Astronomical unit

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astronomical unit
Unit system:Astronomical system of units
(Accepted for use with the SI)
Unit of...length
Symbol:au
Unit conversions
1 au in...is equal to...
   km   149.6×106
   mi   92.956×106
   pc   4.8481×10−6
   ly   15.813×10−6
 
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astronomical unit
Unit system:Astronomical system of units
(Accepted for use with the SI)
Unit of...length
Symbol:au
Unit conversions
1 au in...is equal to...
   km   149.6×106
   mi   92.956×106
   pc   4.8481×10−6
   ly   15.813×10−6

An astronomical unit (abbreviated as au;[1] other abbreviations that are sometimes used include , a.u. and ua[2]) is a unit of length now defined as exactly 149,597,870,700 m (92,955,807.3 mi),[3] or roughly the average EarthSun distance.

Definition[edit]

The astronomical unit was originally defined as the length of the semi-major axis of the Earth's elliptical orbit around the Sun.

In 1976 for greater precision, the International Astronomical Union (IAU) formally adopted the definition that "the astronomical unit of length is that length (A) for which the Gaussian gravitational constant (k) takes the value 0.01720209895 when the units of measurement are the astronomical units of length, mass and time".[4][5][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 0.01720209895 radians per day;[7] or that length such that, when used to describe the positions of the objects in the Solar System, the heliocentric gravitational constant (the product GM) is equal to (0.01720209895)2 au3/d2.

In the IERS numerical standards, the speed of light in a vacuum is defined as c0 = 299,792,458 m/s, in accordance with the SI units. The time to traverse an au is found to be τA = 499.0047838061±0.00000001 s, resulting in the astronomical unit in metres as c0τA = 149,597,870,700±3 m.[8] It is approximately equal to the distance from the Earth to the Sun.

The 1976 value of the astronomical unit was indirectly derived from physical analysis of the motion of the Earth around the Sun, while it had since become possible to measure the distance to celestial bodies directly.[9][10] Furthermore, it was subject to relativity and thus was not constant for all observers. Therefore, in 2012 the IAU redefined the astronomical unit as a conventional unit of length directly tied to the meter, with a length of exactly 149,597,870,700 m and the official abbreviation of au.[9][11]

To gain a sense of scale as to how far one au is, if the Sun were to be scaled down to the size of an NBA basketball (24 cm diameter) then the Earth would be half the diameter of a .177 caliber BB pellet, and at this scale one au is the distance between the two hoops on a basketball court.

Modern determinations[edit]

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: an assembly of calculated positions is often referred to as an ephemeris, in which distances are commonly calculated in astronomical units. One of several ephemeris computation services is provided by the Jet Propulsion Laboratory.[12]

The comparison of the ephemeris with the measured positions leads to a value for the speed of light in astronomical units, which is 173.144 632 6847(69) au/d (TDB).[13] As the speed of light in meters per second (c0) 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):

A = 86\,400 \frac{c_{\rm 0}}{c_{\rm AU}}.

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.[14][15][16]

Usage[edit]

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 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."[17] As such, the meter 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,[18] and "vigorous debate" ensued [19] until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = 149597870700 meters.

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.[20]

History[edit]

According to Archimedes in the Sandreckoner (2.1), Aristarchus of Samos estimated the distance to the Sun to be 10,000 times the Earth's radius (the true value is about 23,000).[21] 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 1,520 Earth radii.[22]

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 80000" 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 80000 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 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,[23][24] 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%).[25] 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.[26]

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 1000 li apart and the assumption that the Earth is flat.[27]

 Solar
parallax
Earth
radii
Archimedes in Sandreckoner
(3rd century BC)
40″10,000
Aristarchus in On Sizes (3rd century BC)380-1,520
Hipparchus (2nd century BC)7′ 490
Ptolemy (2nd century)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
modern8.794143″23,455

In the 2nd century CE, Ptolemy estimated the mean distance of the Sun as 1,210 times the Earth radius.[28][29] 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.[30][31] 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.[30]

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.[32] 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.[33]

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.[34] 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[35]). 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\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,[36] 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 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[37] 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.[38] The various results were collated by Jérôme Lalande to give a figure for the solar parallax of 8.6″.

