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56-555: Lunar Laser Ranging (LLR) is the practice of measuring the distance between the surfaces of the Earth and the Moon using laser ranging . The distance can be calculated from the round-trip time of laser light pulses travelling at the speed of light , which are reflected back to Earth by the Moon's surface or by one of several retroreflectors installed on the Moon. Three were placed by

112-699: A 1 cm weighted rms residual. The range model includes For the terrestrial model, the IERS Conventions (2010) is a source of detailed information. Lunar laser ranging measurement data is available from the Paris Observatory Lunar Analysis Center, the International Laser Ranging Service archives, and the active stations. Some of the findings of this long-term experiment are: Lunar distance (astronomy) Lunar distance

168-441: A lunar eclipse, the lunar distance can be calculated using trigonometry . The earliest accounts of attempts to measure the lunar distance using this technique were by Greek astronomer and mathematician Aristarchus of Samos in the 4th century BC and later by Hipparchus , whose calculations produced a result of 59–67  R 🜨 ( 376 000 –427 000  km or 233 000 –265 000  mi ). This method later found its way into

224-438: A maximal apogee, separated by two weeks, and a maximal perigee and a minimal apogee, also separated by two weeks. The distance to the Moon can be measured to an accuracy of 2 mm over a 1-hour sampling period, which results in an overall uncertainty of a decimeter for the semi-major axis. However, due to its elliptical orbit with varying eccentricity, the instantaneous distance varies with monthly periodicity. Furthermore,

280-401: A re-accumulation of fragments at an initial distance of 3.8  R 🜨 (24,000 km or 15,000 mi). This theory assumes the initial impact to have occurred 4.5 billion years ago. Until the late 1950s all measurements of lunar distance were based on optical angular measurements : the earliest accurate measurement was by Hipparchus in the 2nd century BC. The space age marked

336-543: A repetition rate of 260 pulses per second. After the radio waves echoed off the surface of the Moon, the return signal was detected and the delay time measured. Multiple signals were added together to obtain a reliable signal by superimposing oscilloscope traces onto photographic film. From the measurements, the distance was calculated with an uncertainty of 1.25 km (0.777 mi). These initial experiments were intended to be proof-of-concept experiments and only lasted one day. Follow-on experiments lasting one month produced

392-511: A semi-major axis of 384 402 ± 1.2 km (238,856 ± 0.75 mi), which was the most precise measurement of the lunar distance at the time. An experiment which measured the round-trip time of flight of laser pulses reflected directly off the surface of the Moon was performed in 1962, by a team from Massachusetts Institute of Technology , and a Soviet team at the Crimean Astrophysical Observatory . During

448-411: A turning point when the precision of this value was much improved. During the 1950s and 1960s, there were experiments using radar, lasers, and spacecraft, conducted with the benefit of computer processing and modeling. Some historically significant or otherwise interesting methods of determining the lunar distance: The oldest method of determining the lunar distance involved measuring the angle between

504-399: Is 383,397 km. The actual distance varies over the course of the orbit of the Moon . Values at closest approach ( perigee ) or at farthest ( apogee ) are rarer the more extreme they are. The graph at right shows the distribution of perigee and apogee over six thousand years. Jean Meeus gives the following extreme values for 1500 BC to AD 8000: The instantaneous lunar distance

560-628: Is also believed that this anomalously high rate of recession may continue to accelerate. Theoretically, the lunar distance will continue to increase until the Earth and Moon become tidally locked , as are Pluto and Charon . This would occur when the duration of the lunar orbital period equals the rotational period of Earth, which is estimated to be 47 Earth days. The two bodies would then be at equilibrium, and no further rotational energy would be exchanged. However, models predict that 50 billion years would be required to achieve this configuration, which

616-645: Is also used by the Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) lunar-ranging project. The APOLLO laser has been operational since October 2005, and routinely accomplishes millimeter-level range accuracy between the Earth and the Moon. Observations using the 3.5 m telescope can be carried out remotely by observers using TUI, the Telescope User Interface, via the internet. The SDSS 2.5 m (98 in) telescope

