View source for Lunar distance

{{short description|Distance from center of Earth to center of Moon}}
{{For|the navigation unit|Lunar distance (navigation)}}
{{Infobox unit
| name          = Lunar distance
| image         = Lunar perigee apogee.png
| caption       = A lunar distance, {{cvt|384399|km|mi}}, is the Moon's average distance to Earth. The actual distance varies over the course of [[orbit of the Moon|its orbit]]. The image compares the Moon's [[angular diameter|apparent size]] when it is [[supermoon|nearest]] and farthest from Earth.
| standard      = [[astronomy]]
| quantity      = [[distance]]
| symbol        = LD
| symbol2       = <math display="inline"> \Delta_{\oplus L}</math>
| units1        = [[SI base unit]]
| inunits1      = {{val|3.84399|e=8|u=m}}
| units2        = [[Metric system]]
| inunits2      = {{val|384399|u=km}}
| units3        = [[English units]]
| inunits3      = {{val|238854|u=mile}}
| units4        = [[Astronomical unit]]
| inunits4      = 1/389 ({{val|0.002569|u=au}})
| units5        = [[Lightsecond]]
| inunits5      = {{val|1.28|u=ls}}
}}

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 [[orbit of the Moon|lunar orbit]]. The average lunar distance is approximately {{cvt|385,000|km|mi}}, or 1.28 [[light-second]]s; this is roughly 30 times [[Earth radius|Earth's diameter]]. 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.{{r|NEO}} Lunar semi-major axis is an important astronomical datum. It has implications for testing gravitational theories such as [[general relativity]] {{r|Relativity}} and for refining other astronomical values, such as the [[Earth mass|mass]],{{r|moonbounce}} [[Earth radius|radius]],{{r|Fischer}} and [[Earth's rotation|rotation]] of Earth.{{r|LLR}} 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 [[List of retroreflectors on the Moon|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 {{cvt|3.8|cm|in}} per year, as detected by the [[Lunar Laser Ranging experiment]].{{r|drift1|drift2|drift3}}

==Value==
[[File:Moon's orbit - Variation of distance en.png|thumb|left|upright=1.28|Variation of the distance between the centers of the [[Moon]] and the [[Earth]] over 700 days]]
[[File:Häufigkeitsverteilung von 79535 Perigäen und Apogäen.png|thumb|400px|Distribution of perigee and apogee between 3000 BC and AD 3000]]

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 and semi-minor axes|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 {{cvt|384399|km|mi}}.<ref name="MeeusMorsels1Ch4">Meeus: ''Mathematical Astronomy Morsels.'' Willmann-Bell, Richmond 1997, ISBN 0-943396-51-4, Kapitel 4. See [https://de.wikipedia.org/w/index.php?title=Mondbahn&oldid=237763431 German Wikipedia article] for more information.</ref><ref name=":1">{{cite conference |last1=Williams|first1=James G. |last2=Dickey|first2=Jean O. |date=2002 |title=Lunar geophysics, geodesy and dynamics |url=http://cddis.gsfc.nasa.gov/lw13/lw_proceedings.html |editor=Ron Noomen |editor2=Steven Klosko |editor3=Carey Noll |editor4=Michael Pearlman |book-title=13th International Workshop on Laser Ranging |publisher=[[Goddard Space Flight Center]]}}</ref> 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 {{cvt|385000.6|km|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&nbsp;km.<ref name=SimonEtAl>{{cite journal |last1=J. L. Simon, P. Bretagnon, J. Chapront, M. Chapront-Touze, G. Francou, J. Laskar |title=Numerical expressions for precession formulae and mean elements for the Moon and the planets |journal=Astronomy and Astrophysics |date=Feb 1994 |volume=282 |page=663 |bibcode=1994A&A...282..663S}}</ref>

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:<ref name="MeeusMorsels1Ch2">Meeus: ''Mathematical Astronomy Morsels.'' Willmann-Bell, Richmond 1997, ISBN 0-943396-51-4, Kapitel 2.</ref>

* greatest distance: 406&nbsp;719.97&nbsp;km on January 7, AD 2266
* smallest distance: 356&nbsp;352.93&nbsp;km on November 13, 1054 BC

