THE SAROS CYCLE

If the Moon's orbit around the Earth were in the same plane as the Earth's around the Sun, this is exactly what would happen. However, the Moon's orbit is inclined about 5° to the Earth's orbit. The Moon passes through the ecliptic only twice a month at a pair of points called the nodes. The rest of the time the Moon is either above or below the plane of the Earth's orbit.
 
Since an eclipse can only occur when the Sun, Moon and Earth lie in the same plane (or very close to it), the conditions for a solar eclipse are met when the new Moon takes place near one of the nodes. Contrary to popular belief, solar eclipses are not at all rare. In fact, they are more common than lunar eclipses and outnumber lunar eclipses by almost 5 to 3.

This goes against common experience because lunar eclipses are observed more frequently than solar eclipses. This is explained because solar eclipses are only visible from small regions of the Earth while lunar eclipses are visible from the entire night time hemisphere of the earth. Therefore more people have seen lunar eclipses than solar eclipses, explaining the anomaly.
 
An examination of the geometry of the nodes will give more information on the subject of eclipse recurrence.  Since the Sun and Moon are both of a significant size in the sky, neither has to be exactly at the nodes for an eclipse to occur.  Also, an observer's position on the surface of the Earth introduces a further factor. The combination of these factors makes a solar eclipse possible whenever the Sun is within 18.5° of a node. The Sun travels along the ecliptic at about 1° per day and requires about 37 days to cross through the eclipse zone centered on each node. A New Moon occurs every 29.5 days and therefore at least one solar eclipse must occur during each of the Sun's node crossings.
 
The period during which the Sun is near a node is called an eclipse season and there are two (sometimes three) eclipse seasons each year. If the line of nodes were fixed in space, then eclipse seasons would occur six months apart and at the same time each year. But, the line of the nodes drifts westward at the rate of 19 degrees per year and as a result, eclipse seasons occur every 173.3 days.  Two eclipse seasons make an eclipse year of 346.6 days. This is 18.6 days short of a solar year and is equal to the time required by the Sun to cross the same node twice.
 
In order to find a cycle in the mechanics of solar eclipses, we must look for a synergy between the synodic month and the eclipse year. Now, 19 eclipse years (6585.6 days) are almost exactly equal to 223 synodic months (6585.19 days); they differ by only 11 hours.
 
This coincidence is all the more remarkable when we compare it to a period known as the anomalistic month. This is the time required for the Moon to pass from perigee (closest to earth) through apogee (furthest from earth) and back to perigee and is approximately 27.55 days. The anomalistic month is important because the Moon's distance from earth is vitally important in determining whether an eclipse will be annular or total.
 
This leads to a further coincidence - 239 anomalistic months are also equal to 223 synodic months to within 6 hours.  This is the origin of the famous Saros cycle of 6585.3 days or 18 years, 11 days and 8 hours.
 
Two eclipses separated by one Saros cycle will share very similar characteristics. They occur at the same node (ascending or descending) with the Moon at almost the same distance from Earth and at just about the same time of year (11 days later in the season).  They are however shifted 120 degrees Westwards from the previous eclipse.
 
Because the Saros does not contain an integral number of days, its biggest drawback is that consecutive eclipses will be visible from different parts of the earth.  The 1/3 day change moves the eclipse path about 120° westward with each cycle and the series will return to the same geographic region every 3 Saroses or 56 years and 34 days.
 
A Saros series cannot last indefinitely because the various periods (eclipse year, anomalistic period and synodic period) are not a perfect match with one another.  In particular, 19 eclipse years are 0.5 day longer than the Saros and as a result, the node shifts eastward by about 0.5° with each cycle. After about 70 cycles, the node will have shifted through approximately 35°, or the width of the eclipse window.

Over the course of the next 950 years or so, a central eclipse will occur at each Saros but will move northward by an average of 300 km each time. Halfway through this period, eclipses of long duration will occur near the equator. The last central eclipse of the series will occur near the north pole. The next ten eclipses will be partial with successively smaller magnitudes. Finally, the Saros series will end some 13 centuries after it began at the opposite pole to where it began. A typical series will consist of 70 to 80 eclipses, about 50 of which are central.
 
If a Saros series begins near the ascending node, the first eclipse will be partial from the northern polar region and the sequence described above is reversed. Since at least two solar eclipses occur every year, there are obviously many different Saros series in progress simultaneously.  For example, during the latter half of the twentieth century, there are 41 individual series in progress, 26 of which are producing central eclipses.  As old series end, new series begin and take their place. The total solar eclipses of 1911, 1929. 1947, 1965, 1983, 2001 and 2019 are all members of Saros 127. The series began with a partial eclipse of magnitude 0.034 at the north pole in 991 AD. The 2091 event will be the last central eclipse of the series. Note that the paths of the last four eclipses grow progressively broader as the umbral shadow cone passes closer to the edge of the Earth.  The next eclipse in the series will be a partial eclipse in 2109. Saros 127 will end with a partial eclipse of magnitude 0.026 near the south pole in 2452.

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