Eclipse cycle
Encyclopedia
Eclipse
s may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.
s may occur when the Earth
and the Moon
are aligned with the Sun
, and the shadow of one body cast by the Sun falls on the other. So at new moon
(or rather Dark Moon
), when the Moon is in conjunction with the Sun, the Moon may pass in front of the Sun as seen from a narrow region on the surface of the Earth and cause a solar eclipse
. At full moon
, when the Moon is in opposition to the Sun, the Moon may pass through the shadow of the Earth, and a lunar eclipse
is visible from the night half of the Earth.
Note: Conjunction and opposition of the Moon together have a special name: syzygy
(from Greek
for "junction"), because of the importance of these lunar phase
s.
An eclipse does not happen at every new or full moon, because the plane of the orbit of the Moon around the Earth is tilted with respect to the plane of the orbit of the Earth around the Sun (the ecliptic
): so as seen from the Earth, when the Moon is nearest to the Sun (new moon) or at largest distance (full moon), the three bodies usually are not exactly on the same line.
This inclination
is on average about:
Compare this with the relevant apparent mean diameters:
Therefore, at most new moons the Earth passes too far north or south of the lunar shadow, and at most full moons the Moon misses the shadow of the Earth. Also, at most solar eclipses the apparent angular diameter of the Moon is insufficient to fully obscure the solar disc, unless the Moon is close to perigee. In any case, the alignment must be close to perfect to cause an eclipse.
An eclipse can only occur when the Moon is close to the plane of the orbit of the Earth, i.e. when its ecliptic latitude is small. This happens when the Moon is near one of the two node
s of its orbit on the ecliptic at the time of the syzygy
. Of course, to produce an eclipse, the Sun must also be near a node at that time: the same node for a solar eclipse, or the opposite node for a lunar eclipse.
Eclipses (up to three) occur during an eclipse season
, a one- or two-month period twice a year, around the time when the Sun is near the nodes of the Moon's orbit.
An eclipse does not occur every month, because one month after an eclipse the relative geometry of the Sun, Moon, and Earth has changed.
As seen from the Earth, the time it takes for the Moon to return to a node, the draconic month, is less than the time it takes for the Moon to return to the same ecliptic longitude as the Sun: the synodic month. The main reason is that during the time that the Moon has completed an orbit around the Earth, the Earth (and Moon) have completed about 1/13th of their orbit around the Sun: the Moon has to make up for this in order to come again into conjunction or opposition with the Sun. Secondly, the orbital nodes of the Moon precess
westward in ecliptic longitude, completing a full circle in about 18½ years, so a draconic month is shorter than a sidereal month. In all, the difference in period between synodic and draconic month is nearly 2⅓ days. Likewise, as seen from the Earth, the Sun passes both nodes as it moves along its ecliptic path. The period for the Sun to return to a node is called the eclipse or draconic year: about 346.6201 d, which is about 1/20th year shorter than a sidereal year
because of the precession of the nodes.
If a solar eclipse occurs at one new moon, which must be close to a node, then at the next full moon the Moon is already more than a day past its opposite node, and may or may not miss the Earth's shadow. By the next new moon it is even further ahead of the node, so it is less likely that there will be a solar eclipse somewhere on Earth. By the next month, there will certainly be no event.
However, about 5 or 6 lunations later the new moon will fall close to the opposite node. In that time (half an eclipse year) the Sun will have moved to the opposite node too, so the circumstances will again be suitable for one or more eclipses.
Given an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.
Another thing to consider is that the motion of the Moon is not a perfect circle. Its orbit is distinctly elliptic, so the lunar distance from Earth varies throughout the lunar cycle. 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. This orbital period is called the anomalistic month, and together with the synodic month causes the so-called "full moon cycle
" of about 14 lunations in the timings and appearances of full (and new) Moons. 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 (relative to their mean timing), and changing the apparent lunar angular diameter by about ±6%. An eclipse cycle must comprise close to an integer number of anomalistic months in order to perform well in predicting eclipses.
s 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. These three 2-way relations are not independent (i.e. both the synodic month and eclipse year are dependent on the apparent motion of the Sun, both the draconic month and eclipse year are dependent on the motion of the nodes), and indeed the eclipse year can be described as the beat period
of the synodic and draconic months (i.e. the period of the difference between the synodic and draconic months); in formula:
as can be checked by filling in the numerical values listed above.
