Transit of Venus

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NASA's SDO Satellite Captures First Image of 2012 Venus Transit (Full Disc).jpg
SDO's Ultra-high Definition View of 2012 Venus Transit (171 Angstrom Full Disc).jpg
Images of the transit of Venus, taken from NASA's Solar Dynamics Observatory in 2012: (left) visible light; (right) ultraviolet

A transit of Venus takes place when Venus passes directly between the Sun and the Earth (or any other superior planet), becoming visible against (and hence obscuring a small portion of) the solar disk. During a transit, Venus is visible as a small black circle moving across the face of the Sun.

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Transits of Venus reoccur periodically. A pair of transits takes place eight years apart in December (Gregorian calendar) followed by a gap of 121.5 years, before another pair occurs eight years apart in June, followed by another gap, of 105.5 years. The dates advance by about two days per 243-year cycle. The periodicity is a reflection of the fact that the orbital periods of Earth and Venus are close to 8:13 and 243:395 commensurabilities. The last pairs of transits occurred on 8 June 2004 and 5–6 June 2012. The next pair of transits will occur on 10–11 December 2117 and 8 December 2125.

Transits of Venus were in the past used to determine the size of the Solar System. The 2012 transit has provided research opportunities, particularly in the refinement of techniques to be used in the search for exoplanets.

Conjunctions

Diagram of transits of Venus and the angle between the orbital planes of Venus and Earth Transit diagram angles.png
Diagram of transits of Venus and the angle between the orbital planes of Venus and Earth

The orbit of Venus has an inclination of 3.39° relative to that of the Earth, and so passes under (or over) the Sun when viewed from the Earth. [1] A transit occurs when Venus reaches conjunction with the Sun whilst also passing through the Earth's orbital plane, and passes directly across the face of the Sun.[ citation needed ] [note 1] Sequences of transits usually repeat every 243 years, after which Venus and Earth have returned to nearly the same point in their respective orbits. During the Earth's 243 sidereal orbital periods, which total 88,757.3 days, Venus completes 395 sidereal orbital periods of 224.701 days each, which is equal to 88,756.9 Earth days. This period of time corresponds to 152 synodic periods of Venus. [2]

A pair of transits takes place eight years apart in December, followed by a gap of 121.5 years, before another pair occurs eight years apart in June, followed by another gap, of 105.5 years. Other patterns are possible within the 243-year cycle, because of the slight mismatch between the times when the Earth and Venus arrive at the point of conjunction. Prior to 1518, the pattern of transits was 8, 113.5, and 121.5 years, and the eight inter-transit gaps before the AD 546 transit were 121.5 years apart. The current pattern will continue until 2846, when it will be replaced by a pattern of 105.5, 129.5, and 8 years. Thus, the 243-year cycle is relatively stable, but the number of transits and their timing within the cycle vary over time. [2] Since the 243:395 Earth:Venus commensurability is only approximate, there are different sequences of transits occurring 243 years apart, each extending for several thousand years, which are eventually replaced by other sequences. For instance, there is a series which ended in 541 BC, and the series which includes 2117 only started in AD 1631. [2]

History of observation of the transits

Ancient Indian, Greek, Egyptian, Babylonian, and Chinese observers knew of Venus and recorded the planet's motions. [3] Pythagoras is credited with realizing that the so-called morning and evening stars were really both the planet Venus. There is no evidence that any of these cultures observed planetary transits.[ citation needed ] It has been proposed that frescoes found at the Maya site at Mayapan may contain a pictorial representation of the 12th or 13th century transits. [4]

The Persian polymath Avicenna claimed to have observed Venus as a spot on the Sun. There was a transit on 24 May 1032, but Avicenna did not give the date of his observation, and modern scholars have questioned whether he could have observed the transit from his location; he may have mistaken a sunspot for Venus. He used his alleged transit observation to help establish that Venus was, at least sometimes, below the Sun in Ptolemaic cosmology, [5] i.e., the sphere of Venus comes before the sphere of the Sun when moving out from the Earth in the then prevailing geocentric model. [6] [7]