{|class="wikitable"
|+ Lunar distance expressed in selected units
|-
! Unit
! Mean value
! Uncertainty
|-
|[[meter]]
|{{val|3.84399|e=8}}
|{{val|1.1|u=mm}}{{r|Apache Point}}
|-
|[[kilometer]]
|384,399
|{{val|1.1|u=mm}}{{r|Apache Point}}
|-
|[[mile]]
|{{convert|384399|km|mi|sigfig=6|disp=number}}
|{{convert|1.1|mm|in|sigfig=2|disp=out}}{{r|Apache Point}}
|-
|[[Earth radius]]
|60.32{{r|Dream of West}}
|
|-
|[[Astronomical unit|AU]]
|{{sfrac|388.6}} {{=}} {{val|0.00257}}
{{r|Sea's Edge|Fact Sheet}}
|
|-
|[[light-second]]
|{{val|1.282}}
|{{val|37.5e-12}}{{r|Apache Point}}
|}

*An AU is {{val|389}} Lunar distances.{{r|Groten}}
*A lightyear is 24,611,700 Lunar distances.{{r|Groten}}<ref name="IAU measuring">{{cite web|title=International Astronomical Union {{!}} IAU|url=https://www.iau.org/public/themes/measuring/|website=www.iau.org|access-date=5 May 2019}}</ref>
*[[GEO orbit|Geostationary Earth Orbit]] is {{convert|42,164|km|mi|abbr=on}} from Earth center, or {{sfrac|9.117}} LD {{=}} {{val|0.10968}} LD (or {{val|0.10968}} LDEO<ref>[https://www.kth.se/polopoly_fs/1.1024135.1603899559!/Blue%20Team%20-%20Overall%20Coordinators%20.pdf Design of a Mars Passenger Ship]</ref>)

[[File:Earth and Moon speed of light by James O'Donoghue.gif|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]]
[[File:Osiris-rexearthmoon_rotated2.jpg|thumb|right|upright=4|Photo of [[Earth]] and [[Moon]], taken by the [[OSIRIS-REx]] [[Space probe|probe]]]]
[[File:Orbits schematic edit.png|upright=4|thumb|right|Lunar distance to scale, with [[Medium Earth orbit]]al (MEO) space as pink area for reference]]
{{clear|right}}

== Variation ==
{{wide image|Moon_distance_range_to_scale.svg|800px|Minimum, mean and maximum distances of the Moon from Earth with its angular diameter as seen from Earth's surface, to scale}}
The instantaneous lunar distance is constantly changing. The actual distance between the Moon and Earth can change as quickly as {{val|75|u=meters per second}},{{r|millimeter challenge}} or more than {{convert|1000|km|mi|abbr=on}} in just 6 hours, due to its non-circular orbit.{{r|Zuluaga}} There are other effects that also influence the lunar distance. Some factors are listed in the sections below.

[[File:Lunar distance over anomalistic month.png|thumb|350px|Lunar distance when perigee is at 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 {{val|2|u=mm}} over a 1-hour sampling period,{{r|APOLLO Analysis}} which results in an overall uncertainty of a decimeter for the semi-major axis. However, due to [[Orbit of the Moon|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.{{r|integration}}<ref>{{Cite journal|last1=Park|first1=Ryan S.|last2=Folkner|first2=William M.|last3=Williams|first3=James G.|last4=Boggs|first4=Dale H.|date=2021|title=The JPL Planetary and Lunar Ephemerides DE440 and DE441|journal=The Astronomical Journal|language=en|volume=161|issue=3|pages=105|doi=10.3847/1538-3881/abd414|bibcode=2021AJ....161..105P |issn=1538-3881|doi-access=free}}</ref> The effect of [[Yarkovsky effect|radiation pressure]] from the Sun contributes an amount of ±{{val|3.6|u=mm}} to the lunar distance.{{r|APOLLO Analysis}}

Although the instantaneous uncertainty is a few millimeters, the measured lunar distance can change by more than {{convert|30,000|km|mi|abbr=on}} from the mean value throughout a typical month. These perturbations are well understood{{r|DE431}} and the lunar distance can be accurately modeled over thousands of years.{{r|integration}}

[[File:Moon-Earth distance, Moon phases.gif|thumb|upright=3|left|The Moon's distance from the [[Earth]] and [[Lunar phase|Moon phases]] in 2014.<br>Moon phases: 0 (1) – [[new moon]], 0.25 – first quarter, 0.5 – [[full moon]], 0.75 – last quarter.]]
{{clear}}

