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Eclipse cycle

Animated graph of a the paths of totality of a solar eclipse cycle.

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.

Eclipse conditions

Eclipses may occur when the Earth and the Moon are aligned with the Sun, and the shadow of one body cast by the Sun falls on the other. So at new moon (or rather Dark Moon), when the Moon is in conjunction with the Sun, the Moon may pass in front of the Sun as seen from a narrow region on the surface of the 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 the Earth, and a lunar eclipse is visible from the night half of the Earth.

Note: Conjunction and opposition of the Moon together have a special name: syzygy (from Greek for "junction"), because of the importance of these lunar phases.

An eclipse does not happen at every new or full moon, because the plane of the orbit of the Moon around the Earth is tilted with respect to the plane of the orbit of the Earth around the Sun (the ecliptic): so as seen from the Earth, when the Moon is nearest to the Sun (new moon) or at largest distance (full moon), the three bodies usually are not exactly on the same line.

This inclination is on average about:

I = 5°09'

Compare this with the relevant apparent mean diameters:

Sun: 32' 2"
Moon: 31'37" (as seen from the surface of the Earth right beneath the Moon)
and: 1°23' for the mean diameter of the shadow of the Earth at the mean lunar distance.

Therefore, at most new moons the Earth passes too far north or south of the lunar shadow, and at most full moons the Moon misses the shadow of the Earth. Also, at most solar eclipses the apparent angular diameter of the Moon is insufficient to fully obscure the solar disc, unless the Moon is close to perigee. In any case, the alignment must be close to perfect to cause an eclipse.

An eclipse can only occur when the Moon is close to the plane of the orbit of the Earth, i.e. when its ecliptic latitude is small. This happens when the Moon is near one of the two nodes of its orbit on the ecliptic at the time of the syzygy. Of course, to produce an eclipse, the Sun must also be near a node at that time: the same node for a solar eclipse, or the opposite node for a lunar eclipse.

Recurrence

A symbolic orbital diagram from the view of the Earth at the center, showing the Moon's two nodes where eclipses can occur.

Eclipses (up to three) occur during an eclipse season, a one- or two-month period 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 1/13th 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 18½ years, so a draconic month is shorter than a sidereal month. In all, the difference in period between synodic and draconic month is nearly 2⅓ 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 d, which is about 1/20th 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

These are still rather vague predictions. However we know that if an eclipse occurred at some moment, then there will occur an eclipse again S synodic months later, if that interval is also D draconic months, where D is an integer number (return to same node), or an integer number + ½ (return to opposite node). So an eclipse cycle is any period P for which approximately holds:

P = S×(synodic month length) = D×(Draconic month length)

Given an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.

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 (relative to their mean timing), and changing the apparent lunar angular diameter by about ±6%. An eclipse cycle must comprise close to an integer number of anomalistic months in order to perform well in predicting eclipses.

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)[2]
DM = 27.212220817 days (Draconic month)[3]
AM = 27.55454988 days (Anomalistic month)[4]
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:

$\mbox{EY} = \frac{\mbox{SM}\times\mbox{DM}}{\mbox{SM-DM}}$

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 an approximation 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

2.170391682 = [2;5,1,6,1,1,1,1,1,11,1,...][5]:
Quotients  Convergents
half DM/SM     decimal      named cycle (if any)
2;           2/1    = 2
5           11/5    = 2.2
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
1         9808/4519 = 2.170391679
...


The ratio of synodic months per half eclipse year and per 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
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
1    4519/770    = 5.868831169  4519/385
...


