Astronomical unit

From Infogalactic: the planetary knowledge core
(Redirected from Astronomical Unit)
Jump to: navigation, search

<templatestyles src="Module:Hatnote/styles.css"></templatestyles>

This article is about the unit of length. For constants, see astronomical constant. For units in astronomy, see astronomical system of units.
Astronomical unit
Astronomical unit.png
The red line indicates the Earth–Sun distance, which on average is about 1 astronomical unit
Unit information
Unit system Astronomical system of units
(Accepted for use with the SI)
Unit of length
Symbol au, AU or ua 
Unit conversions
1 au, AU or ua in ... ... is equal to ...
   metric (SI) units    1.4960×1011 m
   imperial & US units    9.2956×107 mi
   astronomical units    4.8481×10−6 pc
   1.5813×10−5 ly

The astronomical unit (symbol au,[1][2] AU[3][4][5] or ua[6]) is a unit of length, roughly the distance from Earth to the Sun. However, that distance varies as Earth orbits the Sun, from a maximum (aphelion) to a minimum (perihelion) and back again once a year. Originally conceived as the average of Earth's aphelion and perihelion, it is now defined as exactly 149597870700 metres (about 150 million kilometres, or 93 million miles).[7] The astronomical unit is used primarily as a convenient yardstick for measuring distances within the Solar System or around other stars. However, it is also a fundamental component in the definition of another unit of astronomical length, the parsec.

Symbol usage

A variety of unit symbols and abbreviations are in use for the astronomical unit. In a 1976 resolution, the International Astronomical Union (IAU) used the symbol A for the AU.[8] In 2006, the International Bureau of Weights and Measures (BIPM) recommended ua as the symbol for the unit.[9] In 2012, the IAU, noting "that various symbols are presently in use for the astronomical unit", recommended the use of the symbol "au".[1] In the 2014 revision of the SI Brochure, the BIPM used the unit symbol "au".[10] The symbol "AU"[3][4][5] and abbreviation a.u.[11][12] are also used. In ISO 80000-3, the symbol of the astronomical unit is "ua".

Development of unit definition

<templatestyles src="Module:Hatnote/styles.css"></templatestyles>

See also: Earth's orbit

Earth's orbit around the Sun is an ellipse. The semi-major axis of this ellipse is defined to be half of the straight line segment that joins the aphelion and perihelion. The centre of the sun lies on this straight line segment, but not at its midpoint. Because 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 on observation. In addition, it mapped out exactly the largest straight-line distance that Earth traverses over the course of a year, defining times and places for observing the largest parallax (apparent shifts of position) in nearby stars. Knowing 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 astronomical unit 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 on 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.[13]

In 1976, in order to establish a yet more precise measure for the astronomical unit, the IAU formally adopted a new definition. Although 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 0.01720209895 when the units of measurement are the astronomical units of length, mass and time".[8][14][15] Equivalently, by this definition, 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 0.01720209895 radians per day;[16] or alternatively that length for which the heliocentric gravitational constant (the product GM☉) is equal to (0.01720209895)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. Because 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. However, for precision the calculations require adjustment for things such as the motions of the probe and object while the photons are transiting. 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 86400 s). By 2009, the IAU had updated its standard measures to reflect improvements, and calculated the speed of light at 173.1446326847(69) AU/d (TDB).[17]

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 other measurements had become too precise for reference to the prototype platinum metre to remain useful. Instead, the metre 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 = 299792458 m/s, a standard also adopted by the IERS numerical standards.[18] From this definition and the 2009 IAU standard, the time for light to traverse an AU is found to be τA = 499.0047838061±0.00000001 s, more than 8 minutes. By simple multiplication then, the best IAU 2009 estimate was A = c0τA = 149597870700±3 m,[19] based on a comparison of JPL and IAA–RAS ephemerides.[20][21][22]

In 2006, the BIPM reported a value of the astronomical unit as 1.49597870691(6)×1011 m.[9] In the 2014 revision of the SI Brochure, the BIPM recognised the IAU's 2012 redefinition of the astronomical unit as 149597870700 m.[10]

This estimate was still derived from observation and measurements subject to error, and based on 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 149597870700 m).[19][23] The new definition also recognizes as a consequence that the astronomical unit is now to play a role of reduced importance, limited in its use to that of a convenience in some applications.[19]

1 astronomical unit   = 149597870700 metres (exactly)
≈ 92.955807 million miles
≈ 4.8481368 millionths of a parsec
≈ 15.812507 millionths of a light-year

This definition makes the speed of light, defined as exactly 299792458 m/s, equal to exactly 299792458 × 86400 ÷ 149597870700 or about 173.144632674240... AU/d, some 60 parts per trillion less than the 2009 estimate.

