# Tropical year

A tropical year (also known as a solar year) is the 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. Because of the precession of the equinoxes, the seasonal cycle does not remain exactly synchronized with the position of the Earth in its orbit around the Sun. As a consequence, the tropical year is about 20 minutes shorter than the time it takes Earth to complete one full orbit around the Sun as measured with respect to the fixed stars (the sidereal year).

Since antiquity, astronomers have progressively refined the definition of the tropical year. The entry for "year, tropical" in the Astronomical Almanac Online Glossary (2015) states:

the period of time for the ecliptic longitude of the Sun to increase 360 degrees. Since the Sun's ecliptic longitude is measured with respect to the equinox, the tropical year comprises a complete cycle of seasons, and its length is approximated in the long term by the civil (Gregorian) calendar. The mean tropical year is approximately 365 days, 5 hours, 48 minutes, 45 seconds.

An equivalent, more descriptive, definition is "The natural basis for computing passing tropical years is the mean longitude of the Sun reckoned from the precessionally moving equinox (the dynamical equinox or equinox of date). Whenever the longitude reaches a multiple of 360 degrees the mean Sun crosses the vernal equinox and a new tropical year begins" (Borkowski 1991, p. 122).

The mean tropical year in 2000 was 365.24219 ephemeris days; each ephemeris day lasting 86,400 SI seconds.[1] This is 365.24217 mean solar days (Richards 2013, p. 587).

## History

### Origin

The word "tropical" comes from the Greek tropikos meaning "turn" (tropic 1992). Thus, the tropics of Cancer and Capricorn mark the extreme north and south latitudes where the Sun can appear directly overhead, and where it appears to "turn" in its annual seasonal motion. Because of this connection between the tropics and the seasonal cycle of the apparent position of the Sun, the word "tropical" also lent its name to the "tropical year". The early Chinese, Hindus, Greeks, and others made approximate measures of the tropical year.

### Early value, precession discovery

In the 2nd century BC Hipparchus measured the time required for the Sun to travel from an equinox to the same equinox again. He reckoned the length of the year to be 1/300 of a day less than 365.25 days (365 days, 5 hours, 55 minutes, 12 seconds, or 365.24667 days). Hipparchus used this method because he was better able to detect the time of the equinoxes, compared to that of the solstices (Meeus & Savoie 1992, p. 40).

Hipparchus also discovered that the equinoctial points moved along the ecliptic (plane of the Earth's orbit, or what Hipparchus would have thought of as the plane of the Sun's orbit about the Earth) in a direction opposite that of the movement of the Sun, a phenomenon that came to be named "precession of the equinoxes". He reckoned the value as 1° per century, a value that was not improved upon until about 1000 years later, by Islamic astronomers. Since this discovery a distinction has been made between the tropical year and the sidereal year (Meeus & Savoie 1992, p. 40).

### Middle Ages and the Renaissance

During the Middle Ages and Renaissance a number of progressively better tables were published that allowed computation of the positions of the Sun, Moon and planets relative to the fixed stars. An important application of these tables was the reform of the calendar.

The Alfonsine Tables, published in 1252, were based on the theories of Ptolemy and were revised and updated after the original publication; the most recent update in 1978 was by the French National Centre for Scientific Research. The length of the tropical year was given as 365 solar days 5 hours 49 minutes 16 seconds (≈ 365.24255 days). This length was used in devising the Gregorian calendar of 1582 (Meeus & Savoie 1992, p. 41).

In the 16th century Copernicus put forward a heliocentric cosmology. Erasmus Reinhold used Copernicus' theory to compute the Prutenic Tables in 1551, and gave a tropical year length of 365 solar days, 5 hours, 55 minutes, 58 seconds (365.24720 days), based on the length of a sidereal year and the presumed rate of precession. This was actually less accurate than the earlier value of the Alfonsine Tables.

Major advances in the 17th century were made by Johannes Kepler and Isaac Newton. In 1609 and 1619 Kepler published his three laws of planetary motion (McCarthy & Seidelmann 2009, p. 26). In 1627, Kepler used the observations of Tycho Brahe and Waltherus to produce the most accurate tables up to that time, the Rudolphine Tables. He evaluated the mean tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds (365.24219 days; Meeus & Savoie 1992, p. 41).

Newton's three laws of dynamics and theory of gravity were published in his Philosophiæ Naturalis Principia Mathematica in 1687. Newton's theoretical and mathematical advances influenced tables by Edmund Halley published in 1693 and 1749 (McCarthy & Seidelmann 2009, pp. 26–28) and provided the underpinnings of all solar system models until Albert Einstein's theory of General relativity in the 20th century.