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672-566: Is also useful in expressing the lunar radius , as well as the distance to the Sun . Millimeter- precision measurements of the lunar distance are made by measuring the time taken for laser beam light to travel between stations on Earth and retroreflectors placed on the Moon. The Moon is spiraling away from Earth at an average rate of 3.8 cm (1.5 in) per year, as detected by the Lunar Laser Ranging experiment . Because of

728-615: Is an astronomical observatory located in the Sacramento Mountains in Sunspot , New Mexico , United States, approximately 18 miles (29 km) south of Cloudcroft . The observatory is operated by New Mexico State University (NMSU) and owned by the Astrophysical Research Consortium (ARC). Access to the telescopes and buildings is private and restricted. The ARC was formed in 1984 with

784-440: 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

840-445: Is commonly used to express the distance to near-Earth object encounters. Lunar semi-major axis is an important astronomical datum; the few-millimeter precision of the range measurements determines semi-major axis to a few decimeters; it has implications for testing gravitational theories such as general relativity and and for refining other astronomical values, such as the mass , radius , and rotation of Earth. The measurement

896-455: Is constantly changing. The actual distance between the Moon and Earth can change as quickly as 75 meters per second , or more than 1,000 km (620 mi) in just 6 hours, due to its non-circular orbit. There are other effects that also influence the lunar distance. Some factors include: The formula of Chapront and Touzé for the distance in kilometres begins with the terms: where G M {\displaystyle G_{M}}

952-525: Is determined to be 384,399.0 km. Due to the modern accessibility of accurate timing devices, high resolution digital cameras, GPS receivers, powerful computers and near-instantaneous communication, it has become possible for amateur astronomers to make high accuracy measurements of the lunar distance. On May 23, 2007, digital photographs of the Moon during a near-occultation of Regulus were taken from two locations, in Greece and England. By measuring

1008-404: Is equivalent in accuracy to determining the distance between Los Angeles and New York to within the width of a human hair. The table below presents a list of active and inactive Lunar Laser Ranging stations on Earth. 1985–2013 532 nm, 200 ps, 150 mJ 1986–2010 2010–present (2021) 532 nm, 70 ps, 75 mJ 532/1064 nm 2021–present (2023) The Lunar Laser Ranging data

1064-403: Is known as lunar parallax . For increased accuracy, the measured angle can be adjusted to account for refraction and distortion of light passing through the atmosphere. Early attempts to measure the distance to the Moon exploited observations of a lunar eclipse combined with knowledge of Earth's radius and an understanding that the Sun is much further than the Moon. By observing the geometry of

1120-485: Is nearly constant throughout the night, but an observer on the surface of Earth is actually 1 Earth radius from the center of Earth. This offset brings them closest to the Moon when it is overhead. Modern cameras have achieved a resolution capable of capturing the Moon with enough precision to detect and measure this tiny variation in apparent size. The results of this experiment were calculated as LD = 60.51 +3.91 −4.19   R 🜨 . The accepted value for that night

1176-525: Is significantly longer than the expected lifetime of the Solar System . Laser measurements show that the average lunar distance is increasing, which implies that the Moon was closer in the past, and that Earth's days were shorter. Fossil studies of mollusk shells from the Campanian era (80 million years ago) show that there were 372 days (of 23 h 33 min) per year during that time, which implies that

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1232-502: Is slowly being transferred to the Moon's orbit. The result is that Earth's rate of spin is gradually decreasing (at a rate of 2.4 milliseconds/century ), and the lunar orbit is gradually expanding. The rate of recession is 3.830 ± 0.008 cm per year . However, it is believed that this rate has recently increased, as a rate of 3.8 cm/year would imply that the Moon is only 1.5 billion years old, whereas scientific consensus supports an age of about 4 billion years. It