Usage and significance

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, so 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 Earth's surface (terrestrial time, TT) are not constant when compared to the motions of the planets: the terrestrial second (TT) appears to be longer during the Northern Hemisphere winter and shorter during the Northern Hemisphere summer when compared to the "planetary second" (conventionally measured in barycentric dynamical time, TDB). This is because the distance between Earth and the Sun is not fixed (it varies between 0.9832898912 and 1.0167103335 AU) and, when Earth is closer to the Sun (perihelion), the Sun's gravitational field is stronger and 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".[24] 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 because it did 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,[25] and "vigorous debate" ensued[26] until in August 2012 the IAU adopted the current definition of 1 astronomical unit = 149597870700 metres.

The astronomical unit is typically used for stellar system scale distances, such as the size of a protostellar disk or the heliocentric distance of an asteroid, whereas other units are used for other distances in astronomy. The astronomical unit is too small to be convenient for interstellar distances, where the parsec and light year are widely used. The parsec (parallax arcsecond) is defined in terms of the astronomical unit, being the distance of an object with a parallax of 1 arcsecond. The light year is often used in popular works, but is not an approved non-SI unit and is rarely used by professional astronomers.[27]

History

According to Archimedes in the Sandreckoner (2.1), Aristarchus of Samos estimated the distance to the Sun to be 10000 times Earth's radius (the true value is about 23000).[28] 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 that Van Helden assumes Aristarchus used for the distance to the Moon, his calculated distance to the Sun would fall between 380 and 1520 Earth radii.[29]

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 4080000 stadia (1903 translation by Edwin Hamilton Gifford), or as 804000000 stadia (edition of Édouard des Places, dated 1974–1991). Using the Greek stadium of 185 to 190 metres,[30][31] the former translation comes to 754800 km to 775200 km, which is far too low, whereas the second translation comes to 148.7 to 152.8 million kilometres (accurate within 2%).[32] Hipparchus also gave an estimate of the distance of Earth from the Sun, 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.[33]

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 Earth is flat.[34]

Distance to the Sun
estimated by		 Estimate in AU
Solar
parallax		 Earth
radii
Archimedes (3rd century BC)
(in The Sand Reckoner)		 40″ 10000 0.426
Aristarchus (3rd century BC)
(in On the Sizes and Distances)  		 – 380-1520 0.016-0.065
Hipparchus (2nd century BC) 7′ 490 0.021
Posidonius (1st century BC)
(quoted by coeval Cleomedes)		 – 10000 0.426
Ptolemy (2nd century) 2′ 50″ 1210 0.052
Godefroy Wendelin (1635) 15″ 14000 0.597
Jeremiah Horrocks (1639) 15″ 14000 0.597
Christiaan Huygens (1659) 8.6″ 24000 1.023
Cassini & Richer (1672) 9<templatestyles src="Sfrac/styles.css" />1/2″ 21700 0.925
Jérôme Lalande (1771) 8.6″ 24000 1.023
Simon Newcomb (1895) 8.80″ 23440 0.9994
Arthur Hinks (1909) 8.807″ 23420 0.9985
H. Spencer Jones (1941) 8.790″ 23466 1.0005
modern Astronomy 8.794143″ 23455 1.0000
Sources:

In the 2nd century CE, Ptolemy estimated the mean distance of the Sun as 1210 times Earth's radius.[35][36] 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<templatestyles src="Sfrac/styles.css" />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.[37][38] He then measured the apparent sizes of the Sun and the Moon and concluded that the apparent diameter of the Sun was equal to the apparent diameter of the Moon at the Moon's greatest distance, and from records of lunar eclipses, he estimated this apparent diameter, as well as the apparent diameter of the shadow cone of Earth traversed by the Moon during a lunar eclipse. Given these data, the distance of the Sun from Earth can be trigonometrically computed to be 1210 Earth radii. This gives a ratio of solar to lunar distance of approximately 19, matching Aristarchus's figure. Although Ptolemy's procedure is theoretically workable, it is very sensitive to small changes in the data, so much so that changing a measurement by a few percent can make the solar distance infinite.[37]

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 1170 Earth radii, whereas in his zij, al-Battānī used a mean solar distance of 1108 Earth radii. Subsequent astronomers, such as al-Bīrūnī, used similar values.[39] Later in Europe, Copernicus and Tycho Brahe also used comparable figures (1142 and 1150 Earth radii), and so Ptolemy's approximate Earth–Sun distance survived through the 16th century.[40]

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

A somewhat more accurate estimate can be obtained by observing the transit of Venus.[41] By measuring the transit in two different locations, one can accurately calculate the parallax of Venus and from the relative distance of Earth and Venus from the Sun, the solar parallax α (which cannot be measured directly[42]). 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 Earth: a solar parallax of 15" is equivalent to an Earth–Sun distance of 13750 Earth radii.

