropriate modeling and sophisticated data analysis, LLR is able to determine many parameters associated with Earth–Moon dynamics, involving the lunar ephemeris, lunar physics, the Moon’s interior, reference frames and Earth orientation parameters. LLR has also become one of the strongest tools for testing Einstein’s theory of general relativity in the solar system. By extending the standard solution, it is possible to solve for parameters related to gravitational physics, like the temporal variation of the gravitational constant, metric parameters as well as the strong equivalence principle, preferred-frame effects and standard-model extensions. This paper provides a review about LLR measurement and analysis. After a short historical overview, we describe the key findings of LLR, the apparatus and technologies involved, the requisite modeling techniques, some recent results and future prospects on all fronts. We expect continued improvements in LLR, maintaining its lead in contributing to science.

The analysis of range or Doppler data between sites on the Earth and Moon requires an accurate computation of the lunar orbit and detailed models of the orientation of the Earth and Moon. Models constructed to understand range and range rate can lack detail, but if they include the largest lunar orbit variations, tracking stations on a rotating Earth, and lunar sites on a synchronously rotating Moon, then they will display the largest effects for orbit elements, Earth orientation, tracking station locations, and lunar site coordinates. The range and range rate are expanded into periodic series. To understand accurate solutions, the largest periodic terms that are sensitive to various solution parameters indicate the sensitivity of data to solution parameters and the time spans needed for their determination. Conclusions include: cylindrical coordinates work well for sites on the rapidly rotating Earth, but Cartesian coordinates are more natural for the synchronously rotating Moon since the series for the three coordinate projections are distinct. For range and range rate data, daily, semimonthly, monthly, and longer periods are present. For Doppler data, the daily periods may be stronger and more useful than the long periods, particularly for terms associated with the terrestrial tracking station. Doppler data do not determine the lander coordinate toward the Earth well. Observational strategies for range and Doppler data are not identical. For all data types, one wishes a variety of hour angles, lunar declinations, times of month, and longer periods. A long span of high-quality range data can improve the lunar orbit, orientation of the Earth’s equator, and physical librations. The locations of new lunar sites or new tracking stations can be determined from shorter spans of data.

Solid body tides on the Moon vary by about ±0.1 m each month. In addition to changes in shape, the Moon's 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 Earth's 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.

The resolution and accuracy of the lunar spherical harmonic gravity field have been dramatically improved as a result of the Gravity Recovery and Interior Laboratory (GRAIL) mission. From the Primary Mission, previous harmonic gravity fields resulted in an average n = 420 surface resolution and a Bouguer spectrum to n = 330. The GRAIL Extended Mission improves the resolution due to a lower average 23 km altitude orbit. As a result, new harmonic degree 900 gravity fields (GL0900C and GL0900D) show nearly a factor of 2 improvement with an average surface resolution n = 870 and the Bouguer spectrum extended to n = 550. Since the minimum spacecraft altitude varies spatially between 3 km and 23 km, the surface resolution is variable from near n = 680 for the central farside to near n = 900 for the polar regions. These gravity fields with 0.8 million parameters are by far the highest-degree fields of any planet ever estimated with a fully dynamic least squares technique using spacecraft tracking data.

Lunar laser ranging (LLR) has made major contributions to our understanding of the Moon's internal structure and the dynamics of the Earth-Moon system. Because of the recent improvements of the ground-based laser ranging facilities, the present LLR measurement accuracy is limited by the retro-reflectors currently on the lunar surface, which are arrays of small corner-cubes. Because of lunar librations, the surfaces of these arrays do not, in general, point directly at the Earth. This effect results in a spread of arrival times, because each cube that comprises the retroreflector is at a slightly different distance from the Earth, leading to the reduced ranging accuracy. Thus, a single, wide aperture corner-cube could have a clear advantage. In addition, after nearly four decades of successful operations the retro-reflectors arrays currently on the Moon started to show performance degradation; as a result, they yield still useful, but much weaker return signals. Thus, fresh and bright instruments on the lunar surface are needed to continue precision LLR measurements. We have developed a new retro-reflector design to enable advanced LLR operations. It is based on a single, hollow corner cube with a large aperture for which preliminary thermal, mechanical, and optical design and analysis have been performed. The new instrument will be able to reach an Earth-Moon range precision of 1-mm in a single pulse while being subjected to significant thermal variations present on the lunar surface, and will have low mass to allow robotic deployment. Here we report on our design results and instrument development effort., Comment: 19 pages, 18 figures, 5 tables