1288-455: Is the mean anomaly (more or less how moon has moved from perigee) and D {\displaystyle D} is the mean elongation (more or less how far it has moved from conjunction with the Sun at new moon). They can be calculated from G M = 134.963 411 38° + 13.064 992 953 630°/d · t D = 297.850 204 20° + 12.190 749 117 502°/d · t where t is the time (in days) since January 1, 2000 (see Epoch (astronomy) ). This shows that

1344-434: Is too weak to see with the human eye. Out of a pulse of 3×10 photons aimed at the reflector, only about 1–5 are received back on Earth, even under good conditions. They can be identified as originating from the laser because the laser is highly monochromatic . As of 2009, the distance to the Moon can be measured with millimeter precision. In a relative sense, this is one of the most precise distance measurements ever made, and

1400-480: Is used for the Sloan Digital Sky Survey , and began operating in 2000. It is a Ritchey-Chretien reflector on an alt-azimuth mount housed under a roll-off enclosure. It was designed with an unusually large 3° field of view to better support its primary task of surveying the entire sky. The NMSU 1.0 m (39 in) telescope is a Ritchey-Chretien reflector set on an alt-azimuth mount, and

1456-750: The Pic du Midi Observatory in France, the Tokyo Astronomical Observatory , and McDonald Observatory in Texas soon followed. The uncrewed Soviet Lunokhod 1 and Lunokhod 2 rovers carried smaller arrays. Reflected signals were initially received from Lunokhod 1 by the Soviet Union up to 1974, but not by western observatories that did not have precise information about location. In 2010 NASA 's Lunar Reconnaissance Orbiter located

1512-411: The parallax between the Moon and the chosen background star, the lunar distance was calculated. A more ambitious project called the "Aristarchus Campaign" was conducted during the lunar eclipse of 15 April 2014. During this event, participants were invited to record a series of five digital photographs from moonrise until culmination (the point of greatest altitude). The method took advantage of

1568-440: The 3.5 m and 0.5 m telescopes comes from the consortium members, but funds for the 2.5 m telescope come from a much wider array of sources. The 1.0 m telescope is supported exclusively by NMSU. The ARC 3.5 m (140 in) telescope is a Ritchey-Chretien reflector on an alt-azimuth mount with instruments attached at several focal points . Construction of the building began in 1985, but full operations of

1624-483: The Apollo missions in 1969, astronauts placed retroreflectors on the surface of the Moon for the purpose of refining the accuracy and precision of this technique. The measurements are ongoing and involve multiple laser facilities. The instantaneous precision of the Lunar Laser Ranging experiments can achieve small millimeter resolution, and is the most reliable method of determining the lunar distance. The semi-major axis

1680-538: The Lunokhod 1 rover on images and in April 2010 a team from University of California ranged the array. Lunokhod 2 's array continues to return signals to Earth. The Lunokhod arrays suffer from decreased performance in direct sunlight—a factor considered in reflector placement during the Apollo missions. The Apollo 15 array is three times the size of the arrays left by the two earlier Apollo missions. Its size made it

1736-443: The Moon and a chosen reference point from multiple locations, simultaneously. The synchronization can be coordinated by making measurements at a pre-determined time, or during an event which is observable to all parties. Before accurate mechanical chronometers, the synchronization event was typically a lunar eclipse , or the moment when the Moon crossed the meridian (if the observers shared the same longitude). This measurement technique

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1792-430: The Moon is calculated approximately using the equation: distance = ( speed of light × duration of delay due to reflection ) / 2 . Since the speed 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

1848-421: The Moon's surface using a laser with a 50J 0.5 millisecond pulse length. Similar measurements were obtained later the same year by a Soviet team at the Crimean Astrophysical Observatory using a Q-switched ruby laser . Shortly thereafter, Princeton University graduate student James Faller proposed placing optical reflectors on the Moon to improve the accuracy of the measurements. This was achieved following

1904-528: The United States' Apollo program ( 11 , 14 , and 15 ), two by the Soviet Lunokhod 1 and 2 missions , and one by India's Chandrayaan-3 mission. Although it is possible to reflect light or radio waves directly from the Moon's surface (a process known as EME ), a much more precise range measurement can be made using retroreflectors, since because of their small size, the temporal spread in