Transits of Venus (1631–2012)
Date(s) of transitsTime (UTC)Notes [8]
StartMidEnd
7 December 163103:5105:1906:47Predicted by Kepler
4 December 1639 14:5718:2521:54First transit to be observed, by Horrocks and Crabtree
6 June 176102:0205:1908:37 Lomonosov, Chappe d'Auteroche, and others observe from Russia; Mason and Dixon observe from the Cape of Good Hope. John Winthrop observes from St. John's, Newfoundland
3–4 June 1769 19:1522:2501:35 Cook sent to Tahiti to observe the transit, Chappe to San José del Cabo, Baja California, and Maximilian Hell to Vardø, Norway.
9 December 1874 01:4904:0706:26 Pietro Tacchini leads expedition to Muddapur, India. A French expedition goes to New Zealand's Campbell Island, and a British expedition travels to Hawaii.
6 December 1882 13:5717:0620:15
8 June 2004 05:1308:2011:26Various media networks globally broadcast live video of the Venus transit.
5–6 June 2012 22:0901:2904:49Visible in its entirety from the Pacific and Eastern Asia, with the beginning of the transit visible from North America and the end visible from Europe. First transit while a spacecraft orbits Venus.

1631 and 1639 transits

William Richard Lavender, Jeremiah Horrocks (1618-1641) (1903), Astley Hall Museum and Art Gallery Lavender - Jeremiah Horrocks (1618-1641).jpg
William Richard Lavender, Jeremiah Horrocks (1618–1641) (1903), Astley Hall Museum and Art Gallery

The German astronomer Johannes Kepler predicted the 1631 transit in 1627, but his methods were not sufficiently accurate to predict that it could not be seen throughout most of Europe. As a consequence, astronomers were unable to use his prediction to observe the event. [9]

The first recorded observation of a transit of Venus was made by the English astronomer Jeremiah Horrocks from his home at Carr House in Much Hoole, near Preston, on 4 December 1639 (24 November O.S.). His friend William Crabtree observed the transit from nearby Broughton. [10] Kepler had predicted transits in 1631 and 1761 and a near miss in 1639. Horrocks corrected Kepler's calculation for the orbit of Venus, realized that transits of Venus would occur in pairs 8 years apart, and so predicted the transit of 1639. [11] Although he was uncertain of the exact time, he calculated that the transit was to begin at approximately 15:00. Horrocks focused the image of the Sun through a simple telescope and onto paper, where he could observe the Sun without damaging his eyesight. After waiting for most of the day, he eventually saw the transit when clouds obscuring the Sun cleared at about 15:15, half an hour before sunset. His observations allowed him to make a well-informed guess for the diameter of Venus and an estimate of the mean distance between the Earth and the Sun (59.4 million mi (95.6 million km; 0.639 AU)). His observations were not published until 1661, well after Horrocks's death. [11] [note 2] Horrocks based his calculation on the (false) presumption that each planet's size was proportional to its rank from the Sun, not on the parallax effect as used by the 1761 and 1769 and following experiments.[ citation needed ]

1761 transit

Measuring Venus transit times to determine solar parallax Venus Transit & Parallax.svg
Measuring Venus transit times to determine solar parallax

In 1663, the Scottish mathematician James Gregory had suggested in his Optica Promota that observations of a transit of Mercury, at widely spaced points on the surface of the Earth, could be used to calculate the solar parallax, and hence the astronomical unit by means of triangulation. Aware of this, the English astronomer Edmond Halley made observations of such a transit on 28 October O.S. 1677 from the island of Saint Helena, but was disappointed to find that only Richard Towneley in the Lancashire town of Burnley, Lancashire had made another accurate observation of the event, whilst Gallet, at Avignon, had simply recorded that it had occurred. Halley was not satisfied that the resulting calculation of the solar parallax of 45" was accurate.[ citation needed ]

In a paper published in 1691, and a more refined one in 1716, Halley proposed that more accurate calculations could be made using measurements of a transit of Venus, although the next such event was not due until 1761 (6 June N.S., 26 May O.S.). [12] In an attempt to observe the first transit of the pair, astronomers from Britain (William Wales and Captain James Cook), Austria (Maximilian Hell), and France (Jean-Baptiste Chappe d'Auteroche and Guillaume Le Gentil) took part in expeditions to places that included Siberia, Newfoundland, and Madagascar. [13] Most of them observed at least part of the transit. Jeremiah Dixon and Charles Mason succeeded in observing the transit at the Cape of Good Hope, [14] but Nevil Maskelyne and Robert Waddington were less successful on Saint Helena, although they put their voyage to good use by trialling the lunar-distance method of finding longitude. [15]