DateMethodA/GmUncertainty
1895aberration149.250.12
1941parallax149.6740.016
1964radar149.59810.001
1976telemetry149.597 8700.000 001
2009telemetry149.597 870 7000.000 000 003

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,[39] which remained in place for the calculation of ephemerides until 1964.[40] The name "astronomical unit" appears first to have been used in 1903.[41]

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.[42] Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931.[35][43]

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.[44]

Developments[edit]

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 meters) can be expressed in terms of other astronomical constants:

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

where G is the Newtonian gravitational constant, M 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,[45] 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.[46] There have also been calls to redefine the astronomical unit in terms of a fixed number of meters.[47]

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.[48][49]

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.[50]

Examples[edit]

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[edit]

Other views[edit]

In 2006 the BIPM defined the astronomical unit as 1.495 978 706 91 (6) × 10^11 m, and recommended "ua" as the symbol for the unit.[2]

See also[edit]

Notes and references[edit]

  1. ^ "RESOLUTION B2 on the re-definition of the astronomical unit of length", RESOLUTION B2, Beijing, Kina: International Astronomical Union, 31 August 2012, retrieved 2013-05-11, "The XXVIII General Assembly of International Astronomical Union recommends [adopted] … that the unique symbol “au” be used for the astronomical unit." 
  2. ^ a b Bureau International des Poids et Mesures (2006), The International System of Units (SI) (8th ed.), Organisation Intergouvernementale de la Convention du Mètre, p. 126 
  3. ^ "RESOLUTION B2 on the re-definition of the astronomical unit of length", RESOLUTION B2, Beijing, Kina: International Astronomical Union, 31 August 2012, retrieved 2012-09-19, "The XXVIII General Assembly of International Astronomical Union recommends [adopted] that the astronomical unit be re-defined to be a conventional unit of length equal to exactly 149 597 870 700 meters, in agreement with the value adopted in IAU 2009 Resolution B2" 
  4. ^ Resolution No. 10 of the XVIth General Assembly of the International Astronomical Union, Grenoble, 1976
  5. ^ 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 3-540-88054-2 
  6. ^ Gareth V Williams (1997), "Astronomical unit", in James H. Shirley, Rhodes Whitmore Fairbridge, Encyclopedia of planetary sciences, Springer, p. 48, ISBN 0-412-06951-2 
  7. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), p. 126, ISBN 92-822-2213-6 
  8. ^ Gérard Petit and Brian Luzum, eds. [clarification needed] (2010). "Table 1.1: IERS numerical standards". IERS technical note no. 36: General definitions and numerical standards. International Earth Rotation and Reference Systems Service.  For complete document see Gérard Petit and Brian Luzum, eds. [clarification needed] (2010). IERS Conventions (2010): IERS technical note no. 36. International Earth Rotation and Reference Systems Service. ISBN 978-3-89888-989-6. 
  9. ^ a b Capitaine, Nicole; Klioner, Sergei; McCarthy, Dennis (2012), "The re-definition of the astronomical unit of length:reasons and consequences", IAU Joint Discussion 7: Space-Time Reference Systems for Future Research, Beijing, China, Bibcode:2012IAUJD...7E..40C, retrieved 16-May-2013 
  10. ^ "Table 6: Units outside the SI that are accepted for use with the SI". The NIST reference on constants, units, and uncertainty: International system of units (SI). NIST, USA. Retrieved 2012-01-16. 
  11. ^ Geoff Brumfiel (14-Sep-2012). "The astronomical unit gets fixed: Earth–Sun distance changes from slippery equation to single number.". Retrieved 14-Sep-2012. 
  12. ^ "HORIZONS System". Solar system dynamics. NASA: Jet Propulsion Laboratory. 4 January 2005. Retrieved 2012-01-16. 
  13. ^ "2009 Selected Astronomical Constants" in The Astronomical Almanac Online, USNOUKHO 
  14. ^ IAU WG on NSFA Current Best Estimates, retrieved 25 September 2009 
