Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles.[1] The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.
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.
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, so a draconic month is shorter than a sidereal month. In all, the difference in period between synodic and draconic month is nearly 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 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.
See also: Saros (astronomy). 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.[2] 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.[2] Total solar eclipses are rare events, although they occur somewhere on Earth every 18 months on average.[3]
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:
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.
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:
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.
See main article: Eclipse season. 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.[4] [5] [6]
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.[2] 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.
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)[7]
DM = 27.212220817 days (Draconic month)[8]
AM = 27.55454988 days (Anomalistic month)[9]
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 x DM | |
| SM-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,...]:[10] 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.
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.[11]
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 − 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 | |
| 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 | |
| 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 |
The next nine cycles, Cartouche through Accuratissima, are all similar, being equal to 52 inex periods plus up to two triads and various numbers of saros periods. This means they all have a near-whole number of anomalistic months. They range from 1505 to 1841 years, and each series lasts for many thousands of years.
Any eclipse can be assigned to a given saros series and inex series. The year of a solar eclipse (in the Gregorian calendar) is then given approximately by:[18]
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).[19] 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.
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.
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, decreseing 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".[20]
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.[21]
One can skew the graph of inex versus saros for solar or lunar eclipses 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).
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:
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.[22] [23]
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 18.03 years. 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 doesn't 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 exammple solar saros 128 gave 20 eclipses with |γ|<0.75 between 1615 and 1958, whereas series 135 gave only nine, between 1872 and 2016.[21]
The time interval between two eclipses in an eclipse cycle is variable. The time of an eclipse can be advanced or delayed by up to ten hours due to the eccentricity of the Moon's orbitthe eclipse will be early when the Moon is going from perigee to apogee, and late when it is going from apogee toward perigee. The time is also delayed because of the eccentricity of the Earth's orbit. Eclipses occur about four hours later in April and four hours earlier in October. This means that the delay varies from eclipse to eclipse in a series. The delay is the sum of two sine-like functions, one based on the time in the anomalistic year and one on the time in the anomalistic month. The periods of these two waves depends on how close the nominal interval between two eclipses in the series is to a whole number of anomalistic years and anomalistic months. In series like the "Immobilis" or the "Accuratissima", which are near whole numbers of both, the delay varies very slowly, so the interval is quite constant. In series like the octon, the Moon's anomaly changes considerably at least twice every three intervals, so the intervals vary considerably.
The "Panorama" can also be related to where on the Earth the shadow of the Moon falls at the central time of the eclipse. If this "maximum eclipse" for a given eclipse is at a particular location, eclipses three saros later will be at a similar latitude (because the saros is close to a whole number of draconic months) and longitude (because a period of three saros is always within a couple hours of being 19755.96 days long, which would change the longitude by about 13° eastward). If instead we increase the saros index at a constant inex index, the intervals are quite variable because the number of anomalistic months or years is not very close to a whole number. This means that although the latitude will be similar (but changing sign), the longitude change can vary by more than 180°. Moving by six inex (a de la Hire cycle) preserves the latitude fairly well but the longitude change is very variable because of the variation of the solar anomaly.
Both the angular size of the Moon in the sky at eclipses at the ascending node and the size of the Sun at those eclipses vary in a sort of sine wave. The sizes at the descending node vary in the same way, but 180° out of phase. The Moon is large at an ascending-node eclipse when its perigee is near the ascending node, so the period for the size of the Moon is the time it takes for the angle between the node and the perigee to go through 360°, or
| 1{1/periodofnode+1/periodofperigee | |
(Note that a plus sign is used because the perigee moves eastward whereas the node moves westward.) A maximum of this is in 2024 (September), explaining why the ascending-node solar eclipse of April 8, 2024, is near perigee and total and the descending-node solar eclipse of October 2, 2024, is near apogee and annular. Although this cycle is about a day less than six years, super-moon eclipses actually occur every three years on average, because there are also the ones at the descending node that occur in between the ones at the ascending node.At lunar eclipses the size of the Moon is 180° out of phase with its size at solar eclipses.
The Sun is large at an ascending-node eclipse when its perigee (the direction toward the Sun when it is closest to the Earth) is near the ascending node, so the period for the size of the Sun is
| 1{1/periodofnode-1/periodofperigee | |
In terms of Delaunay arguments, the Sun is biggest at ascending-node solar eclipses and smallest at descending-node solar eclipses around when l'+D=F (modulo 360°), such as June, 2010. It is smallest at descending-node solar eclipses and biggest at ascending-node solar eclipses 9.3 years later, such as September, 2019.
The lengths of the synodic, draconic, and anomalistic months, the length of the day, and the length of the anomalistic year are all slowly changing. The synodic and draconic months, the day, and the anomalistic year (at least at present) are getting longer, whereas the anomalistic month is getting shorter. The eccentricity of the Earth's orbit is presently decreasing at about one percent per 300 years, thus decreasing the effect of the sun's anomaly. Formulae for the Delaunay arguments show that the lengthening of the synodic month means that eclipses tend to occur later than they would otherwise proportionally to the square of the time separation from now, by about 0.32 hours per millennium squared. The other Delaunay arguments (mean anomaly of the Moon and of the sun and the argument of latitude) will all be increased because of this, but on the other hand the Delaunay arguments are also affected by the fact that the lengths of the draconic month and anomalistic month and year are changing. The net results are:
As an example, from the solar eclipse of April, 1688 BC, to that of April, AD 1623, is 110 inex plus 7 saros (equivalent to a "Palaea-Horologia" plus a "tritrix", 3310.09 Julian years). According to the table above, the Delaunay arguments should change by:
But because of the changing lengths of these, they actually changed by:[22]
Note that in this example, in terms of anomaly (position with respect to perigee) the moon returns to within 1% of an orbit (about 3.4°), rather than 3.2% as predicted using today's values of month lengths.
The fact that the day is getting longer means there are more revolutions of the Earth since some point in the past than what one might calculate from the time and date, and fewer from now to some future time. This effect means eclipses occur earlier in the day or calendar, going in the opposite direction relative to the effect of the lengthening synodic month already mentioned. This effect is known as ΔT. It cannot be calculated exactly but amounts to around 50 minutes per millennium squared.[24] In our example above, this means that although the eclipse in 1688 BC was centred 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).[25]
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.[22]