thumb|right|250px|Paths of partiality, annularity, hybridity, and totality for [[Solar Saros 136|Solar Saros Series 136. The interval between successive eclipses in the series is one saros, approximately 18 years.]]
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
thumb|240px|right|A diagram of a [[solar eclipse (not to scale)]]
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
240px|thumb|A symbolic orbital diagram from the view of the Earth at the center, showing the Moon's two nodes where eclipses can occur.
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 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<!--- 18.59948 a --->, so a draconic month is shorter than a sidereal month. In all, the difference in period between synodic and draconic month is nearly days<!--- 2.31837 d --->. 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 year<!--- 0.05102 ---> 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.
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.
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.
: 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:
:<math>\mbox{EY} = \frac{\mbox{SM} \times \mbox{DM{\mbox{SM}-\mbox{DM</math>
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".
{| class="wikitable sortable"
! Cycle!! Formula!!Days!!Synodic<br>months!!Draconic<br>months!!Anomalistic<br>months!!Eclipse<br>years!!Julian<br>years!!Anomalistic<br>years!!Eclipse<br>seasons!!Node
|-
| fortnight ||19i − s||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)||s||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 + s|| 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|| 1,020.084 || 1,007.411 || 80.084 || 75.999 || 75.997 || 160 || same
|-
| triad || 3i || 31,715.85 || 1,074 || 1,165.500 || 1,151.021 || 91.500 || 86.833 || 86.831 || 183 || alternate
|-
|quarter Palmen cycle||4i - 1s||35,702.48 || 1,209 || 1,312.002 || 1,295.702 || 103.002 || 97.748 || 97.745 || 206 || same
|-
| Mercury cycle|| 2i + 3s || 40,899.87 || 1,385 || 1,502.996 || 1,484.323 || 117.996 || 111.978 || 111.975 || 236 || same
|-
| tritrix || 3i + 3s || 51,471.82 || 1,743 || 1,891.496 || 1,867.997 || 148.496 || 140.922 || 140.918 || 297 || alternate
|-
|de la Hire cycle|| 6i || 63,431.70 || 2,148 || 2,331.001 || 2,302.041 || 183.001 || 173.667 || 173.662 || 366 || same
|-
| trihex || 3i + 6s || 71,227.78 || 2,412 || 2,617.492 || 2,584.973 || 205.492 || 195.011 || 195.006 || 411 || alternate
|-
|Lambert II cycle|| 9i + s || 101,732.88 || 3,445 || 3,738.500 || 3,692.054 || 293.500 || 278.529 || 278.522 || 587 || alternate
|-
|Macdonald cycle|| 6i + 7s || 109,528.95 || 3,709 || 4,024.991 || 3,974.986 || 315.991 || 299.874 || 299.866 || 632 || same
|-
|Utting cycle|| 10i + s || 112,304.83 || 3,803 || 4,127.000 || 4,075.727 || 324.000 || 307.474 || 307.466 || 648 || same
|-
| selebit || 11i + s || 122,876.78 || 4,161 || 4,515.500 || 4,459.401 || 354.500 || 336.418 || 336.409 || 709 || alternate
|-
|Cycle of Hipparchus|| 25i − 21s||126,007.02||4,267||4,630.531||4,573.002||363.531||344.988||344.979||727|| alternate
|-
| Square year|| 12i + s || 133,448.73 || 4,519 || 4,904.000 || 4,843.074 || 385.000 || 365.363 || 365.353 || 770 || same
|-
| Gregoriana || 6i + 11s || 135,870.24 || 4,601 || 4,992.986 || 4,930.955 || 391.986 || 371.992 || 371.983 || 784 || same
|-
| hexdodeka || 6i + 12s || 142,455.56 || 4,824 || 5,234.985 || 5,169.947 || 410.985 || 390.022 || 390.012 || 822 || same
|-