  15. ^ 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 
  16. ^ "The Final Session of the General Assembly", Estrella d'Alva, 14 August 2009: 1 
  17. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), pp. 166–67, ISBN 92-822-2213-6 
  18. ^ 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 
  19. ^ Richard Dodd (2011). "§6.2.3: Astronomical unit: Definition of the astronomical unit, future versions". Using SI Units in Astronomy. Cambridge University Press. p. 76. ISBN 0-521-76917-5.  and also p. 91, Summary and recommendations.
  20. ^ See, for example, the work cited above: Richard Dodd. "§6.2.8: Light year". Using SI Units in Astronomy. p. 82. ISBN 0-521-76917-5. 
  21. ^ Gomez, A. G. (2013) Aristarchos of Samos, the Polymath AuthorHouse, ISBN 9781481789493.
  22. ^ 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 
  23. ^ 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 
  24. ^ 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) 
  25. ^ Rawlins, D. (2008), "Eratothenes' large earth and tiny universe", DIO 14: 3–12 
  26. ^ Toomer, G. J. (1974). "Hipparchus on the distances of the sun and moon". Archive for History of Exact Sciences 14 (2): 126–142. doi:10.1007/BF00329826.  edit
  27. ^ 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 
  28. ^ Goldstein, Bernard R. (1967), "The Arabic Version of Ptolemy's Planetary Hypotheses", Trans. Am. Phil. Soc. 57 (4): 9–12 
  29. ^ 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 
  30. ^ a b pp. 16–19, van Helden 1985
  31. ^ p. 251, Ptolemy's Almagest, translated and annotated by G. J. Toomer, London: Duckworth, 1984, ISBN 0-7156-1588-2
  32. ^ pp. 29–33, van Helden 1985
  33. ^ pp. 41–53, van Helden 1985
  34. ^ An extended historical discussion of this method is provided by Trudy E Bell. "Quest for the astronomical unit". The Bent of Tau Beta Pi, Summer 2004, p. 20. Retrieved 2012-01-16. 
  35. ^ a b Weaver, Harold F. (1943), "The Solar Parallax", Astronomical Society of the Pacific Leaflets 4: 144–51, Bibcode:1943ASPL....4..144W 
  36. ^ Goldstein, S. J., Jr. (1985), "Christiaan Huygens' Measurement of the Distance to the Sun", Observatory 105: 32–33, Bibcode:1985Obs...105...32G 
  37. ^ Halley, E. (1716), "A new Method of determining the Parallax of the Sun, or his Distance from the Earth", Philosophical Transactions of the Royal Society 29: 454–64 
  38. ^ Pogge, Richard (May 2004), How Far to the Sun? The Venus Transits of 1761 & 1769, Ohio State University, retrieved 15 November 2009 
  39. ^ Conférence internationale des étoiles fondamentales, Paris, 18–21 May 1896
  40. ^ Resolution No. 4 of the XIIth General Assembly of the International Astronomical Union, Hamburg, 1964
  41. ^ astronomical unit Merriam-Webster's Online Dictionary
  42. ^ 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 
  43. ^ 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 
  44. ^ Mikhailov, A. A. (1964), "The Constant of Aberration and the Solar Parallax", Sov. Astron. 7 (6): 737–39, Bibcode:1964SvA.....7..737M 
  45. ^ Noerdlinger, 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 
  46. ^ "AU may need to be redefined", New Scientist, 6 February 2008 
  47. ^ Capitaine, N; Guinot, B (2008). "The astronomical units". arXiv:0812.2970v1 [astro-ph].
  48. ^ 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): 363, arXiv:1108.5546, Bibcode:2011CeMDA.111..363F, doi:10.1007/s10569-011-9377-8 
  49. ^ 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. 
  50. ^ Fienga, A.et al. (2011), "The INPOP10a planetary ephemeris and its applications in fundamental physics", Celest. Mech. Dynam. Astron. 111 (3): 363, arXiv:1108.5546, Bibcode:2011CeMDA.111..363F, doi:10.1007/s10569-011-9377-8 
  51. ^ 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. 
  52. ^ "AstDys (90377) Sedna Ephemerides". Department of Mathematics, University of Pisa, Italy. Retrieved 2011-05-05. 
  53. ^ Chris Peat, Spacecraft escaping the Solar System, Heavens-Above, retrieved 25 January 2008 
  54. ^ Voyager 1, Where are the Voyagers – NASA Voyager 1
  55. ^ a b Chebotarev, G.A. (1964), "Gravitational Spheres of the Major Planets, Moon and Sun", Soviet Astronomy 7 (5): 618–622, Bibcode:1964SvA.....7..618C 
  56. ^ [1], Resolutions adopted at the 2012 General Assembly of the IAU

Bibliography[edit]

External links[edit]