### 18th and 19th century

From the time of Hipparchus and Ptolemy, the year was based on two equinoxes (or two solstices) a number of years apart, to average out both observational errors and the effects of nutation (periodic motions of the axis of rotation of the earth, the main cycle being 18.6 years) and the movement of the Sun caused by the gravitational pull of the planets. These effects did not begin to be understood until Newton's time. To model short-term variations of the time between equinoxes (and prevent them from confounding efforts to measure long-term variations) requires either precise observations or an elaborate theory of the apparent motion of the Sun. The necessary theories and mathematical tools came together in the 18th century due to the work of Pierre-Simon de Laplace, Joseph Louis Lagrange, and other specialists in celestial mechanics. They were able to express the mean longitude of the Sun as

L0 = A0 + A1T + A2T2 days

where T is the time in Julian centuries. The inverse of the derivative of L0, dT/dL0 gives the length of the tropical year as a linear function of T. When this is computed, an expression giving the length of the tropical year as a function of T results.

Two equations are given in the table. Both equations estimate that the tropical year gets roughly a half second shorter each century.

Tropical year coefficients
Name Equation Date on which T = 0
Leverrier (Meeus & Savoie 1992, p. 42) Y = 365.24219647 − 6.24×106 T January 0.5, 1900, Ephemeris time
Newcomb (1898) Y = 365.24219879 − 6.14×106 T January 0, 1900, mean time

Newcomb's tables were successful enough that they were used by the joint American-British Astronomical Almanac for the Sun, Mercury, Venus, and Mars through 1983 (Seidelmann 1992, p. 317).

### 20th and 21st centuries

The length of the mean tropical year is derived from a model of the solar system, so any advance that improves the solar system model potentially improves the accuracy of the mean tropical year. Many new observing instruments became available, including

The complexity of the model used for the solar system must be limited to the available computation facilities. In the 1920s punched card equipment came into use by L. J. Comrie in Britain. For the American Ephemeris an electromagnetic computer, the IBM Selective Sequence Electronic Calculator was used since 1948. When modern computers became available, it was possible to compute ephemerides using numerical integration rather than general theories; numerical integration came into use in 1984 for the joint US-UK almanacs (McCarthy & Seidelmann 2009, p. 32).

Einstein's General Theory of Relativity provided a more accurate theory, but the accuracy of theories and observations did not require the refinement provided by this theory (except for the advance of the perihelion of Mercury) until 1984. Time scales incorporated general relativity beginning in the 1970s (McCarthy & Seidelmann 2009, p. 37).

A key development in understanding the tropical year over long periods of time is the discovery that the rate of rotation of the earth, or equivalently, the length of the mean solar day, is not constant. William Ferrel in 1864 and Charles-Eugène Delaunay in 1865 indicated the rotation of the Earth was being retarded by tides. In 1921 William H Shortt invented the Shortt-Synchronome clock, the most accurate commercially produced pendulum clock; it was the first clock capable of measuring variations in the Earth's rotation. The next major time-keeping advance was the quartz clock, first built by Warren Marrison and J. W. Horton in 1927; in the late 1930s quartz clocks began to replace pendulum clocks as time standards (McCarthy & Seidelmann 2009, ch. 9).

A series of experiments beginning in the late 1930s led to the development of the first atomic clock by Louis Essen and J. V. L. Parry in 1955. Their clock was based on a transition in the cesium atom (McCarthy & Seidelmann 2009, pp. 157–9). Due to the accuracy the General Conference on Weights and Measures in 1960 redefined the second in terms of the cesium transition.[1] The atomic second, often called the SI second, was intended to agree with the ephemeris second based on Newcomb's work, which in turn makes it agree with the mean solar second of the mid-19th century (McCarthy & Seidelmann 2009, pp. 81–2, 191–7).

## Time scales

As mentioned in History, advances in time-keeping have resulted in various time scales. One useful time scale is Universal Time, especially the UT1 variant, which is the mean solar time at 0 degrees longitude (the Greenwich meridian). One second of UT is 1/86,400 of a mean solar day. This time scale is known to be somewhat variable. Since all civil calendars count actual solar days, they must ultimately be based on UT (but the actual timing of official midnight is based on UTC).

