Editing Astronomical unit (section)

== History ==
The book ''[[On the Sizes and Distances (Aristarchus)|On the Sizes and Distances of the Sun and Moon]]'', which is ascribed to [[Aristarchus of Samos|Aristarchus]], says the distance to the Sun is 18 to 20 times the [[Lunar distance (astronomy)|distance to the Moon]], whereas the true ratio is about {{val|389.174}}. The latter estimate was based on the angle between the [[Lunar phase|half-moon]] and the Sun, which he estimated as {{val|87|u=deg}} (the true value being close to {{val|89.853|u=deg}}). Depending on the distance that Albert van Helden assumes Aristarchus used for the distance to the Moon, his calculated distance to the Sun would fall between {{val|380}} and {{val|1520}} Earth radii.<ref>{{cite book |last=van&nbsp;Helden |first=Albert |title=Measuring the Universe: Cosmic dimensions from Aristarchus to Halley |place=Chicago |publisher=University of Chicago Press |date=1985 |pages=5–9 |isbn=978-0-226-84882-2}}</ref>

[[Hipparchus]] gave an estimate of the distance of Earth from the Sun, quoted by [[Pappus of Alexandria|Pappus]] as equal to 490 Earth radii. According to the conjectural reconstructions of [[Noel Swerdlow]] and [[G. J. Toomer]], this was derived from his assumption of a "least perceptible" solar parallax of {{val|7|ul=arcminute}}.<ref>{{cite journal |doi=10.1007/BF00329826 |title=Hipparchus on the distances of the sun and moon |journal=Archive for History of Exact Sciences |volume=14 |issue=2 |pages=126–42 |last=Toomer |first=G.J. |date=1974|bibcode=1974AHES...14..126T |s2cid=122093782 }}</ref>

A Chinese mathematical treatise, the ''[[Zhoubi Suanjing]]'' ({{circa|1st century BCE}}), shows how the distance to the Sun can be computed geometrically, using the different lengths of the noontime shadows observed at three places {{val|1000}} [[Li (unit)|''li'']] apart and the assumption that Earth is flat.<ref>{{cite book |first=G. E. R. |last=Lloyd |author-link=G. E. R. Lloyd |title=Adversaries and Authorities: Investigations into Ancient Greek and Chinese Science |publisher=Cambridge University Press |date=1996 |pages=59–60 |isbn=978-0-521-55695-8}}</ref>

According to [[Eusebius]] in the ''[[Praeparatio evangelica]]'' (Book XV, Chapter 53), [[Eratosthenes]] found the distance to the Sun to be "σταδιων μυριαδας τετρακοσιας και οκτωκισμυριας" (literally ''myriads ten hundreds and eighty thousands of stadia'', where in the Greek text the numerals ''[[myriad]]s'', ''ten hundreds'' and ''eighty thousands'' are all [[accusative]] plural, while ''stadia'' is the [[genitive]] plural of ''[[stadion (unit of length)|stadion]]''.) This has been translated either as {{val|4080000}} ({{nowrap|1== ({{val|10000}} × 400) + {{val|80000}}}}) stadia (1903 translation by [[Edwin Hamilton Gifford]]), or as {{val|804000000}} ({{nowrap|1== {{val|10000}} × (400 + {{val|80000}})}}) stadia (edition of [[Édouard des Places]], dated 1974–1991). Using the Greek stadium of 185 to 190 metres,<ref name="Engels1985">{{cite journal |title=The Length of Eratosthenes' Stade |journal=The American Journal of Philology |last=Engels |first=Donald |volume=106 |issue=3 |pages=298–311 |date=1985 |doi=10.2307/295030 |jstor=295030}}</ref><ref name="Gulbekian1987">{{cite journal |url=https://link.springer.com/article/10.1007/BF00417008 |title=The origin and value of the stadion unit used by Eratosthenes in the third century B.C. |journal=Archive for History of Exact Sciences |first=Edward |last=Gulbekian |volume=37 |issue=4 |pages=359–63 |date=1987 |doi=10.1007/BF00417008|s2cid=115314003 }}</ref> the former translation comes to {{val|754800|u=km}} to {{val|775200|u=km}}<!--Depends on if 185 m or 190 m is used-->, which is far too low, whereas the second translation comes to 148.7 to 152.8&nbsp;billion metres (accurate within 2%).<ref>{{cite journal |url=http://www.dioi.org/vols/we0.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.dioi.org/vols/we0.pdf |archive-date=2022-10-09 |url-status=live |title=Eratosthenes' Too-Big Earth & Too-Tiny Universe |journal=DIO |first=D. |last=Rawlins |date=March 2008 |volume=14 |pages=3–12|bibcode=2008DIO....14....3R }}</ref> 

