Lunar Laser Ranging Contributions to Relativity and Geodesy

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.

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 SCF_Lab (Satellite/lunar/GNSS laser ranging/altimetry and Cube/microsat Characterization Facilities Laboratory) is a brand new ISO 7 Clean Room, dedicated to design, characterization and modelling of the space segment of Satellite Laser Ranging (SLR), Lunar Laser Ranging (LLR) and Planetary Laser Ranging and Altimetry (PLRA) for industrial and scientific applications. The role of SCF_Lab is fundamental to study and characterize cube corner retroreflectors (CCRs) necessary for laser ranging. CCRs are special “mirrors” that reflect light in the same direction it came from as it was a normal flat mirror but always orthogonal to the beam no matter where the beam actually came from. It happens thanks to a triple rebound of photons, over the three faces of the corner. In SCF_Lab, thanks to an innovative thermal-vacuum-optical test (SCF-Test), we characterize and model the thermal behaviour and the optical performances of CCRs. The laboratory procedures are approved by the international organization that coordinates all laser ranging activities, the International Laser Ranging Service (ILRS). Our key experimental innovation is the concurrent measurement and modelling of the optical Far Field Diffraction Pattern (FFDP) and the temperature distribution of the SLR/LLR payload of retroreflectors under thermal conditions produced with a solar simulator. The primary target is to provide critical design and diagnostic capabilities for SLR to Galileo and other GNSS (Global Navigation Satellite System) constellations. New retroreflectors designs will be fundamental to improve GNSS orbits, increasing, this manner, accuracy, stability, and distribution of the International Terrestrial Reference Frame (ITRF), in order to provide a more precise definition of the geocenter (origin) and the scale (length unit). The SCF_Lab is also developing, validating and optimizing a next-generation LLR arrays for precision tests of GR with the MoonLIGHT-2 (Moon Laser Instrumentation for General relativity High-accuracy Tests - Phase 2) experiment.

A comparison of polar motion results from three sources [Bureau International de l’Heure (BIH), Defense Mapping Agency Hydrographic/Topographic Center (DMAHTC-Doppler), the International Polar Motion Service (IPMS)] was performed using lunar laser ranging (LLR) data. The rms errors, both of the LLR data and of the determinations of polar motion by the three services, decreased in recent times. The BIH and Doppler polar motion are comparable in quality (12 ± 6 cm for BIH values taken from August 1976 through May 1980, 14 ± 7 cm for Doppler results taken from June 1977 through May 1980). The IPMS errors were substantially larger (33 ± 7 cm for data taken from August 1976 through May 1980). All three analyses give 13 ± 3 cm as an estimate for the combined LLR modeling, fitting, and instrumental error (noise) for the last four years.

Since 1969 Lunar Laser Ranging (LLR) to the Apollo Cube Corner Reflector (CCR) arrays has supplied several significant tests of gravity: Geodetic Precession, the Strong and Weak Equivalence Principle (SEP, WEP), the Parametrized Post Newtonian (PPN) parameter , the time change of the Gravitational constant (G), 1/r<sup>2</sup> deviations and new gravitational theories beyond General Relativity (GR), like the unified braneworld theory (G. Dvali et al., 2003). Now a new generation of LLR can do better using evolved laser retroreflectors, developed from tight collaboration between my institution, INFN–LNF (Istituto Nazionale di Fisica Nucleare – Laboratori Nazionali di Frascati), and Douglas Currie (University of Maryland, USA), one of the fathers of LLR. The new lunar CCR is developing and characterizing at the “Satellite/Lunar laser ranging Characterization Facility” (SCF), in Frascati, performing our new industry standard space test procedure, the “SCF-Test”; this work contains the experimental results of the SCF-Test applied to the new lunar CCR, and all the new payload developments, including the future SCF tests. The International Lunar Network (ILN) research project considers our new retroreflector as one of the possible “Core Instruments”

This paper provides an overview of the Lunar Laser Ranging (LLR) experiments. The measurement principle is explained and its theory is derived. Both contributions, the direct reflected light from retroreflectors as well as the scattered light from the lunar surface are considered. The measurement results from the Sixties until 2007 are then compared between different LLR stations and with the theoretical forecast. The very first experiment was in 1962: a laser beam was directed to the Moon and the scattered light from the lunar surface was detected. The number of received photons was in line with the theory. Then from 1969 the laser beams were directed to retroreflectors placed by Apollo astronauts and Luna space crafts. Retroreflectors are on the one hand reference points for long term measurements; on the other hand they deliver a much stronger return signal compared to the scattered return. But none of the stations could measure the expected amplification of the retroreflectors. The number of received photons remained in line with measurements to the bare surface of the Moon. Therefore either all retroreflectors have degraded such that their return signals fit to the scattered return from the lunar soil or the measurements were indeed taken to the lunar surface only.

High-repetition-rate lunar laser ranging (LLR) has great prospects and significance. We have successfully obtained the effective echo signals of all five corner-cube reflectors (CCRs) on the lunar surface by using a 100 Hz repetition rate. This method can effectively improve the detection ability but has some defects: for example, the main wave and echo signals overlap. In this paper, the frequency selection and signal overlap are theoretically analyzed. The results show that the existing target prediction accuracy can meet the requirement of a 100 Hz repetition rate LLR. In the experiment, the use of a high-repetition-rate pulse laser allowed us to obtain detailed CCR information, such as the column number of CCRs, which will prove that the effective echo signals of LLR are reflected by the CCRs. Finally, we propose to use the resolved data to calculate the precision of inner coincidence and believe the accuracy can be within a millimeter.