1960-475: The average value of this is the inverse of 384,399 km (238,854 mi). On the other hand, the time-averaged distance (rather than the inverse of the average inverse distance) between the centers of Earth and the Moon is 385,000.6 km (239,228.3 mi). One can also model the orbit as an ellipse that is constantly changing, and in this case one can find a formula for the semi-major axis, again involving trigonometric terms. The average value by this method

2016-400: The center of the Earth and the center of the Moon. The orbits of the Moon and planets are integrated numerically along with the orientation of the Moon called physical libration . At the Moon's surface, the beam is about 6.5 kilometers (4.0 mi) wide and scientists liken the task of aiming the beam to using a rifle to hit a moving dime 3 kilometers (1.9 mi) away. The reflected light

2072-413: The distance is perturbed by the gravitational effects of various astronomical bodies – most significantly the Sun and less so Venus and Jupiter. Other forces responsible for minute perturbations are: gravitational attraction to other planets in the Solar System and to asteroids; tidal forces; and relativistic effects. The effect of radiation pressure from the Sun contributes an amount of ± 3.6 mm to

2128-403: The fact that the Moon is actually closest to an observer when it is at its highest point in the sky, compared to when it is on the horizon. Although it appears that the Moon is biggest when it is near the horizon, the opposite is true. This phenomenon is known as the Moon illusion . The reason for the difference in distance is that the distance from the center of the Moon to the center of the Earth

2184-533: The goal of building the 3.5 m telescope. It originally consisted of five institutions: New Mexico State University , University of Washington , University of Chicago , Princeton University , and Washington State University , some of which have since withdrawn. Several additional organizations have joined over time: Johns Hopkins University , University of Colorado , University of Virginia , Georgia State University , University of Oklahoma , University of Wyoming , and Brigham Young University . Funding for

2240-406: The influence of the Sun and other perturbations, the Moon's orbit around the Earth is not a precise ellipse. Nevertheless, different methods have been used to define a semi-major axis . Ernest William Brown provided a formula for the parallax of the Moon as viewed from opposite sides of the Earth, involving trigonometric terms. This is equivalent to a formula for the inverse of the distance, and

2296-536: The installation of a retroreflector array on July 21, 1969 by the crew of Apollo 11 . Two more retroreflector arrays were left by the Apollo 14 and Apollo 15 missions. Successful lunar laser range measurements to the retroreflectors were first reported on Aug. 1, 1969 by the 3.1 m telescope at Lick Observatory . Observations from Air Force Cambridge Research Laboratories Lunar Ranging Observatory in Arizona,

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2352-441: The local meridian, from stations at Greenwich and at Cape of Good Hope . A distance was calculated with an uncertainty of 30 km , and this remained the definitive lunar distance value for the next half century. By recording the instant when the Moon occults a background star, (or similarly, measuring the angle between the Moon and a background star at a predetermined moment) the lunar distance can be determined, as long as

2408-466: 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 through Earth's atmosphere , the location of the observing station and its motion due to crustal motion and tides , and relativistic effects . The distance continually changes for a number of reasons, but averages 385,000.6 km (239,228.3 mi) between

2464-510: The lunar distance was about 60.05  R 🜨 (383,000 km or 238,000 mi). There is geological evidence that the average lunar distance was about 52  R 🜨 (332,000 km or 205,000 mi) during the Precambrian Era ; 2500 million years BP . The widely accepted giant impact hypothesis states that the Moon was created as a result of a catastrophic impact between Earth and another planet, resulting in

2520-414: The lunar distance. Although the instantaneous uncertainty is a few millimeters, the measured lunar distance can change by more than 30,000 km (19,000 mi) from the mean value throughout a typical month. These perturbations are well understood and the lunar distance can be accurately modeled over thousands of years. Through the action of tidal forces , the angular momentum of Earth's rotation

