Astronomical unit

1 astronomical unit =
SI units
149.60×10^6 km 149.60×10^9 m
Astronomical units
4.8481 × 10−6 pc 15.813×10^−6 ly
US customary / Imperial units
92.956×10^6 mi 490.81×10^9 ft

An astronomical unit (abbreviated as AU, au, a.u., or sometimes ua) is a unit of length. In SI units, 1 AU=1.49597871464 × 1011 m.[1] It is defined by the International Astronomical Union as the distance at which a massless particle in an unperturbed circular orbit about the Sun has an orbital period T in days given by T = 2π/k, where k = 0.01720209895 is the Gaussian gravitational constant in astronomical units of length, mass, and time. An approximation to this distance is the mean distance between the Earth and the Sun over one Earth orbit.Cite error: Invalid <ref> tag; refs with no content must have a name[2]

The symbol ua is recommended by the International Bureau of Weights and Measures,[3] but au is more common in Anglosphere countries. The International Astronomical Union recommends au,[4] while international standard ISO 31-1 uses AU. 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.[5] The astronomical constant whose value is one astronomical unit is referred to as unit distance and given the symbol A.

Contents

Definition

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.[6] 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,[3] 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.

Modern determinations

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 light 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).[7] As the speed of light in metres 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 metres (A):

A = 86\,400 \frac{c_{\rm SI}}{c_{\rm AU}}.

The best current (2009) estimate of the International Astronomical Union (IAU) for the value of the astronomical unit in metres is A = 149 597 870 700(3) m, based on a comparison of JPL and IAA–RAS ephemerides.[8][9][10]

Usage

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 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."[11] As such, the metre 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.[12]

History

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 390. His estimate was based on the angle between the half moon and the Sun, which he estimated as 87°.[13]

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 metres,[14][15] 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%).[16] 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.[17]

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.[18]

  Solar
parallax		 Earth
radii
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.[19][20] 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.[21][22] 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.[21]

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.[23] 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.[24]

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[25]). 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 = {1\over{\tan\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 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,[26] 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.

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 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[27] 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.[28] 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 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,[29] which remained in place for the calculation of ephemerides until 1964.[30] The name "astronomical unit" appears first to have been used in 1903.[31]

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.[32] Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931.[25][33]

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.[34]

Developments

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

A^3 = \frac{D^2}{G M_\odot k^2}

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,[35] 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.[36] There have also been calls to redefine the astronomical unit in terms of a fixed number of metres.[37]

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 metres per century.[38] Later estimates based on both radiometric and angular observations lowered this estimate to +7±2 metres per century,[39] but this is still far larger than can be accounted for by solar radiation and current theories of gravitation.[40] The possible variation in the gravitational constant based on radiometric measurements is of the order of parts in 1012 per century, or lower.[41] It has been suggested that the observed increase could be explained by the DGP model.[42]

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.