{|class="wikitable"
! rowspan=2 | Distance to the Sun<br/>estimated by
! colspan=2 | Estimate
! rowspan=2 | In au
! rowspan=2 |Percentage error
|-
! [[Solar parallax|Solar<br/>parallax]]
! [[Earth radii|Earth<br/>radii]]
|-
| [[Aristarchus of Samos|Aristarchus]] {{small|(3rd century BCE)}} {{small|''(in [[On the Sizes and Distances of the Sun and Moon|On Sizes]])''}}&nbsp;&nbsp;
| align=left | {{val|13|u=arcmin}} {{val|24|u=arcsec}}–{{val|7|u=arcmin}} {{val|12|u=arcsec}}
| align=right | {{val|256.5}}–{{val|477.8}}
| align=center | {{val|0.011}}–{{val|0.020}}
| −98.9% to −98%
|-
| [[Archimedes]] {{small|(3rd century BCE)}} {{small|''(in [[The Sand Reckoner]])''}}
| align=left | {{val|21|u=arcsec}}
| align=right | {{val|10000}}
| align=center | {{val|0.426}}
| −57.4%
|-
| [[Hipparchus]] {{small|(2nd century BCE)}}
| align=left | {{val|7|u=arcmin}}
| align=right | {{val|490}}
| align=center | {{val|0.021}}
| −97.9%
|-
| [[Posidonius]] {{small|(1st century BCE)}} {{small|''(quoted by coeval [[Cleomedes]])''}}
| align=left | {{val|21|u=arcsec}}
| align=right | {{val|10000}}
| align=center | {{val|0.426}}
| −57.4%
|-
| [[Ptolemy]] {{small|(2nd century)}}
| align=left | 2′ 50″
| align=right | {{val|1210}}
| align=center | {{val|0.052}}
| −94.8%
|-
| [[Godefroy Wendelin]] {{small|(1635)}}
| align=left | {{val|15|u=arcsec}}
| align=right | {{val|14000}}
| align=center | {{val|0.597}}
| −40.3%
|-
| [[Jeremiah Horrocks]] {{small|(1639)}}
| align=left | {{val|15|u=arcsec}}
| align=right | {{val|14000}}
| align=center | {{val|0.597}}
| −40.3%
|-
| [[Christiaan Huygens]] {{small|(1659)}}
| align=left | {{val|8.2|u=arcsec}}
| align=right | {{val|25086}}<ref name="Goldstein1">{{cite journal |bibcode = 1985Obs...105...32G|title = Christiaan Huygens' measurement of the distance to the Sun|journal = The Observatory|volume = 105|pages = 32|last1 = Goldstein|first1 = S. J.|year = 1985}}</ref>
| align=center | {{val|1.068}}
| +6.8%
|-
| [[Giovanni Domenico Cassini|Cassini]] & [[Jean Richer|Richer]] {{small|(1672)}}
| align=left | {{val|9.5|u=arcsec}}
| align=right | {{val|21700}}
| align=center | {{val|0.925}}
| −7.5%
|-
| [[John Flamsteed|Flamsteed]] {{small|(1672)}}
| align=left | {{val|9.5|u=arcsec}}
| align=right | {{val|21700}}
| align=center | {{val|0.925}}
| −7.5%
|-
| [[Jérôme Lalande]] {{small|(1771)}}
| align=left | {{val|8.6|u=arcsec}}
| align=right | {{val|24000}}
| align=center | {{val|1.023}}
| +2.3%
|-
| [[Simon Newcomb]] {{small|(1895)}}
| align=left | {{val|8.80|u=arcsec}}
| align=right | {{val|23440}}
| align=center | {{val|0.9994}}
| −0.06%
|-
| [[Arthur Robert Hinks|Arthur Hinks]] {{small|(1909)}}
| align=left | {{val|8.807|u=arcsec}}
| align=right | {{val|23420}}
| align=center | {{val|0.9985}}
| −0.15%
|-
| [[Harold Spencer Jones|H. Spencer Jones]] {{small|(1941)}}
| align=left | {{val|8.790|u=arcsec}}
| align=right | {{val|23466}}
| align=center | {{val|1.0005}}
| +0.05%
|-
| Modern [[astronomy]]
| align=left| {{val|8.794143|u=arcsec}}
| align=right | {{val|23455}}
| align=center | {{val|1.0000}}
|}