Eclipse cycles have a period in which a certain number of synodic months closely equals an integer or half-integer number of draconic months: one such period after an eclipse, a syzygy
(new moon
or full moon
) takes place again near a node
of the Moon's orbit on the ecliptic
, and an eclipse can occur again. However,the synodic and draconic months are incommensurate: their ratio is not an integer number. We need to approximate this ratio by common fractions: the numerators and denominators then give the multiples of the two periods - draconic and synodic months - that (approximately) span the same amount of time, representing an eclipse cycle.
These fractions can be found by the method of continued fractions: this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions.
Since there may be an eclipse every half draconic month, we need to find an approximation for the number of half draconic months per synodic month: so the target ratio to approximate is: SM / (DM/2) = 29.530588853 / (27.212220817/2) = 2.170391682
2.170391682 = [2;5,1,6,1,1,1,1,1,11,1,...]:
Quotients Convergents
half DM/SM decimal named cycle (if any)
2; 2/1 = 2
5 11/5 = 2.2
1 13/6 = 2.166666667 semester
6 89/41 = 2.170731707 hepton
1 102/47 = 2.170212766 octon
1 191/88 = 2.170454545 tzolkinex
1 293/135 = 2.170370370 tritos
1 484/223 = 2.170403587 saros
1 777/358 = 2.170391061 inex
11 9031/4161 = 2.170391732
1 9808/4519 = 2.170391679
...
The ratio of synodic months per half eclipse year and per eclipse year yields the same series:
5.868831091 = [5;1,6,1,1,1,1,1,11,1,...]
Quotients Convergents
SM/half EY decimal SM/full EY named cycle
5; 5/1 = 5
1 6/1 = 6 12/1 semester
6 41/7 = 5.857142857 hepton
1 47/8 = 5.875 47/4 octon
1 88/15 = 5.866666667 tzolkinex
1 135/23 = 5.869565217 tritos
1 223/38 = 5.868421053 223/19 saros
1 358/61 = 5.868852459 716/61 inex
11 4161/709 = 5.868829337
1 4519/770 = 5.868831169 4519/385
...
Each of these is an eclipse cycle. Less accurate cycles may be constructed by combinations of these.
Notes:
Fortnight: Half a synodic month. When there is an eclipse, there is a fair chance that at the next syzygy there will be another eclipse: the Sun and Moon will have moved about 15° with respect to the nodes (the Moon being opposite to where it was the previous time), but the luminaries may still be within bounds to make an eclipse.
For example, partial solar eclipse of June 1, 2011
is followed by the total lunar eclipse of' June 16, 2011
and partial solar eclipse of July 1, 2011
.
Synodic Month: Similarly, two events one synodic month apart have the Sun and Moon at two positions on either side of the node, 29° apart: both may cause a partial eclipse.
Pentalunex: 5 synodic months. Successive solar or lunar eclipses may occur 1, 5 or 6 synodic months apart.
Semester: Time between successive eclipse seasons. After 6 (or sometimes 5 or more rarely 7) months, the Sun is at the other node, and eclipses may again occur.
Lunar year: Twelve (synodic) months, a little longer than an eclipse year: the Sun has returned to the node, so eclipses may again occur.
Octon: This is 1/5 of the Metonic cycle, and a fairly decent short eclipse cycle, but poor in anomalistic returns. Each octon in a series is 2 saros apart, always occurring at the same node.
Tzolkinex: Includes a half draconic month, so occurs at alternating nodes and alternates between hemispheres. Each consecutive eclipse is a member of preceding saros series from the one before. Equal to ten tzolk'in
s. Every third tzolkinex in a series is near an integer number of anomalistic months and so will have similar properties.
Sar (Half saros): Includes an odd number of fortnights (223). As a result, eclipses alternate between lunar and solar with each cycle, occurring at the same node and with similar characteristics. A long central total solar eclipse will be followed by a very central total lunar eclipse. A solar eclipse where the moon's penumbra just barely grazes the southern limb of earth will be followed half a saros later by a lunar eclipse where the moon just grazes the southern limb of the earth's penumbra.
Tritos: A mediocre cycle, relates to the saros like the inex. A triple tritos is close to an integer number of anomalistic months and so will have similar properties.
Saros : The best known eclipse cycle, and one of the best for predicting eclipses, in which 223 synodic months equal 242 draconic months with an error of only 51 minutes. It is also close to 239 anomalistic months, which makes the circumstances between two eclipses one saros apart very similar.