The aim of this work is to combine the model of orbital and rotational motion of the Moon developed for DE430 with up-to-date astronomical, geodynamical, and geo- and selenophysical models. The parameters of the orbit and physical libration are determined in this work from lunar laser ranging (LLR) observations made at different observatories in 1970–2013. Parameters of other models are taken from solutions that were obtained independently from LLR. A new implementation of the DE430 lunar model, including the liquid core equations, was done within the EPM ephemeris. The postfit residuals of LLR observations make evident that the terrestrial models and solutions recommended by the IERS Conventions are compatible with the lunar theory. That includes: EGM2008 gravitational potential with conventional corrections and variations from solid and ocean tides; displacement of stations due to solid and ocean loading tides; and precession-nutation model. Usage of these models in the solution for LLR observations has allowed us to reduce the number of parameters to be fit. The fixed model of tidal variations of the geopotential has resulted in a lesser value of Moon’s extra eccentricity rate, as compared to the original DE430 model with two fit parameters. A mixed model of lunar gravitational potential was used, with some coefficients determined from LLR observations, and other taken from the GL660b solution obtained from the GRAIL spacecraft mission. Solutions obtain accurate positions for the ranging stations and the five retroreflectors. Station motion is derived for sites with long data spans. Dissipation is detected at the lunar fluid core-solid mantle boundary demonstrating that a fluid core is present. Tidal dissipation is strong at both Earth and Moon. Consequently, the lunar semimajor axis is expanding by 38.20 mm/yr, the tidal acceleration in mean longitude is $$-25.90 {{}^{\prime \prime }}/\mathrm{cy}^2$$ , and the eccentricity is increasing by $$1.48\times 10^{-11}$$ each year.

Existing capabilities in laser ranging, optical interferometry and metrology, in combination with precision frequency standards, atom-based quantum sensors, and drag-free technologies, are critical for the space-based tests of fundamental physics; as a result, of the recent progress in these disciplines, the entire area is poised for major advances. Thus, accurate ranging to the Moon and Mars will provide significant improvements in several gravity tests, namely the equivalence principle, geodetic precession, PPN parameters $\beta$ and $\gamma$, and possible variation of the gravitational constant $G$. Other tests will become possible with development of an optical architecture that would allow proceeding from meter to centimeter to millimeter range accuracies on interplanetary distances. Motivated by anticipated accuracy gains, we discuss the recent renaissance in lunar laser ranging and consider future relativistic gravity experiments with precision laser ranging over interplanetary distances., Comment: 14 pages, 2 figures, 1 table. To appear in the proceedings of the International Workshop "From Quantum to Cosmos: Fundamental Physics Research in Space", 21-24 May 2006, Warrenton, Virginia, USA http://physics.jpl.nasa.gov/quantum-to-cosmos/

New gravity measurements greatly improve the Moon’s preserved impact basin inventory., Observations from the Gravity Recovery and Interior Laboratory (GRAIL) mission indicate a marked change in the gravitational signature of lunar impact structures at the morphological transition, with increasing diameter, from complex craters to peak-ring basins. At crater diameters larger than ~200 km, a central positive Bouguer anomaly is seen within the innermost peak ring, and an annular negative Bouguer anomaly extends outward from this ring to the outer topographic rim crest. These observations demonstrate that basin-forming impacts remove crustal materials from within the peak ring and thicken the crust between the peak ring and the outer rim crest. A correlation between the diameter of the central Bouguer gravity high and the outer topographic ring diameter for well-preserved basins enables the identification and characterization of basins for which topographic signatures have been obscured by superposed cratering and volcanism. The GRAIL inventory of lunar basins improves upon earlier lists that differed in their totals by more than a factor of 2. The size-frequency distributions of basins on the nearside and farside hemispheres of the Moon differ substantially; the nearside hosts more basins larger than 350 km in diameter, whereas the farside has more smaller basins. Hemispherical differences in target properties, including temperature and porosity, are likely to have contributed to these different distributions. Better understanding of the factors that control basin size will help to constrain models of the original impactor population.

