- Astronomical unit
1 astronomical unit = SI units 149.60×106 km 149.60×109 m Astronomical units 4.8481×10−6 pc 15.813×10−6 ly US customary / Imperial units 92.956×106 mi 490.81×109 ft
The symbol ua is recommended by the International Bureau of Weights and Measures, and the international standard ISO 80000, while au is recommended by the International Astronomical Union, and is more common in Anglosphere countries. In general, capital letters are only used for the symbols of units which are named after individual scientists, while au or a.u. can also mean atomic unit or even arbitrary unit; however, the use of AU to refer to the astronomical unit is widespread. The astronomical constant whose value is one astronomical unit is referred to as unit distance and given the symbol A.
The AU was originally defined as the length of the semi-major axis of the Earth's elliptical orbit around the Sun. In 1976 the International Astronomical Union revised the definition of the AU for greater precision, defining it as that length for which the Gaussian gravitational constant (k) takes the value 0.017 202 098 95 when the units of measurement are the astronomical units of length, mass and time. An equivalent definition is the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with an angular frequency of 0.017 202 098 95 radians per day, or that length for which the heliocentric gravitational constant (the product GM☉) is equal to (0.017 202 098 95)2 AU3/d2. It is approximately equal to the mean Earth–Sun distance.
Precise measurements of the relative positions of the inner planets can be made by radar and by telemetry from space probes. As with all radar measurements, these rely on measuring the time taken for photons to be reflected from an object. These measured positions are then compared with those calculated by the laws of celestial mechanics: the calculated positions are often referred to as an ephemeris, and are usually calculated in astronomical units. The comparison gives the speed of light in astronomical units, which is 173.144 632 6847(69) AU/d (TDB). As the speed of light in meters per second (cSI) is fixed in the International System of Units, this measurement of the speed of light in AU/d (cAU) also determines the value of the astronomical unit in meters (A):
The best current (2009) estimate of the International Astronomical Union (IAU) for the value of the astronomical unit in meters is A = 149 597 870 700(3) m, based on a comparison of JPL and IAA–RAS ephemerides.
By definition, the astronomical unit is dependent on the heliocentric gravitational constant, that is the product of the gravitational constant G and the solar mass M☉. Neither G nor M☉ 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 0.983 289 8912 AU and 1.016 710 3335 AU) 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 meter is defined in terms of the second, and the speed of light is constant for all observers, the terrestrial meter appears to change in length compared to the "planetary meter" on a periodic basis.
The meter 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." As such, the meter is undefined for the purposes of measuring distances within the solar system. The 1976 definition of the astronomical unit is incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but has proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity has been proposed.
According to Archimedes in the Sand Reckoner (1.4), Aristarchus of Samos estimated the distance to the Sun to be 10,000 times the earth's radius. Nevertheless, it has been historically assumed that Aristarchus of Samos estimated the distance to the Sun to be between 18 and 20 times the distance to the moon, whereas the true ratio is about 389. The latter estimate was based on the angle between the half moon and the Sun, which he estimated as 87°. (The true value is roughly 89.85°.)
According to Eusebius of Caesarea in the Praeparatio Evangelica, Eratosthenes found the distance to the sun to be "σταδιων μυριαδας τετρακοσιας και οκτωκισμυριας" (literally "of stadia myriads 400 and 80000"). This has been translated either as 4,080,000 stadia (1903 translation by Edwin Hamilton Gifford), or as 804,000,000 stadia (edition of Édouard des Places, dated 1974-1991). Using the Greek stadium of 185 to 190 meters, the former translation comes to a far too low 755,000 km whereas the second translation comes to 148.7 to 152.8 million km (accurate within 2%). 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.
A Chinese mathematical treatise, the Zhoubi suanjing (ca. 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 1000 li apart and the assumption that the Earth is flat.
Aristarchus (3rd cent. BC) 40″ 10,000 Hipparchus (2nd cent. BC) 7′ 490 Ptolemy (2nd cent.) 2′ 50″ 1,210 Godefroy Wendelin (1635) 15″ 14,000 Jeremiah Horrocks (1639) 15″ 14,000 Christiaan Huygens (1659) 8.6″ 24,000 Cassini & Richer (1672) 9½″ 21,700 Jérôme Lalande (1771) 8.6″ 24,000 Simon Newcomb (1895) 8.80″ 23,440 Arthur Hinks (1909) 8.807″ 23,420 H. Spencer Jones (1941) 8.790″ 23,466 modern 8.794143″ 23,455
In the 2nd century CE, Ptolemy estimated the mean distance of the sun as 1,210 times the Earth radius. 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 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. 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 1,210 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.