In the 2nd century CE, [[Ptolemy]] estimated the mean distance of the Sun as {{val|1210}} times [[Earth's radius]].<ref>{{cite journal |first=Bernard R. |last=Goldstein |title=The Arabic version of Ptolemy's ''planetary hypotheses'' |journal=Trans. Am. Philos. Soc. |volume=57 |issue=4 |date=1967 |pages=9–12 |doi=10.2307/1006040 |jstor=1006040}}</ref><ref>{{cite book |last=van Helden |first=Albert |title=Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley |place=Chicago |publisher=University of Chicago Press |date=1985 |pages=15–27 |isbn=978-0-226-84882-2}}</ref> To determine this value, Ptolemy started by measuring the Moon's parallax, finding what amounted to a horizontal lunar parallax of 1°&nbsp;26′, which was much too large. He then derived a maximum lunar distance of {{sfrac|64|1|6}} Earth radii. Because of cancelling errors in his parallax figure, his theory of the Moon's orbit, and other factors, this figure was approximately correct.<ref name=vh1619>van Helden 1985, pp. 16–19.</ref><ref>''Ptolemy's Almagest'', translated and annotated by G.&nbsp;J. Toomer, London: Duckworth, 1984, p.&nbsp;251. {{ISBN|0-7156-1588-2}}.</ref> He then measured the apparent sizes of the Sun and the Moon and concluded that the apparent diameter of the Sun was equal to the apparent diameter of the Moon at the Moon's greatest distance, and from records of lunar eclipses, he estimated this apparent diameter, as well as the apparent diameter of the shadow cone of Earth traversed by the Moon during a lunar eclipse. Given these data, the distance of the Sun from Earth can be trigonometrically computed to be {{val|1210}} Earth radii. This gives a ratio of solar to lunar distance of approximately 19, matching Aristarchus's figure. Although Ptolemy's procedure is theoretically workable, it is very sensitive to small changes in the data, so much so that changing a measurement by a few per cent can make the solar distance infinite.<ref name=vh1619/>

After Greek astronomy was transmitted to the medieval Islamic world, astronomers made some changes to Ptolemy's cosmological model, but did not greatly change his estimate of the Earth–Sun distance. For example, in his introduction to Ptolemaic astronomy, [[Ahmad ibn Muhammad ibn Kathir al-Farghani|al-Farghānī]] gave a mean solar distance of {{val|1170}} Earth radii, whereas in his ''[[zij]]'', [[Al-Battani|al-Battānī]] used a mean solar distance of {{val|1108}} Earth radii. Subsequent astronomers, such as [[Al-Biruni|al-Bīrūnī]], used similar values.<ref>van Helden 1985, pp. 29–33.</ref> Later in Europe, [[Nicolaus Copernicus|Copernicus]] and [[Tycho Brahe]] also used comparable figures ({{val|1142}} and {{val|1150}} Earth radii), and so Ptolemy's approximate Earth–Sun distance survived through the 16th century.<ref>van Helden 1985, pp. 41–53.</ref>

[[Johannes Kepler]] was the first to realize that Ptolemy's estimate must be significantly too low (according to Kepler, at least by a factor of three) in his ''[[Rudolphine Tables]]'' (1627). [[Kepler's laws of planetary motion]] allowed astronomers to calculate the relative distances of the planets from the Sun, and rekindled interest in measuring the absolute value for Earth (which could then be applied to the other planets). The invention of the [[telescope]] allowed far more accurate measurements of angles than is possible with the naked eye. Flemish astronomer [[Godefroy Wendelin]] repeated Aristarchus’ measurements in 1635, and found that Ptolemy's value was too low by a factor of at least eleven.