=== Tidal dissipation ===
Through the action of [[tidal forces]], the [[angular momentum]] of Earth's rotation is slowly being transferred to the Moon's orbit.{{r|paleoceanography}} The result is that Earth's rate of spin is gradually decreasing (at a rate of {{val|2.4|u=milliseconds/century}}),{{r|space.com}}<ref name=":0">{{Cite journal|last1=Williams|first1=James G.|last2=Boggs|first2=Dale H.|date=2016|title=Secular tidal changes in lunar orbit and Earth rotation|url=https://doi.org/10.1007/s10569-016-9702-3|journal=Celestial Mechanics and Dynamical Astronomy|language=en|volume=126|issue=1|pages=89–129|doi=10.1007/s10569-016-9702-3|bibcode=2016CeMDA.126...89W |s2cid=124256137|issn=1572-9478}}</ref><ref>{{Cite journal|last1=Stephenson|first1=F. R.|last2=Morrison|first2=L. V.|last3=Hohenkerk|first3=C. Y.|date=2016|title=Measurement of the Earth's rotation: 720 BC to AD 2015|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=472|issue=2196|pages=20160404|doi=10.1098/rspa.2016.0404|pmc=5247521|pmid=28119545|bibcode=2016RSPSA.47260404S }}</ref><ref>{{Cite journal|last1=Morrison|first1=L. V.|last2=Stephenson|first2=F. R.|last3=Hohenkerk|first3=C. Y.|last4=Zawilski|first4=M.|date=2021|title=Addendum 2020 to 'Measurement of the Earth's rotation: 720 BC to AD 2015|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=477|issue=2246|pages=20200776|doi=10.1098/rspa.2020.0776|bibcode=2021RSPSA.47700776M |s2cid=231938488|doi-access=free}}</ref> and the lunar orbit is gradually expanding. The rate of recession is {{val|3.830|0.008|u=cm per year}}.{{r|DE431}}<ref name=":0" /> However, it is believed that this rate has recently increased, as a rate of {{val|3.8|u=cm/yr}} would imply that the Moon is only 1.5&nbsp;billion years old, whereas scientific consensus supports an age of about 4 billion years.{{r|Precambrian}} It is also believed that this anomalously high rate of recession may continue to accelerate.{{r|Lunar Orbit Evolution}}

Theoretically, the lunar distance will continue to increase until the Earth and Moon become [[Tidal locking|tidally locked]], as are Pluto and [[Charon (moon)|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&nbsp;billion years would be required to achieve this configuration,{{r|tidal lock}} which is significantly longer than the [[Red giant#The Sun as a red giant|expected lifetime of the Solar System]].

=== Orbital history ===
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 the lunar distance was about {{Earth radius|60.05|link=yes}} (383,000&nbsp;km or 238,000&nbsp;mi).<ref name = "paleoceanography">{{cite journal|last1=Winter|first1=Niels J.|last2=Goderis|first2=Steven|last3=Van Malderen|first3=Stijn J.M.|last4=Sinnesael|first4=Matthias|last5=Vansteenberge|first5=Stef|last6=Snoeck|first6=Christophe|last7=Belza|first7=Joke|last8=Vanhaecke|first8=Frank|last9=Claeys|first9=Philippe|display-authors=3|title=Subdaily-Scale Chemical Variability in a Rudist Shell: Implications for Rudist Paleobiology and the Cretaceous Day-Night Cycle|journal=Paleoceanography and Paleoclimatology|date=18 February 2020|volume=35|issue=2|doi=10.1029/2019PA003723|doi-access=free |hdl=1854/LU-8685501|hdl-access=free}}</ref> There is geological evidence that the average lunar distance was about {{Earth radius|52}} (332,000&nbsp;km or 205,000&nbsp;mi) during the [[Precambrian|Precambrian Era]]; 2500&nbsp;million years [[Before Present|BP]].{{r|Precambrian}}

The widely accepted [[Origin of the Moon|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 {{Earth radius|3.8}} (24,000&nbsp;km or 15,000&nbsp;mi).{{r|Impact}} This theory assumes the initial impact to have occurred 4.5&nbsp;billion years ago.{{r|TheiaPothesis}}

== History of measurement ==
Until the late 1950s most measurements of lunar distance were based on [[optical angular measurement]]s: 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.{{r|DE 102}}

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.{{r|Gutzwiller}} 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.<ref>{{cite web |last1=Stern |first1=David P. |title=Estimating the Moon's Distance |url=https://pwg.gsfc.nasa.gov/stargaze/Shipprc2.htm#:~:text=Aristarchus%20around%20270%20BC%20derived,years%20later%2C%20by%20Eratosthenes%20 |website=pwg.gsfc.nasa.gov |publisher=Nasa |access-date=20 February 2025}}</ref>