Metonic cycle or Enneadecaeteris: This is nearly equal to 19 tropical year
s, but is also 5 "octon" periods and close to 20 eclipse years: so it yields a short series of eclipses on the same calendar date. It consists of 110 hollow months and 125 full months, so nominally 6940 days, and equals 235 lunations with an error of only 7.5 hours.
Inex: By itself a poor cycle, it is very convenient in the classification of eclipse cycles, because after a saros series dies, a new saros series often begins 1 inex later (hence its name: in-ex). One inex after an eclipse, another eclipse takes place at the same longitude, but at the opposite latitude.
Exeligmos: A triple saros, with the advantage that it has nearly an integer number of days, so the next eclipse will be visible at locations near the eclipse that occurred one exeligmos earlier, in contrast to the saros, in which the eclipse occurs about 8 hours later in the day or about 120° to the west of the eclipse that occurred one saros earlier.
Callippic cycle: 441 hollow months and 499 full months; thus 4 Metonic Cycles minus one day or precisely 76 years of 365¼ days. It equals 940 lunations with an error of only 5.9 hours.
Triad: A triple inex, with the advantage that it has nearly an integer number of anomalistic months, which makes the circumstances between two eclipses one Triad apart very similar, but at the opposite latitude. Almost exactly 87 calendar years minus 2 months.
Hipparchic cycle: Not a noteworthy eclipse cycle, but Hipparchus
constructed it to closely match an integer number of synodic and anomalistic months, years (345), and days. By comparing his own eclipse observations with Babylonian records from 345 years earlier, he could verify the accuracy of the various periods that the Chaldeans used.
Babylonian: The ratio 5923 returns to latitude in 5458 months was used by the Chaldeans in their astronomical computations.
Tetradia: Sometimes 4 total lunar eclipses occur in a row with intervals of 6 lunations (semester), and this is called a tetrad. Giovanni Schiaparelli
noticed that there are eras when such tetrads occur comparatively frequently, interrupted by eras when they are rare. This variation takes about 6 centuries. Antonie Pannekoek
(1951) explained this phenomenon and found a period of 591 years. Van den Bergh (1954) from Theodor von Oppolzer
's Canon der Finsternisse found a period of 586 years. This happens to be an eclipse cycle; see Meeus [I] (1997). Recently Tudor Hughes explained the variation from secular changes in the eccentricity
of the Earth's orbit: the period for occurrence of tetrads is variable and currently is about 565 years; see Meeus III (2004) for a detailed discussion.
Eclipse
An eclipse is an astronomical event that occurs when an astronomical object is temporarily obscured, either by passing into the shadow of another body or by having another body pass between it and the viewer...
s may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.
Eclipse conditions
EclipseEclipse
An eclipse is an astronomical event that occurs when an astronomical object is temporarily obscured, either by passing into the shadow of another body or by having another body pass between it and the viewer...
s may occur when the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
and the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
are aligned with the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
, and the shadow of one body cast by the Sun falls on the other. So at new moon
New moon
In astronomical terminology, the new moon is the lunar phase that occurs when the Moon, in its monthly orbital motion around Earth, lies between Earth and the Sun, and is therefore in conjunction with the Sun as seen from Earth...
(or rather Dark Moon
Dark moon
A dark moon describes the Moon during that time that it is invisible against the backdrop of the Sun in the sky. The duration of a dark moon is between 1.5 and 3.5 days, depending on the orientation of the Earth and Sun....
), when the Moon is in conjunction with the Sun, the Moon may pass in front of the Sun as seen from a narrow region on the surface of the Earth and cause a solar eclipse
Solar eclipse
As seen from the Earth, a solar eclipse occurs when the Moon passes between the Sun and the Earth, and the Moon fully or partially blocks the Sun as viewed from a location on Earth. This can happen only during a new moon, when the Sun and the Moon are in conjunction as seen from Earth. At least...
. At full moon
Full moon
Full moon lunar phase that occurs when the Moon is on the opposite side of the Earth from the Sun. More precisely, a full moon occurs when the geocentric apparent longitudes of the Sun and Moon differ by 180 degrees; the Moon is then in opposition with the Sun.Lunar eclipses can only occur at...