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 \pm 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 \pm 1.4 x 10^{-13}, limit deviations from Einstein's general relativity, and show no rate for the gravitational constant $\dot{G}/G$ with uncertainty 9 x 10^{-13} 1/yr., 8 pages, invited talk given at ``35th COSPAR Scientific Assebly,'' July 18-24, 2004, Paris, France

The perihelion of Mercury's orbit precesses due to perturbations from other solar system bodies, solar quadrupole moment (J 2), and relativistic gravitational effects that are proportional to linear combinations of the parametrized post-Newtonian parameters β and γ. The orbits and masses of the solar system bodies are quite well known, and thus the uncertainty in recovering the precession rate of Mercury's perihelion is dominated by the uncertainties in the parameters J 2, β, and γ. Separating the effects due to these parameters is challenging since the secular precession rate has a linear dependence on each parameter. Here we use an analysis of radiometric range measurements to the MESSENGER (MErcury Surface, Space ENvironment, GEochemistry, and Ranging) spacecraft in orbit about Mercury to estimate the precession of Mercury's perihelion. We show that the MESSENGER ranging data allow us to measure not only the secular precession rate of Mercury's perihelion with substantially improved accuracy, but also the periodic perturbation in the argument of perihelion sensitive to β and γ. When combined with the γ estimate from a Shapiro delay experiment from the Cassini mission, we can decouple the effects due to β and J 2 and estimate both parameters, yielding and J 2 = (2.25 ± 0.09) × 10−7. We also estimate the total precession rate of Mercury's perihelion as 575.3100 ± 0.0015''/century and provide estimated contributions and uncertainties due to various perturbing effects.

Analyses of Lunar Laser ranges Shaw a displacement in direction of the Moon's pole of rotation which indicates that strong dissipation is acting on the rotation.

The equivalence principle can be tested using accurate tracking of the Moon, planets, and interplanetary spacecraft. Tests with solar system bodies probe the dependence of the equivalence principle on self-energy. Analysis of lunar laser ranges yields the difference in the ratio of the gravitational and inertial masses for the Earth and Moon of ({-0.7}?1.5)?10-13. In conjunction with laboratory tests of the equivalence principle and spacecraft and VLBI tests of PPN ?, one derives |?-1|?0.0005. Planetary tests are feasible, in particular tests using Mars. Improvements in Doppler accuracies under development may allow tests with interplanetary spacecraft.

To determine the lunar moments of inertia (A (C − A) B and (B − A) C come from Lunar Laser Ranging (LLR) measurement of the lunar orientation Spacecraft or lunar orbit perturbations provide J2 Combining five reported J2 results gives a normalized polar moment of inertia C MR 2 = 0.3929 ± 0.0009 Solid-body tides displace the surface about 0.1 m, but can perturb the orbit of a Moon-orbiting spacecraft. The selenocentric coordinates of four lunar retroreflectors are accurately known and can serve as reference points. The orientation and orbit of the Moon are very well known for the time span of the LLR data.

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 falling towards 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) x 10^{-13} for any possible inequality in the ratios of the gravitational and inertial masses for the Earth and Moon, (m_G/m_I)_E - (m_G/m_I)_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 Equivalence Principle., 13 pages, 2 figures, 3 tables. arXiv admin note: text overlap with arXiv:gr-qc/0507083

Reference frames determined from measurements of extragalactic and solar system objects are used in interplanetary navigation, earth orientation measurements, geodesy, and astrometry. The extragalactic frame is derived solely from Very Long Baseline Interferometric (VLBI) measurements of compact radio sources, while the solar system frame is derived from a variety of range measurements to the inner planets and the moon. We present evidence that, within the spans of presently available observations, each frame exhibits approximately one milliarcsecond (mas) internal precision and is characterized by coordinate drifts which are uncertain by