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 {{Earth radius|62-73}}.<ref>{{cite web |title=How distant is the Moon? |url=https://pwg.gsfc.nasa.gov/stargaze/Shipparc.htm |website=pwg.gsfc.nasa.gov |access-date=20 February 2025}}</ref> This method later found its way into the work of [[Ptolemy]],{{r|Measuring the Universe}} who produced a result of {{Earth radius|{{frac|64|1|6}}}} ({{val|409,000|u=km}} or {{val|253,000|u=mi}}) at its farthest point.{{r|Helden}}

=== 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 [[Parallax#Lunar parallax|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 [[Andrew Claude de la Cherois Crommelin|A.C.D. Crommelin]] observed lunar [[meridian transit]]s 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 (crater)|Mösting A]]) crossed the local meridian, from stations at [[Greenwich]] and at [[Cape of Good Hope]].{{r|Commelin}} A distance was calculated with an uncertainty of {{val|30|u=km}}, and this remained the definitive lunar distance value for the next half century.

=== Occultations ===
By recording the instant when the Moon [[Occultation|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 [[John A. O'Keefe (astronomer)|O'Keefe]] and Anderson calculated the lunar distance by observing four occultations from nine locations in 1952.{{r|O'Keefe}} They calculated a semi-major axis of {{Val|384,407.6|4.7|u=km}} ({{convert|384,407.6|km|mi|sigfig=7|abbr=on|disp=number}} ± {{convert|4.7|km|mi|sigfig=2|abbr=on|disp=out}}). This value was refined in 1962 by [[Irene Fischer]], who incorporated updated [[Geodesy|geodetic]] data to produce a value of {{val|384,403.7|2|u=km}} ({{convert|384,403.7|km|mi|sigfig=7|abbr=on|disp=number}} ± {{convert|2|km|mi|sigfig=1|abbr=on|disp=out}}).{{r|Fischer}}

=== Radar ===
[[Image:First radar return from the Moon - Project Diana 1946.jpg|thumb|Oscilloscope display showing the radar signal.<ref name="Gootée">{{cite journal
  | last = Gootée
  | first = Tom
  | title = Radar reaches the moon
  | journal = Radio News
  | volume = 35
  | issue = 4
  | pages = 25–27
  | publisher = Ziff-Davis Publishing Co. 
  |date=April 1946 
  | url = http://www.americanradiohistory.com/Archive-Radio-News/40s/Radio-News-1946-04-R.pdf
  | issn = 
  | doi = 
  | bibcode = 1946RaNew..35...25G
 | id = 
  | accessdate = September 9, 2014 }}</ref> 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 {{convert|238,000|mi|km|abbr=on}}, 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]].<ref name="SciHi Blog 2022">{{cite web | title=Project Diana hits the Moon… in 1946 | website=SciHi Blog | date=2022-01-10 | url=http://scihi.org/project-diana-moon/ | access-date=2023-01-29}}</ref>

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 {{val|2|u=us}} were broadcast from a {{convert|50|ft|m|sigfig=2|adj=on}} 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.{{r|Radar1}}

The experiment was repeated in 1958 at the [[Royal Radar Establishment]], in England. Radar pulses lasting {{val|5|u=us}} were transmitted with a peak power of 2&nbsp;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 {{convert|1.25|km|mi|sigfig=3|abbr=on}}.{{r|Radar2}}

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 {{val|384,402|1.2|u=km}} ({{convert|384,402|km|mi|sigfig=6|disp=number}} ± {{convert|1.2|km|mi|sigfig=2|disp=out}}),{{r|Radar3}} which was the most precise measurement of the lunar distance at the time.

=== Laser ranging ===
[[Image:Apollo 11 Lunar Laser Ranging Experiment.jpg|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]].{{r|Lunar Laser Ranging}}

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 experiment]]s 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&nbsp;km.<ref name=":1" />

=== Amateur astronomers and citizen scientists ===
{{undue weight|date=September 2020}}
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 [[Occultation|near-occultation]] of [[Regulus]] were taken from two locations, in Greece and England. By measuring the [[parallax]] between the Moon and the chosen background star, the lunar distance was calculated.{{r|Citizen1}}

A more ambitious project called the "Aristarchus Campaign" was conducted during the [[lunar eclipse]] of 15 April 2014.{{r|Zuluaga}} 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 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 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 = {{Earth radius|{{val|60.51|3.91|4.19}}}}. The accepted value for that night was {{Earth radius|60.61}}, which implied a {{Percentage|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:
* Taking pictures of the Moon before it enters the [[Umbra, penumbra and antumbra|penumbra]] and after it is completely eclipsed.
* Measuring, as precisely as possible, the time of the eclipse contacts. 
* Taking good pictures of the partial eclipse when the shape and size of the Earth shadow are clearly visible. 
* Taking a picture of the Moon including, in the same field of view, [[Spica]] and [[Mars]] – from various locations.