The other time scale has two parts. Ephemeris time (ET) is the independent variable in the equations of motion of the solar system, in particular, the equations in use from 1960 to 1984 (McCarthy & Seidelmann 2009, p. 378). That is, the length of the second used in the solar system calculations could be adjusted until the length that gives the best agreement with observations is found. With the introduction of atomic clocks in the 1950s, it was found that ET could be better realized as atomic time. This also means that ET is a uniform time scale, as is atomic time. ET was given a new name, Terrestrial Time (TT), and for most purposes ET = TT = International Atomic Time + 32.184 SI seconds. As of January 2017, TT is ahead of UT1 by 69.184 seconds (International Earth Rotation Service 2017; McCarthy & Seidelmann 2009, pp. 86–7).

As explained below, long-term estimates of the length of the tropical year were used in connection with the reform of the Julian calendar, which resulted in the Gregorian calendar. Participants in that reform were unaware of the non-uniform rotation of the Earth, but now this can be taken into account to some degree. The amount that TT is ahead of UT1 is known as ΔT, or Delta T. The table below gives Morrison and Stephenson's (S & M) 2004 estimates and standard errors (σ) for dates significant in the process of developing the Gregorian calendar.

Event Year Nearest S & M Year ΔT σ
Julian calendar begins −44 0 2h56m20s 4m20s
First Council of Nicaea 325 300 2h8m 2m
Gregorian calendar begins 1582 1600 2m 20s
low-precision extrapolation 4000 4h13m
low-precision extrapolation 10,000 2d11h

The low-precision extrapolations are computed with an expression provided by Morrison & Stephenson (2004)

ΔT in seconds = −20 + 32t2

where t is measured in Julian centuries from 1820. The extrapolation is provided only to show ΔT is not negligible when evaluating the calendar for long periods; Borkowski (1991, p. 126) cautions that "many researchers have attempted to fit a parabola to the measured ΔT values in order to determine the magnitude of the deceleration of the Earth's rotation. The results, when taken together, are rather discouraging."

## Length of tropical year

An oversimplified definition of the tropical year would be the time required for the Sun, beginning at a chosen ecliptic longitude, to make one complete cycle of the seasons and return to the same ecliptic longitude. Before considering an example, the equinox must be examined. There are two important planes in solar system calculations, the plane of the ecliptic (the Earth's orbit around the Sun), and the plane of the celestial equator (the Earth's equator projected into space). These two planes intersect in a line. This direction is given the symbol ♈︎ (the symbol looks like the horns of a ram because it used to be toward the constellation Aries). The opposite direction is given the symbol ♎︎ (because it used to be toward Libra). Because of the precession of the equinoxes and nutation these directions change, compared to the direction of distant stars and galaxies, whose directions have no measurable motion due to their great distance (see International Celestial Reference Frame).

The ecliptic longitude of the Sun is the angle between ♈︎ and the Sun, measured eastward along the ecliptic. This creates a complicated measurement, because as the Sun is moving, the direction the angle is measured from is also moving. It is convenient to have a fixed (with respect to distant stars) direction to measure from; the direction of ♈︎ at noon January 1, 2000 fills this role and is given the symbol ♈︎0.

Using the oversimplified definition, there was an equinox on March 20, 2009, 11:44:43.6 TT. The 2010 March equinox was March 20, 17:33:18.1 TT, which gives a duration of 365 days 5 hours 48 minutes 34.5 seconds (Astronomical Applications Dept., 2009). While the Sun moves, ♈︎ moves in the opposite direction . When the Sun and ♈︎ met at the 2010 March equinox, the Sun had moved east 359°59'09" while ♈︎ had moved west 51" for a total of 360° (all with respect to ♈︎0; Seidelmann 1992, p. 104, expression for pA).

If a different starting longitude for the Sun is chosen, the duration for the Sun to return to the same longitude will be different. This is because although ♈︎ changes at a nearly steady rate[2] there is considerable variation in the angular speed of the Sun. Thus, the 50 or so arcseconds that the Sun does not have to move to complete the tropical year "saves" varying amounts of time depending on the position in the orbit.

### Mean time interval between equinoxes

As already mentioned, there is some choice in the length of the tropical year depending on the point of reference that one selects. But during the period when return of the Sun to a chosen longitude was the method in use by astronomers, one of the equinoxes was usually chosen because it was easier to detect when it occurred. When tropical year measurements from several successive years are compared, variations are found which are due to nutation, and to the planetary perturbations acting on the Sun. Meeus & Savoie (1992, p. 41) provided the following examples of intervals between northward equinoxes:

days hours min s
1985–1986 365 5 48 58
1986–1987 365 5 49 15
1987–1988 365 5 46 38
1988–1989 365 5 49 42
1989–1990 365 5 51 06

Until the beginning of the 19th century, the length of the tropical year was found by comparing equinox dates that were separated by many years; this approach yielded the mean tropical year (Meeus & Savoie 1992, p. 42).

