Editing Great conjunction (section)

== Celestial mechanics ==
{{Main|Celestial mechanics}}
[[File:Jupiter-Saturn-1981.png|thumb|400px|Diagram showing the movements of Jupiter and Saturn during the 1980–81 triple conjunction]]
On average, great conjunction seasons occur once every 19.859 [[Julian year (astronomy)|Julian years]] (each of which is 365.25 days). This number can be calculated by the [[synodic period]] formula

:<math>\frac{1}{\left ( \frac{1}{J} - \frac{1}{S} \right )} \approx 7253.46 \; \mathrm{days} \displaystyle,</math>

in which {{Mvar|J}} and {{Mvar|S}} are the [[orbital period]]s of Jupiter (4332.59 days) and Saturn (10759.22 days), respectively.<ref name="auto3">{{Cite web|url=https://sparky.rice.edu/public-night/jupsat2.html|title=Jupiter-Saturn Conjunction Series|website=sparky.rice.edu}}</ref> This is about 52 days less than 20 years, but in practice, Earth's orbit size can cause great conjunctions to reoccur anytime between 18 years 10 months and 20 years 8 months after the previous one. (See table below.) Since the equivalent periods of other naked-eye planet pairs are all under 900 days, this makes great conjunctions the rarest.{{fact|date=April 2025}}

Occasionally there is more than one great conjunction in a season, which happens whenever they're close enough to [[opposition (astronomy)|opposition]]: this is called a [[triple conjunction]] (which is not exclusive to great conjunctions). In this scenario, Jupiter and Saturn will occupy the same [[right ascension]] on three occasions or same [[ecliptic longitude]] on three occasions, depending on which definition of "conjunction" one uses (this is due to [[apparent retrograde motion]] and happens within months). The most recent triple conjunction occurred in 1980–81<ref name="auto2">{{Cite web|last=Jones|first=Graham|title=The December 2020 Great Conjunction|url=https://www.timeanddate.com/astronomy/planets/great-conjunction|access-date=2020-12-10|website=timeanddate.com|language=en}}</ref> and the next will be in 2238–39.{{fact|date=April 2025}}

The most recent great conjunction occurred on 21 December 2020, and the next will occur on 4 November 2040. During the 2020 great conjunction, the two planets were separated in the sky by 6 [[arcminutes]] at their closest point, which was the closest distance between the two planets since 1623.<ref name="auto1">{{Cite web |last=Hunt |first=Jeffrey L. |date=2020-02-20 |title=1623: The Great Conjunction of Jupiter and Saturn |url=https://whenthecurveslineup.com/2020/02/20/1623-the-great-conjunction-of-jupiter-and-saturn/ |access-date=2020-08-24 |website=When the Curves Line Up |language=en}}</ref> The closeness is the result of the conjunction occurring in the vicinity of one of the two [[ecliptic longitude|longitudes]] where the two orbits appear to intersect when viewed from the Sun (which has a point of view similar to Earth).

{| class=wikitable align=right width=400
|[[File:Jupiter-Saturn-great-conjunctions.png|200px]]<BR>Jupiter and Saturn's great conjunctions repeat every ~120°, Saturn's path relative to Jupiter in blue.
|[[File:Keplers trigon.jpg|200px]]<BR>Kepler's trigon, a diagram of great conjunctions from [[Johannes Kepler]]'s 1606 book ''[[De Stella Nova]]''
|}

Because 19.859 years is equal to 1.674 Jupiter orbits and 0.674 Saturn orbits, three of these periods come close to a whole number of revolutions. As successive great conjunctions occur nearly 120° apart, their appearances form a triangular pattern. In a series, every third conjunction returns after some 60 years to the vicinity of the first. These returns are observed to be shifted by some 8° relative to the fixed stars, so no more than four of them occur in the same [[zodiacal]] constellation. Usually the conjunctions occur in one of the following ''[[Triplicity|triplicities]]'' or ''trigons'' of zodiacal constellations:
#Aries, Sagittarius, and Leo
#Taurus, Capricorn, and Virgo
#Gemini, Aquarius, and Libra
#Cancer, Pisces, and Scorpius
After about 220 years the pattern shifts to the next trigon, and in about 800 or 900 years returns to the first trigon.<ref>If {{mvar|J}} and {{mvar|S}} designate the periods of Jupiter and Saturn then the return takes <math>1/(5/S-2/J).</math> Using sidereal periods, this comes to about 836 years. But the return must be a whole number of conjunction intervals, such as 794, 854, or 913 years. However, for returning to the same [[right ascension]], periods must be used with respect to the [[celestial equinox]], giving about 763 years. See Etz.</ref>

