# Astronomical unit

{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }} Template:Infobox unit The astronomical unit (symbol au;[1] sometimes AU,[2] a.u.[3][4] and ua[5]) is a unit of length, roughly the distance from the Earth to the Sun. However, that distance varies as the Earth orbits the Sun, from a maximum (aphelion) to a minimum (perihelion) and back again once a year. Originally, each distance was measured through observation, and the au was defined as their average, half the sum of the maximum and minimum, making the unit a kind of medium measure for Earth-to-Sun distance. It is now defined as exactly Template:Val metres (about 150 million km, or 93 million miles).[6]

The astronomical unit is used primarily as a convenient yardstick for measuring distances within the Solar System. However, it is also a fundamental component in the definition of another critical unit of astronomical length, the parsec.

## Development of unit definition

The Earth's orbit around the Sun is shaped like an ellipse. The semi-major axis of that ellipse is half of a straight line that crosses the orbit at its extremes, the points of aphelion and perihelion, passing through the center of the sun along its way. Since ellipses are well-understood shapes, measuring the points of its extremes defined the exact shape mathematically, and made possible calculations for the entire orbit as well as predictions based upon observation. In addition, it mapped out exactly the largest straight-line distance the earth traverses over the course of a year, defining times and places for observing the largest parallax effects (apparent shifts of position) in nearby stars. Knowing the earth's shift and a star's shift enabled the star's distance to be calculated. But all measurements are subject to some degree of error or uncertainty, and the uncertainties in the length of the au only increased uncertainties in the stellar distances. Improvements in precision have always been a key to improving astronomical understanding. Throughout the twentieth century, measurements became increasingly precise and sophisticated, and ever more dependent upon accurate observation of the effects described by Einstein's theory of relativity and upon the mathematical tools it used.

Improving measurements were continually checked and cross-checked by means of our understanding of the laws of celestial mechanics, which govern the motions of objects in space. The expected positions and distances of objects at an established time are calculated (in au) from these laws, and assembled into a collection of data called an ephemeris. NASA's Jet Propulsion Laboratory provides one of several ephemeris computation services.[7]

In 1976, in order to establish a yet more precise measure for the au, the International Astronomical Union (IAU) formally adopted a new definition. While directly based on the then-best available observational measurements, the definition was recast in terms of the then-best mathematical derivations from celestial mechanics and planetary ephemerides. It stated that "the astronomical unit of length is that length (A) for which the Gaussian gravitational constant (k) takes the value Template:Val when the units of measurement are the astronomical units of length, mass and time".[8][9][10] Equivalently, one au is the radius of an unperturbed circular Newtonian orbit about the sun of a particle having infinitesimal mass, moving with an angular frequency of Template:Val radians per day;[11] or alternatively that length for which the heliocentric gravitational constant (the product GTemplate:Solar mass) is equal to (Template:Val)2 au3/d2, when the length is used to describe the positions of objects in the Solar System.

Subsequent explorations of the Solar System by space probes made it possible to obtain precise measurements of the relative positions of the inner planets and other objects by means of radar and telemetry. As with all radar measurements, these rely on measuring the time taken for photons to be reflected from an object. Since all photons move at the speed of light in vacuum, a fundamental constant of the universe, the distance of an object from the probe is basically the product of the speed of light and the measured time. For precision though, the calculations require adjustment for things such as the motions of the probe and object while the photons are in transit. In addition, the measurement of the time itself must be translated to a standard scale that accounts for relativistic time dilation. Comparison of the ephemeris positions with time measurements expressed in the TDB scale leads to a value for the speed of light in astronomical units per day (of 86,400 seconds). By 2009, the IAU had updated its standard measures to reflect improvements, and calculated the speed of light at Template:Val TDB.[12]