The following values of time intervals between equinoxes and solstices were provided by Meeus & Savoie (1992, p. 42) for the years 0 and 2000. These are smoothed values which take account of the Earth's orbit being elliptical, using well-known procedures (including solving Kepler's equation). They do not allow for periodic variations due to factors such as the gravitational force of the orbiting Moon and gravitational forces from the other planets. Such perturbations are minor compared to the positional difference resulting from the orbit being elliptical rather than circular.(Meeus 2002, p. 362)

Year 0 Year 2000
Between two Northward equinoxes 365.242137 days 365.242374 days
Between two Northern solstices 365.241726 365.241626
Between two Southward equinoxes 365.242496 365.242018
Between two Southern solstices 365.242883 365.242740
Mean tropical year
365.242310 365.242189

### Mean tropical year current value

The mean tropical year on January 1, 2000 was 365.2421897 or 365 ephemeris days, 5 hours, 48 minutes, 45.19 seconds. This changes slowly; an expression suitable for calculating the length of a tropical year in ephemeris days, between 8000 BC and 12000 AD is

${\displaystyle 365.2421896698-6.15359\times 10^{-6}T-7.29\times 10^{-10}T^{2}+2.64\times 10^{-10}T^{3}}$

where T is in Julian centuries of 36,525 days of 86,400 SI seconds measured from noon January 1, 2000 TT (in negative numbers for dates in the past; McCarthy & Seidelmann 2009, p. 18, calculated from planetary model of Laskar 1986).

Modern astronomers define the tropical year as time for the Sun's mean longitude to increase by 360°. The process for finding an expression for the length of the tropical year is to first find an expression for the Sun's mean longitude (with respect to ♈︎), such as Newcomb's expression given above, or Laskar's expression (1986, p. 64). When viewed over a one-year period, the mean longitude is very nearly a linear function of Terrestrial Time. To find the length of the tropical year, the mean longitude is differentiated, to give the angular speed of the Sun as a function of Terrestrial Time, and this angular speed is used to compute how long it would take for the Sun to move 360° (Meeus & Savoie 1992, p. 42; Astronomical Almanac for the year 2011, L8).

The above formulae give the length of the tropical year in ephemeris days (equal to 86,400 SI seconds), not solar days. It is the number of solar days in a tropical year that is important for keeping the calendar in synch with the seasons (see below).

## Calendar year

The Gregorian calendar, as used for civil and scientific purposes, is an international standard. It is a solar calendar that is designed to maintain synchrony with the mean tropical year (Dobrzycki 1983, p. 123). It has a cycle of 400 years (146,097 days). Each cycle repeats the months, dates, and weekdays. The average year length is 146,097/400 = ​365 97400 = 365.2425 days per year, a close approximation to the mean tropical year (Seidelmann 1992, pp. 576–81).

The Gregorian calendar is a reformed version of the Julian calendar. By the time of the reform in 1582, the date of the vernal equinox had shifted about 10 days, from about March 21 at the time of the First Council of Nicaea in 325, to about March 11. According to North (1983), the real motivation for reform was not primarily a matter of getting agricultural cycles back to where they had once been in the seasonal cycle; the primary concern of Christians was the correct observance of Easter. The rules used to compute the date of Easter used a conventional date for the vernal equinox (March 21), and it was considered important to keep March 21 close to the actual equinox (North 1983, pp. 75–76).

If society in the future still attaches importance to the synchronization between the civil calendar and the seasons, another reform of the calendar will eventually be necessary. According to Blackburn and Holford-Strevens (who used Newcomb's value for the tropical year) if the tropical year remained at its 1900 value of 365.24219878125 days the Gregorian calendar would be 3 days, 17 min, 33 s behind the Sun after 10,000 years. Aggravating this error, the length of the tropical year (measured in Terrestrial Time) is decreasing at a rate of approximately 0.53 s per century. Also, the mean solar day is getting longer at a rate of about 1.5 ms per century. These effects will cause the calendar to be nearly a day behind in 3200. The number of solar days in a "tropical millennium" is decreasing by about 0.06 per millennium (neglecting the oscillatory changes in the real length of the tropical year).[3] This means there should be fewer and fewer leap days as time goes on. A possible reform would be to omit the leap day in 3200, keep 3600 and 4000 as leap years, and thereafter make all centennial years common except 4500, 5000, 5500, 6000, etc. But the quantity ΔT is not sufficiently predictable to form more precise proposals (Blackburn & Holford-Strevens 2003, p. 692).