Conversion factors

See also

Notes and references

  1. ↑ H. Hussmann, F. Sohl, J. Oberst (2009). "§4.2.2.1.3: Astronomical units". In Joachim E Trümper. Astronomy, astrophysics, and cosmology. Volume VI/4B Solar System. Springer. p. 4. ISBN 3540880542. http://books.google.com/books?id=wgydrPWl6XkC&pg=RA1-PA4. 
  2. ↑ 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. 
  3. ↑ 3.0 3.1 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 
  4. ↑ The IAU Style Manual (1989): The Preparation of Astronomical Papers and Reports, p. 23, http://www.iau.org/static/publications/stylemanual1989.pdf 
  5. ↑ See, for example, the Merriam-Webster's Collegiate Dictionary, the American Heritage Dictionary, and the Oxford English Dictionary.
  6. ↑ Resolution No. 10 of the XVIth General Assembly of the International Astronomical Union, Grenoble, 1976
  7. ↑ "2009 Selected Astronomical Constants" in The Astronomical Almanac Online, USNO–UKHO, http://asa.usno.navy.mil/ 
  8. ↑ 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, doi:10.1007/s10569-009-9203-8, http://www.springerlink.com/content/21885q7262104u76/ 
  9. ↑ IAU WG on NSFA Current Best Estimates, http://maia.usno.navy.mil/NSFA/CBE.html, retrieved 2009-09-25 
  10. ↑ "The Final Session of the General Assembly", Estrella d'Alva: 1, 2009-08-14, http://www.astronomy2009.com.br/10.pdf 
  11. ↑ 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 
  12. ↑ Huang, T.-Y.; Han, C.-H.; Yi, Z.-H.; Xu, B.-X. (1995), "What is the astronomical unit of length?", Astron. Astrophys. 298: 629–33, http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1995A%26A...298..629H&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf 
  13. ↑ 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 
  14. ↑ Engels, Donald (1985), "The Length of Eratosthenes' Stade", Am. J. Philol. (The Johns Hopkins University Press) 106 (3): 298–311, doi:10.2307/295030, http://jstor.org/stable/295030 
  15. ↑ Gulbekian, Edward (1987), "The origin and value of the stadion unit used by Eratosthenes in the third century B.C.", Arch. Hist. Exact Sci. 37 (4): 359–63, doi:10.1007/BF00417008 (inactive 2010-01-09) 
  16. ↑ Rawlins, D. (2008), "Eratothenes' large earth and tiny universe", DIO 14: 3–12, http://www.dioi.org/vols/we0.pdf 
  17. ↑ pp. 127, 129, in "Hipparchus on the distances of the sun and moon", G. J. Toomer, Archive for History of Exact Sciences, 14, #2 (June 1974), pp. 126–142, doi:10.1007/BF00329826
  18. ↑ 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 
  19. ↑ Goldstein, Bernard R. (1967), "The Arabic Version of Ptolemy's Planetary Hypotheses", Trans. Am. Phil. Soc. 57 (4): 9–12 
  20. ↑ 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 
  21. ↑ 21.0 21.1 pp. 16–19, van Helden 1985
  22. ↑ p. 251, Ptolemy's Almagest, translated and annotated by G. J. Toomer, London: Duckworth, 1984, ISBN 0-7156-1588-2
  23. ↑ pp. 29–33, van Helden 1985
  24. ↑ pp. 41–53, van Helden 1985
  25. ↑ 25.0 25.1 Weaver, Harold F. (1943), The Solar Parallax "The Solar Parallax", Astron. Soc. Pacific Leaflets 4: 144–51, http://adsabs.harvard.edu/abs/1943ASPL....4..144W The Solar Parallax 
  26. ↑ Goldstein, S. J., Jr. (1985), "Christiaan Huygens' Measurement of the Distance to the Sun", Observatory 105: 32–33, http://articles.adsabs.harvard.edu//full/1985Obs...105...32G/0000033.000.html 
  27. ↑ 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 
  28. ↑ 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 2009-11-15 
  29. ↑ Conférence internationale des étoiles fondamentales, Paris, 18–21 May 1896
  30. ↑ Resolution No. 4 of the XIIth General Assembly of the International Astronomical Union, Hamburg, 1964
  31. ↑ astronomical unit Merriam-Webster's Online Dictionary
  32. ↑ 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, http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1909MNRAS..69..544H&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf 
  33. ↑ 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 
  34. ↑ Mikhailov, A. A. (1964), "The Constant of Aberration and the Solar Parallax", Sov. Astron. 7 (6): 737–39, http://adsabs.harvard.edu/abs/1964SvA.....7..737M 
  35. ↑ Noedlinger, Peter D., "Solar Mass Loss, the Astronomical Unit, and the Scale of the Solar System", Celest. Mech. Dynam. Astron., http://arxiv.org/ftp/arxiv/papers/0801/0801.3807.pdf 
  36. ↑ "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 
  37. ↑ Capitaine, N; Guinot, B (2008). "The astronomical units". arΧiv:0812.2970v1 [astro-ph]. 
  38. ↑ 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, doi:10.1007/s10569-004-0633-z, http://www.springerlink.com/content/g5051650115444k9/ 
  39. ↑ Standish, E. M. (2004), "The Astronomical Unit now", Proc. Int. Astron. Union 2004: 163–79, doi:10.1017/S1743921305001365 
  40. ↑ Lammerzahl, C; Preuss, O; Dittus, H (2006). "Is the physics within the Solar System really understood?". arΧiv:gr-qc/0604052v1 [gr-qc]. 
  41. ↑ Pitjeva, E. V. (2005), "Relativistic effects and solar oblateness from radar observations of planets and spacecraft", Astron. Lett. 31 (5): 340–49, doi:10.1134/1.1922533, http://iau-comm4.jpl.nasa.gov/EVP.pdf 
  42. ↑ 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: 006, doi:10.1088/1475-7516/2005/09/006, JCAP09(2005)006, http://arxiv.org/PS_cache/gr-qc/pdf/0508/0508047v2.pdf 
  43. ↑ Chris Peat, Spacecraft escaping the Solar System, Heavens-Above, http://www.heavens-above.com/solar-escape.asp, retrieved 2008-01-25 
  44. ↑ 44.0 44.1 Chebotarev, G.A. (1964), "Gravitational Spheres of the Major Planets, Moon and Sun", Soviet Astronomy 7 (5): 618–622, http://adsabs.harvard.edu/abs/1964SvA.....7..618C, retrieved 2010-03-24 

Further reading

External links

Units of length used in Astronomy
Astronomical system of units
Earth radius (\scriptstyle R_\oplus or RE) · Light-second (ls) · Solar radius (\scriptstyle R_\odot) · gigametre (Gm) · Astronomical unit (AU) · light-year (ly) · parsec (pc) · kiloparsec (kpc) · megaparsec (Mpc) · gigaparsec (Gpc)
See Also: Cosmic distance ladder · Orders of magnitude (length) · Conversion of units