Eclipse cycle

The conjunction and opposition of the Moon together have a special name: syzygy (Greek for "junction"), because of the importance of these lunar phases.

An eclipse can occur only when the Moon is on or near the plane of Earth's orbit, i.e. when its ecliptic latitude is low.

[2] It is calculated that the Earth will experience a total number of 11,898 solar eclipses between 2000 BCE and 3000 CE.

[2] Total solar eclipses are rare events, although they occur somewhere on Earth every 18 months on average.

[4] For two solar eclipses to be almost identical, the geometric alignment of the Earth, Moon and Sun, as well as some parameters of the lunar orbit should be the same.

The following parameters and criteria must be repeated for the repetition of a solar eclipse: These conditions are related to the three periods of the Moon's orbital motion, viz.

In other words, a particular eclipse will be repeated only if the Moon will complete roughly an integer number of synodic, draconic, and anomalistic periods and the Earth-Sun-Moon geometry will be nearly identical.

Gamma (how far the Moon is north or south of the ecliptic during an eclipse) changes monotonically throughout any single saros series.

The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January).

The following parameters and criteria must be repeated for the repetition of a lunar eclipse: These conditions are related with the three periods of the Moon's orbital motion, viz.

The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January).

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.

The Moon moves faster when it is closer to the Earth (near perigee) and slower when it is near apogee (furthest distance), thus periodically changing the timing of syzygies by up to 14 hours either side (relative to their mean timing), and causing the apparent lunar angular diameter to increase or decrease by about 6%.

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) ): 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: the synodic month when the Moon returns to the Sun, the draconic month when the Moon returns to the node, and the eclipse year when the Sun returns to the node.

This table summarizes the characteristics of various eclipse cycles, and can be computed from the numerical results of the preceding paragraphs; cf.

Every 18 years, the eclipse occurs on average about half a degree further west with respect to the node, but the progression is not uniform.

This graph immediately illuminates that this 1900–2100 period contains an above average number of total lunar eclipses compared to other adjacent centuries.

For example, in the tetrad of 2014-2015 (the so-called Four Blood Moons), the inex numbers were 52, 44, 36, and 28, and the eclipses occurred in April and late September-early October.

[14] One can skew the graph of inex versus saros for solar or lunar eclipses so that the x axis shows the time of year.

There exist formulae for calculating the longitude, latitude, and distance of the Moon and of the Sun using sine and cosine series.

A diagram of inex and saros indices such as the "Panorama" shown above is like a map, and we can consider the values of the Delaunay arguments on it.

The mean argument of latitude, F, is equivalent to 0° or 180° (depending on whether the saros index is even or odd) along the smooth curve going through the centre of the band of eclipses, where gamma is near zero (around inex series 50 at present).

When the inex value is too far from the centre, the eclipses disappear because the Moon is too far north or south of the Sun.

This means it is almost constant when increasing inex by 1 and saros index by 2 (the "Unidos" interval of 65 years).

Contours run at an angle, so that mean anomaly is fairly constant when inex and saros values increase together at a ratio of around 21:24.

This in turn means that the argument of latitude at the actual time of the eclipse will be raised higher in April and lowered in October.

Eclipses that occur when the earth is near perihelion (sun anomaly near zero) are in saros series in which the gamma value changes little every 18.03 years.

This means the Sun's position relative to the node doesn't change as much as for saros series giving eclipses at other times of the year.

Moving by six inex (a de la Hire cycle) preserves the latitude fairly well but the longitude change is very variable because of the variation of the solar anomaly.

Central eclipses in the past and in the future are higher in the graph (lower inex number) than what one would expect from a linear extrapolation.

Paths of partiality, annularity, hybridity, and totality for Solar Saros Series 136 . The interval between successive eclipses in the series is one saros, approximately 18 years.
A diagram of a solar eclipse (not to scale)
A symbolic orbital diagram from the view of the Earth at the center, showing the Moon's two nodes where eclipses can occur.
Histogram of dates of solar eclipses in 21st century. The dates form 35 clusters. Each cluster contains eclipses separated by Metonic cycles of 19 years. Each series contains four or five eclipses, and 46 or 65 or 84 years after the first one another series starts about a day and a half later in the (Julian) year. This means that the clusters slowly move forward to later dates. In a saros series, every 18 years the eclipse moves to the next later cluster. After 631 years (35 saros) it comes back to the original cluster, which by then has moved, in the Julian calendar, to a date about 13 or 14 days later, or about 18 days later in the Gregorian calendar.
Inex and saros for tetrads between AD 1000 and 2500, showing the tetradia
Solar eclipses around the present time. Series of semesters, heptons, and octons are easily visible. Note that the hepton series tend to remain total or annular, because the interval is near a whole number of anomalistic months, whereas in the octon series the type of eclipse changes over a cycle of 3, since the anomaly changes by around 130° each time.
Eclipses between AD 1600 and 2400. One can fairly easily see six of the eclipse cycles mentioned in this article. During some periods there are (non-consecutive) eclipses seven lunar months apart (a change of 69 in inex index). The two eclipses are seen both near the Arctic Circle or both near the Antarctic Circle. The next such period begins in 2098.
Solar eclipses from –11000 to +15000.
Saros and inex values for solar eclipses calculated from approximate date
Time of year for solar eclipses between saros 90 and saros 210
Time of year for solar eclipses between saros 90 and saros 210, but showing only the saros series whose index is divisible by 3. The time of year is related to the anomaly of the Sun. Two of the four eclipses of the year 2000 are indicated, with a line between them which shows (almost exactly) the slope of simultaneity in this graph.
Length of "semester" interval. The length varies considerably, depending on the lunar and solar anomalies of the two eclipses.
Length of Hipparchic intervals ending in 2001-2050 on the same scale as above. The interval is quite constant because it is close to a whole number of anomalistic months (4573.002) and to a whole number of anomalistic years (344.979).
Cosine of mean anomaly of moon at solar eclipses, 2001 through 2040. The curves connect eclipses that are 12 synodic months apart, but do not represent the anomaly between the eclipses. In each such series of four eclipses, the mean anomaly follows a sine wave. The moon is largest when the cosine of the anomaly is 1. On average every 3 years there is a "super moon" eclipse, with anomaly near zero.