The periodicity and recurrence of eclipses is governed by the saros cycle, a period of approximately 6,585.3 days (18 years 11 days 8 hours). It was known to the Chaldeans as a period when lunar eclipses seem to repeat themselves, but the cycle is applicable to solar eclipses as well.
The saros arises from a natural harmony between three of the Moon's orbital periods:
Synodic Month (new moon to new moon) 29.53059 days = 29d 12h 44m Draconic Month (node to node) 27.21222 days = 27d 05h 06m Anomalistic Month (perigee to perigee) 27.55455 days = 27d 13h 19m
One saros is equal to 223 synodic months. However, 242 draconic months and 239 anomalistic months are also equal to this same period (to within a couple hours)!
Any two eclipses separated by one saros cycle share very similar geometries. They occur at the same node with the Moon at nearly the same distance from Earth and at the same time of year. Because the saros period is not equal to a whole number of days, its biggest drawback is that subsequent eclipses are visible from different parts of the globe. The extra 1/3 day displacement means that Earth must rotate an additional ~8 hours or ~120º with each cycle. For solar eclipses, this results in the shifting of each successive eclipse path by ~120º westward. Thus, a saros series returns to about the same geographic region every 3 saroses (54 years and 34 days).
A saros series doesn't last indefinitely because the three lunar months are not perfectly commensurate with one another. In particular, the Moon's node shifts westward by about 0.5º with each cycle. A typical saros series for a solar eclipse begins when new Moon occurs ~18° east of a node. If the first eclipse occurs at the Moon's descending node, the Moon's umbral shadow will pass ~3500 km below Earth and a partial eclipse will be visible from the south polar region. On the following return, the umbra will pass ~300 km closer to Earth and a partial eclipse of slightly larger magnitude will result. After ten or eleven saros cycles (about 200 years), the first central eclipse will occur near the south pole of Earth. Over the course of the next 950 years, a central eclipse occurs every 18.031 years (= saros) but will be displaced northward by an average of ~300 km. Halfway through this period, eclipses of long duration will occur near the equator. The last central eclipse of the series occurs near the north pole. The next approximately ten eclipses will be partial with successively smaller magnitudes. Finally, the saros series will end a dozen or more centuries after it began at the opposite pole. Due to the ellipticity of the orbits of the Earth and Moon, the exact duration and number of eclipses in a complete saros is not constant. A series may last 1226 to 1550 years and is comprised of 69 to 87 eclipses, of which about 40 to 60 are central (i.e., total or annular).
Solar eclipses that take place near the Moon's ascending node have odd saros numbers. Each succeeding eclipse in a series shifts progressively southward with respect to the center of the Earth. On the other hand, solar eclipses occurring near the Moon's descending node have even saros numbers. Each succeeding eclipse in a series shifts progressively northward with respect to the center of the Earth. The numbering system used for the saros series was introduced by the Dutch Astronomer G. van den Bergh in his book Periodicity and Variation of Solar (and Lunar) Eclipses (Tjeenk Willink, Haarlem, Netherlands, 1955). He assigned the number 1 to a pair of solar and lunar eclipse series that were in progress during the second millennium BC.
Understanding the numbering sequence of the saros is complicated by the fact that it does not depend on when a series either begins or ends. Instead, the numbering is determined by the order in which each series peaks. In this context, the peak of a series occurs when the umbral shadow axis passes closest to the center of the Earth. Since the duration of each series varies up to several hundred years and the numbering is keyed to the order in which each series peaks, this explains why the first eclipse of a series which peaks later can actually preceed the first eclipse of a series that peaks earlier. From the solar eclipse catalogs, the column labeled Gamma is the parameter that gives the minimum distance (in Earth radii) of the shadow axis from the center of Earth during each eclipse. Gamma is positive or negative depending on whether the shadow axis passes north or south of Earth's center. Looking at any of the saros catalogs (e.g., Saros 145) one can see how the value of gamma changes with each eclipse in a series. When gamma reaches its minimum (absolute) value, the series is then at its peak. In the case of Saros 145, the peak occurs with the eclipse of 2342 Mar 08 (gamma=0.008).
Since there are two to five solar eclipses every year, there are approximately forty different saros series in progress at any one time. For instance, during the later half of the twentieth century, there are 41 individual series and 26 of them are producing central eclipses. As old series terminate, new ones are beginning and take their places.
To illustrate, the ten central solar eclipses of 1891, 1909, 1927, 1945, 1963, 1981, 1999, 2017, 2035 and 2053 are all members of Saros 145. The series began with a partial eclipse near the north pole in 1639. The first central eclipse of the series was an annular eclipse in 1891. It was followed by another annular in 1909. The next event was the first total eclipse in 1927. The total solar eclipse of 1999 August 11 is number 21 of 77 eclipses in Saros 145, and it is the 5th of 41 total eclipses in the series. Each of the subsequent total eclipses shifts southwards. The last total eclipse occurs in 2648 near the south pole. The last eclipse of the series takes place in 3009. Table of Saros 145 gives details for every eclipse in the series.
The saros cycle for lunar eclipses operates analogously with the solar eclipse saros. For lunar eclipses, the parameter gamma is the Moon's minimum distance measured with respect to the axis of Earth's shadow (units of Earth radii). Note however, that the saros numbering is opposite to that for solar eclipses. Lunar eclipses occurring near the Moon's ascending node have even saros numbers. Each succeeding eclipse in a series shifts progressively southward with respect to the axis of Earth's shadow. Correspondingly, lunar eclipses occurring near the Moon's descending node have odd saros numbers. Each succeeding eclipse in a series shifts progressively northward with respect to the axis of Earth's shadow.
Another significant eclipse cycle is the inex, a period of 358 synodic months (29 years minus about 20 days, or nearly 10,752 days). The inex is useful because it marks the time interval between consecutively numbered saros series. To see a diagram illustrating the relationship between the saros and inex cycles over a period of 26,000 years, visit the Saros-Inex Panorama page.