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.

cycle formula solar days synodic months draconic months anomalistic months eclipse years tropical years
fortnight (38i – 61s)/2 14.77 0.5 0.543 0.536 0.043 0.040
synodic month 38i – 61s 29.53 1 1.085 1.072 0.085 0.081
pentalunex -33i + 53s 147.65 5 5.426 5.359 0.426 0.404
semester 5i – 8s 177.18 6 6.511 6.430 0.511 0.485
lunar year 10i – 16s 354.37 12 13.022 12.861 1.022 0.970
octon 2i – 3s 1387.94 47 51.004 50.371 4.004 3.800
tzolkinex -i + 2s 2598.69 88 95.497 94.311 7.497 7.115
sar (half saros) (0i + s)/2 3292.66 111.5 120.999 119.496 9.499 9.015
tritos i – s 3986.63 135 146.501 144.681 11.501 10.915
saros (s) 0i + s 6585.32 223 241.999 238.992 18.999 18.030
Metonic cycle 10i – 15s 6939.69 235 255.021 251.853 20.021 19.000
inex (i) i ± 0s 10,571.95 358 388.500 383.674 30.500 28.945
exeligmos 0i + 3s 19,755.96 669 725.996 716.976 56.996 54.090
Callippic cycle 40i – 60s 27,758.75 940 1020.084 1007.411 80.084 76.001
triad 3i ± 0s 31,715.85 1074 1165.500 1151.021 91.500 86.835
Hipparchic cycle 25i – 21s 126,007.02 4267 4630.531 4573.002 363.531 344.996
Babylonian 14i + 2s 161,177.95 5458 5922.999 5849.413 464.999 441.291
tetradia (Meeus III) 22i – 4s 206,241.63 6984 7579.008 7484.849 595.008 564.671
tetradia (Meeus [I]) 19i + 2s 214,037.70 7248 7865.500 7767.781 617.500 586.016

Notes:

Fortnight
Half a synodic month. 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, partial solar eclipse of June 1, 2011 is followed by the total lunar eclipse of' June 16, 2011 and partial solar eclipse of July 1, 2011.
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 eclipse.
Pentalunex
5 synodic months. Successive solar or lunar eclipses may occur 1, 5 or 6 synodic months apart.[6]
Semester
Time between successive eclipse seasons. After 6 (or sometimes 5 or more rarely 7) months, the Sun is at the other node, and eclipses may again occur.
Lunar year
Twelve (synodic) months, a little longer than an eclipse year: the Sun has returned to the node, so eclipses may again occur.
Octon
This is 1/5 of the Metonic cycle, and a fairly decent short eclipse cycle, but poor in anomalistic returns. Each octon in a series is 2 saros apart, always occurring at the same node.
Tzolkinex
Includes a half draconic month, so occurs at alternating nodes and alternates between hemispheres. Each consecutive eclipse is a member of preceding saros series from the one before. Equal to ten tzolk'ins. Every third tzolkinex in a series is near an integer number of anomalistic months and so will have similar properties.
Sar (Half saros)
Includes an odd number of fortnights (223). As a result, eclipses alternate between lunar and solar with each cycle, occurring at the same node and with similar characteristics. A long central total solar eclipse will be followed by a very central total lunar eclipse. A solar eclipse where the moon's penumbra just barely grazes the southern limb of earth will be followed half a saros later by a lunar eclipse where the moon just grazes the southern limb of the earth's penumbra.[7]
Tritos
A mediocre cycle, relates to the saros like the inex. A triple tritos is close to an integer number of anomalistic months and so will have similar properties.
Saros
The best known eclipse cycle, and one of the best for predicting eclipses, in which 223 synodic months equal 242 draconic months with an error of only 51 minutes. It is also close to 239 anomalistic months, which makes the circumstances between two eclipses one saros apart very similar.
This is nearly equal to 19 tropical years, but is also 5 "octon" periods and close to 20 eclipse years: so it yields a short series of eclipses on the same calendar date. It consists of 110 hollow months and 125 full months, so nominally 6940 days, and equals 235 lunations with an error of only 7.5 hours.
Inex
By itself a poor cycle, it is very convenient in the classification of eclipse cycles, because after a saros series dies, a new saros series often begins 1 inex later (hence its name: in-ex). One inex after an eclipse, another eclipse takes place at the same longitude, but at the opposite latitude.
Exeligmos
A triple saros, with the advantage that it has nearly an integer number of days, so the next eclipse will be visible at locations near the eclipse that occurred one exeligmos earlier, in contrast to the saros, in which the eclipse occurs about 8 hours later in the day or about 120° to the west of the eclipse that occurred one saros earlier.
Callippic cycle
441 hollow months and 499 full months; thus 4 Metonic Cycles minus one day or precisely 76 years of 365¼ days. It equals 940 lunations with an error of only 5.9 hours.
A triple inex, with the advantage that it has nearly an integer number of anomalistic months, which makes the circumstances between two eclipses one Triad apart very similar, but at the opposite latitude. Almost exactly 87 calendar years minus 2 months.
Hipparchic cycle
Not a noteworthy eclipse cycle, but Hipparchus constructed it to closely match an integer number of synodic and anomalistic months, years (345), and days. By comparing his own eclipse observations with Babylonian records from 345 years earlier, he could verify the accuracy of the various periods that the Chaldeans used.
Babylonian
The ratio 5923 returns to latitude in 5458 months was used by the Chaldeans in their astronomical computations.
Sometimes 4 total lunar eclipses occur in a row with intervals of 6 lunations (semester), and this is called a tetrad. Giovanni Schiaparelli noticed that there are eras when such tetrads occur comparatively frequently, interrupted by eras when they are rare. This variation takes about 6 centuries. Antonie Pannekoek (1951) explained this phenomenon and found a period of 591 years. Van den Bergh (1954) from Theodor von Oppolzer's Canon der Finsternisse found a period of 586 years. This happens to be an eclipse cycle; see Meeus [I] (1997). Recently Tudor Hughes explained the variation from secular changes in the eccentricity of the Earth's orbit: the period for occurrence of tetrads is variable and currently is about 565 years; see Meeus III (2004) for a detailed discussion.

