Eclipse cycle

From Academic Kids

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.

Contents

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 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 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:

<math>\mbox{EY} = \frac{\mbox{SM}\times\mbox{DM}}{\mbox{SM-DM}}<math>

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 draconitic 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 ...
octaeteris 2920 98.881 107.305 105.972 8.424 ...
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 27,759 940.008 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 ...

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
Twelve (synodic) months. 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.
Octaeteris
Eight years of 365 days, equals 99 lunations to within 3.5 days.
Tritos
a mediocre cycle, relates to the saros like the inex.
Saros 
The most well known, and one of the best, eclipse cycles. After this interval, the Moon's eclipses recur with respect to the lunar calendar. 223 synodic months equals 242 draconitic months to within 51 min.
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; thus precisely 19 years of 365 days, and equals 235 lunations to within 7.5 h.
Inex
In itself a poor cycle, it is very convenient in the classification of eclipse cycles. After a saros series dies, a new one begins 1 inex later (hence its name: in-ex). It is the interval after which the Moon's eclipses recur at the same longitudes as the previous cycle but at the opposite latitudes.
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. It is the interval after which the Moon's eclipses recur with respect to the lunar calendar and at the same longitudes as the previous cycle.
Callippic cycle
441 hollow months and 499 full months; thus 4 Metonic Cycles minus one day or precisely 76 years of 365 1/4 days. It equals 940 lunations to within 5.9 h.
Hipparchic cycle
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. It is equal to four Callippic cycles minus one day, and equals 3760 lunations to within 15.5 min.
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.

External links

Search 5,000 years worth of eclipses at: http://www.hermit.org/Eclipse/when_search.shtml

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

Literature

  • 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
  • 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
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