== Calculations ==
The collection of tables that describe the moon's position is called lunar [[ephemeris]]. 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.<ref>{{cite journal |last1=Chapront-Touze |first1=M. |last2=Chapront |first2=J. |title=The lunar ephemeris ELP 2000 |journal=Astronomy and Astrophysics |date=1983 |volume=124 |page=50 |url=https://adsabs.harvard.edu/full/1983A%26A...124...50C |access-date=21 February 2025}}</ref>

The formula for ephemeris [[ELP2000]], by Chapront and Touzé for the distance in kilometers begins with the terms:<ref name="MeeusMorsels1Ch4" />

:<math>
\begin{alignat}{3}
\frac{d}{\mathrm{km}} = 385000.5584 & \ -\ 20905.3550  \cdot \cos(G_M)     \\
                                & \ -   \  3699.1109  \cdot \cos(2D - G_M)      \\
                                & \ -   \  2955.9676  \cdot \cos(2D)            \\
                                & \ -   \ 569.9251    \cdot \cos(2G_M)          \\
                                & \ \pm \ \dotsc
\end{alignat}
</math>
Where <math>G_M</math> is the [[mean anomaly]] (more or less how moon has moved from perigee) and <math>D</math> is the mean [[elongation (astronomy)|elongation]] (more or less how far it has moved from conjunction with the Sun at new moon). They can be calculated from

G{{sub|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 the smallest perigee occurs at either new moon or full moon (ca {{val|356870|u=km}}), as does the greatest apogee (ca {{val|406079|u=km}}), whereas the greatest perigee will be around half-moon (ca {{val|370180|u=km}}), as will be the smallest apogee (ca {{val|404593|u=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 a maximal apogee, separated by two weeks, and a maximal perigee and a minimal apogee, also separated by two weeks.

==See also==
*[[Astronomical unit]]
*[[Ephemeris]]
*[[Jet Propulsion Laboratory Development Ephemeris]]
*[[Lunar Laser Ranging experiment]]
*[[Lunar theory]]
*[[On the Sizes and Distances (Aristarchus)|''On the Sizes and Distances'' (Aristarchus)]]
*[[Orbit of the Moon]]
*[[Prutenic Tables|''Prutenic Tables'' of Erasmus Reinhold]]
*[[Supermoon]]

== References ==
{{reflist|refs=

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<ref name="APOLLO Analysis">{{cite arXiv|last1=Reasenberg|first1=R.D.|last2=Chandler|first2=J.F.|last3=Colmenares|first3=N.R.|last4=Johnson|first4=N.H.|last5=Murphy|first5=T.W.|last6=Shapiro|first6=I.I.|title=Modeling and Analysis of the APOLLO Lunar Laser Ranging Data|eprint=1608.04758|class=astro-ph.IM|display-authors=2|year=2016 }}</ref>

<ref name="Citizen1">{{cite web|last1=Wright|first1=Ernie|title=Overhead view of the Earth-Moon system, to scale Lunar Parallax: Estimating the Moon's Distance|url=http://www.etwright.org/astro/moonpar.html|access-date=29 February 2016}}</ref>

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<!--  <ref name = "corner cube 1">{{cite web|url=http://eclipse.gsfc.nasa.gov/SEhelp/ApolloLaser.html|title=NASA - Accuracy of Eclipse Predictions|website=Eclipse.gsfc.nasa.gov|access-date=2016-02-22}}</ref>

<ref name = "corner cube 2">{{cite web|url=http://www.physics.ucsd.edu/~tmurphy/apollo/lrrr.html|title=Lunar Retroreflectors|website=Physics.ucsd.edu|access-date=2016-02-22}}</ref>-->

