Eclipse cycle (original) (raw)

Eclipses may occur repeatedly, separated by some specific interval of time: this interval is called an eclipse cycle. The series of eclipses is called an eclipse series.

General explanation

Eclipse conditions

Eclipses may occur when the Earth and Moon are on one line 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. At Full Moon, when the Moon is in opposition with the Sun, the Moon may pass through the shadow of the Earth, which 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.

Now 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). 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 diameter of the shadow of the Earth at the position of the Moon.

So at most New Moons the Earth passes too much North or South of the shadow of the Moon, and at most Full Moons the Moon misses the shadow of the Earth. Also most of the time the Moon will not be able to fully cover the Sun, but because of the elliptic orbit it sometimes is nearer and looks bigger. In any case, the alignment must be 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 at the time of the syzygy, the Moon is near one of the two nodes of its orbit on the ecliptic. Of course the Sun is also near a node at that time: the same node in case of a solar eclipse, the opposite node in case of a lunar eclipse.

Recurrence

Now the time it takes for the Moon to return to a node, the so-called draconic month, is less than the time it takes for the Moon to return to the Sun: the synodic month. The reason is that the orbit of the Moon precesses backward with respect to the ecliptic, and makes a full circle in somewhat less than 9 years. The difference in period between synodic and draconic month is about 2 + 1/3 days. Likewise, the Sun passes both nodes as it moves over the ecliptic. The period to return to the same node is called eclipse year, and is about 1/9th year shorter than a sidereal year because of the precession of the nodes of the Moon's orbit in about 9 years.

So if a solar eclipse occurs at one New Moon, so 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, and it is more unlikely that there will be a solar eclipse somewhere at 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 has moved to the opposite node too. Now the circumstances are suitable again for one or more eclipses.

So 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.

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 + 1/2 (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 some 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, which means that the Moon's distance from the Earth varies. This changes the apparent diameter of the Moon, and therefore influences the chances, duration, and appearance 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 perturbations of the orbit may change the times of the syzygies by up to 14 hours, and change the apparent diameter by about 6% in either direction. An eclipse cycle will have to be close to an integer number of anomalistic months for predicting eclipses well.

Numerical values

SM = 29.53059 days (Synodic month)

DM = 27.21222 days (Draconic month)

AM = 27.55455 days (Anomalistic month)

EY = 346.620 days (Eclipse year)

Note that:Good periods can be found from continued fractions:

half draconic months per synodic month:

2.170391... = 2+1/ 2 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 1+1/ 777/358 inex 11+1/ 9031/4161 1+... 9808/4519

synodic months per half eclipse year and per eclipse year yield the same series:

5.868831... = 5+1/ 5 1+1/ 6 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 period. Less accurate periods may be constructed by combination of these.

Eclipse cycles

cycle days synodic draconic anomalistic eclipse yr persistence
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 ...

saros

6585.32

223

241.999

238.992

18.999

...

Metonic cycle

6939.69

235

255.021

251.853

20.021

...

inex

10571.95

358

388.500

383.674

30.500

...

exeligmos

19755.96

669

725.996

716.976

56.996

...

Hipparchos

126007.02

4267

4630.531

4573.002

363.531

...

Babylonian

161177.95

5458

5922.999

5849.413

464.999

...

Comments:

;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 have moved about 15° w.r.t. the nodes (the Moon opposite to where it was the first 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 : 12 months (a lunar year) is a little longer than an eclipse year: the Sun has returned to the node, so more eclipses may occur. ;Octon : This is 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. ;Saros : The most well known, and one of the best, eclipse cycles. ;Metonic cycle : 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. ;Inex : In itself a poor cycle, it is very convenient in the classification of eclipse cycles. After a saros series dies, a new one begings 1 inex later (hence its name: in-ex). ;Exeligmos : 3 saroses: the benefit is, that it is nearly an integer number of days. So the next eclipse will be visible from a location close to the first one; in contrast to the saros, when the eclipse occurs ca. 8h later on the day and about 120° West of the first one. ;Hipparchos : Not a very remarkable eclipse cycle, but Hipparchos constructed it to closely match an integer number of 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.

Frequency

number of eclipses per year; tetrads.

Saros and inex

saros×inex; conjunction points; lifecycles; very long periods.

A comprehensive page with eclipse cycles is at: " class="external">http://www.phys.uu.nl/~vgent/calendar/eclipsecycles.htm

Literature