Venus was generally thought to possess an atmosphere prior to the transit of 1761, but the possibility that it could be detected during a transit seems not to have been considered. The discovery of the planet’s atmosphere has long been attributed to the Russian scientist Mikhail Lomonosov, after he observed the 1761 transit from the Imperial Academy of Sciences of St. Petersburg. [16] The attribution to Lomonosov seems to have arisen from comments made in 1966 by the astronomy writer Willy Ley, who wrote that Lomonosov had inferred the existence of an atmosphere from his observation of a luminous arc. [17] The attribution has since then been questioned. [18]

1769 transit

Diagram from David Rittenhouse's observations of the 1769 transit of Venus Rittenhouse 1769 observation of Venus transit.jpg
Diagram from David Rittenhouse's observations of the 1769 transit of Venus

For the 1769 transit, scientists travelled to places all over the world. The Czech astronomer Christian Mayer was invited by the Russian empress Catherine the Great to observe the transit in Saint Petersburg with Anders Johan Lexell, while other members of the Russian Academy of Sciences went to eight other locations in the Russian Empire under the general coordination of Stepan Rumovsky. [19] King George III of the United Kingdom had the King's Observatory built near his summer residence at Richmond Lodge, so that he and the Astronomer Royal, Stephen Demainbray, could observe the transit. [20] [21] Hell and his assistant János Sajnovics travelled to Vardø, Norway. Wales and Joseph Dymond went to Hudson Bay to observe the event. In Philadelphia, the American Philosophical Society erected three temporary observatories and appointed a committee led by David Rittenhouse. Observations were made by a group led by Dr. Benjamin West in Providence, Rhode Island, [22] Observations were also made from Tahiti by James Cook and Charles Green at a location still known as Point Venus. [23] [note 3]

D'Auteroche went to San José del Cabo in what was then New Spain to observe the transit with two Spanish astronomers (Vicente de Doz and Salvador de Medina). For his trouble he died in an epidemic of yellow fever there shortly after completing his observations. [25] Only 9 of 28 in the entire party returned home alive. [26] [ page needed ] Le Gentil spent over eight years travelling in an attempt to observe either of the transits. Whilst abroad he was declared dead, and as a result he lost his wife and possessions. Upon his return he regained his seat in the French Academy and remarried. [13] Under the influence of the Royal Society, the astronomer Ruđer Bošković travelled to Istanbul, but arrived after the transit had happened.[ citation needed ]

In 1771, using the combined 1761 and 1769 transit data, the French astronomer Jérôme Lalande calculated the astronomical unit to have a value of 153 ± 1 million kilometres (95.07 ± 0.62 million miles). The precision was less than had been hoped for because of the black drop effect. The value obtained was still an improvement on the calculations made by Horrocks. [13] [note 4] Hell published his results in 1770, which included a value for the astronomical unit of 151.7 million kilometres (94.3 million miles). Lalande challenged the accuracy and authenticity of observations obtained by the Hell expedition, but later wrote an article in Journal des sçavans (1778), in which he retracted his comments.[ citation needed ]

1874 and 1882 transits

Observations of the transits of 1874 and 1882 worked to refine the value obtained for the astronomical unit. Three expeditions—from Germany, the United Kingdom, and the United States—were sent to the Kerguelen Archipelago for the 1874 observations. [27] The American astronomer Simon Newcomb combined the data from the last four transits, and he arrived at a value of 149.59 ± 0.31 million kilometres (92.95 ± 0.19 million miles). [13] [note 5]

2004 and 2012 transits

The transit of Venus, June 2012 Transit of Venus - June 2012.svg
The transit of Venus, June 2012

Scientific organisations led by the European Southern Observatory organised a network of amateur astronomers and students to measure Earth's distance from the Sun during the transit of 2004. [28] The participants' observations allowed a calculation of the astronomical unit (AU) of 149,608,708 ± 11,835 kilometres (92,962,541 ± 7,354 miles), which differed from the accepted value by 0.007%. [29]

During the 2004 transit, scientists attempted to measure the loss of light as Venus blocked out some of the Sun's light, in order to refine techniques for discovering extrasolar planets. [30]

