Metonic cycle

For astronomy and calendar studies, the Metonic cycle or Enneadecaeteris (from Ancient Greek: ἐννεακαιδεκαετηρίς, "nineteen years") is a period of very close to 19 years that is nearly a common multiple of the solar year and the synodic (lunar) month. The Greek astronomer Meton of Athens (fifth century BC) observed that a period of 19 years is almost exactly equal to 235 synodic months and, rounded to full days, counts 6,940 days. The difference between the two periods (of 19 solar years and 235 synodic months) is only a few hours, depending on the definition of the year.

Considering a year to be ​119 of this 6,940-day cycle gives a year length of 365 + ​14 + ​176 days (the unrounded cycle is much more accurate), which is about 11 days more than 12 synodic months. To keep a 12-month lunar year in pace with the solar year, an intercalary 13th month would have to be added on seven occasions during the nineteen-year period (235 = 19 × 12 + 7). When Meton introduced the cycle around 432 BC, it was already known by Babylonian astronomers. A mechanical computation of the cycle is built into the Antikythera mechanism.

The cycle was used in the Babylonian calendar, ancient Chinese calendar systems (the 'Rule Cycle' 章) and the medieval computus (i.e., the calculation of the date of Easter). It regulates the 19-year cycle of intercalary months of the modern Hebrew calendar. The start of the Metonic cycle depends on which of these systems is being used; for Easter, the first year of the current Metonic cycle is 2014.

CLM 14456 70v71r
Depiction of the 19 years of the Metonic cycle as a wheel, with the Julian date of the Easter New Moon, from a 9th-century computistic manuscript made in St. Emmeram's Abbey (Clm 14456, fol. 71r)
Christmas full moons 1711-2300
For example, by the 19-year metonic cycle, the full moon repeats on or near Christmas day between 1711 and 2300.[1][2] A small horizontal libration is visible comparing their appearances. A red color shows full moons that are also lunar eclipses.

Mathematical basis

At the time of Meton, axial precession had not yet been discovered, and he could not distinguish between sidereal years (currently: 365.256363 days) and tropical years (currently: 365.242190 days). Most calendars, like the commonly used Gregorian calendar, are based on the tropical year and maintain the seasons at the same calendar times each year. Nineteen tropical years are about two hours shorter than 235 synodic months. The Metonic cycle's error is, therefore, one full day every 219 years, or 12.4 parts per million.

19 tropical years = 6,939.602 days (12 × 354-day years + 7 × 384-day years + 3.6 days).
235 synodic months (lunar phases) = 6,939.688 days (Metonic period by definition).
254 sidereal months (lunar orbits) = 6,939.702 days (19 + 235 = 254).
255 draconic months (lunar nodes) = 6,939.1161 days.

Note that the 19-year cycle is also close (to somewhat more than half a day) to 255 draconic months, so it is also an eclipse cycle, which lasts only for about 4 or 5 recurrences of eclipses. The Octon is ​15 of a Metonic cycle (47 synodic months, 3.8 years), and it recurs about 20 to 25 cycles.

This cycle seems to be a coincidence. The periods of the Moon's orbit around the Earth and the Earth's orbit around the Sun are believed to be independent, and not to have any known physical resonance. An example of a non-coincidental cycle is the orbit of Mercury, with its 3:2 spin-orbit resonance.

A lunar year of 12 synodic months is about 354 days, approximately 11 days short of the "365-day" solar year. Therefore, for a lunisolar calendar, every 2 to 3 years there is a difference of more than a full lunar month between the lunar and solar years, and an extra (embolismic) month needs to be inserted (intercalation). The Athenians initially seem not to have had a regular means of intercalating a 13th month; instead, the question of when to add a month was decided by an official. Meton's discovery made it possible to propose a regular intercalation scheme. The Babylonians seem to have introduced this scheme around 500 BC, thus well before Meton.

Application in traditional calendars

Traditionally, for the Babylonian and Hebrew lunisolar calendars, the years 3, 6, 8, 11, 14, 17, and 19 are the long (13-month) years of the Metonic cycle. This cycle, which can be used to predict eclipses, forms the basis of the Greek and Hebrew calendars, and is used for the computation of the date of Easter each year.

