| Lunar distance | |
|---|---|
 A lunar distance, 384,399 km (238,854 mi), is the Moon's average distance to Earth. The actual distance varies over the course ofits orbit. The image compares the Moon'sapparent size when it isnearest and farthest from Earth. | |
| General information | |
| Unit system | astronomy |
| Unit of | distance |
| Symbol | LD, |
| Conversions | |
| 1 LDin ... | ... is equal to ... |
| SI base unit | =3.84399×108 m |
| Metric system | =384399 km |
| English units | ≈238854.465 miles |
| Astronomical unit | ≈ 1/389.1734 (0.002569 au) |
| Lightsecond | ≈1.282217 ls |
The instantaneousEarth–Moon distance, ordistance to the Moon, is the distance from the center ofEarth to the center of theMoon. In contrast, theLunar distance (LD or), orEarth–Moon characteristic distance, is aunit of measure inastronomy. More technically, it is thesemi-major axis of the geocentriclunar orbit. The average lunar distance is approximately 385,000 km (239,000 mi), or 1.3light-seconds.[1] It is roughly 30 timesEarth's diameter[2] and a non-stop plane flight traveling that distance would take more than two weeks.[3] Around 389 lunar distances make up anastronomical unit (roughly the distance from Earth to the Sun).
Lunar distance is commonly used to express the distance tonear-Earth object encounters.[4] Lunar semi-major axis is an important astronomical datum. It has implications for testing gravitational theories such asgeneral relativity[5] and for refining other astronomical values, such as themass,[6]radius,[7] androtation of Earth.[8] The measurement is also useful in measuring thelunar 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 light to travel between stations on Earth andretroreflectors placed on the Moon. The precision of the range measurements determines the semi-major axis to a few decimeters. The Moon is spiraling away from Earth at an average rate of 3.8 cm (1.5 in) per year, as detected by theLunar Laser Ranging experiment.[9][10][11]
Because of 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 asemi-major axis.Ernest William Brown provided a formula for theparallax of the Moon as viewed from opposite sides of the Earth, involvingtrigonometric terms. This is equivalent to a formula for the inverse of the distance, and the average value of this is the inverse of 384,399 km (238,854 mi).[12][13] 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 is 383,397 km.[14]
The actual distance varies over the course of theorbit 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:[15]
| Unit | Mean value | Uncertainty |
|---|---|---|
| meter | 3.84399×108 | 1.1 mm[16] |
| kilometer | 384,399 | 1.1 mm[16] |
| mile | 238,854 | 0.043 in[16] |
| Earth radius | 60.32[17] | |
| AU | 1/388.6 =0.00257 | |
| light-second | 1.282 | 37.5×10−12[16] |
The instantaneous lunar distance is constantly changing. The actual distance between the Moon and Earth can change as quickly as75 meters per second,[23] or more than 1,000 km (620 mi) in just 6 hours, due to its non-circular orbit.[24] There are other effects that also influence the lunar distance. Some factors are listed in the sections below.
The distance to the Moon can be measured to an accuracy of2 mm over a 1-hour sampling period,[25] which results in an overall uncertainty of a decimeter for the semi-major axis. However, due toits elliptical orbit with varying eccentricity, the instantaneous distance varies with monthly periodicity. Furthermore, 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.[26][27] The effect ofradiation pressure from the Sun contributes an amount of ±3.6 mm to the lunar distance.[25]
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[28] and the lunar distance can be accurately modeled over thousands of years.[26]
Through the action oftidal forces, theangular momentum of Earth's rotation is slowly being transferred to the Moon's orbit.[29] The result is that Earth's rate of spin is gradually decreasing (at a rate of2.4 milliseconds/century),[30][31][32][33] and the lunar orbit is gradually expanding. The rate of recession is3.830±0.008 cm per year.[28][31] However, it is believed that this rate has recently increased, as a rate of3.8 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.[34] It is also believed that this anomalously high rate of recession may continue to accelerate.[35]
Theoretically, the lunar distance will continue to increase until the Earth and Moon becometidally locked, as are Pluto andCharon. 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,[36] which is significantly longer than theexpected 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 theCampanian era (80 million years ago) show that there were 372 days (of 23 h 33 min) per year during that time, which implies that the lunar distance was about 60.05 R🜨 (383,000 km or 238,000 mi).[29] There is geological evidence that the average lunar distance was about 52 R🜨 (332,000 km or 205,000 mi) during thePrecambrian Era; 2500 million yearsBP.[34]
The widely acceptedgiant impact hypothesis states that the Moon was created as a result of a catastrophic impact between Earth and another planet, resulting in a re-accumulation of fragments at an initial distance of 3.8 R🜨 (24,000 km or 15,000 mi).[37] This theory assumes the initial impact to have occurred 4.5 billion years ago.[38]
Until the late 1950s most measurements of lunar distance were based onoptical angular measurements: the earliest accurate measurement was byAristarchus of Samos, and laterHipparchus in the 2nd century BC. The space age marked 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.[39]
Some historically significant or otherwise interesting methods of determining the lunar distance:
The earliest account of attempts to measure the lunar distance using an eclipse were by Greek astronomer and mathematician Aristarchus in 270 BC.[40] He exploited observations of alunar eclipse combined with knowledge of Earth's radius and an understanding that the Sun is much further than the Moon. By observing the duration of an eclipse, which is about 4 hours, and comparing that to the orbital period of the moon (28 days), the circumference of the moon's orbit was determined.[41]
Later, in 129 BC,Hipparchus performed a calculation based on observing a solar eclipse from two separate locations. In one location, the eclipse was complete, but in another, the sun was partially visible. Usingtrigonometry, his calculations produced a result of 62-73 R🜨.[42] This method later found its way into the work ofPtolemy,[43] who produced a result of64+1⁄6 R🜨 (409000 km or253000 mi) at its farthest point.[44]
Early methods involved measuring the angle between 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, occultation, or the moment when the Moon crossed the meridian (if the observers shared the same longitude). This measurement technique is known aslunar parallax.