## Notes

1. ^ a b "The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom." 13th CGPM (1967/68, Resolution 1; CR, 103 and Metrologia, 1968, 4, 43), as quoted in Bureau International des Poids et Mesures 2006, 133)
2. ^ The expression for pA in Seidelmann 1992, p. 104 shows the progression per 365 days of TT increases steadily from 50.25865 arcseconds (January 1, 2009 – January 1, 2010) to 50.25889 arcseconds (December 1, 2009 – December 1, 2010).
3. ^ 365242×1.5/8640000.

## References

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• Astronomical almanac for the year 2011. Washington: Astronomical Almanac Office US Naval Observatory. 2010.
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• Blackburn, B.; Holford-Strevens, L. (2003). The Oxford companion to the year. Oxford University Press. Corrected reprint of 1999.
• Borkowski, K.M. (1991). "The tropical year and the solar calendar" (PDF). Journal of the Royal Astronomical Society of Canada. 85 (3): 121–130. Bibcode:1991JRASC..85..121B.
• "The International System of Units". Bureau International des Poids et Mesures. 2006. Archived from the original on October 1, 2009.
• Butrica, A.J. (1996). "NASA SP-4218: To see the unseen—A history of planetary radar astronomy". NASA. Archived from the original on August 23, 2007.
• Dershowitz, N.; Reingold, E.M. (2008). Calendrical calculations (3rd ed.). Cambridge University Press. ISBN 978-0-521-70238-6.
• Dobrzycki, J. (1983). "Astronomical aspects of the calendar reform". In Coyne, G.V.; Hoskin, M.A.; Pedersen, O. (eds.). Gregorian reform of the calendar. Vatican Observatory.
• International Earth Rotation Service (February 2, 2017). "Bulletin A 30(5)". US Naval Observatory.
• Jet Propulsion Laboratory (2005). DSN: History. NASA.
• Laskar, J. (1986). "Secular terms of classical planetary theories using the results of general theory". Astronomy and Astrophysics. 157: 59–70. Bibcode:1986A&A...157...59L. ISSN 0004-6361. Note: In the article at this URL page 68 should be put before page 66.
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• Meeus, J.; Savoie, D. (1992). "The history of the tropical year". Journal of the British Astronomical Association. 102 (1): 40–42. Bibcode:1992JBAA..102...40M.
• Meeus, Jean (2002). More astronomical astronomy morsels. Richmond, VA: Willmann-Bell. ISBN 0-943396-74-3. Contains updates to Meeus & Savoie 1992.
• Morrison, L.V.; Stephenson, F.R. (2004). "Historical values of the Earth's clock error ΔT and the calculation of eclipses". Journal for the History of Astronomy. 35, Part 3 (120): 327–336. Bibcode:2004JHA....35..327M.
• Newcomb, S. (1898). Tables of the four inner planets. Astronomical papers prepared for the use of the American ephemeris and nautical almanac. 6 (2nd ed.). Washington: Bureau of Equipment, Navy Department.
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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.

Calendar

A calendar is a system of organizing days for social, religious, commercial or administrative purposes. This is done by giving names to periods of time, typically days, weeks, months and years. A date is the designation of a single, specific day within such a system. A calendar is also a physical record (often paper) of such a system. A calendar can also mean a list of planned events, such as a court calendar or a partly or fully chronological list of documents, such as a calendar of wills.

Periods in a calendar (such as years and months) are usually, though not necessarily, synchronised with the cycle of the sun or the moon. The most common type of pre-modern calendar was the lunisolar calendar, a lunar calendar that occasionally adds one intercalary month to remain synchronised with the solar year over the long term.

The term calendar is taken from calendae, the term for the first day of the month in the Roman calendar, related to the verb calare "to call out", referring to the "calling" of the new moon when it was first seen.

Latin calendarium meant "account book, register" (as accounts were settled and debts were collected on the calends of each month). The Latin term was adopted in Old French as calendier and from there in Middle English as calender by the 13th century (the spelling calendar is early modern). A calendar can be on paper or electronic device.