<ref name="DE 102">{{cite journal|last1=Newhall|first1=X.X|last2=Standish|first2=E.M|last3=Williams|first3=J. G.|title=DE 102 - A numerically integrated ephemeris of the moon and planets spanning forty-four centuries|journal=Astronomy and Astrophysics|date=Aug 1983|volume=125|issue=1|pages=150–167|url=http://articles.adsabs.harvard.edu/full/1983A%26A...125..150N?iframe=true&width=100%&height=100%|access-date=28 February 2016|issn=0004-6361|bibcode = 1983A&A...125..150N }}</ref>

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<ref name = "drift2">{{cite book|title=Solar System Dynamics|author=C.D. Murray & S.F. Dermott|publisher=Cambridge University Press|date=1999|page=184}}</ref>

<ref name = "drift3">{{cite book|last = Dickinson|first = Terence|author-link = Terence Dickinson|title = From the Big Bang to Planet X|publisher = [[Camden House]]|date = 1993|location = Camden East, Ontario|pages = 79–81|isbn = 978-0-921820-71-0 }}</ref>

<ref name="Fact Sheet">{{cite web|last1=Williams|first1=Dr. David R.|title=Planetary Fact Sheet - Ratio to Earth Values|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/planet_table_ratio.html|publisher=NASA Goddard Space Flight Center|access-date=28 February 2016|date=18 November 2015}}</ref>

<ref name="Fischer">{{cite journal|last1=Fischer|first1=Irene|title=Parallax of the moon in terms of a world geodetic system|journal=The Astronomical Journal|url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?db_key=AST&bibcode=1962aj.....67..373f&letter=0&classic=YES&defaultprint=YES&whole_paper=YES&page=374&epage=374&send=Send+PDF&filetype=.pdf|date=August 1962|volume=67|pages=373|doi=10.1086/108742|bibcode = 1962AJ.....67..373F }}</ref>

<ref name="Groten">{{cite journal|last1=Groten|first1=Erwin|title=Fundamental Parameters and Current (2004) Best Estimates of the Parameters of Common Relevance to Astronomy, Geodesy, and Geodynamics by Erwin Groten, IPGD, Darmstadt|journal=Journal of Geodesy|date=1 April 2004|volume=77|issue=10–11|pages=724–797|doi=10.1007/s00190-003-0373-y|url=http://www.iag-aig.org/attach/e354a3264d1e420ea0a9920fe762f2a0/51-groten.pdf|access-date=2 March 2016|bibcode = 2004JGeod..77..724.|s2cid=16907886}}</ref>

<ref name=Gutzwiller>{{cite journal|doi=10.1103/RevModPhys.70.589|title=Moon–Earth–Sun: The oldest three-body problem|date=1998|author=Gutzwiller, Martin C.|journal=Reviews of Modern Physics|volume=70|issue=2|pages=589–639|bibcode=1998RvMP...70..589G}}</ref>

<ref name="Helden">{{cite book|last1=Helden|first1=Albert van|title=Measuring the universe : cosmic dimensions from Aristarchus to Halley|date=1986|publisher=University of Chicago Press|location=Chicago|isbn=978-0-226-84882-2|page=16|edition=Repr.}}</ref>

<ref name="Impact">{{cite journal|last1=Canup|first1=R. M.|author1-link=Robin Canup |title=Forming a Moon with an Earth-like Composition via a Giant Impact|journal=Science|date=17 October 2012|volume=338|issue=6110|pages=1052–1055|doi=10.1126/science.1226073|bibcode = 2012Sci...338.1052C|pmid=23076098|pmc=6476314}}</ref>

<ref name="integration">{{cite journal|last1=Vitagliano|first1=Aldo|title=Numerical integration for the real time production of fundamental ephemerides over a wide time span|journal=Celestial Mechanics and Dynamical Astronomy|date=1997|volume=66|issue=3|pages=293–308|doi=10.1007/BF00049383|url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?db_key=AST&bibcode=1997CeMDA..66..293V&letter=0&classic=YES&defaultprint=YES&whole_paper=YES&page=293&epage=293&send=Send+PDF&filetype=.pdf|bibcode = 1996CeMDA..66..293V|s2cid=119510653}}</ref>