The three points of the triangle revolve in the same direction as the planets at the rate of approximately one-sixth of a revolution per four centuries, thus creating especially close conjunctions on an approximately four-century cycle. Currently the longitudes of close great conjunctions are about 307.4 and 127.4 degrees, in Capricornus and Cancer respectively. 

In astrology, one of the [[Classical element|four elements]] was ascribed to each triangular pattern. Particular importance was accorded to the occurrence of a great conjunction in a new trigon, which is bound to happen after some 240 years at most.<ref name=Etz>Etz D., (2000), [http://adsabs.harvard.edu/full/2000JRASC..94..174E ''Conjunctions of Jupiter and Saturn''], Journal of the Royal Astronomical Society of Canada, Vol. 94, p.174</ref> Even greater importance was attributed to the beginning of a new cycle after all fours trigons had been visited.
Medieval astrologers usually gave 960 years as the duration of the full cycle, perhaps because in some cases it took 240 years to pass from one trigon to the next.<ref name=Etz/> If a cycle is defined by when the conjunctions return to the same [[right ascension]] rather than to the same constellation, then because of [[axial precession]] the cycle is less than 800 years.<ref>See previous footnote.</ref> Use of the [[Alphonsine tables]] apparently led to the use of precessing signs, and [[Kepler]] gave a value of 794 years (40 conjunctions).<ref name=Etz/><ref name="Woody"/>

Despite mathematical errors and some disagreement among astrologers about when [[Triplicity|trigons]] began, belief in the significance of such events generated a stream of publications that grew steadily until the end of the 16th century. As the great conjunction of 1583 was last in the water trigon it was widely supposed to herald apocalyptic changes; a [[papal bull]] against divination was issued in 1586 but as nothing significant happened by the feared event of 1603, public interest rapidly died. By the start of the next trigon, modern scientific consensus had condemned astrology as pseudoscience, and astronomers no longer perceived planetary alignments as omens.<ref name=Thomas>Keith Thomas, ''Religion and the Decline of Magic: Studies in Popular Beliefs in Sixteenth and Seventeenth-Century England'' ([[Oxford University Press]], 1971) p. 414-415, {{ISBN|9780195213607}}</ref> However, in the year 1962, when all five planets formed a cluster 17° wide, there was considerable concern.<ref>{{cite web |title=Planets for dessert |url=https://science.nasa.gov/science-news/science-at-nasa/2000/ast30mar_1m |publisher=NASA Science}}</ref>

[[File:Great conjunction separations.svg|thumb|500px|Cumulative distribution of separation distance of great conjunctions between AD 1200 and AD 2400 (blue), and an approximation formula (red)]]
Saturn's [[orbital inclination|orbit plane]] is inclined 2.485 degrees relative to Earth's, and Jupiter's is inclined 1.303 degrees. The [[ascending node]]s of both planets are similar (100.6 degrees for Jupiter and 113.7 degrees for Saturn), meaning if Saturn is above or below Earth's orbital plane Jupiter usually is too. Because these nodes align so well it would be expected that no closest approach will ever be much worse than the difference between the two inclinations. Indeed, between year 1 and 3000, the maximum conjunction distances were 1.3 degrees in 1306 and 1940. Conjunctions in both years occurred when the planets were tilted most out of the plane: longitude 206 degrees (therefore above the plane) in 1306, and longitude 39 degrees (therefore below the plane) in 1940.<ref name="auto3" />