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Transit of Venus

From Wikipedia, the free encyclopedia
Astronomical transit of Venus across the Sun
This article is about the astronomical phenomenon. For other uses, seeTransit of Venus (disambiguation).

The sun in orange, as seen from the visible spectrum, with Venus in the top left quadrant
Images of thetransit of Venus, taken fromNASA'sSolar Dynamics Observatory in 2012: (left) visible light; (right)ultraviolet

Atransit of Venus takes place whenVenus passes directly between theSun and the Earth (or any othersuperior planet), becoming visible against (and hence obscuring a small portion of) thesolar disk. During atransit, Venus is visible as a small black circle moving across the face of the Sun.

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 theorbital periods of Earth and Venusare close to 8:13 and 243:395commensurabilities. The last pairs of transits occurred on8 June 2004 and5–6 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 the first significantly accurately measurable occurrences, providing highly accuratesolar parallax measurements, to determine accurately the distance of Earth to Venus, allowing the calculation of the byKepler's third law proportionateastronomical unit and the distances of the other bodies of theSolar System. The 2012 transit has provided research opportunities, particularly in the refinement of techniques to be used in the search forexoplanets.

Conjunctions

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Diagram of transits of Venus and the angle between the orbital planes of Venus and Earth

Theorbit of Venus has aninclination 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 reachesconjunction with the Sun whilst also passing through the Earth'sorbital 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 243sidereal 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 152synodic 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

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AncientIndian,Greek,Egyptian,Babylonian, andChinese observers knew of Venus and recorded the planet's motions.[3]Pythagoras is credited with realizing that the so-calledmorning andevening 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 thatfrescoes found at theMaya site atMayapan may contain a pictorial representation of the 12th or 13th century transits.[4]

The Persian polymathAvicenna 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 asunspot for Venus. He used his alleged transit observation to help establish that Venus was, at least sometimes, below the Sun inPtolemaic cosmology,[5] i.e., the sphere of Venus comes before the sphere of the Sun when moving out from the Earth in the then prevailinggeocentric model.[6][7]

Transits of Venus (1631–2012)
Date(s) of transitsTime (UTC)Notes[8]
StartMidEnd
7 December 163103:5105:1906:47Predicted byKepler
4 December 163914:5718:2521:54First transit to be recorded, byHorrocks andCrabtree
6 June 176102:0205:1908:37Lomonosov,Chappe d'Auteroche, and others observe from Russia;Mason andDixon observe from the Cape of Good Hope.John Winthrop observes from St. John's, Newfoundland
3–4 June 176919:1522:2501:35Cooksent to Tahiti toobserve the transit,Chappe toSan José del Cabo,Baja California, andMaximilian Hell toVardø, Norway.
9 December 187401:4904:0706:26Pietro Tacchini leads expedition to Muddapur,India. AFrench expedition goes to New Zealand's Campbell Island, and aBritish expedition travels to Hawaii. This is also the first ever transitfilmed.
6 December 188213:5717:0620:15
8 June 200405:1308:2011:26Various media networks globally broadcast live video of the Venus transit.
5–6 June 201222:0901:2904:49Visible 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

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Main article:Transit of Venus, 1639
William Richard Lavender,Jeremiah Horrocks (1618–1641) (1903),Astley Hall Museum and Art Gallery

The German astronomerJohannes 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 astronomerJeremiah Horrocks from his home atCarr House inMuch Hoole, nearPreston, on 4 December 1639 (24 November O.S.). His friendWilliam Crabtree observed the transit from nearbyBroughton.[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 simpletelescope 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

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Measuring Venus transit times to determinesolar parallax