The 2012 transit of Venus provided scientists with research opportunities as well, in particular in regard to the study of exoplanets. The event additionally was the first of its kind to be documented from space, photographed aboard the International Space Station by NASA astronaut Don Pettit. The measurement of the dips in a star's brightness during a transit is one observation that can help astronomers find exoplanets. Unlike the 2004 Venus transit, the 2012 transit occurred during an active phase of the 11-year activity cycle of the Sun, and it gave astronomers an opportunity to practise picking up a planet's signal around a "spotty" variable star. Measurements made of the apparent diameter of a planet such as Venus during a transit allows scientists to estimate exoplanet sizes. Observation made of the atmosphere of Venus from Earth-based telescopes and the Venus Express gave scientists a better opportunity to understand the intermediate level of Venus's atmosphere than was possible from either viewpoint alone, and provided new information about the climate of the planet. Spectrographic data of the atmosphere of Venus can be compared to studies of the atmospheres of exoplanets. The Hubble Space Telescope used the Moon as a mirror to study light from the atmosphere of Venus, and so determine its composition. [31] [32]

Future transits

Date(s) of
transit		Time (UTC)Notes [8]
StartMidEnd
10–11 December 211723:5802:4805:38Visible in entirety in eastern China, Korea, Japan, south of Russian Far East, Taiwan, Indonesia, and Australia. Partly visible in Central Asia, the Middle East, south part of Russia, in India, most of Africa, and on extreme U.S. West Coast.
8 December 212513:1516:0118:48Visible in entirety in South America and the eastern U.S. Partly visible in Western U.S., Europe, Africa, and Oceania.
11 June 224708:4211:3314:25Visible in entirety in Africa, Europe, and the Middle East. Partly visible in East Asia and Indonesia, and in North and South America.
9 June 225501:0804:3808:08Visible in entirety in Russia, India, China, and western Australia. Partly visible in Africa, Europe, and the western U.S.
12–13 December 236022:3201:4404:56Visible in entirety in Australia and most of Indonesia. Partly visible in Asia, Africa, and the western half of the Americas.
10 December 236812:2914:4517:01Visible in entirety in South America, western Africa, and the U.S. East Coast. Partly visible in Europe, the western U.S., and the Middle East.
12 June 249011:3914:1716:55Visible in entirety through most of the Americas, western Africa, and Europe. Partly visible in eastern Africa, the Middle East, and Asia.
10 June 249803:4807:2511:02Visible in entirety through most of Europe, Asia, the Middle East, and eastern Africa. Partly visible in eastern Americas, Indonesia, and Australia.

Transits usually occur in pairs, because the length of eight Earth years is almost the same as 13 years on Venus. This approximate conjunction is not precise enough to produce a triplet, as Venus arrives 22 hours earlier each time. The last transit not to be part of a pair was in 1396 (the planet passed slightly above the disc of the Sun in 1388); [33] the next one will be in 3089.[ citation needed ]

After 243 years the transits of Venus return. The 1874 transit is a member of the 243-year cycle #1. The 1882 transit is a member of #2. The 2004 transit is a member of #3, and the 2012 transit is a member of #4. The 2117 transit is a member of #1, and so on. However, the ascending node (December transits) of the orbit of Venus moves backwards after each 243 years so the transit of 2854 is the last member of series #3 instead of series #1. The descending node (June transits) moves forwards, so the transit of 3705 is the last member of #2.[ citation needed ]

Over longer periods of time, new series of transits will start and old series will end. Unlike the saros series for lunar eclipses, it is possible for a transit series to restart after a hiatus. The transit series also vary much more in length than the saros series.[ citation needed ]

Grazing and simultaneous transits

Sometimes Venus only grazes the Sun during a transit. In this case it is possible that in some areas of the Earth a full transit can be seen while in other regions there is only a partial transit (no second or third contact). The last transit of this type was on 6 December 1631, and the next such transit will occur on 13 December 2611. It is also possible that a transit of Venus can be seen in some parts of the world as a partial transit, while in others Venus misses the Sun. Such a transit last occurred on 19 November 541 BC, and the next transit of this type will occur on 14 December 2854. [2] These effects are due to parallax, since the size of the Earth affords different points of view with slightly different lines of sight to Venus and the Sun. It can be demonstrated by closing an eye and holding a finger in front of a smaller more distant object; when the viewer opens the other eye and closes the first, the finger will no longer be in front of the object.[ citation needed ]