, when the Moon is in opposition to the Sun, the Moon may pass through the shadow of the Earth, and a lunar eclipse
Lunar eclipse
A lunar eclipse occurs when the Moon passes behind the Earth so that the Earth blocks the Sun's rays from striking the Moon. This can occur only when the Sun, Earth, and Moon are aligned exactly, or very closely so, with the Earth in the middle. Hence, a lunar eclipse can only occur the night of a...
is visible from the night half of the Earth.
Note: Conjunction and opposition of the Moon together have a special name: syzygy
Syzygy (astronomy)
In astronomy, a syzygy is a straight line configuration of three celestial bodies in a gravitational system. The word is usually used in reference to the Sun, the Earth and either the Moon or a planet, where the latter is in conjunction or opposition. Solar and lunar eclipses occur at times of...
(from Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
for "junction"), because of the importance of these lunar phase
Lunar phase
A lunar phase or phase of the moon is the appearance of the illuminated portion of the Moon as seen by an observer, usually on Earth. The lunar phases change cyclically as the Moon orbits the Earth, according to the changing relative positions of the Earth, Moon, and Sun...
s.
An eclipse does not happen at every new or full moon, because the plane of the orbit of the Moon around the Earth is tilted with respect to the plane of the orbit of the Earth around the Sun (the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
): so as seen from the Earth, when the Moon is nearest to the Sun (new moon) or at largest distance (full moon), the three bodies usually are not exactly on the same line.
This inclination
Inclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...
is on average about:
- I = 5°09'
Compare this with the relevant apparent mean diameters:
- Sun: 32' 2"
- Moon: 31'37" (as seen from the surface of the Earth right beneath the Moon)
- and: 1°23' for the mean diameter of the shadow of the Earth at the mean lunar distance.
Therefore, at most new moons the Earth passes too far north or south of the lunar shadow, and at most full moons the Moon misses the shadow of the Earth. Also, at most solar eclipses the apparent angular diameter of the Moon is insufficient to fully obscure the solar disc, unless the Moon is close to perigee. In any case, the alignment must be close to perfect to cause an eclipse.
An eclipse can only occur when the Moon is close to the plane of the orbit of the Earth, i.e. when its ecliptic latitude is small. This happens when the Moon is near one of the two node
Lunar node
The lunar nodes are the orbital nodes of the Moon, that is, the points where the orbit of the Moon crosses the ecliptic . The ascending node is where the moon crosses to the north of the ecliptic...
s of its orbit on the ecliptic at the time of the syzygy
Syzygy (astronomy)
In astronomy, a syzygy is a straight line configuration of three celestial bodies in a gravitational system. The word is usually used in reference to the Sun, the Earth and either the Moon or a planet, where the latter is in conjunction or opposition. Solar and lunar eclipses occur at times of...
. Of course, to produce an eclipse, the Sun must also be near a node at that time: the same node for a solar eclipse, or the opposite node for a lunar eclipse.
Recurrence
Eclipse season
Eclipse seasons are the only times during a year eclipses can occur, due to the 5° inclination of the moon's orbit. Each season lasts for approximately 33 days and repeats just short of six months, thus there are always two full eclipse seasons each year. 2 to 3 eclipses always occur each eclipse...
, a one- or two-month period twice a year, around the time when the Sun is near the nodes of the Moon's orbit.
An eclipse does not occur every month, because one month after an eclipse the relative geometry of the Sun, Moon, and Earth has changed.
As seen from the Earth, the time it takes for the Moon to return to a node, the draconic month, is less than the time it takes for the Moon to return to the same ecliptic longitude as the Sun: the synodic month. The main reason is that during the time that the Moon has completed an orbit around the Earth, the Earth (and Moon) have completed about 1/13th of their orbit around the Sun: the Moon has to make up for this in order to come again into conjunction or opposition with the Sun. Secondly, the orbital nodes of the Moon precess
Lunar precession
Precession is the rotation of a plane with respect to a reference plane. The orbit of the Moon has two important such precessional motions....
westward in ecliptic longitude, completing a full circle in about 18½ years, so a draconic month is shorter than a sidereal month. In all, the difference in period between synodic and draconic month is nearly 2⅓ days. Likewise, as seen from the Earth, the Sun passes both nodes as it moves along its ecliptic path. The period for the Sun to return to a node is called the eclipse or draconic year: about 346.6201 d, which is about 1/20th year shorter than a sidereal year
Sidereal year
A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...
because of the precession of the nodes.