|Grattan Guinness cycle||12i - 4s||142,809.92||4,836|| 5,248.007 || 5,182.807 || 412.007 || 390.992 || 390.982 || 824 || same
|-
| Hipparchian|| 14i + 2s || 161,177.95 || 5,458 || 5,922.999 || 5,849.413 || 464.999 || 441.281 || 441.270 || 930 || same
|-
|Basic period|| 18i || 190,295.11 || 6,444 || 6,993.002 || 6,906.123 || 549.002 || 521.000 || 520.986 || 1,098 || same
|-
| Chalepe || 18i + 2s || 203,465.76 || 6,890 || 7,476.999 || 7,384.107 || 586.999 || 557.059 || 557.044 || 1,174 || same
|-
|tetradia (Meeus III)||22i − 4s||206,241.63 || 6,984 || 7,579.008 || 7,484.849 || 595.008 || 564.659 || 564.644 || 1,190 || same
|-
|tetradia (Meeus I) ||19i + 2s|| 214,037.71 || 7,248 || 7,865.499 || 7,767.781 || 617.499 || 586.003 || 585.988 || 1,235 || alternate
|-
|hyper exeligmos|| 24i + 12s || 332,750.68 || 11,268 || 12,227.987 || 12,076.070 || 959.987 || 911.022 || 910.998 || 1,920 || same
|-
| cartouche || 52i || 549,741.44 || 18,616 || 20,202.006 || 19,951.022 || 1,586.006 || 1,505.110 || 1,505.070 || 3,172 || same
|-
|Palaea-Horologia || 55i + 3s|| 601,213.26 || 20,359 || 22,093.502 || 21,819.019|| 1,734.502 || 1,646.032 || 1,645.989 || 3,469 || alternate
|-
| hybridia || 55i + 4s || 607,798.58 || 20,582 || 22,335.501 || 22,058.012|| 1,753.501 || 1,664.062 || 1,664.018 || 3,507 || alternate
|-
| Selenid 1 || 55i + 5s || 614,383.90 || 20,805 || 22,577.499 || 22,297.004 || 1,772.499 || 1,682.091 || 1,682.047 || 3,545 || alternate
|-
| Proxima || 58i + 5s || 646,099.75 || 21,879 || 23,743.000 || 23,448.024 || 1,864.000 || 1,768.925 || 1,768.878 || 3,728 || same
|-
| heliotrope || 58i + 6s || 652,685.07 || 22,102 || 23,984.998 || 23,687.016|| 1,882.998 || 1,786.954 || 1,786.907 || 3,766 || same
|-
| Megalosaros|| 58i + 7s || 659,270.40 || 22,325 || 24,226.997 || 23,926.009 || 1,901.997 || 1,804.984 || 1,804.936 || 3,804 || same
|-
| immobilis || 58i + 8s || 665,855.72 || 22,548 || 24,468.996 || 24,165.001 || 1,920.996 || 1,823.014 || 1,822.966 || 3,842 || same
|-
|accuratissima|| 58i + 9s || 672,441.04 || 22,771 || 24,710.994 || 24,403.993 || 1,939.994 || 1,841.043 || 1,840.995 || 3,880 || alternate
|-
|Mackay cycle|| 76i + 9s || 862736.15 || 29,215 || 31,703.996 || 31,310.116 || 2,488.996 || 2,362.043 || 2,361.981 || 4,978 || alternate
|-
| Selenid 2 || 95i + 11s || 1,076,773.86 || 36,463 || 39,569.496 || 39,077.897 || 3,106.496 || 2,948.046 || 2,947.968 || 6,213 || alternate
|-
| Horologia || 110i + 7s || 1,209,011.84 || 40,941 || 44,429.003 || 43,877.031 || 3,488.003 || 3,310.094 || 3,310.007 || 6,976 || same
|}
Notes
;Fortnight: Half a synodic month (29.53 days). When there is an eclipse, there is a fair chance that at the next syzygy there will be another eclipse: the Sun and Moon will have moved about 15° with respect to the nodes (the Moon being opposite to where it was the previous time), but the luminaries may still be within bounds to make an eclipse. For example, the penumbral lunar eclipse of May 26, 2002 is followed by the annular solar eclipse of June 10, 2002 and penumbral lunar eclipse of June 24, 2002. The shortest lunar fortnight between a new moon and a full moon lasts only about 13 days and 21.5 hours, while the longest such lunar fortnight lasts about 15 days and 14.5 hours. (Due to evection, these values are different going from quarter moon to quarter moon. The shortest lunar fortnight between first and last quarter moons lasts only about 13 days and 12 hours, while the longest lasts about 16 days and 2 hours.)
