# Eclipse cycle

Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.

## Eclipse conditions

Eclipses may occur when Earth and the Moon are aligned with the Sun, and the shadow of one body projected by the Sun falls on the other. So at new moon, when the Moon is in conjunction with the Sun, the Moon may pass in front of the Sun as viewed from a narrow region on the surface of Earth and cause a solar eclipse. At full moon, when the Moon is in opposition to the Sun, the Moon may pass through the shadow of Earth, and a lunar eclipse is visible from the night half of Earth. The conjunction and opposition of the Moon together have a special name: syzygy (Greek for "junction"), because of the importance of these lunar phases.
An eclipse does not occur at every new or full moon, because the plane of the Moon's orbit around Earth is tilted with respect to the plane of Earth's orbit around the Sun (the ecliptic): so as viewed from Earth, when the Moon appears nearest the Sun (at new moon) or furthest from it (at full moon), the three bodies are usually not exactly on the same line.
This inclination is on average about 5° 9′, much larger than the apparent mean diameter of the Sun (32′ 2″), the Moon as viewed from Earth's surface directly below the Moon (31′ 37″), and Earth's shadow at the mean lunar distance (1° 23′).
Therefore, at most new moons, Earth passes too far north or south of the lunar shadow, and at most full moons, the Moon misses Earth's shadow. Also, at most solar eclipses, the apparent angular diameter of the Moon is insufficient to fully occlude the solar disc, unless the Moon is around its perigee, i.e. nearer Earth and apparently larger than average. In any case, the alignment must be almost perfect to cause an eclipse.
An eclipse can occur only when the Moon is on or near the plane of Earth's orbit, i.e. when its ecliptic latitude is low. This happens when the Moon is around either of the two orbital nodes on the ecliptic at the time of the syzygy. Of course, to produce an eclipse, the Sun must also be around a node at that time – the same node for a solar eclipse or the opposite node for a lunar eclipse.

## Recurrences

Up to three eclipses may occur during an eclipse season, a one- or two-month period that happens twice a year, around the time when the Sun is near the nodes of the Moon's orbit.
An eclipse does not occur every month, because one month after an eclipse the relative geometry of the Sun, Moon, and Earth has changed.
As seen from the Earth, the time it takes for the Moon to return to a node, the draconic month, is less than the time it takes for the Moon to return to the same ecliptic longitude as the Sun: the synodic month. The main reason is that during the time that the Moon has completed an orbit around the Earth, the Earth (and Moon) have completed about 1⁄13 of their orbit around the Sun: the Moon has to make up for this in order to come again into conjunction or opposition with the Sun. Secondly, the orbital nodes of the Moon precess westward in ecliptic longitude, completing a full circle in about 18.60 years, so a draconic month is shorter than a sidereal month. In all, the difference in period between synodic and draconic month is nearly 2+1⁄3 days. Likewise, as seen from the Earth, the Sun passes both nodes as it moves along its ecliptic path. The period for the Sun to return to a node is called the eclipse or draconic year: about 346.6201 days, which is about 1⁄20 year shorter than a sidereal year because of the precession of the nodes.
If a solar eclipse occurs at one new moon, which must be close to a node, then at the next full moon the Moon is already more than a day past its opposite node, and may or may not miss the Earth's shadow. By the next new moon it is even further ahead of the node, so it is less likely that there will be a solar eclipse somewhere on Earth. By the next month, there will certainly be no event.
However, about 5 or 6 lunations later the new moon will fall close to the opposite node. In that time (half an eclipse year) the Sun will have moved to the opposite node too, so the circumstances will again be suitable for one or more eclipses.

## Periodicity

The periodicity of solar eclipses is the interval between any two solar eclipses in succession, which will be either 1, 5, or 6 synodic months. It is calculated that the Earth will experience a total number of 11,898 solar eclipses between 2000 BCE and 3000 CE. A particular solar eclipse will be repeated approximately after every 18 years 11 days and 8 hours (6,585.32 days) of period, but not in the same geographical region. A particular geographical region will experience a particular solar eclipse in every 54 years 34 days period. Total solar eclipses are rare events, although they occur somewhere on Earth every 18 months on average.