The Babylonians applied the 19-year cycle since the late sixth century BC. As they measured the moon's motion against the stars, the 235:19 relationship may originally have referred to sidereal years, instead of tropical years as it has been used for various calendars.

According to Livy, the king of Rome Numa Pompilius (753-673 BC) inserted intercalary months in such a way that in the twentieth year the days should fall in with the same position of the sun from which they had started.[3] As the twentieth year takes place nineteen years after the first year, this seems to indicate that the Metonic cycle was applied to Numa's calendar.

Apollo was said to have visited the Hyperboreans once every 19 years, presumably at the high point of the cycle.

The Runic calendar is a perpetual calendar based on the 19-year-long Metonic cycle. Also known as a Rune staff or Runic Almanac, it appears to have been a medieval Swedish invention. This calendar does not rely on knowledge of the duration of the tropical year or of the occurrence of leap years. It is set at the beginning of each year by observing the first full moon after the winter solstice. The oldest one known, and the only one from the Middle Ages, is the Nyköping staff, which is believed to date from the 13th century.

The Bahá'í calendar, established during the middle of the 19th century, is also based on cycles of 19 years.

Further details

The Metonic cycle is related to two less accurate subcycles:

  • 8 years = 99 lunations (an Octaeteris) to within 1.5 days, i.e. an error of one day in 5 years; and
  • 11 years = 136 lunations within 1.5 days, i.e. an error of one day in 7.3 years.

By combining appropriate numbers of 11-year and 19-year periods, it is possible to generate ever more accurate cycles. For example, simple arithmetic shows that:

  • 687 tropical years = 250,921.39 days;
  • 8,497 lunations = 250,921.41 days.

This gives an error of only about half an hour in 687 years (2.5 seconds a year), although this is subject to secular variation in the length of the tropical year and the lunation.

Meton of Athens approximated the cycle to a whole number (6,940) of days, obtained by 125 long months of 30 days and 110 short months of 29 days. During the next century, Callippus developed the Callippic cycle of four 19-year periods for a 76-year cycle with a mean year of exactly 365.25 days.

See also

References

  1. ^ Rare Full Moon on Christmas Day, NASA
  2. ^ Ask Tom: How unusual is a full moon on Christmas Day?
  3. ^ Livy, Ab Urbe Condita, I, XIX, 6.
  • Mathematical Astronomy Morsels, Jean Meeus, Willmann-Bell, Inc., 1997 (Chapter 9, p. 51, Table 9.A Some eclipse Periodicities)

External links

August 1970 lunar eclipse

A partial lunar eclipse took place on August 17, 1970. The Earth's shadow on the moon was clearly visible in this eclipse, with 41% of the Moon in shadow; the partial eclipse lasted for 2 hours and 11 minutes.

August 1988 lunar eclipse

A partial lunar eclipse took place on August 27, 1988, the second of two lunar eclipses in 1988.

August 2008 lunar eclipse

A partial lunar eclipse took place on August 16, 2008, the second of two lunar eclipses in 2008, with the first being a total eclipse on February 20, 2008. The next lunar eclipse was a penumbral eclipse occurring on February 9, 2009, while the next total lunar eclipse occurred on December 21, 2010.

Buddhist calendar

The Buddhist calendar is a set of lunisolar calendars primarily used in mainland Southeast Asian countries of Cambodia, Laos, Myanmar and Thailand as well as in Sri Lanka and Chinese populations of Malaysia and Singapore for religious or official occasions. While the calendars share a common lineage, they also have minor but important variations such as intercalation schedules, month names and numbering, use of cycles, etc. In Thailand, the name Buddhist Era is a year numbering system shared by the traditional Thai lunisolar calendar and by the Thai solar calendar.

The Southeast Asian lunisolar calendars are largely based on an older version of the Hindu calendar, which uses the sidereal year as the solar year. One major difference is that the Southeast Asian systems, unlike their Indian cousins, do not use apparent reckoning to stay in sync with the sidereal year. Instead, they employ their versions of the Metonic cycle. However, since the Metonic cycle is not very accurate for sidereal years, the Southeast Asian calendar is slowly drifting out of sync with the sidereal, approximately one day every 100 years. Yet no coordinated structural reforms of the lunisolar calendar have been undertaken.