:For more information see eclipse season.
;Synodic month: Similarly, two events one synodic month apart have the Sun and Moon at two positions on either side of the node, 29° apart: both may cause a partial solar eclipse. For a lunar eclipse, it is a penumbral lunar eclipse.
;Pentalunex: 5 synodic months. Successive solar or lunar eclipses may occur 1, 5 or 6 synodic months apart.
thumb|600px| [[Histogram of dates of solar eclipses in 21st century. The dates form 35 clusters. Each cluster contains eclipses separated by Metonic cycles of 19 years. Each series contains four or five eclipses, and 46 or 65 or 84 years after the first one another series starts about a day and a half later in the (Julian) year. This means that the clusters slowly move forward to later dates. In a saros series, every 18 years the eclipse moves to the next later cluster. After 631 years (35 saros) it comes back to the original cluster, which by then has moved, in the Julian calendar, to a date about 13 or 14 days later, or about 18 days later in the Gregorian calendar.]]
;Metonic cycle or enneadecaeteris: Nearly 6940 days, but as an eclipse cycle can be taken as 235 synodic months. This is just an hour and a half less than 19 years of days. It is also 5 "octon" periods and close to 20 eclipse years, so it yields a short series of four or five eclipses on the same calendar date or on two calendar dates. It is equivalent to 110 "hollow months" of 29 days and 125 "full months" of 30 days. Twenty four Metonic cycles of 235 months makes an eclipse cycle of 456 years, equivalent to 17 inex periods minus two saros periods.
;Semanex: Equal to a whole number of weeks plus a hundredth of a day, so consecutive eclipses of the cycle are usually on the same day of the week. Each eclipse in this period is a member of a preceding saros series, always occurring on alternating nodes. so that eclipses synchronize with the timing of Mercury's position in its orbit during each period, equaling 112 years minus one week or 1385 lunations.
;Hexdodeka: Equal to six Unidos or two Trihex. Useful for giving accurate calculations of the timing of lunisolar syzygies.
;Immobilis: Equals 58 inex plus 8 saros (one saros more than a Megalosaros), which is exactly 1879 lunar years. Always occurs on the same node. Very close to a whole number of anomalistic months, although 43 inex minus 5 saros (14279 months, 1154.5 years) is even closer.
:year = 28.945 × number of the saros series + 18.030 × number of the inex series − 2882.55
When this is greater than 1, the integer part gives the year AD, but when it is negative the year BC is obtained by taking the integer part and adding 2. For instance, the eclipse in saros series 0 and inex series 0 was in the middle of 2884 BC.
A "panorama" of solar eclipses arranged by saros and inex has been produced by Luca Quaglia and John Tilley showing 61775 solar eclipses from 11001 BC to AD 15000 (see below).
Each column of the graph is a complete Saros series which progresses smoothly from partial eclipses into total or annular eclipses and back into partials. Each graph row represents an inex series. Since a saros, of 223 synodic months, is slightly less than a whole number of draconic months, the early eclipses in a saros series (in the upper part of the diagram) occur after the Moon goes through its node (the beginning and end of a draconic month), while the later eclipses (in the lower part) occur before the Moon goes through its node. Every 18 years, the eclipse occurs on average about half a degree further west with respect to the node, but the progression is not uniform.
800px|thumb|Solar eclipses from –11000 to +15000.
thumb|800px|Saros and inex values for solar eclipses calculated from approximate date
Saros and inex number can be calculated for an eclipse near a given date. One can also find the approximate date of solar eclipses at distant dates by first determining one in an inex series such as series 50. This can be done by adding or subtracting some multiple of 28.9450 Gregorian years from the solar eclipse of 10 May, 2013, or 28.9444 Julian years from the Julian date of 27 April, 2013. Once such an eclipse has been found, others around the same time can be found using the short cycles. For lunar eclipses, the anchor dates May 4, 2004 or Julian April 21 may be used.