Existing capabilities of laser ranging, optical interferometry, and metrology, in combination with precision frequency standards, atom-based quantum sensors, and drag-free technologies, are critical for space-based tests of fundamental physics; as a result of the recent progress in these disciplines, the entire area is poised for major advances. Thus, accurate ranging to the Moon and Mars will provide significant improvements in several gravity tests, namely the equivalence principle, geodetic precession, PPN parameters β and γ, and possible variation of the gravitational constant G. Other tests will become possible with the development of an optical architecture that allows one to proceed from meter to centimeter to millimeter range accuracies on interplanetary distances. Motivated by anticipated accuracy gains, we discuss the recent renaissance in lunar laser ranging and consider future relativistic gravity experiments with precision laser ranging over interplanetary distances.

The utility of a Global Geodetic Observing System is not limited to our home planet, but also extends to scientific studies of the planets and their moons (including our own Moon), observations beyond the solar system, and the exploration of space in general. Examples of the first two of these include Planetary Geodesy and Radio Science, where the GGOS infrastructure is a requirement for making and interpreting measurements for these sciences; we include a section on each. The exploration of space in general involves spacecraft in Earth orbit and beyond. Tracking these spacecraft from Earth depends critically on the GGOS infrastructure. In the section on inter-planetary navigation we describe the current and future requirements of GGOS for this application.

Lunar Laser Ranging (LLR), which has been carried out for more than 35 years, is used to determine many parameters within the Earth-Moon system. This includes coordinates of terrestrial ranging stations and that of lunar retro-reflectors, as well as lunar orbit, gravity field, and its tidal acceleration. LLR data analysis also performs a number of gravitational physics experiments such as test of the equivalence principle, search for time variation of the gravitational constant, and determines value of several metric gravity parameters. These gravitational physics parameters cause both secular and periodic effects on the lunar orbit that are detectable with LLR. Furthermore, LLR contributes to the determination of Earth orientation parameters (EOP) such as nutation, precession (including relativistic precession), polar motion, and UT1. The corresponding LLR EOP series is three decades long. LLR can be used for the realization of both the terrestrial and selenocentric reference frames. The realization of a dynamically defined inertial reference frame, in contrast to the kinematically realized frame of VLBI, offers new possibilities for mutual cross-checking and confirmation. Finally, LLR also investigates the processes related to the Moon’s interior dynamics.

[1] Very precise measurements of the lunar gravity field could detect a solid inner core. For synchronous rotation, the equator planes of the inner core and mantle should be tilted with respect to the ecliptic plane and precess along that plane with the same 18.6 yr period. Generally, two different tilts would cause a static inner core gravity field to appear as small periodic (27.212 d) variations in the mantle-referenced C21 and S21 coefficients. Tidal variations also contribute to the C21 variation so a much improved k2 Love number would be required. Model computations suggest that the inner core signature is likely to be very small requiring sensitive gravity measurements. In principle, a signature analogous to the Moon's should be present for other synchronous satellites with interior liquid layers and also Mercury.

A decreasing gravitational constant, G, coupled with angular momentum conservation is expected to increrase a planetary semimajor axis, a, as \dot a/a=-\dot G/G. Analysis of lunar laser ranging data strongly limits such temporal variations and constrains a local (~1 AU) scale expansion of the solar system as \dot a/a=-\dot G/G =-(4\pm9)\times10^{-13} yr^{-1}, including that due to cosmological effects.

International audience; The IAU Working Group on Precession and the Equinox looked at several solutions for replacing the precession part of the IAU 2000A precession nutation model, which is not consistent with dynamical theory. These comparisons show that the (Capitaine et al., Astron. Astrophys., 412, 2003a) precession theory, P03, is both consistent with dynamical theory and the solution most compatible with the IAU 2000A nutation model. Thus, the working group recommends the adoption of the P03 precession theory for use with the IAU 2000A nutation. The two greatest sources of uncertainty in the precession theory are the rate of change of the Earth's dynamical flattening, DeltaJ 2, and the precession rates (i.e. the constants of integration used in deriving the precession). The combined uncertainties limit the accuracy in the precession theory to approximately 2 mas cent-2.