Meanwhile, in 1983, the International Committee for Weights and Measures (CIPM) modified the International System of Units (SI, or "modern" metric system) to make the metre independent of physical objects entirely, because measurement errors had become too large for the prototype metre to be useful any more. Instead, it was redefined in terms of the speed of light in vacuum, which could be independently determined at need. The speed of light could then be expressed exactly as c0 = Template:Val, a standard also adopted by the IERS numerical standards.[13] From this definition and the 2009 IAU standard, the time for light to traverse an au is found to be τA = Template:Val seconds, more than 8 minutes. By simple multiplication then, the best IAU 2009 estimate was A = c0τA = Template:Val metres,[14] based on a comparison of JPL and IAA–RAS ephemerides.[15][16][17]

This estimate was still derived from observation and measurements subject to error, and based in techniques that did not yet standardize all relativistic effects, and thus were not constant for all observers. In 2012, finding that the equalization of relativity alone would make the definition overly complex, the IAU simply used the 2009 estimate to redefine the astronomical unit as a conventional unit of length directly tied to the metre (exactly Template:Val) and assigned it the official symbol au.[14][18] The new definition also recognizes as a consequence that the au unit is now to play a role of reduced importance, limited in its use to that of a convenience in some applications.[14]

 1 astronomical unit = Template:Val metres (exactly) ≈ Template:Val million miles ≈ Template:Val millionths of a parsec ≈ Template:Val millionths of a light-year

## Usage

With the definitions used before 2012, the astronomical unit was dependent on the heliocentric gravitational constant, that is the product of the gravitational constant G and the solar mass Template:Solar mass. Neither G nor Template:Solar mass can be measured to high accuracy in SI units, but the value of their product is known very precisely from observing the relative positions of planets (Kepler's Third Law expressed in terms of Newtonian gravitation). Only the product is required to calculate planetary positions for an ephemeris, which explains why ephemerides are calculated in astronomical units and not in SI units.

The calculation of ephemerides also requires a consideration of the effects of general relativity. In particular, time intervals measured on the surface of the Earth (terrestrial time, TT) are not constant when compared to the motions of the planets: the terrestrial second (TT) appears to be longer in Northern Hemisphere winter and shorter in Northern Hemisphere summer when compared to the "planetary second" (conventionally measured in barycentric dynamical time, TDB). This is because the distance between the Earth and the Sun is not fixed (it varies between Template:Val and Template:Val) and, when the Earth is closer to the Sun (perihelion), the Sun's gravitational field is stronger and the Earth is moving faster along its orbital path. As the metre is defined in terms of the second, and the speed of light is constant for all observers, the terrestrial metre appears to change in length compared to the "planetary metre" on a periodic basis.

The metre is defined to be a unit of proper length, but the SI definition does not specify the metric tensor to be used in determining it. Indeed, the International Committee for Weights and Measures (CIPM) notes that "its definition applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored."[19] As such, the metre is undefined for the purposes of measuring distances within the Solar System. The 1976 definition of the astronomical unit was incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed,[20] and "vigorous debate" ensued [21] until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = Template:Val metres.

The au is too small for interstellar distances, where the parsec is commonly used. See the article cosmic distance ladder. The light year is often used in popular works, but is not an approved non-SI unit.[22]

## History

According to Archimedes in the Sandreckoner (2.1), Aristarchus of Samos estimated the distance to the Sun to be Template:Val times the Earth's radius (the true value is about Template:Val).[23] However, the book On the Sizes and Distances of the Sun and Moon, which has long been ascribed to Aristarchus, says that he calculated the distance to the Sun to be between 18 and 20 times the distance to the Moon, whereas the true ratio is about 389.174. The latter estimate was based on the angle between the half moon and the Sun, which he estimated as 87° (the true value being close to 89.853°). Depending on the distance Van Helden assumes Aristarchus used for the distance to the Moon, his calculated distance to the Sun would fall between 380 and Template:Val Earth radii.[24]