Saros and inex numbers are also defined for lunar eclipses. A solar eclipse of given saros and inex series will be preceded a fortnight earlier by a lunar eclipse whose saros number is 26 lower and whose inex number is 18 higher, or it will be followed a fortnight later by a lunar eclipse whose saros number is 12 higher and whose inex number is 43 lower. As with solar eclipses, the Gregorian year of a lunar eclipse can be calculated as:
:year = 28.945 × number of the saros series + 18.030 × number of the inex series − 2454.564
Lunar eclipses can also be plotted in a similar diagram, this diagram covering 1000 AD to 2500 AD. The yellow diagonal band represents all the eclipses from 1900 to 2100. This graph immediately illuminates that this 1900–2100 period contains an above average number of total lunar eclipses compared to other adjacent centuries.
640px|thumb
This is related to the fact that tetrads (see above) are more common at present than at other periods. Tetrads occur when four lunar eclipses occur at four lunar inex numbers, decreasing by 8 (that is, a semester apart), which are in the range giving fairly central eclipses (small gamma), and furthermore the eclipses take place around halfway between the Earth's perihelion and aphelion. For example, in the tetrad of 2014-2015 (the so-called Four Blood Moons), the inex numbers were 52, 44, 36, and 28, and the eclipses occurred in April and late September-early October. Normally the absolute value of gamma decreases and then increases, but because in April the Sun is further east than its mean longitude, and in September/October further west than its mean longitude, the absolute values of gamma in the first and fourth eclipse are decreased, while the absolute values in the second and third are increased. The result is that all four gamma values are small enough to lead to total lunar eclipses. The phenomenon of the Moon "catching up" with the Sun (or the point opposite the Sun), which is usually not at its mean longitude, has been called a "stern chase".
Inex series move slowly through the year, each eclipse occurring about 20 days earlier in the year, 29 years later. This means that over a period of 18.2 inex cycles (526 years) the date moves around the whole year. But because the perihelion of Earth's orbit is slowly moving as well, the inex series that are now producing tetrads will again be halfway between Earth's perihelion and aphelion in about 586 years.
thumb|550px|Time of year for solar eclipses between saros 90 and saros 210. The blue patches repeat at intervals of about 1640 years.
The graph of inex versus saros for solar or lunar eclipses can be skewed so that the x axis shows the time of year. (An eclipse which is two saros series and one inex series later than another will be only 1.8 days later in the year in the Gregorian calendar.) This shows the 586-year oscillations as oscillations that go up around perihelion and down around aphelion (see graph).
Properties of eclipses
The properties of eclipses, such as the timing, the distance or size of the Moon and Sun, or the distance the Moon passes north or south of the line between the Sun and the Earth, depend on the details of the orbits of the Moon and the Earth. There exist formulae for calculating the longitude, latitude, and distance of the Moon and of the Sun using sine and cosine series. The arguments of the sine and cosine functions depend on only four values, the Delaunay arguments:
- D, the mean elongation (angle between the Sun and Moon longitudes)
- F, the mean argument of latitude (the angle between the Moon and the ascending node)
- l, the mean anomaly of the Moon (how far the Moon is from perigee)
- l', the mean anomaly of the Sun (or of the Earth)
These four arguments are basically linear functions of time but with slowly varying higher-order terms. A diagram of inex and saros indices such as the "Panorama" shown above is like a map, and we can consider the values of the Delaunay arguments on it. The mean elongation, D, goes through 360° 223 times when the inex value goes up by 1, and 358 times when the saros value goes up by 1. It is thus equivalent to 0°, by definition, at each combination of solar saros index and inex index, because solar eclipses occur when the elongation is zero. From D one can find the actual elapsed time from some reference time such as J2000, which is like a linear function of inex and saros but with a deviation that grows quadratically with distance from the reference time, amounting to about 19 minutes at a distance of 1000 years. The mean argument of latitude, F, is equivalent to 0° or 180° (depending on whether the saros index is even or odd) along the smooth curve going through the centre of the band of eclipses, where gamma is near zero (around inex series 50 at present). F decreases as we go away from this curve towards higher inex series, and increases on the other side, by about 0.5° per inex series. When the inex value is too far from the centre, the eclipses disappear because the Moon is too far north or south of the Sun. The mean anomaly of the Sun is a smooth function, increasing by about 10° when increasing inex by 1 in a saros series and decreasing by about 20° when increasing saros index by 1 in an inex series. This means it is almost constant when increasing inex by 1 and saros index by 2 (the "Unidos" interval of 65 years). The above graph showing the time of year of eclipses basically shows the solar anomaly, since the perihelion moves by only one day per century in the Julian calendar, or 1.7 days per century in the Gregorian calendar. The mean anomaly of the Moon is more complicated. If we look at the eclipses whose saros index is divisible by 3, then the mean anomaly is a smooth function of inex and saros values. Contours run at an angle, so that mean anomaly is fairly constant when inex and saros values increase together at a ratio of around 21:24. The function varies slowly, changing by only 7.4° when changing the saros index by 3 at a constant inex value. A similar smooth function obtains for eclipses with saros modulo 3 equal to 1, but shifted by about 120°, and for saros modulo 3 equal to 2, shifted by 120° the other way.
