Metonic cycle

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Heliocentric Solar System

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 remarkable for being 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 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 slightly 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 Hebrew calendar.

Mathematical basis[edit]

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 do not 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[edit]

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.

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[edit]

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.

In the Gregorian Calendar after 10 x 76 years all the phases of the moon falls three days later in the year, but after three years this will be just (the same) three days earlier! So we get a long period of 10 x 76 + 3 = 763 years. After this period the phases of the Moon falls (nearly) exactly on the same day in the year and also the eclipses does this! See for instance -1469, -706, 57, 820 (Julian Calendar) followed in the Gregorian Calendar by 1583, 2346, 3109 and 3872. Source http://eclipse.gsfc.nasa.gov/phase/phasecat.html Solar eclipses in this years are on:

Julian calendar

-1469(Jan  6  10:08 P  falls from the next 763-year period everytime in the previous year and therefore will this eclipse now not be futher regarded)
      Jun  2  12:41 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/-1499--1400/-1469-06-02.gif saros 4
      Jul  1  19:52 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/-1499--1400/-1469-07-02.gif saros 42
      Nov 26  14:52 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/-1499--1400/-1469-11-27.gif saros 9    
          
-706  Jun 26  15:33 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/-0799--0700/-706-06-26.gif  saros 64
      Nov 20  15:38 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/-0799--0700/-706-11-20.gif  saros 31   
      Dec 20  05:42 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/-0799--0700/-706-12-20.gif  saros 69                                        
  57  Jun 20  18:20 A  http://eclipse.gsfc.nasa.gov/5MCSEmap/0001-0100/57-06-20.gif  saros 86  
      Nov 15  08:30 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/0001-0100/57-11-15.gif  saros 53    
      Dec 14  19:37 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/0001-0100/57-12-14.gif  saros 91 
 820  Jun 14  10:40 A  http://eclipse.gsfc.nasa.gov/5MCSEmap/0801-0900/820-06-14.gif saros 108    
      Dec  9  09:55 P  http://eclipse.gsfc.nasa.gov/5MCSEmap/0801-0900/820-12-09.gif saros 113  

and so on in the Gregorian calendar

1583  Jun 19  19:43 T  http://eclipse.gsfc.nasa.gov/5MCSEmap/1501-1600/1583-06-19.gif saros 130
      Dec 14  04:58 A  http://eclipse.gsfc.nasa.gov/5MCSEmap/1501-1600/1583-12-14.gif saros 135  
2346  Jun 20  12:49 T  http://eclipse.gsfc.nasa.gov/5MCSEmap/2301-2400/2346-06-20.gif saros 152     
      Dec 13  20:48 A  http://eclipse.gsfc.nasa.gov/5MCSEmap/2301-2400/2346-12-13.gif saros 157
3109  Jun 20  16:32 A  04m55s   gamma -0.2644                                         saros 174
      Dec 14  06:22 T  04m40s   gamma  0.3188                                         saros 179
3872  Jun 19  05:56 A                                                                 saros 196
      Dec 13  19:40 H                                                                 saros 201 
         

After every 763 year you must add 22 to the # of the saros series. See also 2148 http://eclipse.gsfc.nasa.gov/5MCSEmap/2101-2200/2148-08-14.gif saros 157, after 5 x 763 = 3815 years followed by 5963 Aug 16 H2 01m52s gamma -0.1291 saros [157 + (5 x 22) =] 267.

Sources http://eclipse.gsfc.nasa.gov/SEcat5/SEcatalog.html , for the year 3109 http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros174.html and http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros179.html , and for the year 5963 the very last eclipse on http://eclipse.gsfc.nasa.gov/SEcatmax/SE-3999-6000MaxH.html

See also for instance http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros004.html http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros026.html http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros048.html http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros070.html http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros092.html http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros114.html http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros136.html http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros158.html and http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros180.html and then especially the column Ecl. Type on all this given links.

For all the lunar eclipses is this always exactly the same!

Other but less accurate time periods[edit]

After (5 x 76) + 11 = 391 years the phases of the Moon falls (nearly) exactly on the same days in the year again, also the eclipse does this, but regaring the eclipses they are less accurate after 391 years than after the just above described time period of 763 years.

Also a 399-year periond exists in the Gregorian Calendar but then are the eclipses often a month later and seldom on the same day (read: two days later) in the year. Is it New Moon on Sunday 1 January then after this 399 years will New Moon averaged be on Monday 3 January and after again 399 years averaged be on Tuesday 5 January then averaged be on Wednesday 7 January and so on. It is known that in the Gregorian Calendar after 400 years the days always fall on the same day of the week. In the Julian Calendar you must instead of averaged add two days, now averaged just deduct one day.

There exists also a long period of 6x(4x19) years = 456 years. After each 456 years in the Gregorian Calendar the phases of the Moon will fall averaged two days later in the year, and in the Julian Calendar the phases of the Moon fell after each 456 years averaged just 1,5 days earlier in the year. See for instance 1601 and 2057 (Gregorian Calendar) and see also for instance 1045 and 1501 (Julian Calendar). In the Gregorian Calendar even the days falls on the same day of the week after this 456 years. Even the eclipses are then the same two days later (Gregorian Calendar) or just exactly the same 1,5 days earlier in the year (Julian Calendar). The gammas of these eclipses are alternately + or - and also the solar eclipses are then alternately total or annular. Therefore it is better to double this period to 912 years. However also partial solar- and lunar eclipses are then possible. But the differences of the gammas of this eclipses are after each 912 years always small! Source: http://eclipse.gsfc.nasa.gov/phase/phasecat.html

The most longest time period in the Gregorian Calendar is exactly 1200 years. See for instance 1600 and 2800, or 2000 and 3200 in the source stated above. And also now the phases of the Moon will fall averaged two days later in the year after each 1200 years! Even the lunar- and solar eclipses are now often on this same two days later. But regarding the solar eclipses it appears that a better period is 2400 years (2x 1200 years) because after 1200 years you will get the same regarding solar eclipses after the above described 456 years. In the Julian Calendar there exist a very long period of 2000 years and also even the lunar- and solar eclipses are now often just two days earlier in the year. See for instance -900 and 1100, or -500 and 1500 in the source stated above.

And see finally also this both tables of solar eclipses:

after each (15 x 76) + 14 = 1154 years:

-1879 Jul 04 saros series  16 A nn gamma -0.1592 06m28s
-0725 Jun 26 saros series  54 T n- gamma -0.3288 07m18s
 0429 Jun 17 saros series  92 A p- gamma -0.5325 07m19s
 1583 Jun 19 saros series 130 T p- gamma -0.5581 06m23s
 2737 Jun 20 saros series 168 A p- gamma -0.7282 03m14s

after each 1154 + 763 = 1917 years:

-1819 Apr 03  saros series   0  A -p gamma 0.8990 04m48s
 0098 Mar 21  saros series  60  A -p gamma 0.8008 01m16s
 2015 Mar 20  saros series 120  T -t gamma 0.9454 02m47s
 3932 Mar 20  saros series 180  P    gamma 1.4305  

Source: http://eclipse.gsfc.nasa.gov/SEcat5/SEcatalog.html

When were this both series begun and when will this both series end?

See also[edit]

External links[edit]

References[edit]

  • Mathematical Astronomy Morsels, Jean Meeus, Willmann-Bell, Inc., 1997 (Chapter 9, p. 51, Table 9.A Some eclipse Periodicities)