The frequency of solar and lunar eclipses
- Key People:
- al-Kāshī
- Anaxagoras
- Georg von Peuerbach
- Related Topics:
- totality
- inex period
- contact
- total eclipse
- partial eclipse
A solar eclipse, especially a total one, can be seen from only a limited part of Earth, whereas the eclipsed Moon can be seen at the time of the eclipse wherever the Moon is above the horizon.
In most calendar years there are two lunar eclipses; in some years one or three or none occur. Solar eclipses occur two to five times a year, five being exceptional; there last were five in 1935, and there will not be five again until 2206. The average number of total solar eclipses in a century is 66 for Earth as a whole.
Numbers of solar eclipses that have taken place or are predicted to take place during the 20th to 25th centuries are:
- 1901–2000: 228 eclipses, of which 145 were central (i.e., total or annular);
- 2001–2100: 224 eclipses, 144 central;
- 2101–2200: 235 eclipses, 151 central;
- 2201–2300: 248 eclipses, 156 central;
- 2301–2400: 248 eclipses, 160 central;
- 2401–2500: 237 eclipses, 153 central.
Any point on Earth may on the average experience no more than one total solar eclipse in three to four centuries. The situation is quite different for lunar eclipses. An observer remaining at the same place (and granted cloudless skies) could see 19 or 20 lunar eclipses in 18 years. Over that period three or four total eclipses and six or seven partial eclipses may be visible from beginning to end, and five total eclipses and four or five partial eclipses may be at least partially visible. All these numbers can be worked out from the geometry of the eclipses. A total lunar eclipse can last as long as an hour and three-quarters, but for a solar total eclipse maximum duration of totality is only 71/2 minutes. This difference results from the fact that the Moon’s diameter is much smaller than the extension of Earth’s shadow at the Moon’s distance from Earth, but the Moon can be only a little greater in apparent size than the Sun.
Cycles of eclipses
The eclipses of the Sun and the Moon occur at new moon and full moon, respectively, so that one basic time period involved in the occurrence of eclipses is the synodic month—i.e., the interval between successive new moons, as seen from Earth.
A solar eclipse does not occur at every new moon, nor does a lunar eclipse occur at every full moon, because the Moon’s orbital plane is inclined to the ecliptic, the plane of the orbit of Earth around the Sun. The angle between the planes is about 5°; thus, the Moon can pass well above or below the Sun. The line of intersection of the planes is called the line of the nodes, being the two points where the Moon’s orbit intersects the ecliptic plane. The ascending node is the point where the Moon crosses the ecliptic from south to north, and the descending node is where it crosses from north to south. The nodes move along the ecliptic from east to west as seen from Earth, completing a revolution in 18.6 years. The Moon’s revolution from one node to the same node again (called the draconic month, 27.212220 days) takes somewhat less time than a revolution from new moon to new moon (the synodic month, 29.530589 days). For a solar or lunar eclipse to occur, the Moon has to be near one of the nodes of its orbit. The draconic month is therefore the other basic period of eclipses.
Resonance between these two periods results in an interval called the saros, after which time the Moon and the Sun return very nearly to the same relative positions. The saros was known to the ancient Babylonians. It comprises 223 synodic months—that is, 6,585.321124 days, or 241.9986 draconic months. This latter value is nearly a whole number, so the new moon is in almost the same position (i.e., very near a node) at the beginning and end of a saros. The saros lasts 18 years 111/3 days or 18 years 101/3 days if five leap years fall within the period. Thus, there is usually a close resemblance between an eclipse and the one taking place 18 years and 11 days earlier or later. Because the date differs by only about 11 days in the calendar year, the latitudes on Earth of the two eclipses will be about the same, as will the relative apparent sizes of the Sun and Moon. The saros period also comprises 238.992 anomalistic months, again nearly a whole number. In one anomalistic month, the Moon describes its orbit from perigee to perigee, the point at which it is nearest to Earth. Thus, the Moon’s distance from Earth is the same after a whole number of anomalistic months and very nearly the same after one saros. The saros period is therefore extremely useful for the prediction of both solar and lunar eclipses.
Because of the extra one-third day (and thus an additional eight hours of Earth’s rotation) in the saros, the eclipse recurs each time approximately 120° farther west on the surface of Earth. After three saroses, or 54 years and about a month, the longitude is repeated.
There is a regular shift on Earth to the north or to the south of successive eclipse tracks from one saros to the next. The eclipses occurring when the Moon is near its ascending node shift to the south; those happening when it is near its descending node shift to the north. A saros series of eclipses begins its life at one pole of Earth and ends it at the other. A saros series lasts between 1,226 and 1,550 years and comprises 69 to 87 eclipses. As old series finish, new ones begin; about 42 of these series are in progress at any given time.
Two consecutive saros series are separated by the inex, a period of 29 years minus 20 days—that is, 358 synodic months—after which time the new moon has come from one node to the opposite node. A group of inex periods lasts about 23,000 years, with about 70 groups coexisting at any one time, each group comprising an average of 780 eclipses. All other cycles in eclipses are combinations of the saros and the inex.
Prediction and calculation of solar and lunar eclipses
The problem may be divided into two parts. The first is to find out when an eclipse will occur, the other to determine when and where it will be visible.