For increased accuracy, the measured angle can be adjusted to account for refraction and distortion of light passing through the atmosphere.
An expedition by French astronomerA.C.D. Crommelin observed lunarmeridian 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 the local meridian, from stations atGreenwich and atCape of Good Hope.[45] A distance was calculated with an uncertainty of30 km, and this remained the definitive lunar distance value for the next half century.
By recording the instant when the Moonoccults 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 the measurements are taken from multiple locations of known separation.
AstronomersO'Keefe and Anderson calculated the lunar distance by observing four occultations from nine locations in 1952.[46] They calculated a semi-major axis of384407.6±4.7 km (238,859.8 ± 2.9 mi). This value was refined in 1962 byIrene Fischer, who incorporated updatedgeodetic data to produce a value of384403.7±2 km (238,857.4 ± 1 mi).[7]
The distance to the moon was directly measured by means of radar first in 1946 as part ofProject Diana.[48]
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 lasting2 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 was detected and the delay time measured. From that measurement, the distance could be calculated. In practice, however, thesignal-to-noise ratio was so low that an accurate measurement could not be reliably produced.[49]
The experiment was repeated in 1958 at theRoyal Radar Establishment, in England. Radar pulses lasting5 μs were transmitted with a peak power of 2 megawatts, at 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).[50]
These initial experiments were intended to be proof-of-concept experiments and only lasted one day. Follow-on experiments lasting one month produced a semi-major axis of384402±1.2 km (238,856 ± 0.75 mi),[51] which was the most precise measurement of the lunar distance at the time.
An experiment which measured the round-triptime of flight of laser pulses reflected directly off the surface of the Moon was performed in 1962, by a team fromMassachusetts Institute of Technology, and a Soviet team at theCrimean Astrophysical Observatory.[52]
During the Apollo missions in 1969, astronauts placedretroreflectors 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 theLunar Laser Ranging experiments can achieve small millimeter resolution, and is the most reliable method of determining the lunar distance. The semi-major axis is determined to be 384,399.0 km.[13]
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 anear-occultation ofRegulus were taken from two locations, in Greece and England. By measuring theparallax between the Moon and the chosen background star, the lunar distance was calculated.[53]
A more ambitious project called the "Aristarchus Campaign" was conducted during thelunar eclipse of 15 April 2014.[24] During this event, participants were invited to record a series of five digital photographs from moonrise untilculmination (the point of greatest altitude).
The method took advantage of 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 theMoon illusion. The reason for the difference in distance is that the distance from the center of the Moon to the center of the Earth 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 was 60.61 R🜨, which implied a3% 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:
The collection of tables that describe the moon's position is called lunarephemeris. Modern methods compute the ephemeris using equations which accommodate for the known perturbation effects. These include gravitational forces of the Earth, Sun, and other planets, and also minor variation due to tidal forces, relativistic effects, and changes within the solar system.[54]
The formula for ephemerisELP2000, by Chapront and Touzé for the distance in kilometers begins with the terms:[12]
Where is themean anomaly (more or less how moon has moved from perigee) and is the meanelongation (more or less how far it has moved from conjunction with the Sun at new moon). They can be calculated from
GM= 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 (seeEpoch (astronomy)).This shows that the smallest perigee occurs at either new moon or full moon (ca356870 km), as does the greatest apogee (ca406079 km), whereas the greatest perigee will be around half-moon (ca370180 km), as will be the smallest apogee (ca404593 km). The exact values will be slightly different due to other terms. Twice in everyfull moon cycle of about 411 days there will be a minimal perigee and a maximal apogee, separated by two weeks, and a maximal perigee and a minimal apogee, also separated by two weeks.
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