If a solar eclipse occurs at one new moon, which must be close to a node, then at the next full moon the Moon is already more than a day past its opposite node, and may or may not miss the Earth's shadow. By the next new moon it is even further ahead of the node, so it is less likely that there will be a solar eclipse somewhere on Earth. By the next month, there will certainly be no event.
However, about 5 or 6 lunations later the new moon will fall close to the opposite node. In that time (half an eclipse year) the Sun will have moved to the opposite node too, so the circumstances will again be suitable for one or more eclipses.
Periodicity
These are still rather vague predictions. However we know that if an eclipse occurred at some moment, then there will occur an eclipse again S synodic months later, if that interval is also D draconic months, where D is an integer number (return to same node), or an integer number + ½ (return to opposite node). So an eclipse cycle is any period P for which approximately holds:- P = S×(synodic month length) = D×(Draconic month length)
Given an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.
Another thing to consider is that the motion of the Moon is not a perfect circle. Its orbit is distinctly elliptic, so the lunar distance from Earth varies throughout the lunar cycle. 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. This orbital period is called the anomalistic month, and together with the synodic month causes the so-called "full moon cycle
Full moon cycle
The full moon cycle is a cycle of about 14 lunations over which full moons vary in apparent size and age . The sequence is*Full moon big - *Full moon young -...
" of about 14 lunations in the timings and appearances of full (and new) Moons. 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 (relative to their mean timing), and changing the apparent lunar angular diameter by about ±6%. An eclipse cycle must comprise close to an integer number of anomalistic months in order to perform well in predicting eclipses.
Numerical values
These are the lengths of the various types of monthMonth
A month is a unit of time, used with calendars, which was first used and invented in Mesopotamia, as a natural period related to the motion of the Moon; month and Moon are cognates. The traditional concept arose with the cycle of moon phases; such months are synodic months and last approximately...
s as discussed above (according to the lunar ephemeris
Ephemeris
An ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...
ELP2000-85, valid for the epoch
Epoch (astronomy)
In astronomy, an epoch is a moment in time used as a reference point for some time-varying astronomical quantity, such as celestial coordinates, or elliptical orbital elements of a celestial body, where these are subject to perturbations and vary with time...
J2000.0; taken from (e.g.) Meeus (1991) ):
- SM = 29.530588853 days (Synodic month)
- DM = 27.212220817 days (Draconic month)
- AM = 27.55454988 days (Anomalistic month)
- EY = 346.620076 days (Eclipse year)
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. These three 2-way relations are not independent (i.e. both the synodic month and eclipse year are dependent on the apparent motion of the Sun, both the draconic month and eclipse year are dependent on the motion of the nodes), and indeed the eclipse year can be described as the beat period
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....
of the synodic and draconic months (i.e. the period of the difference between the synodic and draconic months); in formula:
as can be checked by filling in the numerical values listed above.
Eclipse cycles have a period in which a certain number of synodic months closely equals an integer or half-integer number of draconic months: one such period after an eclipse, a syzygy
Syzygy (astronomy)
In astronomy, a syzygy is a straight line configuration of three celestial bodies in a gravitational system. The word is usually used in reference to the Sun, the Earth and either the Moon or a planet, where the latter is in conjunction or opposition. Solar and lunar eclipses occur at times of...
(new moon
New moon
In astronomical terminology, the new moon is the lunar phase that occurs when the Moon, in its monthly orbital motion around Earth, lies between Earth and the Sun, and is therefore in conjunction with the Sun as seen from Earth...
or full moon
Full moon
Full moon lunar phase that occurs when the Moon is on the opposite side of the Earth from the Sun. More precisely, a full moon occurs when the geocentric apparent longitudes of the Sun and Moon differ by 180 degrees; the Moon is then in opposition with the Sun.Lunar eclipses can only occur at...
) takes place again near a node
Lunar node
The lunar nodes are the orbital nodes of the Moon, that is, the points where the orbit of the Moon crosses the ecliptic . The ascending node is where the moon crosses to the north of the ecliptic...
of the Moon's orbit on the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
, and an eclipse can occur again. However,the synodic and draconic months are incommensurate: their ratio is not an integer number. We need to approximate this ratio by common fractions: the numerators and denominators then give the multiples of the two periods - draconic and synodic months - that (approximately) span the same amount of time, representing an eclipse cycle.
These fractions can be found by the method of continued fractions: this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions.