### Repetition of solar eclipses

For two solar eclipses to be almost identical, the geometric alignment of the Earth, Moon and Sun, as well as some parameters of the lunar orbit should be the same. The following parameters and criteria must be repeated for the repetition of a solar eclipse:
* The Moon must be in new phase.
* The longitude of perigee or apogee of the Moon must be the same.
* The longitude of the ascending node or descending node must be the same.
* The Earth will be nearly the same distance from the Sun, and tilted to it in nearly the same orientation.
These conditions are related to the three periods of the Moon's orbital motion, viz. the synodic month, anomalistic month and draconic month, and to the anomalistic year. In other words, a particular eclipse will be repeated only if the Moon will complete roughly an integer number of synodic, draconic, and anomalistic periods and the Earth-Sun-Moon geometry will be nearly identical. The Moon will be at the same node and the same distance from the Earth. This happens after the period called the saros. Gamma (how far the Moon is north or south of the ecliptic during an eclipse) changes monotonically throughout any single saros series. The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January). When the Earth is near its average distance (March to April or September to October), the change in gamma is average.

### Repetition of lunar eclipses

For the repetition of a lunar eclipse, the geometric alignment of the Moon, Earth and Sun, as well as some parameters of the lunar orbit should be repeated. The following parameters and criteria must be repeated for the repetition of a lunar eclipse:
* The Moon must be in full phase.
* The longitude of perigee or apogee of the Moon must be the same.
* The longitude of the ascending node or descending node must be the same.
* The Earth will be nearly the same distance from the Sun, and tilted to it in nearly the same orientation.
These conditions are related with the three periods of the Moon's orbital motion, viz. the synodic month, anomalistic month and draconic month. In other words, a particular eclipse will be repeated only if the Moon will complete roughly an integer number of synodic, draconic, and anomalistic periods (223, 242, and 239) and the Earth-Sun-Moon geometry will be nearly identical to that eclipse. The Moon will be at the same node and the same distance from the Earth. Gamma changes monotonically throughout any single Saros series. The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January). When the Earth is near its average distance (March to April or September to October), the change in gamma is average.

### Effect of Eccentricity

Another thing to consider is that the motion of the Moon is not a perfect circle. Its orbit is distinctly elliptic, so the lunar distance from Earth varies throughout the lunar cycle. This varying distance changes the apparent diameter of the Moon, and therefore influences the chances, duration, and type (partial, annular, total, mixed) of an eclipse. This orbital period is called the anomalistic month, and together with the synodic month causes the so-called "full moon cycle" of about 14 lunations in the timings and appearances of full (and new) Moons. The Moon moves faster when it is closer to the Earth (near perigee) and slower when it is near apogee (furthest distance), thus periodically changing the timing of syzygies by up to 14 hours either side (relative to their mean timing), and causing the apparent lunar angular diameter to increase or decrease by about 6%. An eclipse cycle must comprise close to an integer number of anomalistic months in order to perform well in predicting eclipses.
If the Earth had a perfectly circular orbit centered around the Sun, and the Moon's orbit was also perfectly circular and centered around the Earth, and both orbits were coplanar (on the same plane) with each other, then two eclipses would happen every lunar month (29.53 days). A lunar eclipse would occur at every full moon, a solar eclipse every new moon, and all solar eclipses would be the same type. In fact the distances between the Earth and Moon and that of the Earth and the Sun vary because both the Earth and the Moon have elliptic orbits. Also, both the orbits are not on the same plane. The Moon's orbit is inclined about 5.14° to Earth's orbit around the Sun. So the Moon's orbit crosses the ecliptic at two points or nodes. If a New Moon takes place within about 17° of a node, then a solar eclipse will be visible from some location on Earth.
At an average angular velocity of 0.99° per day, the Sun takes 34.5 days to cross the 34° wide eclipse zone centered on each node. Because the Moon's orbit with respect to the Sun has a mean duration of 29.53 days, there will always be one and possibly two solar eclipses during each 34.5-day interval when the Sun passes through the nodal eclipse zones. These time periods are called eclipse seasons. Either two or three eclipses happen each eclipse season. During the eclipse season, the inclination of the Moon's orbit is low, hence the Sun, Moon, and Earth become aligned straight enough (in syzygy) for an eclipse to occur.