Christiaan Huygens believed that the distance was even greater: by comparing the apparent sizes of Venus and Mars, he estimated a value of about 24000 Earth radii,[43] 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 9<templatestyles src="Sfrac/styles.css" />1/2", equivalent to an Earth–Sun distance of about 22000 Earth radii. They were also the first astronomers to have access to an accurate and reliable value for the radius of Earth, which had been measured by their colleague Jean Picard in 1669 as 3269 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[44] 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.[45] 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. 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,[46] which remained in place for the calculation of ephemerides until 1964.[47] The name "astronomical unit" appears first to have been used in 1903.[48]

The discovery of the near-Earth asteroid 433 Eros and its passage near Earth in 1900–1901 allowed a considerable improvement in parallax measurement.[49] Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931.[42][50]

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.[51]

Developments

The astronomical unit is used as the baseline of the triangle to measure stellar parallaxes (distances in the image are not to scale).

The unit distance A (the value of the astronomical unit in metres) 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,[52] 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.[53]

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.[54][55]

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.[56]

Examples

The following table contains some distances given in astronomical units. It includes some examples with distances that are normally not given in astronomical units, because they are either too short or far too long. Distances normally change over time. Examples are listed by increasing distance.

Object Length or distance (AU) Range Comment and reference point Refs
Earth's circumference 0.0003 – circumference of Earth at the equator (about 40075 km or 24901 mi) –
Light-second 0.002  – distance light travels in one second –
Lunar distance 0.0026 – average distance from Earth (which the Apollo missions took about 3 days to travel) –
Solar radius 0.005  – radius of the Sun (695500 km, 432450 mi, ~110 times the radius of Earth or 10 times the average radius of Jupiter) –
Lagrangian point 0.01   – The Lagrangian point L2 is about 1500000 km (930000 mi) from Earth. Unmanned space missions, such as the James Webb Space Telescope, Planck and Gaia take advantage of this sun-shielded location. [57]
Light-minute 0.12   – distance light travels in one minute –
Mercury 0.39   – average distance from the Sun –
Venus 0.72   – average distance from the Sun –
Earth 1.00   – average distance of Earth's orbit from the Sun (sunlight travels for 8 minutes and 19 seconds before reaching Earth) –
Mars 1.52   – average distance from the Sun –
Ceres 2.77   – average distance from the Sun. The only dwarf planet in the asteroid belt. –
Jupiter 5.20   – average distance from the Sun –
Betelgeuse 5.5    – star's mean diameter (It is a red supergiant with about 1000 solar radii.) –
Light-hour 7.2    – distance light travels in one hour –
NML Cygni 7.67   – radius of one of the largest known stars –
Saturn 9.58   – average distance from the Sun –
Uranus 19.23   – average distance from the Sun –
Neptune 30.10   – average distance from the Sun –
Kuiper belt 30      – begins at roughly that distance from the Sun [58]
New Horizons 32.92   – spacecraft's distance from the Sun, as of 16 July 2015 [59]
Pluto 39.3    – average distance from the Sun (It varies by 9.6 AU due to orbital eccentricity.) –
Scattered disc 45      – roughly begins at that distance from the Sun (it overlaps with the Kuiper belt.) –
Kuiper belt 50      ± 3 ends at that distance from the Sun –
Eris 67.8    – its semi-major axis –
90377 Sedna 76      – closest distance from the Sun (perihelion) –
90377 Sedna 87      – distance from the Sun as of 2012 (It is an object of the scattered disc and takes 11400 years to orbit the Sun.) [60]
Termination shock 94      – distance from the Sun of boundary between solar winds/interstellar winds/interstellar medium –
Eris 96.4    – distance from the Sun as of 2014 (Eris and its moon are currently the most distant known objects in the Solar System apart from long-period comets and space probes, and roughly three times as far as Pluto.) [61]
Heliosheath 100      – the region of the heliosphere beyond the termination shock, where the solar wind is slowed down, more turbulent and compressed due to the interstellar medium –
Voyager 1 125      – As of August 2013, the space probe is the furthest human-made object from the Sun. It is traveling at about 3.5 astronomical units per year. [62]
Light-day 173      – distance light travels in one day –
90377 Sedna 942      – farthest distance from the Sun (aphelion) –
Hills cloud 2000      ± 1000 beginning of Hills cloud (It is the inner part of the Oort cloud and shaped like a disc or doughnut.) –
Hills cloud 20000      – end of the inner Oort cloud, beginning of outer Oort cloud, which is weakly bound to the Sun and believed to have a spherical shape –
Light-year 63241      – distance light travels in one Julian year (365.25 days) –
Oort cloud 75000      ± 25000 distance of the outer limit of Oort cloud from the Sun (estimated, corresponds to 1.2 light-years) –
Parsec 206265      – one parsec (The parsec is defined in terms of the astronomical unit, is used to measure distances beyond the scope of the Solar System and is about 3.26 light-years.) [63]
Hill/Roche sphere 230000      – maximum extent of the Sun's gravitational field, beyond this is true interstellar medium (~3.6 light-years) [64]
Proxima Centauri 268000      ± 126 distance to the nearest star to the Solar System –
Sirius 544000      – distance to the brightest star seen in Earth's night sky (~8.6 light-years) –
Betelgeuse 40663000      – distance to Betelgeuse in the constellation of Orion (~643 light-years) –
Galactic Centre 1700000000      – distance from the Sun to the centre of the Milky Way –
Note: figures in this table are generally rounded, estimates, often rough estimates, and may considerably differ from other sources. Table also includes other units of length for comparison.