Calendar year

Generally speaking, a calendar year begins on the New Year's Day of the given calendar system and ends on the day before the following New Year's Day, and thus consists of a whole number of days. A year can also be measured by starting on any other named day of the calendar, and ending on the day before this named day in the following year. This may be termed a "year's time", but not a "calendar year". To reconcile the calendar year with the astronomical cycle (which has a fractional number of days) certain years contain extra days ("leap days" or "intercalary days").

The Gregorian year, which is in use in most of the world, begins on January 1 and ends on December 31. It has a length of 365 days in an ordinary year, with 8,760 hours, 525,600 minutes, or 31,536,000 seconds; but 366 days in a leap year, with 8,784 hours, 527,040 minutes, or 31,622,400 seconds. With 97 leap years every 400 years, the year has an average length of 365.2425 days. Other formula-based calendars can have lengths which are further out of step with the solar cycle: for example, the Julian calendar has an average length of 365.25 days, and the Hebrew calendar has an average length of 365.2468 days. The Islamic calendar is a lunar calendar consisting of 12 months in a year of 354 or 355 days.

The astronomer's mean tropical year, which is averaged over equinoxes and solstices, is currently 365.24219 days, slightly shorter than the average length of the year in most calendars, but the astronomer's value changes over time, so John Herschel's suggested correction to the Gregorian calendar may become unnecessary by the year 4000.

Common year

A common year is a calendar year with 365 days, as distinguished from a leap year, which has 366. More generally, a common year is one without intercalation. The Gregorian calendar, (like the earlier Julian calendar), employs both common years and leap years to keep the calendar aligned with the tropical year, which does not contain an exact number of days.

The common year of 365 days has 52 weeks and one day, hence a common year always begins and ends on the same day of the week (for example, January 1 and December 31 fell on a Sunday in 2017) and the year following a common year will start on the subsequent day of the week. In common years, February has four weeks, so March will begin on the same day of the week. November will also begin on this day.

In the Gregorian calendar, 303 of every 400 years are common years. By comparison, in the Julian calendar, 300 out of every 400 years are common years, and in the Revised Julian calendar (used by Greece) 682 out of every 900 years are common years.

Euctemon

Euctemon (Greek: Εὐκτήμων, gen. Εὐκτήμωνος; fl. 432 BC) was an Athenian astronomer. He was a contemporary of Meton and worked closely with this astronomer. Little is known of his work apart from his partnership with Meton and what is mentioned by Ptolemy. With Meton, he made a series of observations of the solstices (the points at which the sun is at greatest distance from the equator) in order to determine the length of the tropical year. Geminus and Ptolemy quote him as a source on the rising and setting of the stars. Pausanius's Description of Greece names Damon and Philogenes and Euctemon's children.The lunar crater Euctemon is named after him.

Gregorian calendar

The Gregorian calendar is the calendar used in most of the world. It is named after Pope Gregory XIII, who introduced it in October 1582. The calendar spaces leap years to make the average year 365.2425 days long, approximating the 365.2422-day tropical year that is determined by the Earth's revolution around the Sun. The rule for leap years is:

Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 are not leap years, but the year 2000 is.

The calendar was developed as a correction to the Julian calendar, shortening the average year by 0.0075 days to stop the drift of the calendar with respect to the equinoxes. To deal with the 10 days' difference (between calendar and reality) that this drift had already reached, the date was advanced so that 4 October 1582 was followed by 15 October 1582. There was no discontinuity in the cycle of weekdays or of the Anno Domini calendar era. The reform also altered the lunar cycle used by the Church to calculate the date for Easter (computus), restoring it to the time of the year as originally celebrated by the early Church.

The reform was adopted initially by the Catholic countries of Europe and their overseas possessions. Over the next three centuries, the Protestant and Eastern Orthodox countries also moved to what they called the Improved calendar, with Greece being the last European country to adopt the calendar in 1923. To unambiguously specify a date during the transition period, dual dating is sometimes used to specify both Old Style and New Style dates. Due to globalization in the 20th century, the calendar has also been adopted by most non-Western countries for civil purposes. The calendar era carries the alternative secular name of "Common Era".

Hipparchic cycle

The Greek astronomer Hipparchus introduced two cycles that have been named after him in later literature.

Intercalation (timekeeping)

Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.

Julian calendar

The Julian calendar, proposed by Julius Caesar in 46 BC (708 AUC), was a reform of the Roman calendar. It took effect on 1 January 45 BC (709 AUC), by edict. It was the predominant calendar in the Roman world, most of Europe, and in European settlements in the Americas and elsewhere, until it was refined and gradually replaced by the Gregorian calendar, promulgated in 1582 by Pope Gregory XIII.

