Astronomical unit | |
---|---|
The grey line indicates the Earth–Sun distance, which on average is about 1 astronomical unit. | |
General information | |
Unit system | Astronomical system of units (Accepted for use with the SI) |
Unit of | length |
Symbol | au or AU or AU |
Conversions | |
1 au or AU or AU in ... | ... is equal to ... |
metric (SI) units | 1.495978707×10^{11} m |
imperial & US units | 9.2956×10^{7} mi |
astronomical units | 4.8481×10^{−6} pc 1.5813×10^{−5} ly |
The astronomical unit (symbol: au,^{[1]}^{[2]}^{[3]} or AU or AU) is a unit of length, roughly the distance from Earth to the Sun and equal to about 150 million kilometres (93 million miles) or ~8 light minutes. The actual distance varies by about 3% as Earth orbits the Sun, from a maximum (aphelion) to a minimum (perihelion) and back again once each year. The AU was originally conceived as the average of Earth's aphelion and perihelion; however, since 2012 it has been defined as exactly 149597870700 m.^{[4]}
The astronomical unit is used primarily for measuring distances within the Solar System or around other stars. It is also a fundamental component in the definition of another unit of astronomical length, the parsec.^{[5]}
History of symbol usage
A variety of unit symbols and abbreviations have been in use for the astronomical unit. In a 1976 resolution, the International Astronomical Union (IAU) had used the symbol A to denote a length equal to the astronomical unit.^{[6]} In the astronomical literature, the symbol AU was (and remains) common. In 2006, the International Bureau of Weights and Measures (BIPM) had recommended ua as the symbol for the unit.^{[7]} In the non-normative Annex C to ISO 80000-3:2006 (now withdrawn), the symbol of the astronomical unit is "ua".
In 2012, the IAU, noting "that various symbols are presently in use for the astronomical unit", recommended the use of the symbol "au".^{[1]} The scientific journals published by the American Astronomical Society and the Royal Astronomical Society subsequently adopted this symbol.^{[3]}^{[8]} In the 2014 revision and 2019 edition of the SI Brochure, the BIPM used the unit symbol "au".^{[9]}^{[10]} ISO 80000-3:2019, which replaces ISO 80000-3:2006, does not mention the astronomical unit.^{[11]}^{[12]}
Development of unit definition
Earth's orbit around the Sun is an ellipse. The semi-major axis of this elliptic orbit is defined to be half of the straight line segment that joins the perihelion and aphelion. 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 improved 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 HORIZONS System 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".^{[6]}^{[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} au^{3}/d^{2}, 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 calculated as 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 Barycentric Dynamical Time (TDB) 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 CIPM modified the International System of Units (SI) to make the metre defined as the distance travelled in a vacuum by light in 1 / 299792458 second. This replaced the previous definition, valid between 1960 and 1983, which was that the metre equalled a certain number of wavelengths of a certain emission line of krypton-86. (The reason for the change was an improved method of measuring the speed of light.) The speed of light could then be expressed exactly as c_{0} = 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 astronomical unit is found to be τ_{A} = 499.0047838061±0.00000001 s, which is slightly more than 8 minutes 19 seconds. By multiplication, the best IAU 2009 estimate was A = c_{0}τ_{A} = 149597870700±3 m,^{[19]} based on a comparison of Jet Propulsion Laboratory and IAA–RAS ephemerides.^{[20]}^{[21]}^{[22]}
In 2006, the BIPM reported a value of the astronomical unit as 1.49597870691(6)×10^{11} m.^{[7]} In the 2014 revision of the SI Brochure, the BIPM recognised the IAU's 2012 redefinition of the astronomical unit as 149597870700 m.^{[9]}
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) ≈ 92955807 miles ≈ 499.00478384 light-seconds ≈ 4.8481368×10^{−6} parsecs ≈ 1.5812507×10^{−5} light-years
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 separately, 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 with the motions of the planets: the terrestrial second (TT) appears to be longer near January and shorter near July when compared with the "planetary second" (conventionally measured in 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 with 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 August 2012 when 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″. 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]}
When simulating a numerical model of the Solar System, the astronomical unit provides an appropriate scale that minimizes (overflow, underflow and truncation) errors in floating point calculations.