See also

References

  1. ↑ 1.0 1.1 Lua error in package.lua at line 80: module 'strict' not found.
  2. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  3. ↑ 3.0 3.1 Lua error in package.lua at line 80: module 'strict' not found.
  4. ↑ 4.0 4.1 Lua error in package.lua at line 80: module 'strict' not found.
  5. ↑ 5.0 5.1 Lua error in package.lua at line 80: module 'strict' not found.
  6. ↑ ISO 80000-3, Quantities and units - Space and time
  7. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  8. ↑ 8.0 8.1 Resolution No. 10 of the XVIth General Assembly of the International Astronomical Union, Grenoble, 1976
  9. ↑ 9.0 9.1 Lua error in package.lua at line 80: module 'strict' not found.
  10. ↑ 10.0 10.1 Lua error in package.lua at line 80: module 'strict' not found.
  11. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  12. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  13. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  14. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  15. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  16. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  17. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  18. ↑ Lua error in package.lua at line 80: module 'strict' not found. For complete document see Lua error in package.lua at line 80: module 'strict' not found.
  19. ↑ 19.0 19.1 19.2 Lua error in package.lua at line 80: module 'strict' not found.
  20. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  21. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  22. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  23. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  24. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  25. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  26. ↑ Lua error in package.lua at line 80: module 'strict' not found. and also p. 91, Summary and recommendations.
  27. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  28. ↑ Gomez, A. G. (2013) Aristarchos of Samos, the Polymath AuthorHouse, ISBN 978-1481789493.
  29. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  30. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  31. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  32. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  33. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  34. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  35. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  36. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  37. ↑ 37.0 37.1 pp. 16–19, van Helden 1985
  38. ↑ p. 251, Ptolemy's Almagest, translated and annotated by G. J. Toomer, London: Duckworth, 1984, ISBN 0-7156-1588-2
  39. ↑ pp. 29–33, van Helden 1985
  40. ↑ pp. 41–53, van Helden 1985
  41. ↑ An extended historical discussion of this method is provided by Lua error in package.lua at line 80: module 'strict' not found.
  42. ↑ 42.0 42.1 Lua error in package.lua at line 80: module 'strict' not found.
  43. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  44. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  45. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  46. ↑ Conférence internationale des étoiles fondamentales, Paris, 18–21 May 1896
  47. ↑ Resolution No. 4 of the XIIth General Assembly of the International Astronomical Union, Hamburg, 1964
  48. ↑ astronomical unit Merriam-Webster's Online Dictionary
  49. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  50. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  51. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  52. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  53. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  54. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  55. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  56. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  57. ↑ http://www.esa.int What are Lagrange points, 21 June 2013
  58. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  59. ↑ As of 16 July 2015 [1]
  60. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  61. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  62. ↑ Voyager 1, Where are the Voyagers – NASA Voyager 1
  63. ↑ http://www.iau.org, Measuring the Universe–The IAU and astronomical units
  64. ↑ Lua error in package.lua at line 80: module 'strict' not found.

Further reading

  • Lua error in package.lua at line 80: module 'strict' not found.

External links

Retrieved from "https://infogalactic.com/w/index.php?title=Astronomical_unit&oldid=479"