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. Time series of the highly accurate measurements of the distance between the Earth and Moon provide unique information used to determine whether, in accordance with the Equivalence Principle (EP), both of these celestial bodies are falling towards the Sun at the same rate, despite their different masses, compositions, and gravitational self-energies. Current LLR solutions give $(-1.0 \pm 1.4) \times 10^{-13}$ for any possible inequality in the ratios of the gravitational and inertial masses for the Earth and Moon, $\Delta(M_G/M_I)$. This result, in combination with laboratory experiments on the weak equivalence principle, yields a strong equivalence principle (SEP) test of $\Delta(M_G/M_I)_{\tt SEP} = (-2.0 \pm 2.0) \times 10^{-13}$. Such an accurate result allows other tests of gravitational theories. The result of the SEP test translates into a value for the corresponding SEP violation parameter $\eta$ of $(4.4 \pm 4.5)\times10^{-4}$, where $\eta = 4\beta -\gamma -3$ and both $\gamma$ and $\beta$ are parametrized post-Newtonian (PPN) parameters. The PPN parameter $\beta$ is determined to be $\beta - 1 = (1.2 \pm 1.1) \times 10^{-4}$. Focusing on the tests of the EP, we discuss the existing data, and characterize the modeling and data analysis techniques. The robustness of the LLR solutions is demonstrated with several different approaches that are presented in the text. We emphasize that near-term improvements in the LLR ranging accuracy will further advance the research of relativistic gravity in the solar system, and, most notably, will continue to provide highly accurate tests of the Equivalence Principle., Comment: 50 pages, 18 figures, 4 tables

Accurate analysis of precision ranges to the Moon has provided several tests of gravitational theory including the Equivalence Principle, geodetic precession, parameterized post-Newtonian (PPN) parameters $\gamma$ and $\beta$, and the constancy of the gravitational constant {\it G}. Since the beginning of the experiment in 1969, the uncertainties of these tests have decreased considerably as data accuracies have improved and data time span has lengthened. We are exploring the modeling improvements necessary to proceed from cm to mm range accuracies enabled by the new Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) currently under development in New Mexico. This facility will be able to make a significant contribution to the solar system tests of fundamental and gravitational physics. In particular, the Weak and Strong Equivalence Principle tests would have a sensitivity approaching 10$^{-14}$, yielding sensitivity for the SEP violation parameter $\eta$ of $\sim 3\times 10^{-5}$, $v^2/c^2$ general relativistic effects would be tested to better than 0.1%, and measurements of the relative change in the gravitational constant, $\dot{G}/G$, would be $\sim0.1$% the inverse age of the universe. Having this expected accuracy in mind, we discusses the current techniques, methods and existing physical models used to process the LLR data. We also identify the challenges for modeling and data analysis that the LLR community faces today in order to take full advantage of the new APOLLO ranging station., Comment: 15 pages, 3 figures, talk presented at 2003 NASA/JPL Workshop on Fundamental Physics in Space, April 14-16, 2003, Oxnard, CA

On 21 July 1969, during the first manned lunar mission, Apollo 11, the first retroreflector array was placed on the moon, enabling highly accurate measurements of the Earthmoon separation by means of laser ranging. Lunar laser ranging (LLR) turns the Earthmoon system into a laboratory for a broad range of investigations, including astronomy, lunar science, gravitational physics, geodesy, and geodynamics. Contributions from LLR include the three-orders-of-magnitude improvement in accuracy in the lunar ephemeris, a several-orders-of-magnitude improvement in the measurement of the variations in the moon's rotation, and the verification of the principle of equivalence for massive bodies with unprecedented accuracy. Lunar laser ranging analysis has provided measurements of the Earth's precession, the moon's tidal acceleration, and lunar rotational dissipation. These scientific results, current technological developments, and prospects for the future are discussed here.

