The instantaneous Earth–Moon distance, or distance to the Moon, is the distance from the center of Earth to the center of the Moon. In contrast, the Lunar distance (LD or <math display="inline"> \Delta_{\oplus L}</math>), or Earth–Moon characteristic distance, is a unit of measure in astronomy. More technically, it is the semi-major axis of the geocentric lunar orbit. The average lunar distance is approximately , or 1.3 light-seconds. It is roughly 30 times Earth's diameter; to cover the same distance on earth would take a non-stop airplane flight more than two weeks. Around 389 lunar distances make up an astronomical unit (roughly the distance from Earth to the Sun).
Lunar distance is commonly used to express the distance to near-Earth object encounters. Lunar semi-major axis is an important astronomical datum. It has implications for testing gravitational theories such as general relativity and for refining other astronomical values, such as the mass, radius, and rotation of Earth. The measurement is also useful in measuring 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 light to travel between stations on Earth and retroreflectors 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 per year, as detected by the Lunar Laser Ranging experiment.
Value
thumb|left|upright=1.28|Variation of the distance between the centers of the [[Moon and the Earth over 700 days]]
thumb|400px|Distribution of perigee and apogee between 3000 BC and AD 3000<!-- Should consider replacing BC/AD with BCE/CE or other notations? -->
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 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 the average value of this is the inverse of . 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 . 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.
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:
- greatest distance: 406 719.97 km on January 7, AD 2266
- smallest distance: 356 352.93 km on November 13, 1054 BC
{|class="wikitable"
|+ Lunar distance expressed in selected units
|-
! Unit
! Mean value
! Uncertainty
|-
|meter
|
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|kilometer
|384,399
|
|-
|mile
|
|
|-
|Earth radius
|60.32
|
|-
|AU
|
|
|-
|light-second
|
|
|}
- An AU is Lunar distances.
- A lightyear is 24,611,700 Lunar distances.
- Geostationary Earth Orbit is from Earth center, or LD LD (or LDEO)
thumb|right|upright=4|Distance between the [[Earth and Moon (mean value) – sizes and distance to scale – with travel time at speed of light animated]]
thumb|right|upright=4|Photo of [[Earth and Moon, taken by the OSIRIS-REx probe]]
upright=4|thumb|right|Lunar distance to scale, with [[Medium Earth orbital (MEO) space as pink area for reference]]
Variation
The instantaneous lunar distance is constantly changing. The actual distance between the Moon and Earth can change as quickly as , or more than in just 6 hours, due to its non-circular orbit. There are other effects that also influence the lunar distance. Some factors are listed in the sections below.
thumb|350px|Lunar distance when perigee is at [[Syzygy (astronomy)|syzygy (full moon or new moon, in blue) or at half moon (red). The horizontal lines (extending exactly half a mean anomalistic month to each side of perigee) are the respective averages over one mean anomalistic month, and are almost identical.]]
Perturbations and eccentricity
The distance to the Moon can be measured to an accuracy of 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, 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 ± to the lunar distance.
Although the instantaneous uncertainty is a few millimeters, the measured lunar distance can change by more than 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.
thumb|upright=3|left|The Moon's distance from the [[Earth and Moon phases in 2014.<br />Moon phases: 0 (1) – new moon, 0.25 – first quarter, 0.5 – full moon, 0.75 – last quarter.]]
Tidal dissipation
Through the action of tidal forces, the angular momentum of Earth's rotation 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 ), and the lunar orbit is gradually expanding. The rate of recession is . There is geological evidence that the average lunar distance was about (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 a re-accumulation of fragments at an initial distance of (24,000 km or 15,000 mi). This theory assumes the initial impact to have occurred 4.5 billion years ago.
History of measurement
Until the late 1950s most measurements of lunar distance were based on optical angular measurements: the earliest accurate measurement was by Aristarchus of Samos, and later Hipparchus 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.
Some historically significant or otherwise interesting methods of determining the lunar distance:
Eclipses
The earliest account of attempts to measure the lunar distance using an eclipse were by Greek astronomer and mathematician Aristarchus in 270 BC. He 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 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.
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. Using trigonometry, his calculations produced a result of . This method later found its way into the work of Ptolemy, who produced a result of ( or ) at its farthest point.
Parallax
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 as lunar parallax.
For increased accuracy, the measured angle can be adjusted to account for refraction and distortion of light passing through the atmosphere.
Meridian crossing
An expedition by French astronomer A.C.D. Crommelin observed lunar meridian transits on the same night from two 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 at Greenwich and at Cape of Good Hope. A distance was calculated with an uncertainty of , and this remained the definitive lunar distance value for the next half century.
Occultations
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 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 ( ± ). This value was refined in 1962 by Irene Fischer, who incorporated updated geodetic data to produce a value of ( ± ).
Radar
thumb|Oscilloscope display showing the radar signal. The large pulse on the left is the transmitted signal, the small pulse on the right is the return signal from the Moon. The horizontal axis is time, but is calibrated in miles. It can be seen that the measured range is , approximately the distance from the Earth to the Moon.
The distance to the moon was directly 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 were broadcast from a 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, 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 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 .
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 of ( ± ), which was the most precise measurement of the lunar distance at the time.
Laser ranging
thumb|Lunar Laser Ranging Experiment from the Apollo 11 mission
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 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 is determined to be 384,399.0 km.
The formula for ephemeris ELP2000, by Chapront and Touzé for the distance in kilometers begins with the terms:
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External links
- Wolfram Alpha widget – Current Moon Earth distance