Today, the traditional Buddhist lunisolar calendar is used mainly for Theravada Buddhist festivals, and no longer has the official calendar status anywhere. The Thai Buddhist Era, a renumbered Gregorian calendar, is the official calendar in Thailand.

Burmese calendar

The Burmese calendar (Burmese: မြန်မာသက္ကရာဇ်, pronounced [mjəmà θɛʔkəɹɪʔ], or ကောဇာသက္ကရာဇ်, [kɔ́zà θɛʔkəɹɪʔ]; Burmese Era (BE) or Myanmar Era (ME)) is a lunisolar calendar in which the months are based on lunar months and years are based on sidereal years. The calendar is largely based on an older version of the Hindu calendar, though unlike the Indian systems, it employs a version of the Metonic cycle. The calendar therefore has to reconcile the sidereal years of the Hindu calendar with the Metonic cycle's near tropical years by adding intercalary months and days at irregular intervals.

The calendar has been used continuously in various Burmese states since its purported launch in 640 CE in the Sri Ksetra Kingdom, also called the Pyu era. It was also used as the official calendar in other mainland Southeast Asian kingdoms of Arakan, Lan Na, Xishuangbanna, Lan Xang, Siam, and Cambodia down to the late 19th century.

Today the calendar is used only in Myanmar as the traditional civil calendar, alongside the Buddhist calendar. It is still used to mark traditional holidays such as the Burmese New Year, and other traditional festivals, many of which are Burmese Buddhist in nature.

Callippic cycle

For astronomy and calendar studies, the Callippic cycle (or Calippic) is a particular approximate common multiple of the year (specifically the tropical year) and the synodic month, that was proposed by Callippus during 330 BC. It is a period of 76 years, as an improvement of the 19-year Metonic cycle.

A century before Callippus, Meton had discovered the cycle in which 19 years equals 235 lunations. If we assume a year is about ​365 1⁄4 days, 19 years total about 6940 days, which exceeds 235 lunations by almost a third of a day, and 19 tropical years by four tenths of a day. It implicitly gave the solar year a duration of ​6940⁄19 = 365 + ​5⁄19 = 365 + ​1⁄4 + ​1⁄76 days = 365 d 6 h 18 min 56 s. Callippus accepted the 19-year cycle, but held that the duration of the year was more closely ​365 1⁄4 days (= 365 d 6 h), so he multiplied the 19-year cycle by 4 to obtain an integer number of days, and then omitted 1 day from the last 19-year cycle. Thus, he computed a cycle of 76 years that consists of 940 lunations and 27,759 days, which has been named the Callippic cycle after him.Although the cycle's error has been computed as one full day in 553 years, or 4.95 parts per million, in actuality 27,759 days in 76 years has a mean year of exactly ​365 1⁄4 days, which relative to the mean northward equinoctial year is about 11 minutes too long per year, in other words the cycle drifts another day late per ​130 10⁄11 years, which is considerably worse than the drift of the unrounded Metonic cycle. If the Callippic cycle is considered as closer to its unrounded length of ​27,758 3⁄4 days (based on 940 lunations) then its accuracy is essentially the same as the unrounded Metonic cycle (within a few seconds per year). If it is considered as 940 lunations less one day then the Callippic mean year will be shortened by ​1⁄76 of a day (18 minutes 57 seconds), making it grossly too short, and it will also grossly drift ahead with respect to the mean lunar cycle at the rate of ​1⁄940 of a day (1 minute 31 seconds) per lunar month. If the cycle length is truncated to 27,758 days then the mean year is 365 days 5 hours 41 minutes 3 seconds, or almost 8 minutes too brief per year, and it will drift ahead of the mean lunar cycle by about ​(​3⁄4)⁄940 day (1 minute 9 seconds) per lunar month. Altogether, the purported accuracy of this cycle is not impressive, but it is of historical interest.The first year of the first Callippic cycle began at the summer solstice of 330 BC (28 June in the proleptic Julian calendar), and was subsequently used by later astronomers. In Ptolemy's Almagest, for example, he cites (Almagest VII 3, H25) observations by Timocharis during the 47th year of the first Callippic cycle (283 BC), when on the eighth of Anthesterion, the Pleiades star cluster was occulted by the Moon.The Callippic calendar originally used the names of months from the Attic calendar. Later astronomers, such as Hipparchus, preferred other calendars, including the ancient Egyptian calendar. Also Hipparchus invented his own Hipparchic calendar cycle as an improvement upon the Callippic cycle. Ptolemy's Almagest provided some conversions between the Callippic and Egyptian calendars, such as that Anthesterion 8, 47th year of the first Callippic period was equivalent to day 29 of the month of Athyr, during year 465 of Nabonassar. However, the original, complete form of the Callippic calendar is no longer known.