A somewhat more accurate estimate can be obtained by observing the [[transit of Venus]].<ref name=Bell>
{{cite magazine
 |first=Trudy E. |last=Bell
 |date=Summer 2004
 |title=Quest for the astronomical unit
 |magazine=The Bent of Tau Beta Pi
 |page=20
 |url=http://www.tbp.org/pages/publications/bent/features/su04bell.pdf
 |access-date=16 January 2012 |url-status=dead
 |archive-url=https://web.archive.org/web/20120324164801/http://www.tbp.org/pages/publications/bent/features/su04bell.pdf
 |archive-date=24 March 2012
 |postscript=none
}} – provides an extended historical discussion of the [[transit of Venus]] method.</ref> By measuring the transit in two different locations, one can accurately calculate the parallax of Venus and from the relative distance of Earth and Venus from the Sun, the [[Parallax#Solar parallax|solar parallax]] {{mvar|α}} (which cannot be measured directly due to the brightness of the Sun<ref name="Weaver">{{cite report |last=Weaver |first=Harold F. |title=The Solar Parallax |bibcode=1943ASPL....4..144W |date=March 1943 |journal=Astronomical Society of the Pacific Leaflets |volume=4 |issue=169 |pages=144–51}}</ref>). [[Jeremiah Horrocks]] had attempted to produce an estimate based on his observation of the [[1639 transit of Venus|1639 transit]] (published in 1662), giving a solar parallax of {{val|15|ul=arcsecond}}, similar to Wendelin's figure. The solar parallax is related to the Earth–Sun distance as measured in Earth radii by
: <math>A = \cot\alpha \approx 1\,\textrm{radian}/\alpha.</math>

The smaller the solar parallax, the greater the distance between the Sun and Earth: a solar parallax of {{val|15|u=arcsecond}} is equivalent to an Earth–Sun distance of {{val|13750}} Earth radii.

[[Christiaan Huygens]] believed that the distance was even greater: by comparing the apparent sizes of Venus and [[Mars]], he estimated a value of about {{val|24000}} Earth radii,<ref name="Goldstein1"/> equivalent to a solar parallax of {{val|8.6|u=arcsecond}}. Although Huygens' estimate is remarkably close to modern values, it is often discounted by historians of astronomy because of the many unproven (and incorrect) assumptions he had to make for his method to work; the accuracy of his value seems to be based more on luck than good measurement, with his various errors cancelling each other out.

[[File:Venustransit 2004-06-08 07-44.jpg|thumb|right|Transits of Venus across the face of the Sun were, for a long time, the best method of measuring the astronomical unit, despite the difficulties (here, the so-called "[[black drop effect]]") and the rarity of observations.]]
[[Jean Richer]] and [[Giovanni Domenico Cassini]] measured the parallax of Mars between [[Paris]] and [[Cayenne]] in [[French Guiana]] when Mars was at its closest to Earth in 1672. They arrived at a figure for the solar parallax of {{val|9.5|u=arcsecond}}, equivalent to an Earth–Sun distance of about {{val|22000}} Earth radii. They were also the first astronomers to have access to an accurate and reliable value for the radius of Earth, which had been measured by their colleague [[Jean Picard]] in 1669 as {{val|3269000}} ''[[toise]]s''. This same year saw another estimate for the astronomical unit by [[John Flamsteed]], which accomplished it alone by measuring the [[Mars|martian]] [[diurnal parallax]].<ref>Van Helden, A. (2010). Measuring the universe: cosmic dimensions from Aristarchus to Halley. University of Chicago Press. Ch. 12.</ref> Another colleague, [[Ole Rømer]], discovered the finite speed of light in 1676: the speed was so great that it was usually quoted as the time required for light to travel from the Sun to the Earth, or "light time per unit distance", a convention that is still followed by astronomers today.

A better method for observing Venus transits was devised by [[James Gregory (astronomer and mathematician)|James Gregory]] and published in his ''[[Optica Promata]]'' (1663). It was strongly advocated by [[Edmond Halley]]<ref>{{cite journal |last=Halley |first=E. |author-link=Edmond Halley |date=1716 |title=A new method of determining the parallax of the Sun, or his distance from the Earth |journal=Philosophical Transactions of the Royal Society |volume=29 |issue=338–350 |pages=454–64 |doi=10.1098/rstl.1714.0056 |s2cid=186214749 |doi-access=free }}</ref> and was applied to the transits of Venus observed in 1761 and 1769, and then again in 1874 and 1882. Transits of Venus occur in pairs, but less than one pair every century, and observing the transits in 1761 and 1769 was an unprecedented international scientific operation including observations by James Cook and Charles Green from Tahiti. Despite the [[Seven Years' War]], dozens of astronomers were dispatched to observing points around the world at great expense and personal danger: several of them died in the endeavour.<ref>{{cite web |last=Pogge |first=Richard |title=How far to the Sun? The Venus transits of 1761 & 1769 |url=http://www.astronomy.ohio-state.edu/~pogge/Ast161/Unit4/venussun.html |date=May 2004 |publisher=Ohio State University |department=Astronomy |access-date=15 November 2009}}</ref> The various results were collated by [[Jérôme Lalande]] to give a figure for the solar parallax of {{val|8.6|u=arcsecond}}. [[Karl Rudolph Powalky]] had made an estimate of {{val|8.83|u=arcsecond}} in 1864.<ref>{{cite journal|last=Newcomb|first=Simon|date=1871|title=The Solar Parallax|url=http://www.nature.com/articles/005060a0|journal=Nature|language=en|volume=5|issue=108|pages=60–61|doi=10.1038/005060a0|bibcode=1871Natur...5...60N|s2cid=4001378|issn=0028-0836}}</ref>