Since there may be an eclipse every half draconic month, we need to find an approximation for the number of half draconic months per synodic month: so the target ratio to approximate is: SM / (DM/2) = 29.530588853 / (27.212220817/2) = 2.170391682
2.170391682 = [2;5,1,6,1,1,1,1,1,11,1,...]:
Quotients Convergents
half DM/SM decimal named cycle (if any)
2; 2/1 = 2
5 11/5 = 2.2
1 13/6 = 2.166666667 semester
6 89/41 = 2.170731707 hepton
1 102/47 = 2.170212766 octon
1 191/88 = 2.170454545 tzolkinex
1 293/135 = 2.170370370 tritos
Tritos
The tritos is an eclipse cycle of 3986.63 days. It corresponds to:*135 synodic months*146.50144 draconic months*11.50144 eclipse years *144.68135 anomalistic months....
1 484/223 = 2.170403587 saros
1 777/358 = 2.170391061 inex
Inex
The inex is an eclipse cycle of 10,571.95 days . The cycle was first described by Crommelin in 1901, but was named by George van den Bergh who studied it half a century later...
11 9031/4161 = 2.170391732
1 9808/4519 = 2.170391679
...
The ratio of synodic months per half eclipse year and per eclipse year yields the same series:
5.868831091 = [5;1,6,1,1,1,1,1,11,1,...]
Quotients Convergents
SM/half EY decimal SM/full EY named cycle
5; 5/1 = 5
1 6/1 = 6 12/1 semester
6 41/7 = 5.857142857 hepton
1 47/8 = 5.875 47/4 octon
1 88/15 = 5.866666667 tzolkinex
1 135/23 = 5.869565217 tritos
Tritos
The tritos is an eclipse cycle of 3986.63 days. It corresponds to:*135 synodic months*146.50144 draconic months*11.50144 eclipse years *144.68135 anomalistic months....
1 223/38 = 5.868421053 223/19 saros
Saros cycle
The saros is a period of 223 synodic months , that can be used to predict eclipses of the Sun and Moon. One saros after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle...
1 358/61 = 5.868852459 716/61 inex
Inex
The inex is an eclipse cycle of 10,571.95 days . The cycle was first described by Crommelin in 1901, but was named by George van den Bergh who studied it half a century later...
11 4161/709 = 5.868829337
1 4519/770 = 5.868831169 4519/385
...
Each of these is an eclipse cycle. Less accurate cycles may be constructed by combinations of these.
Eclipse cycles
This table summarizes the characteristics of various eclipse cycles, and can be computed from the numerical results of the preceding paragraphs; cf. Meeus (1997) Ch.9 . More details are given in the comments below, and several notable cycles have their own pages.| cycle | formula | solar days | synodic months | draconic months | anomalistic months | eclipse years | tropical years |
|---|---|---|---|---|---|---|---|
| fortnight | (38i – 61s)/2 | 14.77 | 0.5 | 0.543 | 0.536 | 0.043 | 0.040 |
| synodic month | 38i – 61s | 29.53 | 1 | 1.085 | 1.072 | 0.085 | 0.081 |
| pentalunex | -33i + 53s | 147.65 | 5 | 5.426 | 5.359 | 0.426 | 0.404 |
| semester | 5i – 8s | 177.18 | 6 | 6.511 | 6.430 | 0.511 | 0.485 |
| lunar year | 10i – 16s | 354.37 | 12 | 13.022 | 12.861 | 1.022 | 0.970 |
| octon | 2i – 3s | 1387.94 | 47 | 51.004 | 50.371 | 4.004 | 3.800 |
| tzolkinex | -i + 2s | 2598.69 | 88 | 95.497 | 94.311 | 7.497 | 7.115 |
| sar (half saros) | (0i + s)/2 | 3292.66 | 111.5 | 120.999 | 119.496 | 9.499 | 9.015 |
| tritos Tritos The tritos is an eclipse cycle of 3986.63 days. It corresponds to:*135 synodic months*146.50144 draconic months*11.50144 eclipse years *144.68135 anomalistic months.... |
i – s | 3986.63 | 135 | 146.501 | 144.681 | 11.501 | 10.915 |
| saros (s) | 0i + s | 6585.32 | 223 | 241.999 | 238.992 | 18.999 | 18.030 |
| Metonic cycle Metonic cycle In astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the solar year and the synodic month... |
10i – 15s | 6939.69 | 235 | 255.021 | 251.853 | 20.021 | 19.000 |
| inex Inex The inex is an eclipse cycle of 10,571.95 days . The cycle was first described by Crommelin in 1901, but was named by George van den Bergh who studied it half a century later... (i) |
i ± 0s | 10,571.95 | 358 | 388.500 | 383.674 | 30.500 | 28.945 |