Chula Sakarat

Chula Sakarat or Chulasakarat (Pali: Culāsakaraj; Burmese: ကောဇာသက္ကရာဇ်, pronounced [kɔ́zà θɛʔkəɹɪʔ]; Khmer: ចុល្លសករាជ "Chulasakarach"; Thai: จุลศักราช, RTGS: Chunlasakkarat, pronounced [t͡ɕũn˧.lä˥.säk̚˩.kä˩.räːt̚˨˩], abbrv. จ.ศ. Choso) is a lunisolar calendar derived from the Burmese calendar, whose variants were in use by most mainland Southeast Asian kingdoms down to the late 19th century. The calendar is largely based on an older version of the Hindu calendar though unlike the Indian systems, it employs a version of the Metonic cycle. The calendar therefore has to reconcile the sidereal years of the Hindu calendar with Metonic cycle's tropical years by adding intercalary months and intercalary days on irregular intervals.

Although the name Culāsakaraj is a generic term meaning "Lesser Era" in Pali, the term Chula Sakarat is often associated with the various versions of the calendar used in regions that make up modern-day Thailand. The calendar is used in Myanmar and the Sipsong Panna area of China. In Thailand, it is only used in academia for Thai history studies.

February 1970 lunar eclipse

A partial lunar eclipse took place on February 21, 1970.

February 1989 lunar eclipse

A total lunar eclipse took place on February 20, 1989, the first of two total lunar eclipses in 1989.

January 2037 lunar eclipse

A total lunar eclipse will take place on January 31, 2037. The Moon will be plunged into darkness for 1 hour and 4 minutes, in a deep total eclipse which will see the Moon 21% of its diameter inside the Earth's umbral shadow. The visual effect of this depends on the state of the Earth's atmosphere, but the Moon may be stained a deep red colour for observers in north and west North America, most of Asia, Australia and New Zealand. The partial eclipse will last for 3 hours and 18 minutes in total. It occurs during a supermoon (perigee), and blue moon (second full moon of month), just like the eclipse of January 31, 2018, one metonic cycle (19 years) previous.

March 2026 lunar eclipse

A total lunar eclipse will take place on March 3, 2026.

Meton of Athens

Meton of Athens (Greek: Μέτων ὁ Ἀθηναῖος; gen.: Μέτωνος) was a Greek mathematician, astronomer, geometer, and engineer who lived in Athens in the 5th century BC. He is best known for calculations involving the eponymous 19-year Metonic cycle which he introduced in 432 BC into the lunisolar Attic calendar.

September 2006 lunar eclipse

A partial lunar eclipse took place on September 7, 2006, the second of two lunar eclipses in 2006.

September 2025 lunar eclipse

A total lunar eclipse will take place on September 7, 2025.

Solar eclipse of December 15, 1982

A partial solar eclipse occurred on December 15, 1982. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A partial solar eclipse occurs in the polar regions of the Earth when the center of the Moon's shadow misses the Earth.

Solar eclipse of July 22, 2028

A total solar eclipse will occur on July 22, 2028.

The central line of the path of the eclipse will cross the Australian continent from the Kimberley region in the north west and continue in a south-easterly direction through Western Australia, the Northern Territory, south-west Queensland and New South Wales, close to the towns of Wyndham, Kununurra, Tennant Creek, Birdsville, Bourke and Dubbo, and continuing on through the centre of Sydney, where the eclipse will have a duration of over three minutes. It will also cross Dunedin, New Zealand.

Solar eclipse of May 11, 1975

A partial solar eclipse occurred on May 11, 1975. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A partial solar eclipse occurs in the polar regions of the Earth when the center of the Moon's shadow misses the Earth.

Solar eclipse of May 9, 2032

An annular solar eclipse will occur on May 9, 2032. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide.

Solar eclipse of October 3, 2043

An annular solar eclipse will occur on October 3, 2043. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide.

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