Although it is possible to reflect light or radio waves directly from the Moon's surface (a process known asEME), a much more precise range measurement can be made using retroreflectors, since because of their small size, the temporal spread in the reflected signal is much smaller[4] and because the return will be more evenly reflected with less diffusion.
The uncrewed SovietLunokhod 1 andLunokhod 2 rovers carried smaller arrays. Reflected signals were initially received fromLunokhod 1 by the Soviet Union up to 1974, but not by western observatories that did not have precise information about location. In 2010NASA'sLunar Reconnaissance Orbiter located the Lunokhod 1 rover on images and in April 2010 a team from University of California ranged the array.[10]Lunokhod 2's array continues to return signals to Earth.[11] The Lunokhod arrays suffer from decreased performance in direct sunlight—a factor considered in reflector placement during the Apollo missions.[12]
The Apollo 15 array is three times the size of the arrays left by the two earlier Apollo missions. Its size made it the target of three-quarters of the sample measurements taken in the first 25 years of the experiment. Improvements in technology since then have resulted in greater use of the smaller arrays, by sites such as theCôte d'Azur Observatory inNice, France; and theApache Point Observatory Lunar Laser-ranging Operation (APOLLO) at theApache Point Observatory inNew Mexico.
In the 2010s severalnew retroreflectors were planned. TheMoonLIGHT reflector, which was to be placed by the privateMX-1E lander, was designed to increase measurement accuracy up to 100 times over existing systems.[13][14][15] MX-1E was set to launch in July 2020,[16] however, as of February 2020, the launch of the MX-1E has been canceled.[17] India'sChandrayaan-3 lunar lander successfully placed a sixth reflector on the Moon in August 2023.[3]
Annotated image of the near side of the Moon showing the location of retroreflectors left on the surface by Apollo and Lunokhod missions[18]
The distance to the Moon is calculatedapproximately using the equation:distance = (speed of light ×duration of delay due to reflection) / 2. Since thespeed of light is a defined constant, conversion between distance and time of flight can be made without ambiguity.
To compute the lunar distance precisely, many factors must be considered in addition to the round-trip time of about 2.5 seconds. These factors include the location of the Moon in the sky, the relative motion of Earth and the Moon, Earth's rotation,lunar libration,polar motion,weather, speed of light in various parts of air, propagation delay throughEarth's atmosphere, the location of the observing station and its motion due tocrustal motion andtides, andrelativistic effects.[19][20] The distance continually changes for a number of reasons, but averages 385,000.6 km (239,228.3 mi) between the center of the Earth and the center of the Moon.[21] The orbits of the Moon and planets are integrated numerically along with the orientation of the Moon called physicallibration.[22]
At the Moon's surface, the beam is about 6.5 kilometers (4.0 mi) wide[23][i] and scientists liken the task of aiming the beam to using a rifle to hit a movingdime 3 kilometers (1.9 mi) away. The reflected light is too weak to see with the human eye. Out of a pulse of 3×1017 photons[24] aimed at the reflector, only about 1–5 are received back on Earth, even under good conditions.[25] They can be identified as originating from the laser because the laser is highlymonochromatic.
As of 2009, the distance to the Moon can be measured with millimeter precision.[26] In a relative sense, this is one of the most precise distance measurements ever made, and is equivalent in accuracy to determining the distance between Los Angeles and New York to within the width of a human hair.
The Lunar Laser Ranging data is collected in order to extract numerical values for a number of parameters. Analyzing the range data involves dynamics, terrestrial geophysics, and lunar geophysics. The modeling problem involves two aspects: an accurate computation of the lunar orbit and lunar orientation, and an accurate model for the time of flight from an observing station to a retroreflector and back to the station. Modern Lunar Laser Ranging data can be fit with a 1 cm weighted rms residual.
The center of Earth to center of Moon distance is computed by a program that numerically integrates the lunar and planetary orbits accounting for the gravitational attraction of the Sun, planets, and a selection of asteroids.[35][22]
The same program integrates the 3-axis orientation of the Moon called physicalLibration.
Tides in the solid Earth and seasonal motion of the solid Earth with respect to itscenter of mass.
Relativistic transformation of time and space coordinates from a frame moving with the station to a frame fixed with respect to the solar system center of mass. Lorentz contraction of the Earth is part of this transformation.
Delay in the Earth's atmosphere.
Relativistic delay due to the gravity fields of the Sun, Earth, and Moon.
The position of the retroreflector accounting for orientation of the Moon and solid-body tides.
Lunar laser ranging measurement data is available from the Paris Observatory Lunar Analysis Center,[38] the International Laser Ranging Service archives,[39][40] and the active stations. Some of the findings of thislong-term experiment are:[21]
Gauge freedom plays a major role in a correct physical interpretation of the relativistic effects in the Earth-Moon system observed with LLR technique.[48]
The likelihood of anyNordtvedt effect (a hypothetical differential acceleration of the Moon and Earth towards the Sun caused by their different degrees of compactness) has been ruled out to high precision,[49][47][50] strongly supporting thestrong equivalence principle.
^During the round-trip time, an Earth observer will have moved by around1 km (depending on their latitude). This has been presented, incorrectly, as a 'disproof' of the ranging experiment, the claim being that the beam to such a small reflector cannot hit such a moving target. However the size of the beam is far larger than any movement, especially for the returned beam.
^Chapront, J.; Chapront-Touzé, M.; Francou, G. (1999). "Determination of the lunar orbital and rotational parameters and of the ecliptic reference system orientation from LLR measurements and IERS data".Astronomy and Astrophysics.343:624–633.Bibcode:1999A&A...343..624C.
^Müller, Jürgen; Murphy, Thomas W.; Schreiber, Ulrich; Shelus, Peter J.; Torre, Jean-Marie; Williams, James G.; Boggs, Dale H.; Bouquillon, Sebastien; Bourgoin, Adrien; Hofmann, Franz (2019). "Lunar Laser Ranging: a tool for general relativity, lunar geophysics and Earth science".Journal of Geodesy.93 (11):2195–2210.Bibcode:2019JGeod..93.2195M.doi:10.1007/s00190-019-01296-0.ISSN1432-1394.S2CID202641440.
^abcWilliams, James G.; Dickey, Jean O. (2002).Lunar Geophysics, Geodesy, and Dynamics(PDF). 13th International Workshop on Laser Ranging. 7–11 October 2002. Washington, D. C.