The Julian calendar is still used in parts of the Eastern Orthodox Church, in parts of Oriental Orthodoxy and Anabaptism, as well as by the Berbers.

During the 20th and 21st centuries, the date according to the Julian calendar is 13 days behind the Gregorian date, and after the year 2100 will be one day more.

Julian year (astronomy)

In astronomy, a Julian year (symbol: a) is a unit of measurement of time defined as exactly 365.25 days of 86400 SI seconds each. The length of the Julian year is the average length of the year in the Julian calendar that was used in Western societies until some centuries ago, and from which the unit is named. Nevertheless, because astronomical Julian years are measuring duration rather than designating dates, this Julian year does not correspond to years in the Julian calendar or any other calendar. Nor does it correspond to the many other ways of defining a year.

Leap year

A leap year (also known as an intercalary year or bissextile year) is a calendar year containing one additional day (or, in the case of lunisolar calendars, a month) added to keep the calendar year synchronized with the astronomical or seasonal year. Because seasons and astronomical events do not repeat in a whole number of days, calendars that have the same number of days in each year drift over time with respect to the event that the year is supposed to track. By inserting (also called intercalating) an additional day or month into the year, the drift can be corrected. A year that is not a leap year is called a common year.

For example, in the Gregorian calendar, each leap year has 366 days instead of 365, by extending February to 29 days rather than the common 28. These extra days occur in years which are multiples of four (with the exception of centennial years not divisible by 400). Similarly, in the lunisolar Hebrew calendar, Adar Aleph, a 13th lunar month, is added seven times every 19 years to the twelve lunar months in its common years to keep its calendar year from drifting through the seasons. In the Bahá'í Calendar, a leap day is added when needed to ensure that the following year begins on the vernal equinox.

The name "leap year" probably comes from the fact that while a fixed date in the Gregorian calendar normally advances one day of the week from one year to the next, the day of the week in the 12 months following the leap day (from March 1 through February 28 of the following year) will advance two days due to the extra day (thus "leaping over" one of the days in the week). For example, Christmas Day (December 25) fell on a Sunday in 2016, Monday in 2017, and Tuesday in 2018, then will fall on Wednesday in 2019 but then "leaps" over Thursday to fall on a Friday in 2020.

The length of a day is also occasionally changed by the insertion of leap seconds into Coordinated Universal Time (UTC), owing to the variability of Earth's rotational period. Unlike leap days, leap seconds are not introduced on a regular schedule, since the variability in the length of the day is not entirely predictable.

Light-year

The light-year is a unit of length used to express astronomical distances and measures about 9.46 trillion kilometres (9.46 x 1012 km) or 5.88 trillion miles (5.88 x 1012 mi). As defined by the International Astronomical Union (IAU), a light-year is the distance that light travels in vacuum in one Julian year (365.25 days). Because it includes the word "year", the term light-year is sometimes misinterpreted as a unit of time.

The light-year is most often used when expressing distances to stars and other distances on a galactic scale, especially in nonspecialist and popular science publications. The unit most commonly used in professional astrometry is the parsec (symbol: pc, about 3.26 light-years; the distance at which one astronomical unit subtends an angle of one second of arc).

Lunisolar calendar

A lunisolar calendar is a calendar in many cultures whose date indicates both the Moon phase and the time of the solar year. If the solar year is defined as a tropical year, then a lunisolar calendar will give an indication of the season; if it is taken as a sidereal year, then the calendar will predict the constellation near which the full moon may occur. As with all calendars which divide the year into months there is an additional requirement that the year have a whole number of months. In this case ordinary years consist of twelve months but every second or third year is an embolismic year, which adds a thirteenth intercalary, embolismic, or leap month.

Minute

The minute is a unit of time or angle. As a unit of time, the minute is most of times equal to ​1⁄60 (the first sexagesimal fraction) of an hour, or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). As a unit of angle, the minute of arc is equal to ​1⁄60 of a degree, or 60 seconds (of arc). Although not an SI unit for either time or angle, the minute is accepted for use with SI units for both. The SI symbols for minute or minutes are min for time measurement, and the prime symbol after a number, e.g. 5′, for angle measurement. The prime is also sometimes used informally to denote minutes of time.

Runic calendar

A Runic calendar (also Rune staff or Runic Almanac) is a perpetual calendar, variants of which have been used in Northern Europe until the 19th century.