Lorentz symmetry violations can be parametrized by an effective field theory framework that contains both general relativity and the standard model of particle physics called the standard-model extension (SME). We present new constraints on pure gravity SME coefficients obtained by analyzing lunar laser ranging (LLR) observations. We use a new numerical lunar ephemeris computed in the SME framework and we perform a LLR data analysis using a set of 20 721 normal points covering the period of August, 1969 to December, 2013. We emphasize that linear combination of SME coefficients to which LLR data are sensitive and not the same as those fitted in previous postfit residuals analysis using LLR observations and based on theoretical grounds. We found no evidence for Lorentz violation at the level of 10^{-8} for s[over ¯]^{TX}, 10^{-12} for s[over ¯]^{XY} and s[over ¯]^{XZ}, 10^{-11} for s[over ¯]^{XX}-s[over ¯]^{YY} and s[over ¯]^{XX}+s[over ¯]^{YY}-2s[over ¯]^{ZZ}-4.5s[over ¯]^{YZ}, and 10^{-9} for s[over ¯]^{TY}+0.43s[over ¯]^{TZ}. We improve previous constraints on SME coefficient by a factor up to 5 and 800 compared to postfit residuals analysis of respectively binary pulsars and LLR observations.

The lunar laser range data give information on the rotations of the Earth and the Moon about their centres of mass, as well as on the lunar orbit. A vital question for the proposed EROLD campaign is how well the coordinates of the pole and changes in Earth rotation can be determined. We have simulated the likely outcome of EROLD for this problem using the observatories which are expected to participate. The dependence of the results on station location, lunar declination, averaging interval and, data loss due to weather and other such factors is examined. We find that Universal Time and polar motion are usually obtainable to better than measuring accuracy if the averaging interval is two days. It is very important for the Australian station to be ranging when the Moon is at northern declinations. The effect of plate motion on the results is estimated.

Additional results are presented concerning a study that consider improvements over present Earth Rotation Parameter (ERP) determination methods by directly combining observations from various space geodetic systems in one adjustment. Earlier results are extended, showing that in addition to slight improvements in accuracy substantial (a factor of three or more) improvements in precision and significant reductions in correlations between various parameters can be obtained (by combining Lunar Laser Ranging (LLR), Satellite Laser Ranging (SLR) to Lageos, and Very Long Baseline Interferometry (VLBI) data in one adjustment) as compared to results from individual systems. Smaller improvements are also seen over the weighted means of the individual system results. Although data transmission would not be significantly reduced, negligible additional computer time would be required if (standardized) normal equations were available from individual solutions. Suggestions for future work and implications for the new International Earth Rotation Service (IERS) are also presented.

Lunar laser ranging and other new space measurement techniques have now brought us the capability of determining position on the Earth and the Earth-Moon distance to 10 cm, spatial orientation of the Earth to a centisecond of arc and sidereal time to a millisecond (Stolz et al 1976). The sceptic may say that these figures are optimistic but clearly it is only a matter of time until even these thresholds are broken.

We present data from the Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) covering the 15 yr span from 2006 April through the end of 2020. APOLLO measures the Earth–Moon separation by recording the round-trip travel time of photons from the Apache Point Observatory to five retro-reflector arrays on the Moon. The APOLLO data set, combined with the 50 yr archive of measurements from other lunar laser ranging (LLR) stations, can be used to probe fundamental physics such as gravity and Lorentz symmetry, as well as properties of the Moon itself. We show that range measurements performed by APOLLO since 2006 have a median nightly accuracy of 1.7 mm, which is significantly better than other LLR stations.

Measurements of lunar gravity and rotation are important because they provide information of the physical state of the lunar interior. Previously only passive LLR (Lunar Laser Ranging) using CCR (corner cube reflectors) has been applied for the detailed study of lunar librations, rotation variability. As for candidate instruments for SELENE-2 (forthcoming lunar landing mission by JAXA) and future lunar missions, we propose VLBI (inverse-VLBI and differential VLBI) for gravity measurement to constrain tidal Love number, LLR and ILOM (In-situ Lunar Orientation Measurement) for libration measurements.

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.

We review recent developments and results in testing general relativity (GR) at cosmological scales. The subject has witnessed rapid growth during the last two decades with the aim of addressing the question of cosmic acceleration and the dark energy associated with it. However, with the advent of precision cosmology, it has also become a well-motivated endeavor by itself to test gravitational physics at cosmic scales. We overview cosmological probes of gravity, formalisms and parameterizations for testing deviations from GR at cosmological scales, selected modified gravity (MG) theories, gravitational screening mechanisms, and computer codes developed for these tests. We then provide summaries of recent cosmological constraints on MG parameters and selected MG models. We supplement these cosmological constraints with a summary of implications from the recent binary neutron star merger event. Next, we summarize some results on MG parameter forecasts with and without astrophysical systematics that will dominate the uncertainties. The review aims at providing an overall picture of the subject and an entry point to students and researchers interested in joining the field. It can also serve as a quick reference to recent results and constraints on testing gravity at cosmological scales.