The simultaneous occurrence of transits of Mercury and Venus does occur, but extremely infrequently. Such an event last occurred on 22 September 373,173 BC and will next occur on 26 July 69,163, [34] and again on 29 March 224,504. [35] The simultaneous occurrence of a solar eclipse and a transit of Venus is currently possible, but very rare. The next solar eclipse occurring during a transit of Venus will be on 5 April 15,232. [36]

The Canadian rock band Three Days Grace titled their fourth studio album Transit of Venus and announced the album title and release date on June 5, 2012, the date of the last transit of Venus. The album's first song, "Sign of the Times", references the transit in the lyric "Venus is passing by".

The progressive rock band Big Big Train have a song titled "The Transit of Venus Across the Sun". It is the fifth track on their ninth album Folklore (Big Big Train album).

The Transit of Venus March was written by John Philip Sousa in 1883 to commemorate the 1882 transit.

See also

Notes

  1. ↑ Although the inclination between these two orbital planes is only 3.4°, Venus can be as far as 9.6° from the Sun when viewed from the Earth at inferior conjunction.[ citation needed ]
  2. ↑ His estimation for the distance from Earth to the Sun was about two thirds of the actual distance of 93 million mi (150 million km), but was a more accurate figure than any suggested up to that time. [11]
  3. ↑ The observations of the transit on Tahiti occurred during the first voyage of James Cook, after which Cook explored New Zealand and Australia. This was one of five expeditions organised by the Royal Society and Maskelyne. [24]
  4. ↑ The black drop effect distorts the image of Venus at the observed edge of the Sun. The effect is caused by turbulence in the Earth's atmosphere, imperfections in the viewing apparatus, or the extreme change in brightness at the edge of the Sun.[ citation needed ]
  5. ↑ The need for parallax calculations has been superseded, as modern techniques, such as the use of radio telemetry from space probes and of radar measurements of the distances to planets and asteroids in the Solar System, have allowed a value for the AU to be calculated to a precision of about ±30 metres (98 ft). [13]

