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Exeligmos: the Triple Saros cycle of eclipses

Continued from the Saros cycle: Unfortunately, 18 years and 11 days after an eclipse the next eclipse in that particular Saros series occurs an additional 8 hours later (since the Saros cycle is 18 years 11 1/3 days long), rarely again in light or dark as it did previously and thus usually invisible at the same location (It does, however, repeat at the same time of day 120° to the west or 1/3 of the world away, since the Earth rotates 1/3 of 360° in 1/3 of a day.)

Because of the extra 1/3 day in the 18 year 11 1/3 day Saros cycle however, eclipses do occur at the same time and place after three complete Saros cycles. This period of almost exactly 54 years and 34 days (19,755.96 days) is known as the Triple Saros or exeligmos (Greek "turn of the wheel"). So by recording all eclipses in a 54-year period, early astronomer/astrologers could predict future eclipses at their location with certainty.

Here's how that happens (the 3 eclipse criteria are repeated below for clarity):

1) The Moon must be at its new (solar eclipse) or full (lunar eclipse) phase. The new Moon is between the Earth and the Sun, and at full Moon the Earth is between the Sun and Moon. One of these geometries must obtain for the Sun’s light to be blocked at all.

2) Since the Moon’s orbital plane around the Earth and the Earth’s orbital plane around the Sun are inclined to each other at about 5°, the Moon must be at or near the intersection of these two planes (a node) for either its shadow to fall on the Earth (at its descending/south node) or for the Earth’s shadow to fall on the Moon (at its ascending/north node); i.e., the Moon must be on the line joining the Earth to the Sun for eclipses to occur (see the two diagrams below).

3) Since the Moon’s orbit is an ellipse (as Kepler demonstrated for all orbiting bodies, page 35), and because the Moon’s shadow is only long enough to reach the Earth (solar eclipse) when it is closest to the Earth (perigee), the Moon must be at or near perigee for a solar eclipse to occur. Also for lunar eclipses, the Earth’s shadow is only long enough to reach the Moon when Earth is closest to the Moon. Depending on the Earth-Sun distance (226,420-252,724 miles) at eclipse, a solar eclipse can be complete or partial.

As the Moon orbits the Earth, a new or full Moon eventually occurs near perigee fulfilling the first and third eclipse criteria. During the Moon's journey around the Earth those two conditions from time-to-time simultaneously reoccur. Eventually at such a reoccurrence the Moon is close enough to a node to satisfy the second eclipse criterion at the same time, and a partial eclipse just barely occurs. 18 years, 11 and 1/3 days later, because of the Saros cycle, a slightly more complete eclipse will occur. Then for a few centuries, at the end of every Saros interval, ever more complete eclipses occur.

In the above diagram the central five arrowed moon paths represent five Saros events. As the Moon orbits the Earth after an eclipse, eclipses at successive Saros intervals gradually become complete (central three paths) and then partial again. Finally, after more than one thousand years the second eclipse criterion above (being at a node) fails, ending this "Saros series". Every Saros series is comprised of 69-87 eclipses over 1226-1550 years. Although 2 to 5 solar eclipses occur every year, only those separated by multiples of 18 years 11 days and 8 hours are in the same series. Thus about 40 different Saros series are always in progress.

Both the Saros and the Exeligmos cycles were known to the Greeks and the Babylonians as early as 100 BC since eclipses in these cycles could be predicted—or were at least demonstrable—by the Antikythera mechanism created no later than 100 BC. Both cycles also appear in a list of eclipse dates on a Babylonian clay tablet from 490-374 BC, although I have been unable to confirm when that table was actually created.





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© Carl Woebcke, Exeligmos: the Triple Saros Cycle, 1991-2016. All rights reserved.