According to Eusebius of Caesarea in the Praeparatio Evangelica (Book XV, Chapter 53), Eratosthenes found the distance to the Sun to be "σταδιων μυριαδας τετρακοσιας και οκτωκισμυριας" (literally "of stadia myriads 400 and Template:Val" but with the additional note that in the Greek text the grammatical agreement is between myriads (not stadia) on the one hand and both 400 and Template:Val on the other, as in Greek, unlike English, all three or all four if one were to include stadia, words are inflected). This has been translated either as Template:Val stadia (1903 translation by Edwin Hamilton Gifford), or as Template:Val stadia (edition of Édouard des Places, dated 1974–1991). Using the Greek stadium of 185 to 190 metres,[25][26] the former translation comes to a far too low Template:Val whereas the second translation comes to 148.7 to 152.8 million kilometres (accurate within 2%).[27] Hipparchus also gave an estimate of the distance of the Sun from the Earth, quoted by 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 7 arc minutes.[28]

A Chinese mathematical treatise, the Zhoubi suanjing (c. 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 Template:Val li apart and the assumption that the Earth is flat.[29]

Solar
parallax
Earth
Archimedes in Sandreckoner
(3rd century BC)
40″ Template:Val
Aristarchus in On Sizes (3rd century BC) 380-Template:Val
Hipparchus (2nd century BC) 7′  490
Posidonius (1st century BC) quoted in Cleomedes (1st century) Template:Val
Ptolemy (2nd century) 2′ 50″ Template:Val
Godefroy Wendelin (1635) 15″ Template:Val
Jeremiah Horrocks (1639) 15″ Template:Val
Christiaan Huygens (1659) 8.6″ Template:Val
Cassini & Richer (1672) Template:Frac Template:Val
Jérôme Lalande (1771) 8.6″ Template:Val
Simon Newcomb (1895) 8.80″ Template:Val
Arthur Hinks (1909) 8.807″ Template:Val
H. Spencer Jones (1941) 8.790″ Template:Val
modern Template:Val Template:Val

In the 2nd century CE, Ptolemy estimated the mean distance of the Sun as Template:Val times the Earth radius.[30][31] To determine this value, Ptolemy started by measuring the Moon's parallax, finding what amounted to a horizontal lunar parallax of 1° 26′, which was much too large. He then derived a maximum lunar distance of Template:Frac 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.[32][33] 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 the Earth traversed by the Moon during a lunar eclipse. Given these data, the distance of the Sun from the Earth can be trigonometrically computed to be Template:Val 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 percent can make the solar distance infinite.[32]

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, al-Farghānī gave a mean solar distance of Template:Val Earth radii, while in his zij, al-Battānī used a mean solar distance of Template:Val Earth radii. Subsequent astronomers, such as al-Bīrūnī, used similar values.[34] Later in Europe, Copernicus and Tycho Brahe also used comparable figures (Template:Val and Template:Val Earth radii), and so Ptolemy's approximate Earth–Sun distance survived through the 16th century.[35]

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 the 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.[36] By measuring the transit in two different locations, one can accurately calculate the parallax of Venus and from the relative distance of the Earth and Venus from the Sun, the solar parallax α (which cannot be measured directly[37]). Jeremiah Horrocks had attempted to produce an estimate based on his observation of the 1639 transit (published in 1662), giving a solar parallax of 15 arcseconds, similar to Wendelin's figure. The solar parallax is related to the Earth–Sun distance as measured in Earth radii by

${\displaystyle A={\cot \alpha }.}$

The smaller the solar parallax, the greater the distance between the Sun and the Earth: a solar parallax of 15" is equivalent to an Earth–Sun distance of Template:Val Earth radii.

Christiaan Huygens believed the distance was even greater: by comparing the apparent sizes of Venus and Mars, he estimated a value of about Template:Val Earth radii,[38] equivalent to a solar parallax of 8.6". 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.

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 Template:Frac", equivalent to an Earth–Sun distance of about Template:Val Earth radii. They were also the first astronomers to have access to an accurate and reliable value for the radius of the Earth, which had been measured by their colleague Jean Picard in 1669 as Template:Val thousand toises. 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 and published in his Optica Promata (1663). It was strongly advocated by Edmond Halley[39] 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. 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.[40] The various results were collated by Jérôme Lalande to give a figure for the solar parallax of 8.6″.