| exeligmos Exeligmos An exeligmos is a period of 54 years, 33 days that can be used to predict successive eclipses with similar properties and location. For a solar eclipse, every exeligmos a solar eclipse of similar characteristics will occur close to the eclipse before it... |
0i + 3s | 19,755.96 | 669 | 725.996 | 716.976 | 56.996 | 54.090 |
| Callippic cycle Callippic cycle In astronomy and calendar studies, the Callippic cycle is a particular approximate common multiple of the year and the synodic month, that was proposed by Callippus in 330 BC... |
40i – 60s | 27,758.75 | 940 | 1020.084 | 1007.411 | 80.084 | 76.001 |
| triad | 3i ± 0s | 31,715.85 | 1074 | 1165.500 | 1151.021 | 91.500 | 86.835 |
| Hipparchic cycle Hipparchic cycle The Greek astronomer Hipparchus introduced two cycles that have been named after him in later literature.The first is described in Ptolemy's Almagest IV.2... |
25i – 21s | 126,007.02 | 4267 | 4630.531 | 4573.002 | 363.531 | 344.996 |
| Babylonian | 14i + 2s | 161,177.95 | 5458 | 5922.999 | 5849.413 | 464.999 | 441.291 |
| tetradia (Meeus III) | 22i – 4s | 206,241.63 | 6984 | 7579.008 | 7484.849 | 595.008 | 564.671 |
| tetradia (Meeus [I]) | 19i + 2s | 214,037.70 | 7248 | 7865.500 | 7767.781 | 617.500 | 586.016 |
Notes:
Fortnight: Half a synodic month. When there is an eclipse, there is a fair chance that at the next syzygy there will be another eclipse: the Sun and Moon will have moved about 15° with respect to the nodes (the Moon being opposite to where it was the previous time), but the luminaries may still be within bounds to make an eclipse.
For example, partial solar eclipse of June 1, 2011
Solar eclipse of June 1, 2011
A partial solar eclipse occurred on June 1, 2011. This eclipse is the second of four partial solar eclipses in 2011, with the others occurring on January 4, 2011, July 1, 2011, and November 25, 2011....
is followed by the total lunar eclipse of' June 16, 2011
June 2011 lunar eclipse
A total lunar eclipse took place on June 15, 2011. It was the first of two such eclipses in 2011. The second will occur on December 10, 2011.This was a relatively rare central lunar eclipse, in which the center point of Earth's shadow passes across the Moon. The last time a lunar eclipse was closer...
and partial solar eclipse of July 1, 2011
Solar eclipse of July 1, 2011
A partial solar eclipse occurred on July 1, 2011. This is the first solar eclipse of saros series 156, only visible as a partial solar eclipse in a small area south of South Africa and north of Antarctica. At greatest eclipse, the magnitude is just 0.097. It is the first new saros series to begin...
.
Synodic Month: Similarly, two events one synodic month apart have the Sun and Moon at two positions on either side of the node, 29° apart: both may cause a partial eclipse.
Pentalunex: 5 synodic months. Successive solar or lunar eclipses may occur 1, 5 or 6 synodic months apart.
Semester: Time between successive eclipse seasons. After 6 (or sometimes 5 or more rarely 7) months, the Sun is at the other node, and eclipses may again occur.
Lunar year: Twelve (synodic) months, a little longer than an eclipse year: the Sun has returned to the node, so eclipses may again occur.
Octon: This is 1/5 of the Metonic cycle, and a fairly decent short eclipse cycle, but poor in anomalistic returns. Each octon in a series is 2 saros apart, always occurring at the same node.
Tzolkinex: Includes a half draconic month, so occurs at alternating nodes and alternates between hemispheres. Each consecutive eclipse is a member of preceding saros series from the one before. Equal to ten tzolk'in
Tzolk'in
Tzolk'in is the name bestowed by Mayanists on the 260-day Mesoamerican calendar used by the Maya civilization of pre-Columbian Mesoamerica.The tzolk'in, the basic cycle of the Maya calendar, is a pre-eminent...
s. Every third tzolkinex in a series is near an integer number of anomalistic months and so will have similar properties.