## Numerical values

These are the lengths of the various types of months as discussed above (according to the lunar ephemeris ELP2000-85, valid for the epoch J2000.0; taken from (e.g.) Meeus (1991) ):
    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:
    ${\mbox{EY}}={\frac {{\mbox{SM}}\times {\mbox{DM}}}{{\mbox{SM}}-{\mbox{DM}}}}$
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:
The ratio of synodic months per half eclipse year yields the same series:
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 number of days given is the average. The actual number of days and fractions of days between two eclipses 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".
| Cycle | Formula | Days | Synodic months | Draconic months | Anomalistic months | Eclipse years | Julian years | Anomalistic years | Eclipse seasons | Node |
| fortnight | 19i − 30+1⁄2s | 14.77 | 0.5 | 0.543 | 0.536 | 0.043 | 0.040 | 0.040 | 0.086 | alternate |
| synodic month | 38i − 61s | 29.53 | 1 | 1.085 | 1.072 | 0.085 | 0.081 | 0.081 | 0.17 | same |
| pentalunex | 53s − 33i | 147.65 | 5 | 5.426 | 5.359 | 0.426 | 0.404 | 0.404 | 0.852 | alternate |
| semester | 5i − 8s | 177.18 | 6 | 6.511 | 6.430 | 0.511 | 0.485 | 0.485 | 1 | alternate |
| lunar year | 10i − 16s | 354.37 | 12 | 13.022 | 12.861 | 1.022 | 0.970 | 0.970 | 2 | same |
| hexon | 13s - 8i | 1,033.57 | 35 | 37.982 | 37.510 | 2.982 | 2.830 | 2.830 | 6 | same |
| hepton | 5s − 3i | 1,210.75 | 41 | 44.493 | 43.940 | 3.493 | 3.315 | 3.315 | 7 | alternate |
| octon | 2i − 3s | 1,387.94 | 47 | 51.004 | 50.371 | 4.004 | 3.800 | 3.800 | 8 | same |
| tzolkinex | 2s − i | 2,598.69 | 88 | 95.497 | 94.311 | 7.497 | 7.115 | 7.115 | 15 | alternate |
| Hibbardina | 31s − 19i | 3,277.90 | 111 | 120.457 | 118.960 | 9.457 | 8.974 | 8.974 | 19 | alternate |
| sar (half saros) | 1⁄2s | 3,292.66 | 111.5 | 120.999 | 119.496 | 9.499 | 9.015 | 9.015 | 19 | same |
| tritos | i − s | 3,986.63 | 135 | 146.501 | 144.681 | 11.501 | 10.915 | 10.915 | 23 | alternate |
| saros (s) | s | 6,585.32 | 223 | 241.999 | 238.992 | 18.999 | 18.030 | 18.029 | 38 | same |
| Metonic cycle | 10i − 15s | 6,939.69 | 235 | 255.021 | 251.853 | 20.021 | 19.000 | 18.999 | 40 | same |
| semanex | 3s - i | 9,184.01 | 311 | 337.496 | 333.303 | 26.496 | 25.145 | 25.144 | 53 | alternate |
| thix | 4i - 5s | 9,361.20 | 317 | 344.007 | 339.733 | 27.007 | 25.630 | 25.629 | 54 | same |
| inex (i) | i | 10,571.95 | 358 | 388.500 | 383.674 | 30.500 | 28.944 | 28.944 | 61 | alternate |
| exeligmos | 3s | 19,755.96 | 669 | 725.996 | 716.976 | 56.996 | 54.089 | 54.087 | 114 | same |
| Aubrey cycle | i + 1⁄2s | 20,449.93 | 692.5 | 751.498 | 742.162 | 58.998 | 55.989 | 55.987 | 118 | alternate |
| unidos | i + 2s | 23,742.59 | 804 | 872.497 | 861.658 | 68.497 | 65.004 | 65.002 | 137 | alternate |
| Callippic cycle | 40i − 60s | 27,758.75 | 940 | 1020.084 | 1007.411 | 80.084 | 75.999 | 75.997 | 160 | same |
| triad | 3i | 31,715.85 | 1074 | 1165.500 | 1151.021 | 91.500 | 86.833 | 86.831 | 183 | alternate |
| quarter Palmen cycle | 4i - 1s | 35,702.48 | 1209 | 1312.002 | 1295.702 | 103.002 | 97.748 | 97.745 | 206 | same |
| Mercury cycle | 2i + 3s | 40,899.87 | 1385 | 1502.996 | 1484.323 | 117.996 | 111.978 | 111.975 | 236 | same |
| tritrix | 3i + 3s | 51,471.82 | 1743 | 1891.496 | 1867.997 | 148.496 | 140.922 | 140.918 | 297 | alternate |