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 1,170 Earth radii, while in his zij, al-Battānī used a mean solar distance of 1,108 Earth radii. Subsequent astronomers, such as al-Bīrūnī, used similar values. Later in Europe, Copernicus and Tycho Brahe also used comparable figures (1,142 Earth radii and 1,150 Earth radii), and so Ptolemy's approximate Earth-Sun distance survived through the 16th century.
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. 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). 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
- A = cot α.
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 13,750 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 24,000 Earth radii, 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 based more on luck than good measurement, with his various errors cancelling each other out.
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 9½", equivalent to an Earth–Sun distance of about 22,000 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 3,269 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 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. 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 1964 radar 149.5981 0.001 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 8.794143″), 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 kilometers. 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, which remained in place for the calculation of ephemerides until 1964. The name "astronomical unit" appears first to have been used in 1903.
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. Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931.
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.
The unit distance A (the value of the astronomical unit in meters) can be expressed in terms of other astronomical constants:
where G is the Newtonian gravitational constant, M☉ is the solar mass, k is the Gaussian gravitational constant and D is the time period of one day. The Sun is constantly losing mass by radiating away energy, 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. There have also been calls to redefine the astronomical unit in terms of a fixed number of meters.
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 GM☉ 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.
Another explanation for the earth recession rate could be solar tidal friction. The moon's recession rate is governed by similar tidal friction with the earth. Takaho Miura of Hirosaki University in Japan suggested this possibility in 2009.
Later estimates based on both radiometric and angular observations lowered this estimate to +7±2 meters per century, but this is still far larger than can be accounted for by solar radiation and current theories of gravitation. The possible variation in the gravitational constant based on radiometric measurements is of the order of parts in 1012 per century, or lower. It has been suggested that the observed increase could be explained by the DGP model.
- The Moon is 0.0026 ± 0.0001 AU from the Earth
- The Earth is 1.00 ± 0.02 AU from the Sun
- Mars is 1.52 ± 0.14 AU from the Sun
- Jupiter is 5.20 ± 0.05 AU from the Sun
- 8.4–9.8 AU: The radius of VY Canis Majoris, the largest known star.
- New Horizons is 21 AU from the sun as of August 2011 as it makes it way to Pluto for a flyby.
- Pluto is 39.5 ± 9.8 AU from the Sun
- The Kuiper Belt begins at roughly 30 AU 
- Beginning of Scattered disk at 45 AU (10 AU overlap with Kuiper Belt)
- Ending of Kuiper Belt at 50-55 AU
- 90377 Sedna's orbit ranges between 76 and 942 AU from the Sun; Sedna is currently (as of 2009[update]) about 88 AU from the Sun
- 94 AU: Termination shock between Solar winds/Interstellar winds/Interstellar medium
- 96.7 AU: The distance of dwarf planet Eris from the sun, as of 2009. Eris and its moon are currently the most distant known objects in the Solar System apart from long-period comets and space probes.
- 100 AU: Heliosheath
- 118 AU: As of August 2011[update], Voyager 1 is the furthest human-made object from the Sun: it is currently travelling at about 3½ AU/yr
- 100-1000 AU: Mostly populated by objects from the Scattered Disc
- 1000-3000 AU: Beginning of Hills cloud/"Inner Oort Cloud"
- 20,000 AU: Ending of Hills Cloud/"Inner Oort Cloud", beginning of "Outer Oort Cloud"
- 50,000 AU: possible closest estimate of the "Outer Oort Cloud" limits (0.8 ly)
- 100,000 AU: possible farthest estimate of the "Outer Oort Cloud" limits (1.6 ly)
- 230,000 AU: maximum extent of influence of the Sun's gravitational field (Hill/Roche sphere)--beyond this is true interstellar medium. This distance is 1.1 parsecs (3.6 light-years).