References

1. ^ properly, these are periods, not cycles
2. ^ Meeus (1991) form. 47.1
3. ^ Meeus (1991) ch. 49 p.334
4. ^ Meeus (1991) form. 48.1
5. ^ 2.170391682 = 2 + 0.170391682 ; 1/0.170391682 = 5 + 0.868831085... ; 1/0.868831085... = 1 + 0.15097171... ; 1/0.15097171 = 6 + 0.6237575... ; etc. ; Evaluating: 1/6 + 1 = 7/6; 6/7 + 5 = 41/7 ; 7/41 + 2 = 89/41
6. ^ A Catalogue of Eclipse Cycles, Robert Harry van Gent
7. ^ A Catalogue of Eclipse Cycles, Robert Harry van Gent
• S. Newcomb (1882): On the recurrence of solar eclipses. Astron.Pap.Am.Eph. vol.I pt.I . Bureau of Navigation, Navy Dept., Washington 1882
• J.N. Stockwell (1901): Eclips-cycles. Astron.J. 504 [vol.xx1(24)], 14-Aug-1901
• A.C.D. Crommelin (1901): The 29-year eclipse cycle. Observatory xxiv nr.310, 379, Oct-1901
• A. Pannekoek (1951): Periodicities in Lunar Eclipses. Proc. Kon. Ned. Acad. Wetensch. Ser.B vol.54 pp. 30..41 (1951)
• G. van den Bergh (1954): Eclipses in the second millennium B.C. Tjeenk Willink & Zn NV, Haarlem 1954
• G. van den Bergh (1955): Periodicity and Variation of Solar (and Lunar) Eclipses, 2 vols. Tjeenk Willink & Zn NV, Haarlem 1955
• Jean Meeus (1991): Astronomical Algorithms (1st ed.). Willmann-Bell, Richmond VA 1991; ISBN 0-943396-35-2
• Jean Meeus (1997): Mathematical Astronomy Morsels [I], Ch.9 Solar Eclipses: Some Periodicities (pp. 49..55). Willmann-Bell, Richmond VA 1997; ISBN 0-943396-51-4
• Jean Meeus (2004): Mathematical Astronomy Morsels III, Ch.21 Lunar Tetrads (pp. 123..140). Willmann-Bell, Richmond VA 2004; ISBN 0-943396-81-6

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