The scientific areas which used data from the Lunar Laser Ranging Experiment, collected from measurements to the Apollo 11, 14, and 15 and Lunakhod 2, include lunar science (i.e., studies of variations in the lunar angular orientation from that for uniform rotation, lunar tidal displacements, and the lunar mass distribution), geodynamics, astrometry, and gravitational physics. This paper argues that the placement of microwave and optical transponders on the moon would improve the accuracy of laser range measurements by nearly two orders of magnitude and would simplify the measurements. The K-band microwave transponders would be operated at the lunar base and at two remote sites on the moon surface, yielding much improved lunar libration and tidal displacement measurements. A two-wavelength laser transponder also would be operated at the lunar base, allowing accurate tropospheric propagation corrections to be made. This would introduce major improvements in measurements of the lunar orbit and of the earth's rotation, and in tests of general relativity.

Small torques by the planets can alter the direction of the Earth's rotation axis in space. These direct planetary-induced nutations have been computed using a numerical technique. 62 nutation terms with periods between 0.2 and 883 yr are presented. The maximum,nutation is 0.5 marcsec. Also given are planetary-induced rates and accelerations of precession and obliquity.

The precession and nutation of the Earth's equator arise from solar, lunar, and planetary torques on the oblate Earth. The mean lunar orbit plane is nearly coincident with the ecliptic plane. A small tilt out of the ecliptic is caused by planetary perturbations and the Earth's gravitational harmonic J(sub 2). These planetary perturbations on the lunar orbit result in torques on the oblate Earth which contribute to precession, obliquity rate, and nutation while the J(sub 2) perturbations contribute to precession and nutation. Small additional contributions to the secular rates arise from tidal effects and planetary torques on the Earth's bulge. The total correction to the obliquity rate is -0.024 sec/century, it is an observable motion in space (the much larger conventional obliquity rate is wholly from the motion of the ecliptic, not the equator), and it is not present in the IAU-adopted expressions for the orientation of the Earth's equator. The effects have generally been allowed for in past nutation theories and some precession theories. For the planetary effect, the contributions to the 18.6 yr nutation are -0.03 mas (milliarcseconds) for the in-phase Delta(psi) plus out-of-phase contributions of 0.14 mas in Delta(psi) and -0.03 mas in Delta(sub epsilon). The latter terms demonstrate that out-of-phase contributions can arise by means other than dissipation. The sum of the contributions to the precession rate is considered and the inferred value of the moment of inertia combination (C-A)/C, which is used to scale the coefficients in the nutation series, is evaluated. Using an updated value for the precession rate, the rigid body (C-A)/C =0.003 273 763 4 which, in combination with a satellite-derived J(sub 2), gives a normalized polar moment of inertia C/MR(exp 2) = 0.330 700 7. The planetary contributions to the precession and obliquity rates are not constant for long times causing accelerations in both quantities. Acceleration in precession also arises from tides and changing J(sub 2) Contributions from the improved theory, masses, ecliptic motion, and measured values of the precession rate and obliquity are combined to give expressions (polynomials in time) for precession, obliquity, and Greenwich Mean Sidereal Time.

Analysis of 5.5 years of Lageos satellite range data reveal significant residual nodal signatures: an acceleration and annual and semiannual periods. These signatures primarily reflect variations in the zonal gravitational harmonic J2 coefficient and hence the polar moment of inertia. The implied decrease of J2 = -3 x 10 to the -11th/yr is consistent with both historical observations of the nontidal acceleration of the earth's rotation and models of viscous rebound of the solid earth from the decrease in load due to the last deglaciation.