<ref name="LLR">{{cite journal|last1=Dickey|first1=J. O.|last2=Bender|first2=P. L.|last3=Faller|first3=J. E.|last4=Newhall|first4=X X|last5=Ricklefs|first5=R. L.|last6=Ries|first6=J. G.|last7=Shelus|first7=P. J.|last8=Veillet|first8=C.|last9=Whipple|first9=A. L.|last10=Wiant|first10=J. R.|last11=Williams|first11=J. G.|last12=Yoder|first12=C. F.|display-authors=2|title=Lunar Laser Ranging: A Continuing Legacy of the Apollo Program|url=http://www.physics.ucsd.edu/~tmurphy/apollo/doc/Dickey.pdf|journal=Science|date=22 July 1994|volume=265|issue=5171|pages=482–490|doi=10.1126/science.265.5171.482|bibcode = 1994Sci...265..482D|pmid=17781305|s2cid=10157934}}</ref>

<ref name="Lunar Laser Ranging">{{cite journal|author1=Bender, P. L.|author2=Currie, D. G.|author3=Dicke, R. H.|author4=Eckhardt, D. H.|author5=Faller, J. E.|author6=Kaula, W. M.|author7=Mullholland, J. D.|author8=Plotkin, H. H.|author9=Silverberg, E. C.|author10=Wilkinson, D. T.|author11=Williams, J. G.|author12=Alley, C. O.|date=October 19, 1973|title=The Lunar Laser Ranging Experiment|journal=[[Science (journal)|Science]]|volume=182|issue=4109|pmid=17749298|bibcode=1973Sci...182..229B|doi=10.1126/science.182.4109.229|pages=229–238|s2cid=32027563|access-date=April 27, 2013|url=http://www.physics.ucsd.edu/~tmurphy/apollo/doc/Bender.pdf|display-authors=3}}</ref>

<ref name = "Lunar Orbit Evolution">{{Citation|author =Bills, B.G.|author2 =Ray, R.D.|name-list-style =amp|title =Lunar Orbital Evolution: A Synthesis of Recent Results|journal =Geophysical Research Letters|volume =26|pages =3045–3048|date =1999|doi = 10.1029/1999GL008348|issue=19|bibcode = 1999GeoRL..26.3045B|doi-access =free }}</ref>

<ref name = "Measuring the Universe">{{citation|contribution=3.2 Aristarchus, Hipparchus, and Ptolemy|pages=27–35|title=Measuring the Universe: The Cosmological Distance Ladder|first=Stephen|last=Webb|publisher=Springer|date=1999|isbn=978-1-85233-106-1}}. See in particular p.&nbsp;33: "Almost everything we know about Hipparchus comes down to us by way of Ptolemy."</ref>

<ref name="millimeter challenge">{{cite journal|last1=Murphy|first1=T W|title=Lunar laser ranging: the millimeter challenge|journal=Reports on Progress in Physics|date=1 July 2013|volume=76|issue=7|page=2|doi=10.1088/0034-4885/76/7/076901|pmid=23764926|url=http://physics.ucsd.edu/~tmurphy/papers/rop-llr.pdf|arxiv=1309.6294|bibcode=2013RPPh...76g6901M|s2cid=15744316 }}</ref>

<ref name="moonbounce">{{cite journal|last1=Shuch|first1=H. Paul|title=Measuring the mass of the earth: the ultimate moonbounce experiment|journal=Proceedings, 25th Conference of the Central States VHF Society|date=July 1991|pages=25–30|url=http://www.setileague.org/articles/ham/masserth.pdf|access-date=28 February 2016}}</ref>

<ref name="NEO">{{cite web|url=http://neo.jpl.nasa.gov/ca/|title=NEO Earth Close Approaches|website=Neo.jpl.nasa.gov|access-date=2016-02-22|url-status=dead|archive-url=https://web.archive.org/web/20140307125729/http://neo.jpl.nasa.gov/ca/|archive-date=2014-03-07 }}</ref>

<ref name="O'Keefe">{{cite journal|last1=O'Keefe|first1=J. A.|last2=Anderson|first2=J. P.|title=The earth's equatorial radius and the distance of the moon|journal=Astronomical Journal|date=1952|volume=57|pages=108–121|url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?db_key=AST&bibcode=1952AJ.....57..108O&letter=0&classic=YES&defaultprint=YES&whole_paper=YES&page=108&epage=108&send=Send+PDF&filetype=.pdf|bibcode = 1952AJ.....57..108O|doi = 10.1086/106720 }}</ref>

