Eclipse cycle

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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.

Contents 1 Eclipse conditions 2 Recurrence 3 Periodicity 4 Numerical values 5 Eclipse cycles 6 See also 7 References 8 External links

[edit] Eclipse conditions

Eclipses may occur when the Earth and 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 Moon is insufficient to fully obscure the solar disc, unless 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 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, 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.

[edit] Recurrence

An eclipse does not occur every month, because one month after an eclipse the relative geometry of the Sun, Moon, and Earth have 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 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 in 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 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 over 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.

Eclipses can occur in a one- or two-month period twice a year, around the time when the Sun is near the nodes of the Moon's orbit.

[edit] 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. Moon moves faster when it is closer to 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.

[edit] 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)^[1]

DM = 27.212220817 days (Draconic month)^[2]

AM = 27.55454988 days (Anomalistic month)^[3]

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: month when the Moon returns to the node,the node. These 3-way relations are not independent, and indeed the eclipse year can be described as the beat period of the synodic and anomalistic months; in formula:

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 fractions]]: this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions. The target ratio is 29.530588853/ (27.212220817/2) = 2.170391682

   2.170391682 =                  half draconic months/synodic months:
   2+1/                                              2/1
       5+1/                                         11/5
           1+1/                                     13/6        semester
               6+1/                                 89/41
                   1+1/                            102/47
                       1+1/                        191/88
                           1+1/                    293/135      tritos
                               1+1/                484/223      Saros
                                   67+1/            777/358      Inex
                                       15+1/      9031/4161
                                            1+... 9808/4519

The ratio of synodic months per half eclipse year and per eclipse year yields the same series:

   5.868831091 =                        synodic months/half ,   /full eclipse years
   5+1/                                              5/1
       1+1/                                          6/1              semester
           6+1/                                     41/7
               1+1/                                 47/8      47/4
                   1+1/                             88/15
                       1+1/                        135/23             tritos
                           1+1/                    223/38    223/19   Saros
                               1+1/                358/61             Inex
                                   11+1/          4161/709
                                        1+...     4519/770  4519/385

Each of these is an eclipse cycle. Less accurate cycles may be constructed by combinations of these.

[edit] 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	number of days	number of synodic months	number of draconic months	number of anomalistic months	number of eclipse years
fortnight	14.77	0.5	0.543	0.536	0.043
month	29.53	1	1.085	1.072	0.085
semester	177.18	6	6.511	6.430	0.511
lunar year	354.37	12	13.022	12.861	1.022
octon	1387.94	47	51.004	50.371	4.004
octaeteris	2923.53	99	107.399	106.100	8.434
tritos	3986.63	135	146.501	144.681	11.501
Saros	6585.32	223	241.999	238.992	18.999
Metonic cycle	6939.69	235	255.021	251.853	20.021
Inex	10,571.95	358	388.500	383.674	30.500
exeligmos	19,755.96	669	725.996	716.976	56.996
Callippic cycle	27,758.75	940.008	1020.093	1007.420	80.085
Hipparchic cycle	126,007.02	4267	4630.531	4573.002	363.531
Babylonian	161,177.95	5458	5922.999	5849.413	464.999
tetradia	214,038.72	7248	7865.500	7767.781	617.500

Notes:

Fortnight

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 total lunar eclipse of 15 May 2003 was followed by the partial solar eclipse of 31 May 2003.

Month

Similarly, two events one month apart have the Sun and Moon at two positions on either side of the node, 29° apart: both may cause a partial eclipse.

Semester

or "eclipse season". After 6 (or sometimes 5 or 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. One octon after an eclipse from some Saros series, an eclipse from the Saros series with the next higher number occurs.

Octaeteris

An old calendar cycle rather than an eclipse cycle, in which 8 years equal 99 lunations with an error of about 1.5 days.

Tritos

A mediocre cycle, relates to the Saros like the Inex.

Saros

The best known eclipse cycle, and one of the best for predicting eclipses, which 223 synodic months equal 242 Draconic months with an error of only 51 minutes.

Metonic cycle or Enneadecaeteris

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 h.

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.

Tetradia

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. 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 (2004) for a detailed discussion.

[edit] See also

• Solar eclipse
• Lunar eclipse
• Saros cycle

[edit] References

1. ^ Meeus (1991) form. 47.1
2. ^ Meeus (1991) ch. 49 p.334
3. ^ Meeus (1991) form. 48.1

• 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, Ch.9 Solar Eclipses: Some Periodicities. Willmann-Bell, Richmond VA 1997; ISBN
• Jean Meeus (2004): Mathematical Astronomy Morsels III, Ch.21 Lunar Tetrads (pp.123..140). Willmann-Bell, Richmond VA 2004; ISBN 0-943396-81-6

[edit] External links

• Search 5,000 years worth of eclipses
• A catalogue of eclipse cycles