History
The book On the Sizes and Distances of the Sun and Moon, which is ascribed to Aristarchus, says the distance to the Sun is 18 to 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 1,520 Earth radii.^{[28]}
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 Édourad des Places^{ [de]}, dated 1974–1991). Using the Greek stadium of 185 to 190 metres,^{[29]}^{[30]} 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%).^{[31]} 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′.^{[32]}
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 1,000 li apart and the assumption that Earth is flat.^{[33]}
Distance to the Sun estimated by |
Estimate | In AU | |
---|---|---|---|
Solar parallax |
Earth radii | ||
Aristarchus (3rd century BCE) (in On Sizes) |
13′ 24″–7′ 12″ | 256.5–477.8 | 0.011–0.020 |
Archimedes (3rd century BCE) (in The Sand Reckoner) |
21″ | 10000 | 0.426 |
Hipparchus (2nd century BCE) | 7′ | 490 | 0.021 |
Posidonius (1st century BCE) (quoted by coeval Cleomedes) |
21″ | 10000 | 0.426 |
Ptolemy (2nd century) | 2' 50″ | 1,210 | 0.052 |
Godefroy Wendelin (1635) | 15″ | 14000 | 0.597 |
Jeremiah Horrocks (1639) | 15″ | 14000 | 0.597 |
Christiaan Huygens (1659) | 8.2″ | 25086^{[34]} | 1.068 |
Cassini & Richer (1672) | 9.5″ | 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 |
In the 2nd century CE, Ptolemy estimated the mean distance of the Sun as 1,210 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+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 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 per cent 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 1,170 Earth radii, whereas in his zij, al-Battānī used a mean solar distance of 1,108 Earth radii. Subsequent astronomers, such as al-Bīrūnī, used similar values.^{[39]} Later in Europe, Copernicus and Tycho Brahe also used comparable figures (1,142 and 1,150 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 due to the brightness of the Sun^{[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″, similar to Wendelin’s figure. The solar parallax is related to the Earth–Sun distance as measured in Earth radii by
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,^{[34]} 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.
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 9.5″, 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 3269000 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^{[43]} 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 including observations by James Cook and Charles Green from Tahiti. 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.^{[44]} The various results were collated by Jérôme Lalande to give a figure for the solar parallax of 8.6″. Karl Rudolph Powalky had made an estimate of 8.83″ in 1864.^{[45]}
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 | 149.597870 | 0.000001 |
2009 | telemetry | 149.597870700 | 0.000000003 |
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]}^{[failed verification]}
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 unit distance A (the value of the astronomical unit in metres) can be expressed in terms of other astronomical constants:
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 G×M_{☉} 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 has not been 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 |
---|---|---|---|---|
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, a hundred times the radius of Earth or ten times the average radius of Jupiter) | – |
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 | – |
Jupiter | 5.2 | – | average distance from the Sun | – |
Light-hour | 7.2 | – | distance light travels in one hour | – |
Saturn | 9.5 | – | average distance from the Sun | – |
Uranus | 19.2 | – | average distance from the Sun | – |
Kuiper belt | 30 | – | Inner edge begins at approximately 30 AU | ^{[57]} |
Neptune | 30.1 | – | average distance from the Sun | – |
Eris | 67.8 | – | average distance from the Sun | – |
Voyager 2 | 122 | – | distance from the Sun in 2019 | ^{[58]} |
Voyager 1 | 149 | – | distance from the Sun in 2020 | ^{[58]} |
Light-day | 173 | – | distance light travels in one day | – |
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: 1 pc = 1 AU/tan(1″) | ^{[5]}^{[59]} |
Proxima Centauri | 268000 | ± 126 | distance to the nearest star to the Solar System | – |
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
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... recommends ... 5. that the unique symbol "au" be used for the astronomical unit.
- ^ "Monthly Notices of the Royal Astronomical Society: Instructions for Authors". Oxford Journals. Retrieved 20 March 2015.
The units of length/distance are Å, nm, μm, mm, cm, m, km, au, light-year, pc.
- ^ ^{a} ^{b} "Manuscript Preparation: AJ & ApJ Author Instructions". American Astronomical Society. Archived from the original on 21 February 2016. Retrieved 29 October 2016.
Use standard abbreviations for ... natural units (e.g., au, pc, cm).
- ^ On the re-definition of the astronomical unit of length (PDF). XXVIII General Assembly of International Astronomical Union. Beijing: International Astronomical Union. 31 August 2012. Resolution B2.
... recommends [adopted] that the astronomical unit be re-defined to be a conventional unit of length equal to exactly 149,597,870,700 metres, in agreement with the value adopted in IAU 2009 Resolution B2
- ^ ^{a} ^{b} Luque, B.; Ballesteros, F.J. (2019). "Title: To the Sun and beyond". Nature Physics. 15: 1302. doi:10.1038/s41567-019-0685-3.
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The units of length/distance are Å, nm, µm, mm, cm, m, km, au, light-year, pc.
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- ^ Richard Dodd (2011). "§ 6.2.8: Light-year". Using SI Units in Astronomy. p. 82. ISBN 978-0-521-76917-4.
- ^ van Helden, Albert (1985). Measuring the Universe: Cosmic dimensions from Aristarchus to Halley. Chicago: University of Chicago Press. pp. 5–9. ISBN 978-0-226-84882-2.
- ^ Engels, Donald (1985). "The Length of Eratosthenes' Stade". The American Journal of Philology. 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 17 January 2021).CS1 maint: DOI inactive as of January 2021 (link)
- ^ Rawlins, D. (March 2008). "Eratosthenes' Too-Big Earth & Too-Tiny Universe" (PDF). DIO. 14: 3–12. Bibcode:2008DIO....14....3R.
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- ^ ^{a} ^{b} Most distant space probes.
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Further reading
- Williams, D.; Davies, R. D. (1968). "A radio method for determining the astronomical unit". Monthly Notices of the Royal Astronomical Society. 140 (4): 537. Bibcode:1968MNRAS.140..537W. doi:10.1093/mnras/140.4.537.
External links
- 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