For this purpose it is convenient first to consider Earth as fixed and to suppose an observer is looking out from its centre. To this observer, labeled O in the of the celestial sphere, the Sun and Moon appear projected on the celestial sphere. While this sphere appears to rotate daily, as measured by the positions of the stars, around the axis PP′ (Earth’s axis of rotation), the Sun’s disk, S, appears to travel slowly along the great circle EE′ (the ecliptic), making a complete revolution in one year. At the same time, the Moon’s disk, M, travels along its own great-circle path, LL′, once during a lunar month. The angular diameters of the Sun’s and the Moon’s disks, S and M, are each about 0.5° but vary slightly.
Every month, the Moon’s disk moving along its path, LL′, will overtake the more slowly moving Sun once, at the moment of the new moon. Usually the Moon’s disk will pass above or below the Sun’s disk. Overlapping of the two results in an eclipse of the Sun, which can happen only when the new moon occurs at the same time that the Sun is near the ascending node or descending node, [nodeascnd] and [nodedescd], respectively, of the Moon’s orbit. Because the nodes are 180° apart, eclipses occur in the so-called eclipse seasons, six months apart.
In the of the celestial sphere, the projection of Earth’s umbra is shown as a disk, U, at the distance of the Moon’s orbit. At that distance the shadow’s disk subtends an angle of about 1.4°; its centre is always opposite the Sun’s disk and travels along the ecliptic, EE′. A lunar eclipse occurs whenever the shadow’s disk overlaps the Moon’s disk; this happens only when the shadow’s disk is near one of the nodes and the Sun is near the opposite node. The Sun’s passage through the lunar nodes is thus the critical time for both solar and lunar eclipses. The plane of the Moon’s path, LL′, is not fixed, and its nodes move slowly along the ecliptic in the direction indicated by the arrows, making a complete revolution in about 19 years. The interval between two successive passages of the Sun through one of the nodes is termed an eclipse year, and, since the Moon’s node moves so as to meet the advancing Sun, this interval is about 18.6 days less than a tropical (or ordinary) year.
In the of the Moon’s ascending node, this region is depicted as seen from the centre of the celestial sphere and is shown much enlarged. The node is kept fixed, and the apparent motions of the Sun and the Moon (top portion of the figure) are shown relative to the node. To the observer on Earth at the centre of the sphere, the Sun’s disk will travel along the ecliptic, EE′, and the Moon’s disk along its designated path, LL′. The Sun is so distant compared with the size of Earth that, from all places on Earth’s surface, the Sun is seen nearly in the same position as it would be from the very centre. On the other hand, the Moon is relatively near, and so its projected position on the celestial sphere is different for various places of observation on Earth. In fact, it may be displaced as much as 1° from the position in which it is seen from the centre of Earth. If the radius of the Moon’s disk is enlarged by 1°, a “Moon circle,” C, is obtained that encloses all possible positions of the Moon’s disk seen from anywhere on Earth. Conversely, if any disk of the Moon’s size is placed inside this Moon circle, there is a place on Earth from which the Moon is seen in that position.
Accordingly, an eclipse of the Sun occurs somewhere on Earth whenever the Moon overtakes the Sun in such a position that the Moon circle, C, passes over the Sun’s disk; when the latter is entirely covered by the Moon circle, the eclipse will be total or annular. From the top portion of the of the Moon’s ascending node, it is evident that a solar eclipse will take place if a new moon occurs while the Sun moves from position S1 to position S4. This period is the eclipse season; it starts 19 days before the Sun passes through a lunar node and ends 19 days thereafter. There are two complete eclipse seasons, one at each node, during a calendar year. Because there is a new moon every month, at least one solar eclipse, and occasionally two, occurs during each eclipse season. A fifth solar eclipse during a calendar year is possible because part of a third eclipse season may occur at the beginning of January or at the end of December.
The bottom portion of the of the Moon’s ascending node illustrates the condition necessary for a lunar eclipse. If a full moon occurs within 13 days of the passage of the Sun though a lunar node—and thus of the Earth’s umbral disk, U, through the opposite node—the Moon will be eclipsed. (In the figure the umbral disk passes through the ascending node.) Most eclipse seasons, but not all, will thus also contain a lunar eclipse. When two eclipse seasons and a partial third season fall in a calendar year, there may be three lunar eclipses in that year. Eclipses of the Sun are evidently more frequent than those of the Moon. Solar eclipses, however, can be seen from only a very limited region of Earth, whereas lunar eclipses are visible from an entire hemisphere.
During a solar eclipse the shadow cones—the umbra and penumbra—of the Moon sweep across the face of Earth (see the of an eclipse of the Sun), while, at the same time, Earth is rotating on its axis. Within the narrow area covered by the umbra, the eclipse is total. Within the wider surrounding region covered by the penumbra, the eclipse is partial.
Astronomical ephemerides, or tables, that are published annually for the year ahead provide maps tracing the paths of the more important eclipses in considerable detail, as well as data for accurate calculation of the times of contact at any given observing location on Earth. Calculations are made some years ahead in Terrestrial Time (TT), which is defined by the orbital motion of Earth and the other planets. At the time of the eclipse, the correction is made to Universal Time (UT), which is defined by the rotation of Earth and is not rigorously uniform.
Modern computers make it possible to predict solar eclipses several years ahead with high accuracy. By means of the same calculational methods, eclipses can be “predicted backward” in time. The generation of the times and observational locations for ancient eclipses has been valuable in historical and scientific research (see below Eclipses in history).