Date Method A/Gm Uncertainty
1895 aberration 149.25 0.12
1941 parallax 149.674 0.016
1976 telemetry 149.597 870 0.000 001
2009 telemetry 149.597 870 700 0.000 000 003

Another method involved determining the constant of aberration, and Simon Newcomb gave great weight to this method when deriving his widely accepted value of 8.80″ for the solar parallax (close to the modern value of Template:Val″), 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 kilometres. 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 constants in 1896,[41] which remained in place for the calculation of ephemerides until 1964.[42] The name "astronomical unit" appears first to have been used in 1903.[43]

The discovery of the near-Earth asteroid 433 Eros and its passage near the Earth in 1900–1901 allowed a considerable improvement in parallax measurement.[44] Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931.[37][45]

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.[46]

## Developments

The astronomical distance unit parsec uses the au as a baseline and an angle of one arcsecond for parallax. 1 au and 1 pc not to scale. (See also stellar parallax)

The unit distance A (the value of the astronomical unit in metres) can be expressed in terms of other astronomical constants:

${\displaystyle A^{3}={\frac {GM_{\odot }D^{2}}{k^{2}}}}$

where G is the Newtonian gravitational constant, Template:Solar mass is the solar mass, k is the numerical value of Gaussian gravitational constant and D is the time period of one day. The Sun is constantly losing mass by radiating away energy,[47] so the orbits of the planets are steadily expanding outward from the Sun. This has led to calls to abandon the astronomical unit as a unit of measurement.[48]

As the speed of light has an exact defined value in SI units and the Gaussian gravitational constant k is fixed in the astronomical system of units, measuring the light time per unit distance is exactly equivalent to measuring the product GTemplate:Solar mass in SI units. Hence, it is possible to construct ephemerides entirely in SI units, which is increasingly becoming the norm.

A 2004 analysis of radiometric measurements in the inner Solar System suggested that the secular increase in the unit distance was much larger than can be accounted for by solar radiation, +15±4 meters per century.[49][50]

The measurements of the secular variations of the astronomical unit are not confirmed by other authors and are quite controversial. Furthermore, since 2010, the astronomical unit is not yet estimated by the planetary ephemerides.[51]

## Examples

The distances are approximate mean distances. It has to be taken into consideration that the distances between celestial bodies change in time due to their orbits and other factors.

## Other views

In 2006 the BIPM defined the astronomical unit as Template:Val, and recommended "ua" as the symbol for the unit.[5] In the 2014 revision of the SI Brochure, the BIPM recognised the IAU's 2012 redefinition of the astronomical unit as Template:Val, with the unit symbol "au".[58]

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## Notes and references

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8. Resolution No. 10 of the XVIth General Assembly of the International Astronomical Union, Grenoble, 1976
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11. Template:SIbrochure8th
12. "2009 Selected Astronomical Constants" in {{#invoke:citation/CS1|citation |CitationClass=citation }}
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21. {{#invoke:citation/CS1|citation |CitationClass=citation }} and also p. 91, Summary and recommendations.
22. See, for example, the work cited above: {{#invoke:citation/CS1|citation |CitationClass=citation }}
23. Gomez, A. G. (2013) Aristarchos of Samos, the Polymath AuthorHouse, ISBN 9781481789493.
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30. {{#invoke:citation/CS1|citation |CitationClass=citation }}
31. {{#invoke:citation/CS1|citation |CitationClass=citation }}
32. pp. 16–19, van Helden 1985
33. p. 251, Ptolemy's Almagest, translated and annotated by G. J. Toomer, London: Duckworth, 1984, ISBN 0-7156-1588-2
34. pp. 29–33, van Helden 1985
35. pp. 41–53, van Helden 1985
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41. Conférence internationale des étoiles fondamentales, Paris, 18–21 May 1896
42. Resolution No. 4 of the XIIth General Assembly of the International Astronomical Union, Hamburg, 1964
43. astronomical unit Merriam-Webster's Online Dictionary
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56. Voyager 1, Where are the Voyagers – NASA Voyager 1
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