| de la Hire cycle | 6i | 63,431.70 | 2148 | 2331.001 | 2302.041 | 183.001 | 173.667 | 173.662 | 366 | same |
| trihex | 3i + 6s | 71,227.78 | 2412 | 2617.492 | 2584.973 | 205.492 | 195.011 | 195.006 | 411 | alternate |
| Lambert II cycle | 9i + s | 101,732.88 | 3445 | 3738.500 | 3692.054 | 293.500 | 278.529 | 278.522 | 587 | alternate |
| Macdonald cycle | 6i + 7s | 109,528.95 | 3709 | 4024.991 | 3974.986 | 315.991 | 299.874 | 299.866 | 632 | same |
| Utting cycle | 10i + s | 112,304.83 | 3803 | 4127.000 | 4075.727 | 324.000 | 307.474 | 307.466 | 648 | same |
| selebit | 11i + s | 122,876.78 | 4161 | 4515.500 | 4459.401 | 354.500 | 336.418 | 336.409 | 709 | alternate |
| Cycle of Hipparchus | 25i − 21s | 126,007.02 | 4267 | 4630.531 | 4573.002 | 363.531 | 344.988 | 344.979 | 727 | alternate |
| Square year | 12i + s | 133,448.73 | 4519 | 4904.000 | 4843.074 | 385.000 | 365.363 | 365.353 | 770 | same |
| Gregoriana | 6i + 11s | 135,870.24 | 4601 | 4992.986 | 4930.955 | 391.986 | 371.992 | 371.983 | 784 | same |
| hexdodeka | 6i + 12s | 142,455.56 | 4824 | 5234.985 | 5169.947 | 410.985 | 390.022 | 390.012 | 822 | same |
| Grattan Guinness cycle | 12i - 4s | 142,809.92 | 4836 | 5248.007 | 5182.807 | 412.007 | 390.992 | 390.982 | 824 | same |
| Hipparchian | 14i + 2s | 161,177.95 | 5458 | 5922.999 | 5849.413 | 464.999 | 441.281 | 441.270 | 930 | same |
| Basic period | 18i | 190,295.11 | 6444 | 6993.002 | 6906.123 | 549.002 | 521.000 | 520.986 | 1098 | same |
| Chalepe | 18i + 2s | 203,465.76 | 6890 | 7476.999 | 7,384.107 | 586.999 | 557.059 | 557.044 | 1174 | same |
| tetradia (Meeus III) | 22i − 4s | 206,241.63 | 6984 | 7579.008 | 7484.849 | 595.008 | 564.659 | 564.644 | 1190 | same |
| tetradia (Meeus I) | 19i + 2s | 214,037.71 | 7248 | 7865.499 | 7767.781 | 617.499 | 586.003 | 585.988 | 1235 | alternate |
| hyper exeligmos | 24i + 12s | 332,750.68 | 11268 | 12227.987 | 12076.070 | 959.987 | 911.022 | 910.998 | 1920 | same |
| cartouche | 52i | 549,741.44 | 18616 | 20202.006 | 19951.022 | 1586.006 | 1505.110 | 1505.070 | 3172 | same |
| Palaea-Horologia | 55i + 3s | 601,213.26 | 20359 | 22093.502 | 21819.019 | 1734.502 | 1646.032 | 1645.989 | 3469 | alternate |
| hybridia | 55i + 4s | 607,798.58 | 20582 | 22335.501 | 22058.012 | 1753.501 | 1664.062 | 1664.018 | 3507 | alternate |
| Selenid 1 | 55i + 5s | 614,383.90 | 20805 | 22577.499 | 22297.004 | 1772.499 | 1682.091 | 1682.047 | 3545 | alternate |
| Proxima | 58i + 5s | 646,099.75 | 21879 | 23743.000 | 23448.024 | 1864.000 | 1768.925 | 1768.878 | 3728 | same |
| heliotrope | 58i + 6s | 652,685.07 | 22102 | 23984.998 | 23687.016 | 1882.998 | 1786.954 | 1786.907 | 3766 | same |
| Megalosaros | 58i + 7s | 659,270.40 | 22325 | 24226.997 | 23926.009 | 1901.997 | 1804.984 | 1804.936 | 3804 | same |
| immobilis | 58i + 8s | 665,855.72 | 22548 | 24468.996 | 24165.001 | 1920.996 | 1823.014 | 1822.966 | 3842 | same |
| accuratissima | 58i + 9s | 672,441.04 | 22771 | 24710.994 | 24403.993 | 1939.994 | 1841.043 | 1840.995 | 3880 | alternate |
| Mackay cycle | 76i + 9s | 862736.15 | 29215 | 31703.996 | 31310.116 | 2488.996 | 2362.043 | 2361.981 | 4978 | alternate |
| Selenid 2 | 95i + 11s | 1,076,773.86 | 36463 | 39569.496 | 39077.897 | 3106.496 | 2948.046 | 2947.968 | 6213 | alternate |
| Horologia | 110i + 7s | 1,209,011.84 | 40941 | 44429.003 | 43877.031 | 3488.003 | 3310.094 | 3310.007 | 6976 | same |