<ref name = "Precambrian">{{cite journal|last1=Walker|first1=James C. G.|last2=Zahnle|first2=Kevin J.|title=Lunar nodal tide and distance to the Moon during the Precambrian|journal=Nature|date=17 April 1986|volume=320|issue=6063|pages=600–602|doi=10.1038/320600a0|pmid=11540876|bibcode = 1986Natur.320..600W|hdl=2027.42/62576|s2cid=4350312|url=https://deepblue.lib.umich.edu/bitstream/2027.42/62576/1/320600a0.pdf|hdl-access=free}}</ref>

<ref name="Relativity">{{cite journal|last1=Williams|first1=J. G.|last2=Newhall|first2=X. X.|last3=Dickey|first3=J. O.|title=Relativity parameters determined from lunar laser ranging|journal=Physical Review D|date=15 June 1996|volume=53|issue=12|pages=6730–6739|doi=10.1103/PhysRevD.53.6730|url=http://physics.ucsd.edu/~tmurphy/apollo/doc/JGW96.pdf|bibcode = 1996PhRvD..53.6730W|pmid=10019959 }}</ref>

<ref name="Sea's Edge">{{cite book|last1=Leslie|first1=William T. Fox |others=Illustrated by Clare Walker|title=At the sea's edge : an introduction to coastal oceanography for the amateur naturalist|date=1983|publisher=Prentice-Hall|location=Englewood Cliffs, N.J.|isbn=978-0130497833|page=[https://archive.org/details/atseasedgeintrod0000foxw/page/101 101]|url=https://archive.org/details/atseasedgeintrod0000foxw/page/101}}</ref>

<ref name="tidal lock">{{cite web|last1=Cain|first1=Fraser|title=WHEN WILL EARTH LOCK TO THE MOON?|url=http://www.universetoday.com/128350/will-earth-lock-moon/|website=Universe Today|access-date=1 September 2016|date=2016-04-12}}</ref>

<ref name="TheiaPothesis">
{{cite web|title = The Theia Hypothesis: New Evidence Emerges that Earth and Moon Were Once the Same|publisher = The Daily Galaxy|url = http://www.dailygalaxy.com/my_weblog/2007/07/the-theia-hypot.html|date = 2007-07-05|access-date = 2013-11-13
}}
</ref>

<ref name="Radar1">{{cite journal|last1=Yaplee|first1=B. S.|last2=Roman|first2=N. G.|last3=Scanlan|first3=T. F.|last4=Craig|first4=K. J.|title=A lunar radar study at 10-cm wavelength|journal=Paris Symposium on Radio Astronomy|date=30 July – 6 August 1958|volume=IAU Symposium no. 9|issue=9|pages=19|bibcode=1959IAUS....9...19Y}}</ref>

<ref name="Radar2">{{cite journal|last1=Hey|first1=J. S.|last2=Hughes|first2=V. A.|title=Radar observation of the moon at 10-cm wavelength|journal=Paris Symposium on Radio Astronomy|volume=9|pages=13–18|date=30 July – 6 August 1958|issue=9|bibcode=1959IAUS....9...13H|doi=10.1017/s007418090005049x|doi-access=free}}</ref>

<ref name = "Radar3">{{cite journal|last1=Yaplee|first1=B. S.|last2=Knowles|first2=S. H.|last3=Shapiro|first3=A.|last4=Craig|first4=K. J.|last5=Brouwer|first5=D.|display-authors=2|title=The mean distance to the Moon as determined by radar|journal=Symposium - International Astronomical Union|date=January 1965|volume=21|page=2|doi=10.1017/S0074180900104826|bibcode=1965IAUS...21...81Y|doi-access=free}}</ref>

<ref name="space.com">{{cite web|last1=Choi|first1=Charles Q.|title=Moon Facts: Fun Information About the Earth's Moon|url=http://www.space.com/55-earths-moon-formation-composition-and-orbit.html|website=Space.com|publisher=TechMediaNetworks, Inc|access-date=3 March 2016|date=19 November 2014}}</ref>

<ref name="Zuluaga">{{cite arXiv |eprint=1405.4580 |last1=Zuluaga |first1=Jorge I. |last2=Figueroa |first2=Juan C. |last3=Ferrin |first3=Ignacio |title=The simplest method to measure the geocentric lunar distance: a case of citizen science |date=19 May 2014|class=physics.pop-ph }}{{page needed |date=May 2023}}</ref>

}}

== External links ==
* [http://www.wolframalpha.com/widgets/view.jsp?id=8618ef756ac52a32062790f016d40230 Wolfram Alpha widget – Current Moon Earth distance]

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[[Category:Orbit of the Moon]]
[[Category:Units of measurement in astronomy]]
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