In recent years, a revolution in astronomical position measurements has been taking place with the advent of modern space techniques. These new techniques, which supplement the traditional astrometric measurements, include laser ranging to the moon and artificial satellites, very-long-baseline interferometry (VLBI) of galactic and extra-galactic radio sources and spacecraft, radio tracking of satellites, and radar-ranging and spacecraft tracking during planetary encounters. Impressive accuracies have been achieved and further improvements are forthcoming. Each technique can be expected to establish its own reference frame which is derived from observations of a particular class of objects. The celestial and terrestrial coordinate systems are related through adopted constants and definitions. Contemporary astronomy has led to the development of three principal celestial coordinate systems: the optical frame (FK4/FK5) based on positions of galactic stars; the planetary/lunar ephemeris frame based on the major celestial bodies of the solar system; and the radio frame constructed from observations of extragalactic radio sources (quasars). Each frame is rotated with respect to others; furthermore, the optical frame offset is time variable. It is important that all frames be interconnected and unified. The optical frame is being connected to the radio frame by VLBI observations of radio emitting stars. The radio frame is being tied to the ephemeris frame in several ways – one is via differential VLBI measurements between quasars and planet-orbiting spacecraft.

Tidal dissipation causes a misalignment of the tidal bulge of the Earth with the Earth-Moon direction. The bulge exerts a secular torque which expands the lunar orbit while slowing Earth rotation. Because the lunar orbit is neither circular nor coplanar with the Earth’s equator, the tide can be separated into several distinct frequency components with bands near 0, 1, and 2 cycles/day. The tidally induced secular acceleration ri of the lunar mean longitude is particularly sensitive to the dominant semi-diurnal M 2 constituent, with smaller contributions from the diurnal O 1 and semi-diurnal N 2 components.

The orbit of Asteroid 1976 AA is described, with attention given to calculations of its period and its distance from earth, both of which could be accurately and quickly determined by measuring the minor planet's position over wide ranges of hour angle on one to three nights. The geometry of the asteroid's orbit is compared to that of earth's orbit, and the periodicity of the minor planet's approaches to earth is projected. The motion of 1976 AA over an interval of seven centuries into both past and future is also studied; the possibility of its libration with respect to earth or to Venus is examined. Some data on closest approaches of the asteroid to Mars and Venus, as well as to earth, are given.

An analysis of six years of lunar-laser-ranging data gives a zero amplitude for the Nordtvedt term in the earth-moon distance yielding the Nordtvedt parameter eta = 0.00 plus or minus 0.03. Thus, earth's gravitational self-energy contributes equally, plus or minus 3%, to its inertial mass and passive gravitational mass. At the 70% confidence level this result is only consistent with the Brans-Dicke theory for omega greater than 29. We obtain the absolute value of beta - 1 less than about 0.02 to 0.05 for five-parameter parametrized post-Newtonian theories of gravitation with energy-momentum conservation.

The analysis of lunar laser ranging data requires very accurate calculations of the lunar physical librations. Libration terms are given which arise from the additive and planetary terms in the lunar theory. The large size of the recently discovered terms due to third degree gravitational harmonics will allow some of these harmonics to be measured by laser ranging to the moon. Numerical integration promises to be an effective method of calculating librations. Comparison of numerical integrations with analytic series indicates that the calculation of the series due to third and fourth degree harmonics is not yet as accurate as the more extensively developed second degree terms.

Lunar laser ranging (LLR) data are obtained on the basis of the timing of laser pulses travelling from observatories on earth to retroreflectors placed on the moon's surface during the Apollo program. The modeling and analysis of the LLR data can provide valuable insights into earth's dynamics. The feasibility to model accurately the lunar orbit over the full 13-year observation span makes it possible to conduct relatively long-term studies of variations in the earth's rotation. A description is provided of general analysis techniques, and the calculation of universal time (UT1) from LLR is discussed. Attention is also given to a summary of intercomparisons with different techniques, polar motion results and intercomparisons, and a polar motion error analysis.

The orbital evolution of (1685) Toro has been investigated over 5000 yr. The asteroid exhibits a locked resonance through an 8:5 commensurability with the earth for the first 3400 yr of integration and a locked resonance with Venus through a 13:5 commensurability for the last 1000 yr. Control of these resonances passes from the earth to Venus as the distance from the earth's orbit gets greater and the distance to Venus' orbit gets closer as a consequence of the precession of the argument of perihelion. The double resonance may give Toro the ability to avoid collisions with the earth and Venus at the present time but this is still uncertain. Mars perturbations could remove Toro from the deepest part of the resonance in about 3,000,000 yrs.