- Proxima Centauri (the nearest star to Earth, excluding our own Sun) is ~268 000 AU from the Sun
- The mean diameter of Betelgeuse is 5.5 AU (822 800 000 km)
- The distance from the Sun to the center of the Milky Way is approximately 1.7×109 AU
- 1 AU = 149,597,870.700 ± 0.003 km ≈ 92,955,807.273 ± 0.002 mi ≈ 8.317 light minutes ≈ 499 light-seconds
- 1 light-second ≈ 0.002 AU
- 1 gigameter ≈ 0.0067 AU
- 1 light-year ≈ 63,241 AU
- 1 parsec = 648,000/π ≈ 206,264.8 AU
Notes and references
- ^ a b IAU WG on NSFA Current Best Estimates, http://maia.usno.navy.mil/NSFA/CBE.html, retrieved 25 September 2009
- ^ a b International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), p. 126, ISBN 92-822-2213-6, http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf
- ^ The IAU Style Manual (1989): The Preparation of Astronomical Papers and Reports, p. 23, http://www.iau.org/static/publications/stylemanual1989.pdf
- ^ See, for example, the Merriam-Webster's Collegiate Dictionary, the American Heritage Dictionary, and the Oxford English Dictionary.
- ^ Resolution No. 10 of the XVIth General Assembly of the International Astronomical Union, Grenoble, 1976
- ^ H. Hussmann, F. Sohl, J. Oberst (2009), "§22.214.171.124.3: Astronomical units", in Joachim E Trümper, Astronomy, astrophysics, and cosmology. Volume VI/4B Solar System, Springer, p. 4, ISBN 3540880542, http://books.google.com/?id=wgydrPWl6XkC&pg=RA1-PA4
- ^ Gareth V Williams (1997), "Astronomical unit", in James H. Shirley, Rhodes Whitmore Fairbridge, Encyclopedia of planetary sciences, Springer, p. 48, ISBN 0412069512, http://books.google.com/books?id=dw2GadaPkYcC&pg=PA48
- ^ "2009 Selected Astronomical Constants" in The Astronomical Almanac Online, USNO–UKHO, http://asa.usno.navy.mil/
- ^ Pitjeva, E. V.; Standish, E. M. (2009), "Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit", Celest. Mech. Dynam. Astron. 103 (4): 365–72, Bibcode 2009CeMDA.103..365P, doi:10.1007/s10569-009-9203-8, http://www.springerlink.com/content/21885q7262104u76/
- ^ "The Final Session of the General Assembly", Estrella d'Alva: 1, 14 August 2009, http://www.astronomy2009.com.br/10.pdf
- ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), pp. 166–67, ISBN 92-822-2213-6, http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf
- ^ Huang, T.-Y.; Han, C.-H.; Yi, Z.-H.; Xu, B.-X.; Han; Yi; Xu (1995), "What is the astronomical unit of length?", Astron. Astrophys. 298: 629–33, Bibcode 1995A&A...298..629H
- ^ Gomez, A. G. (2011) 'Aristarchos of Samos the Polymath', Journal of Scientific and Mathematical Research.
- ^ Van Helden, Albert (1985), Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley, Chicago: University of Chicago Press, pp. 5–9, ISBN 0-226-84882-5
- ^ Engels, Donald (1985), "The Length of Eratosthenes' Stade", Am. J. Philol. (The Johns Hopkins University Press) 106 (3): 298–311, doi:10.2307/295030, JSTOR 295030
- ^ Gulbekian, Edward (1987), "The origin and value of the stadion unit used by Eratosthenes in the third century B.C", Archive for History of Exact Sciences 37 (4): 359–63, doi:10.1007/BF00417008 (inactive 2010-01-09), http://www.springerlink.com/content/n7n7u7tj18676374/
- ^ Rawlins, D. (2008), "Eratothenes' large earth and tiny universe", DIO 14: 3–12, http://www.dioi.org/vols/we0.pdf
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- ^ Lloyd, G. E. R. (1996), Adversaries and Authorities: Investigations into Ancient Greek and Chinese Science, Cambridge University Press, pp. 59–60, ISBN 0-521-55695-3
- ^ Goldstein, Bernard R. (1967), "The Arabic Version of Ptolemy's Planetary Hypotheses", Trans. Am. Phil. Soc. 57 (4): 9–12
- ^ van Helden, Albert (1985), Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley, Chicago: University of Chicago Press, pp. 15–27, ISBN 0-226-84882-5
- ^ a b pp. 16–19, van Helden 1985
- ^ p. 251, Ptolemy's Almagest, translated and annotated by G. J. Toomer, London: Duckworth, 1984, ISBN 0-7156-1588-2
- ^ pp. 29–33, van Helden 1985
- ^ pp. 41–53, van Helden 1985