{| class="wikitable" style="float:right; margin:0 0 0 0.5em;"
|-
! Date
! Method
! ''A''/Gm
! Uncertainty
|-
| 1895
| aberration
| {{val|149.25}}
| {{val|0.12}}
|-
| 1941
| parallax
| {{val|149.674}}
| {{val|0.016}}
|-
| 1964
| radar
| {{val|149.5981}}
| {{val|0.001}}
|-
| 1976
| telemetry
| {{val|149.597870}}
| {{val|0.000001}}
|-
| 2009
| telemetry
| {{val|149.597870700}}
| {{val|0.000000003}}
|}
Another method involved determining the constant of [[aberration of light|aberration]]. [[Simon Newcomb]] gave great weight to this method when deriving his widely accepted value of {{val|8.80|u=arcsecond}} for the solar parallax (close to the modern value of {{val|8.794143|u=arcsec}}), although Newcomb also used data from the transits of Venus. Newcomb also collaborated with [[A. A. Michelson]] to measure the speed of light with Earth-based equipment; combined with the constant of aberration (which is related to the light time per unit distance), this gave the first direct measurement of the Earth–Sun distance in metres. Newcomb's value for the solar parallax (and for the constant of aberration and the Gaussian gravitational constant) were incorporated into the first international system of [[astronomical constant]]s in 1896,<ref>Conférence internationale des étoiles fondamentales, Paris, 18–21 May 1896</ref> which remained in place for the calculation of ephemerides until 1964.<ref>{{cite conference |title=On the system of astronomical constants |conference= XIIth General Assembly of the International Astronomical Union |publisher=International Astronomical Union |place=Hamburg, Germany |date=1964 | url =http://www.iau.org/static/resolutions/IAU1964_French.pdf|url-status=dead |archive-url=https://web.archive.org/web/20250311181040/https://www.iau.org/static/resolutions/IAU1964_French.pdf |archive-date=11 March 2025}}</ref> The name "astronomical unit" appears first to have been used in 1903.<ref>[http://www.merriam-webster.com/dictionary/astronomical%20unit "astronomical unit"], ''Merriam-Webster's Online Dictionary''</ref>{{failed verification|date=June 2019}}

The discovery of the [[near-Earth asteroid]] [[433 Eros]] and its passage near Earth in 1900–1901 allowed a considerable improvement in parallax measurement.<ref>{{cite journal |last=Hinks |first=Arthur R. |author-link=Arthur Robert Hinks |title=Solar parallax papers No. 7: The general solution from the photographic right ascensions of Eros, at the opposition of 1900 |journal=Monthly Notices of the Royal Astronomical Society |volume=69 |issue=7 |pages=544–67 |date=1909 |bibcode=1909MNRAS..69..544H |doi=10.1093/mnras/69.7.544|url=https://zenodo.org/record/1431881 |doi-access=free }}</ref> Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931.<ref name="Weaver"/><ref>{{cite journal |last=Spencer Jones |first=H. |author-link=Harold Spencer Jones |title=The solar parallax and the mass of the Moon from observations of Eros at the opposition of 1931 |journal=Mem. R. Astron. Soc. |volume=66 |date=1941 |pages=11–66 |issn=0369-1829 }}</ref>

Direct radar measurements of the distances to Venus and Mars became available in the early 1960s. Along with improved measurements of the speed of light, these showed that Newcomb's values for the solar parallax and the constant of aberration were inconsistent with one another.<ref>{{cite journal |last=Mikhailov |first=A. A. |date=1964 |title=The Constant of Aberration and the Solar Parallax |bibcode=1964SvA.....7..737M |journal=Sov. Astron. |volume=7 |issue=6 |pages=737–39 }}</ref>