Sar (Half saros): Includes an odd number of fortnights (223). As a result, eclipses alternate between lunar and solar with each cycle, occurring at the same node and with similar characteristics. A long central total solar eclipse will be followed by a very central total lunar eclipse. A solar eclipse where the moon's penumbra just barely grazes the southern limb of earth will be followed half a saros later by a lunar eclipse where the moon just grazes the southern limb of the earth's penumbra.
Tritos: A mediocre cycle, relates to the saros like the inex. A triple tritos is close to an integer number of anomalistic months and so will have similar properties.
Saros : The best known eclipse cycle, and one of the best for predicting eclipses, in which 223 synodic months equal 242 draconic months with an error of only 51 minutes. It is also close to 239 anomalistic months, which makes the circumstances between two eclipses one saros apart very similar.
Metonic cycle or Enneadecaeteris: This is nearly equal to 19 tropical year
Tropical year
A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...
s, but is also 5 "octon" periods and close to 20 eclipse years: so it yields a short series of eclipses on the same calendar date. It consists of 110 hollow months and 125 full months, so nominally 6940 days, and equals 235 lunations with an error of only 7.5 hours.
Inex: By itself a poor cycle, it is very convenient in the classification of eclipse cycles, because after a saros series dies, a new saros series often begins 1 inex later (hence its name: in-ex). One inex after an eclipse, another eclipse takes place at the same longitude, but at the opposite latitude.
Exeligmos: A triple saros, with the advantage that it has nearly an integer number of days, so the next eclipse will be visible at locations near the eclipse that occurred one exeligmos earlier, in contrast to the saros, in which the eclipse occurs about 8 hours later in the day or about 120° to the west of the eclipse that occurred one saros earlier.
Callippic cycle: 441 hollow months and 499 full months; thus 4 Metonic Cycles minus one day or precisely 76 years of 365¼ days. It equals 940 lunations with an error of only 5.9 hours.
Triad: A triple inex, with the advantage that it has nearly an integer number of anomalistic months, which makes the circumstances between two eclipses one Triad apart very similar, but at the opposite latitude. Almost exactly 87 calendar years minus 2 months.
Hipparchic cycle: Not a noteworthy eclipse cycle, but Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
constructed it to closely match an integer number of synodic and anomalistic months, years (345), and days. By comparing his own eclipse observations with Babylonian records from 345 years earlier, he could verify the accuracy of the various periods that the Chaldeans used.
Babylonian: The ratio 5923 returns to latitude in 5458 months was used by the Chaldeans in their astronomical computations.
Tetradia: Sometimes 4 total lunar eclipses occur in a row with intervals of 6 lunations (semester), and this is called a tetrad. Giovanni Schiaparelli
Giovanni Schiaparelli
Giovanni Virginio Schiaparelli was an Italian astronomer and science historian. He studied at the University of Turin and Berlin Observatory. In 1859-1860 he worked in Pulkovo Observatory and then worked for over forty years at Brera Observatory...
noticed that there are eras when such tetrads occur comparatively frequently, interrupted by eras when they are rare. This variation takes about 6 centuries. Antonie Pannekoek
Antonie Pannekoek
Antonie Pannekoek was a Dutch astronomer and Marxist theorist. He was one of the main theorists of council communism .- Biography :...
(1951) explained this phenomenon and found a period of 591 years. Van den Bergh (1954) from Theodor von Oppolzer
Theodor von Oppolzer
Theodor von Oppolzer was an Austrian astronomer and mathematician.The son of the physician Johann Ritter von Oppolzer, Theodor was born in Prague, at the time part of the Austrian Empire. He completed his graduate studies in medicine at the University of Vienna, gaining a Ph.D. in 1865...
's Canon der Finsternisse found a period of 586 years. This happens to be an eclipse cycle; see Meeus [I] (1997). Recently Tudor Hughes explained the variation from secular changes in the eccentricity
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...
of the Earth's orbit: the period for occurrence of tetrads is variable and currently is about 565 years; see Meeus III (2004) for a detailed discussion.
External links
- A Catalogue of Eclipse Cycles (more comprehensive than the above)
- Search 5,000 years worth of eclipses
- Eclipses, Cosmic Clockwork of the Ancients