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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Eclipse cycle | 3/12 | https://en.wikipedia.org/wiki/Eclipse_cycle | reference | science, encyclopedia | 2026-05-05T11:12:34.827205+00:00 | kb-cron |
SM = 29.530588853 days (Synodic month) DM = 27.212220817 days (Draconic month) AM = 27.55454988 days (Anomalistic month) EY = 346.620076 days (Eclipse year) Note that there are three main moving points: the Sun, the Moon, and the (ascending) node; and that there are three main periods, when each of the three possible pairs of moving points meet one another: the synodic month when the Moon returns to the Sun, the draconic month when the Moon returns to the node, and the eclipse year when the Sun returns to the node. These three 2-way relations are not independent (i.e. both the synodic month and eclipse year are dependent on the apparent motion of the Sun, both the draconic month and eclipse year are dependent on the motion of the nodes), and indeed the eclipse year can be described as the beat period of the synodic and draconic months (i.e. the period of the difference between the synodic and draconic months); in formula:
EY
=
SM
×
DM
SM
−
DM
{\displaystyle {\mbox{EY}}={\frac {{\mbox{SM}}\times {\mbox{DM}}}{{\mbox{SM}}-{\mbox{DM}}}}}
as can be checked by filling in the numerical values listed above. Eclipse cycles have a period in which a certain number of synodic months closely equals an integer or half-integer number of draconic months: one such period after an eclipse, a syzygy (new moon or full moon) takes place again near a node of the Moon's orbit on the ecliptic, and an eclipse can occur again. However, the synodic and draconic months are incommensurate: their ratio is not an integer number. We need to approximate this ratio by common fractions: the numerators and denominators then give the multiples of the two periods – draconic and synodic months – that (approximately) span the same amount of time, representing an eclipse cycle. These fractions can be found by the method of continued fractions: this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions. Since there may be an eclipse every half draconic month, we need to find approximations for the number of half draconic months per synodic month: so the target ratio to approximate is: SM / (DM/2) = 29.530588853 / (27.212220817/2) = 2.170391682 The continued fractions expansion for this ratio is:
2.170391682 = [2;5,1,6,1,1,1,1,1,11,1,...]: Quotients Convergents half DM/SM decimal named cycle (if any) 2; 2/1 = 2 synodic month 5 11/5 = 2.2 pentalunex 1 13/6 = 2.166666667 semester 6 89/41 = 2.170731707 hepton 1 102/47 = 2.170212766 octon 1 191/88 = 2.170454545 tzolkinex 1 293/135 = 2.170370370 tritos 1 484/223 = 2.170403587 saros 1 777/358 = 2.170391061 inex 11 9031/4161 = 2.170391732 selebit 1 9808/4519 = 2.170391679 square year ...
The ratio of synodic months per half eclipse year yields the same series:
5.868831091 = [5;1,6,1,1,1,1,1,11,1,...] Quotients Convergents SM/half EY decimal SM/full EY named cycle 5; 5/1 = 5 pentalunex 1 6/1 = 6 12/1 semester 6 41/7 = 5.857142857 hepton 1 47/8 = 5.875 47/4 octon 1 88/15 = 5.866666667 tzolkinex 1 135/23 = 5.869565217 tritos 1 223/38 = 5.868421053 223/19 saros 1 358/61 = 5.868852459 716/61 inex 11 4161/709 = 5.868829337 selebit 1 4519/770 = 5.868831169 4519/385 square year ...
Each of these is an eclipse cycle. Less accurate cycles may be constructed by combinations of these.
== Eclipse cycles == This table summarizes the characteristics of various eclipse cycles, and can be computed from the numerical results of the preceding paragraphs; cf. Meeus (1997) Ch.9. More details are given in the comments below, and several notable cycles have their own pages. Many other cycles have been noted, some of which have been named. The numbers in the table are the average values. The actual length of time between two eclipses in an eclipse cycle varies because of the variation in the speed of the Moon and of the Sun in the sky. The variation is less if the number of anomalistic months is near a whole number, and if the number of anomalistic years is near a whole number. (See graphs lower down of semester and Hipparchic cycle.) Any eclipse cycle, and indeed the interval between any two eclipses, can be expressed as a combination of saros (s) and inex (i) intervals. These are listed in the column "formula".