2576-473: The measurements are taken from multiple locations of known separation. Astronomers O'Keefe and Anderson calculated the lunar distance by observing four occultations from nine locations in 1952. They calculated a semi-major axis of 384 407 .6 ± 4.7 km (238,859.8 ± 2.9 mi). This value was refined in 1962 by Irene Fischer , who incorporated updated geodetic data to produce a value of 384 403 .7 ± 2 km (238,857.4 ± 1 mi). The distance to

2632-472: The moon was measured by means of radar first in 1946 as part of Project Diana . Later, an experiment was conducted in 1957 at the U.S. Naval Research Laboratory that used the echo from radar signals to determine the Earth-Moon distance. Radar pulses lasting 2 μs were broadcast from a 50-foot (15 m) diameter radio dish. After the radio waves echoed off the surface of the Moon, the return signal

2688-569: The reflected signal is much smaller and because the return will be more evenly reflected with less diffusion. Laser ranging measurements can also be made with retroreflectors installed on Moon-orbiting satellites such as the LRO . The first successful lunar ranging tests were carried out in 1962 when Louis Smullin and Giorgio Fiocco from the Massachusetts Institute of Technology succeeded in observing laser pulses reflected from

2744-407: The smallest perigee occurs at either new moon or full moon (ca 356870 km), as does the greatest apogee (ca 406079 km), whereas the greatest perigee will be around half-moon (ca 370180 km), as will be the smallest apogee (ca 404593 km). The exact values will be slightly different due to other terms. Twice in every full moon cycle of about 411 days there will be a minimal perigee and

2800-709: 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 the Côte d'Azur Observatory in Nice , France; and the Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) at the Apache Point Observatory in New Mexico . In the 2010s several new retroreflectors were planned. The MoonLIGHT reflector, which

2856-404: The telescope were delayed until November 1994 due to problems with fabricating the primary mirror . From 1991 until early 1993, the telescope was fitted with a 1.8 m mirror, now located at Rothney Astrophysical Observatory under a cost-sharing agreement . There are a variety of optical and near-infrared instruments available for the 3.5 m telescope, including: The 3.5 m telescope

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2912-429: The work of Ptolemy , who produced a result of 64 + 1 ⁄ 6   R 🜨 ( 409 000  km or 253 000  mi ) at its farthest point. An expedition by French astronomer A.C.D. Crommelin observed lunar meridian transits on the same night from two different locations. Careful measurements from 1905 to 1910 measured the angle of elevation at the moment when a specific lunar crater ( Mösting A ) crossed

2968-418: Was 60.61  R 🜨 , which implied a 3% accuracy. The benefit of this method is that the only measuring equipment needed is a modern digital camera (equipped with an accurate clock, and a GPS receiver). Other experimental methods of measuring the lunar distance that can be performed by amateur astronomers involve: Apache Point Observatory The Apache Point Observatory (APO; obs. code : 705 )

3024-642: Was completed in 1994. A 2048 × 2048 CCD mounted at the Nasmyth focus provides at 15.7 arcminute view of the sky. The ARC Small Aperture Telescope (ARCSAT) was previously called the Photometric Telescope (PT) when it was part of the SDSS project. It is a 0.5 m (20 in) reflecting telescope on an equatorial mount , with a single CCD camera cooled by a CryoTiger unit. It was built in 1991, moved from its previous location in 1998, and used by

3080-532: Was detected and the delay time measured. From that measurement, the distance could be calculated. In practice, however, the signal-to-noise ratio was so low that an accurate measurement could not be reliably produced. The experiment was repeated in 1958 at the Royal Radar Establishment , in England. Radar pulses lasting 5 μs were transmitted with a peak power of 2 megawatts, at

3136-620: Was to be placed by the private MX-1E lander, was designed to increase measurement accuracy up to 100 times over existing systems. MX-1E was set to launch in July 2020, however, as of February 2020, the launch of the MX-1E has been canceled. India's Chandrayaan-3 lunar lander successfully placed a sixth reflector on the Moon in August 2023. MoonLIGHT will be launched in early 2024 with a Commercial Lunar Payload Services (CLPS) mission. The distance to

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