In 1663, the Scottish mathematicianJames Gregory had suggested in hisOptica Promota that observations of atransit of Mercury, at widely spaced points on the surface of the Earth, could be used to calculate thesolar parallax, and hence theastronomical unit by means oftriangulation. Aware of this, the English astronomerEdmond Halley made observations of such a transit on 28 OctoberO.S. 1677 from the island ofSaint Helena, but was disappointed to find that onlyRichard Towneley in theLancashire town ofBurnley, Lancashire had made another accurate observation of the event, whilst Gallet, atAvignon, 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 JuneN.S., 26 May O.S.).[12] In an attempt to observe the first transit of the pair, astronomers from Britain (William Wales andCaptain James Cook), Austria (Maximilian Hell), and France (Jean-Baptiste Chappe d'Auteroche andGuillaume Le Gentil) took part in expeditions to places that includedSiberia,Newfoundland, and Madagascar.[13] Most of them observed at least part of the transit.Jeremiah Dixon andCharles Mason succeeded in observing the transit at theCape of Good Hope,[14] butNevil Maskelyne andRobert Waddington were less successful on Saint Helena, although they put their voyage to good use by trialling thelunar-distance method of finding longitude.[15]

Venus was generally thought to possessan 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 scientistMikhail Lomonosov, after he observed the 1761 transit from theImperial Academy of Sciences of St. Petersburg.[16] The attribution to Lomonosov seems to have arisen from comments made in 1966 by the astronomy writerWilly 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

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Diagram fromDavid Rittenhouse's observations of the 1769 transit of Venus

For the 1769 transit, scientists travelled to places all over the world. The Czech astronomerChristian Mayer was invited by the Russian empressCatherine the Great to observe the transit inSaint Petersburg withAnders Johan Lexell, while other members of theRussian Academy of Sciences went to eight other locations in theRussian Empire under the general coordination ofStepan Rumovsky.[19] KingGeorge III of the United Kingdom had theKing's Observatory built near his summer residence atRichmond Lodge, so that he and theAstronomer Royal,Stephen Demainbray, could observe the transit.[20][21] Hell and his assistantJános Sajnovics travelled toVardø, Norway. Wales andJoseph Dymond went toHudson Bay to observe the event. InPhiladelphia, theAmerican Philosophical Society erected three temporary observatories and appointed a committee led byDavid Rittenhouse. Observations were made by a group led by Dr. Benjamin West inProvidence, Rhode Island,[22]Observations were also made from Tahiti by Captain James Cook and Charles Green at a location still known asPoint Venus.[23][note 3]

D'Auteroche went to San José del Cabo in what was thenNew Spain to observe the transit with two Spanish astronomers (Vicente de Doz and Salvador de Medina). For his trouble he died in an epidemic ofyellow 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 theFrench Academy and remarried.[13] Under the influence of theRoyal Society, the astronomerRuđer Bošković travelled toIstanbul, but arrived after the transit had happened.[citation needed]

In 1771, using the combined 1761 and 1769 transit data, the French astronomerJérôme 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 theblack 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 inJournal des sçavans (1778), in which he retracted his comments.[citation needed]

1874 and 1882 transits

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Observations of thetransits of 1874 and1882 worked to refine the value obtained for the astronomical unit. Three expeditions—from Germany, the United Kingdom, and the United States—were sent to theKerguelen Archipelago for the 1874 observations.[27] The American astronomerSimon 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

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The transit of Venus, June 2012

Scientific organisations led by theEuropean Southern Observatory organised a network of amateur astronomers and students to measure Earth's distance from the Sun during thetransit 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 discoveringextrasolar planets.[30]

The2012 transit of Venus provided scientists with research opportunities as well, in particular in regard to the study ofexoplanets. The event additionally was the first of its kind to be documented from space, photographed aboard the International Space Station by NASA astronautDon 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 theVenus 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 theclimate of the planet. Spectrographic data of the atmosphere of Venus can be compared to studies of the atmospheres of exoplanets. TheHubble Space Telescope used theMoon as a mirror to study light from the atmosphere of Venus, and so determine its composition.[31][32]

Future transits

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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, the Philippines, and Australia. Partly visible in Central Asia, the Middle East, south part of Russia, in India, most of Africa, and on extreme West Coast of North America.
8 December 212513:1516:0118:48Visible in entirety in South America and the eastern North America. Partly visible in Western North America, 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 western North America.
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 North American East Coast. Partly visible in Europe, the western North America, 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 thesaros 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

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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 (nosecond 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 toparallax, 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 September373,173 BC and will next occur on 26 July69,163,[34] and again on 27–28 March 224,508.[35] The simultaneous occurrence of asolar 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]

In popular culture

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The Canadian rock bandThree Days Grace titled their fourth studio albumTransit 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 bandBig Big Train have a song titled "The Transit of Venus Across the Sun". It is the fifth track on their ninth albumFolklore (Big Big Train album).