[[File:Solar eclipse time of year for saros index divisible by 3.png|thumb|550px|Time of year for solar eclipses between saros 90 and saros 210, but showing only the saros series whose index is divisible by 3. The time of year is related to the anomaly of the Sun. Two of the four eclipses of the year 2000 are indicated, with a line between them which shows (almost exactly) the slope of simultaneity in this graph.
]]
The upshot is that the properties vary slowly over the diagram in any of the three sets of saros series. The accompanying graph shows just the saros series that have saros index modulo 3 equal to zero. The blue areas are where the mean anomaly of the Moon is near 0°, meaning that the Moon is near perigee at the time of the eclipse, and therefore relatively large, favoring total eclipses. In the red area, the Moon is generally further from the Earth, and the eclipses are annular. We can also see the effect of the Sun's anomaly. Eclipses in July, when the Sun is further from the Earth, are more likely to be total, so the blue area extends over a greater range of inex index than for eclipses in January.
The waviness seen in the graph is also due to the Sun's anomaly. In April the Sun is further east than if its longitude progressed evenly, and in October it is further west, and this means that in April the Moon catches up with the Sun relatively late, and in October relatively early. This in turn means that the argument of latitude at the actual time of the eclipse will be raised higher in April and lowered in October. Eclipses (either partial or not) with low inex index (near the upper edge in the "Panorama" graph) fail to occur in April because syzygy occurs too far to the east of the node, but more eclipses occur at high inex values in April because syzygy is not so far west of the node. The opposite applies to October. It also means that in April ascending-node solar eclipses will cast their shadow further north (such as the solar eclipse of April 8, 2024), and descending-node eclipses further south. The opposite is the case in October.
Eclipses that occur when the earth is near perihelion (sun anomaly near zero) are in saros series in which the gamma value changes little every 18 years and 11 days. The reason for this is that from one eclipse to the next in the saros series, the day in the year advances by about 11 days, but the Sun's position moves eastward by more than what it does for that change of day in year at other times. This means the Sun's position relative to the node does not change as much as for saros series giving eclipses at other times of the year. In the first half of the 21st century, solar saros series showing this slow rate of change of gamma include 122 (giving an eclipse on January 6, 2019), 132 (January 5, 2038), 141 (January 15, 2010), and 151 (January 4, 2011). Sometimes this phenomenon leads to a saros series giving a large number of central eclipses, for example solar saros 128 gave 20 eclipses with |γ|<0.75 between 1615 and 1958, whereas series 135 gave only nine, between 1872 and 2016. In our example above, this means that although the eclipse in 1688 BC was centered on March 16 at 00:15:31 in Dynamic time, it actually occurred before midnight and therefore on March 15 (using time based on the location of present-day Greenwich, and using the proleptic Julian calendar).
The fact that the argument of latitude is decreased explains why one sees a curvature in the "Panorama" above. Central eclipses in the past and in the future are higher in the graph (lower inex number) than what one would expect from a linear extrapolation. This is because the ratio of the length of a synodic month to the length of a draconic month is getting smaller. Although both are getting longer, the draconic month is doing so more quickly because the rate at which the node moves west is decreasing.