- ^ a b Weaver, Harold F. (1943), "The Solar Parallax", Astron. Soc. Pacific Leaflets 4: 144–51, Bibcode 1943ASPL....4..144W
- ^ Goldstein, S. J., Jr. (1985), "Christiaan Huygens' Measurement of the Distance to the Sun", Observatory 105: 32–33, Bibcode 1985Obs...105...32G
- ^ Halley, E. (1716), "A new Method of determining the Parallax of the Sun, or his Distance from the Earth", Phil. Trans. R. Soc. London 29: 454–64, http://www.dsellers.demon.co.uk/venus/ven_ch8.htm
- ^ Pogge, Richard (May 2004), How Far to the Sun? The Venus Transits of 1761 & 1769, Ohio State University, http://www.astronomy.ohio-state.edu/~pogge/Ast161/Unit4/venussun.html, retrieved 15 November 2009
- ^ Conférence internationale des étoiles fondamentales, Paris, 18–21 May 1896
- ^ Resolution No. 4 of the XIIth General Assembly of the International Astronomical Union, Hamburg, 1964
- ^ astronomical unit Merriam-Webster's Online Dictionary
- ^ Hinks, Arthur R. (1909), "Solar Parallax Papers No. 7: The General Solution from the Photographic Right Ascensions of Eros, at the Opposition of 1900", Month. Not. R. Astron. Soc. 69 (7): 544–67, Bibcode 1909MNRAS..69..544H
- ^ Spencer Jones, H. (1941), "The Solar Parallax and the Mass of the Moon from Observations of Eros at the Opposition of 1931", Mem. R. Astron. Soc. 66: 11–66
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- ^ Noedlinger, Peter D. (2008), "Solar Mass Loss, the Astronomical Unit, and the Scale of the Solar System", Celest. Mech. Dynam. Astron. 0801: 3807, arXiv:0801.3807, Bibcode 2008arXiv0801.3807N
- ^ "AU may need to be redefined", New Scientist, 6 February 2008, http://space.newscientist.com/article/dn13286-astronomical-unit-may-need-to-be-redefined.html?feedId=online-news_rss20
- ^ Capitaine, N; Guinot, B (2008). "The astronomical units". arXiv:0812.2970v1 [astro-ph].
- ^ Krasinsky, G. A.; Brumberg, V. A. (2004), "Secular increase of astronomical unit from analysis of the major planet motions, and its interpretation", Celest. Mech. Dynam. Astron. 90 (3–4): 267, Bibcode 2004CeMDA..90..267K, doi:10.1007/s10569-004-0633-z, http://www.springerlink.com/content/g5051650115444k9/
- ^ John D. Anderson and Michael Martin Nieto (2009), "Astrometric Solar-System Anomalies;§2: Increase in the astronomical unit", American Astronomical Society 261: 0702, arXiv:0907.2469, Bibcode 2009IAU...261.0702A.
- ^ Why is the earth moving away from the sun?, http://www.newscientist.com/article/dn17228-why-is-the-earth-moving-away-from-the-sun.html
- ^ Standish, E. M. (2004), "The Astronomical Unit now", Proc. Int. Astron. Union 2004: 163–79, doi:10.1017/S1743921305001365
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- ^ Iorio, Lorenzo (2005), "Secular increase of the astronomical unit and perihelion precessions as tests of the Dvali–Gabadadze–Porrati multi-dimensional braneworld scenario", J. Cosmol. Astroparticle Phys. 2005 (9): 006, arXiv:gr-qc/0508047, Bibcode 2005JCAP...09..006I, doi:10.1088/1475-7516/2005/09/006
- ^ Alan Stern; Colwell, Joshua E. (1997), "Collisional Erosion in the Primordial Edgeworth-Kuiper Belt and the Generation of the 30–50 AU Kuiper Gap", The Astrophysical Journal 490 (2): 879–882, Bibcode 1997ApJ...490..879S, doi:10.1086/304912, http://www.iop.org/EJ/article/0004-637X/490/2/879/36659.html.
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- Williams, D.; Davies, R. D. (1968), "A radio method for determining the astronomical unit", Monthly Notices of the Royal Astronomical Society 140: 537, Bibcode 1968MNRAS.140..537W
- The IAU and astronomical units
- Recommendations concerning Units (HTML version of the IAU Style Manual)
- Chasing Venus, Observing the Transits of Venus
- Transit of Venus
Units of length used in Astronomy Astronomical system of units See Also: Cosmic distance ladder · Orders of magnitude (length) · Conversion of units SI units Base units Derived units Accepted for use
See also Book:International System of Units · Category:SI base units
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