TheTransit of Venus March was written byJohn Philip Sousa in 1883 to commemorate the 1882 transit.

See also

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Notes

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  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 thefirst voyage of James Cook, after which Cook exploredNew Zealand andAustralia. 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 radiotelemetry fromspace probes and ofradar measurements of the distances to planets andasteroids in theSolar System, have allowed a value for the AU to be calculated to a precision of about ±30 metres (98 ft).[13]

References

[edit]
  1. ^"Venus and Earth Compared".Venus Express.European Space Agency. 1 September 2019. Archived fromthe original on 5 April 2022. Retrieved7 March 2024.
  2. ^abcdEspenak, Fred (11 February 2004)."Transits of Venus, Six Millennium Catalog: 2000 BCE to 4000 CE".Eclipse.NASA. Archived fromthe original on 24 June 2011. Retrieved7 March 2024.
  3. ^Rincon, Paul (7 November 2005)."Planet Venus: Earth's 'evil twin'". BBC. Retrieved25 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. ^ab"Transits of Venus: 1000AD–2700AD".HM Nautical Almanac Office. 3 May 2011. Archived fromthe original on 19 April 2017. Retrieved7 March 2024.
  9. ^van Gent, Robert H."Transit of Venus Bibliography". Retrieved11 September 2009.
  10. ^Kollerstrom, Nicholas (2004)."William Crabtree's Venus transit observation"(PDF).Proceedings IAU Colloquium No. 196, 2004. International Astronomical Union. Retrieved10 May 2012.
  11. ^abcMarston, 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. ^abcdePogge, Prof. Richard."Lecture 26: How far to the Sun? The Venus Transits of 1761 & 1769". Retrieved25 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. Retrieved22 February 2012. (Subscription,Wikipedia Library access orUK 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 MemoriamArchived 5 December 2014 at theWayback Machine French_Academy_of_Sciences
  26. ^Anderson 2012.
  27. ^"1874 transit".The Royal Society.Royal Society. Archived fromthe original on 2 April 2015. Retrieved6 March 2015.
  28. ^"The Venus Transit 2004".European Southern Observatory. 12 February 2013. Archived fromthe original on 11 February 2024. Retrieved6 June 2012.
  29. ^"Summing Up the Unique Venus Transit 2004 (VT-2004) Programme".European Southern Observatory. 2 November 2004. Retrieved6 June 2012.
  30. ^McKee, Maggie (6 June 2004)."Extrasolar planet hunters eye Venus transit".New Scientist. Archived fromthe original on 13 April 2016. Retrieved12 March 2024.
  31. ^Wall, Michael (16 May 2012)."Venus Transit On June 5 May Bring New Alien Planet Discoveries".The Huffington Post. Retrieved21 May 2012.
  32. ^Tanga, Paolo."The Venus Twilight Experiment: Refraction and scattering phenomena during the transit of Venus on June 5–6, 2012". Archived fromthe original on 23 October 2016. Retrieved23 October 2016.
  33. ^Steve, Bell (10 June 2004)."The Cyclical Nature of the Transits of Venus".Transits of Venus.HM Nautical Almanac Office. Archived fromthe original on 1 October 2006. Retrieved8 March 2024.
  34. ^Espenak, Fred (21 April 2005)."Transits of Mercury, Seven Century Catalog: 1601 CE to 2300 CE". NASA. Archived fromthe original on 28 September 2006. Retrieved27 September 2006.
  35. ^Meeus & Vitagliano 2004, p. 133.
  36. ^Meeus & Vitagliano 2004, p. 134.

Sources

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Further reading

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External links

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