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References

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  3. ↑ Rincon, Paul (7 November 2005). "Planet Venus: Earth's 'evil twin'". BBC. Retrieved 25 September 2006.
  4. ↑ Galindo Trejo & Allen 2005, p. 124.
  5. ↑ Sally P. Ragep (2007). "Ibn Sīnā: Abū ʿAlī al-Ḥusayn ibn ʿAbdallāh ibn Sīnā". In Thomas Hockey (ed.). The Biographical Encyclopedia of Astronomers. Springer Science+Business Media. pp. 570–572.
  6. ↑ Goldstein, Bernard R. (1969). "Some Medieval Reports of Venus and Mercury Transits". Centaurus. 14 (1): 49–59. Bibcode:1969Cent...14...49G. doi:10.1111/j.1600-0498.1969.tb00135.x.
  7. ↑ Goldstein, Bernard R. (March 1972). "Theory and Observation in Medieval Astronomy". Isis . 63 (1): 39–47 [44]. Bibcode:1972Isis...63...39G. doi:10.1086/350839. S2CID   120700705.
  8. 1 2 "Transits of Venus: 1000AD–2700AD". HM Nautical Almanac Office. 3 May 2011. Archived from the original on 19 April 2017. Retrieved 7 March 2024.
  9. ↑ van Gent, Robert H. "Transit of Venus Bibliography" . Retrieved 11 September 2009.
  10. ↑ Kollerstrom, Nicholas (2004). "William Crabtree's Venus transit observation" (PDF). Proceedings IAU Colloquium No. 196, 2004. International Astronomical Union. Retrieved 10 May 2012.
  11. 1 2 3 Marston, Paul (2004). Jeremiah Horrocks—young genius and first Venus transit observer. University of Central Lancashire. pp. 14–37.
  12. ↑ Teets, D.A. (2003). "Transits of Venus and the Astronomical Unit". Mathematics Magazine. 76 (5): 335–348. doi:10.1080/0025570X.2003.11953207. JSTOR   3654879. S2CID   54867823.
  13. 1 2 3 4 5 Pogge, Prof. Richard. "Lecture 26: How far to the Sun? The Venus Transits of 1761 & 1769" . Retrieved 25 September 2006.
  14. ↑ Howse, Derek (2004). "Oxford Dictionary of National Biography: Jeremiah Dixon" . Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/37360 . Retrieved 22 February 2012.(Subscription or UK public library membership required.)
  15. ↑ Howse 1989, pp. 38–39.
  16. ↑ Vladimir Shiltsev (1970). "Lomonosov's Discovery of Venus Atmosphere in 1761: English Translation of Original Publication with Commentaries". arXiv: 1206.3489 [physics.hist-ph].
  17. ↑ Marov, Mikhail Ya. (2004). "Mikhail Lomonosov and the discovery of the atmosphere of Venus during the 1761 transit". Proceedings of the International Astronomical Union. 2004: 209–219. Bibcode:2005tvnv.conf..209M. doi: 10.1017/S1743921305001390 .
  18. ↑ Sheehan, Jay; Sheehan, William (2012). "Lomonosov, the Discovery of Venus's Atmosphere, and Eighteenth-century Transits of Venus". Journal of Astronomical History and Heritage. 15 (1): 3. Bibcode:2012JAHH...15....3P. doi:10.3724/SP.J.1440-2807.2012.01.01. S2CID   55848433.
  19. ↑ Mayer, Christian; Parsons, James (1764). "An Account of the Transit of Venus: In a Letter to Charles Morton, M. D. Secret. R. S. from Christian Mayer, S. J. Translated from the Latin by James Parsons, M. D". Philosophical Transactions of the Royal Society. 54: 163–164. Bibcode:1764RSPT...54..163M. doi: 10.1098/rstl.1764.0030 .
  20. ↑ McLaughlin, Stewart (1992). "The Early History of Kew Observatory". Richmond History: Journal of the Richmond Local History Society . 13: 48–49. ISSN   0263-0958.
  21. ↑ Mayes, Julian (2004). "The History of Kew Observatory". Richmond History: Journal of the Richmond Local History Society . 25: 44–57. ISSN   0263-0958.
  22. ↑ Catherine B. Hurst, Choosing Providence, 23 March 2012. Retrieved 20 April 2016.
  23. ↑ See, for example, Stanley, David (2004). Moon Handbooks South Pacific (8th ed.). Avalon Travel Publishing. p.  175. ISBN   978-1-56691-411-6. point venus cook.
  24. ↑ Rhys, Ernest, ed. (1999). The Voyages of Captain Cook. Wordsworth Editions Ltd. pp. 29–30. ISBN   978-1-84022-100-8.
  25. ↑ In Memoriam Archived 5 December 2014 at the Wayback Machine French_Academy_of_Sciences
  26. ↑ Anderson 2012.
  27. ↑ "1874 transit". The Royal Society. Royal Society. Archived from the original on 2 April 2015. Retrieved 6 March 2015.
  28. ↑ "The Venus Transit 2004". European Southern Observatory. 12 February 2013. Archived from the original on 11 February 2024. Retrieved 6 June 2012.
  29. ↑ "Summing Up the Unique Venus Transit 2004 (VT-2004) Programme". European Southern Observatory. 2 November 2004. Retrieved 6 June 2012.
  30. ↑ McKee, Maggie (6 June 2004). "Extrasolar planet hunters eye Venus transit". New Scientist . Archived from the original on 13 April 2016. Retrieved 12 March 2024.
  31. ↑ Wall, Michael (16 May 2012). "Venus Transit On June 5 May Bring New Alien Planet Discoveries". The Huffington Post . Retrieved 21 May 2012.
  32. ↑ Tanga, Paolo. "The Venus Twilight Experiment: Refraction and scattering phenomena during the transit of Venus on June 5–6, 2012". Archived from the original on 23 October 2016. Retrieved 23 October 2016.
  33. ↑ Steve, Bell (10 June 2004). "The Cyclical Nature of the Transits of Venus". Transits of Venus. HM Nautical Almanac Office. Archived from the original on 1 October 2006. Retrieved 8 March 2024.
  34. ↑ Espenak, Fred (21 April 2005). "Transits of Mercury, Seven Century Catalog: 1601 CE to 2300 CE". NASA. Archived from the original on 28 September 2006. Retrieved 27 September 2006.
  35. ↑ Meeus & Vitagliano 2004, p. 132.
  36. ↑ Meeus & Vitagliano 2004, p. 134.

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