The calendar is based on the 19-year-long Metonic cycle, correlating the Sun and the Moon. Runic calendars were written on parchment or carved onto staves of wood, bone, or horn. The oldest one known, and the only one from the Middle Ages, is the Nyköping staff from Sweden, believed to date from the 13th century. Most of the several thousand which survive are wooden calendars dating from the 16th and the 17th centuries. During the 18th century, the Runic calendars had a renaissance, and around 1800, such calendars were made in the form of tobacco boxes in brass.

A typical Runic calendar consisted of several horizontal lines of symbols, one above the other.

Special days like solstices, equinoxes, and celebrations (including Christian holidays and feasts) were marked with additional lines of symbols.

The calendar does not prove knowledge of the length 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 first full moon also marked the date of Disting, a pagan feast and a fair day.

Second

The second is the base unit of time in the International System of Units (SI), commonly understood and historically defined as ​1⁄86400 of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds each. Analog clocks and watches often have sixty tick marks on their faces, representing seconds, and a "second hand" to mark the passage of time in seconds. Digital clocks and watches often have a two-digit seconds counter. The second is also part of several other units of measurement like meters per second for velocity, meters per second per second for acceleration, and per second for frequency.

Although the historical definition of the unit was based on this division of the Earth's rotation cycle, the formal definition in the International System of Units (SI) is a much steadier timekeeper: 1 second is defined to be exactly "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom" (at a temperature of 0 K).

Because the Earth's rotation varies and is also slowing ever so slightly, a leap second is periodically added to clock time to keep clocks in sync with Earth's rotation.

Multiples of seconds are usually counted in hours and minutes. Fractions of a second are usually counted in tenths or hundredths. In scientific work, small fractions of a second are counted in milliseconds (thousandths), microseconds (millionths), nanoseconds (billionths), and sometimes smaller units of a second.

An everyday experience with small fractions of a second is a 1-gigahertz microprocessor which has a cycle time of 1 nanosecond. Camera shutter speeds usually range from ​1⁄60 second to ​1⁄250 second.

Sexagesimal divisions of the day from a calendar based on astronomical observation have existed since the third millennium BC, though they were not seconds as we know them today. Small divisions of time could not be counted back then, so such divisions were figurative. The first timekeepers that could count seconds accurately were pendulum clocks invented in the 17th century. Starting in the 1950s, atomic clocks became better timekeepers than earth's rotation, and they continue to set the standard today.

Sidereal year

A sidereal year (from Latin sidus "asterism, star") 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 equals 365.256 363 004 SI days for the J2000.0 epoch.The sidereal year differs from the tropical year, "the period of time required for the ecliptic longitude of the sun to increase 360 degrees", due to the precession of the equinoxes.

The sidereal year is 20 min 24.5 s longer than the mean tropical year at J2000.0 (365.242 190 402 SI days).Before the discovery of the precession of the equinoxes by Hipparchus in the Hellenistic period, the difference between sidereal and tropical year was unknown. For naked-eye observation, the shift of the constellations relative to the equinoxes only becomes apparent over centuries or "ages", and pre-modern calendars such as Hesiod's Works and Days would give the times of the year for sowing, harvest, and so on by reference to the first visibility of stars, effectively using the sidereal year. The South and Southeast Asian solar New Year, based on Indic influences, is traditionally reckoned by the sun's entry into Aries and thus the sidereal year, but is also supposed to align with the spring equinox and have relevance to the harvesting and planting season and thus the tropical year. As these have diverged, in some countries and cultures the date has been fixed according to the tropical year while in others the astronomical calculation and sidereal year is still used.

Twenty-Eight Mansions

The Twenty-Eight Mansions (Chinese: 二十八宿; pinyin: Èrshíbā Xiù), hsiu, xiu or sieu are part of the Chinese constellations system. They can be considered as the equivalent to the zodiacal constellations in Western astronomy, though the Twenty-eight Mansions reflect the movement of the Moon through a sidereal month rather than the Sun in a tropical year.

The lunar mansion system was in use in other parts of East Asia, such as ancient Japan; the Bansenshukai, written by Fujibayashi Yasutake, mentions the system several times and includes an image of the twenty-eight mansions.Another similar system, called Nakshatra, is used in traditional Indian astronomy.

Year

A year is the orbital period of the Earth moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the hours of daylight, and, consequently, vegetation and soil fertility. The current year is 2019.

In temperate and subpolar regions around the planet, four seasons are generally recognized: spring, summer, autumn, and winter. In tropical and subtropical regions, several geographical sectors do not present defined seasons; but in the seasonal tropics, the annual wet and dry seasons are recognized and tracked.

A calendar year is an approximation of the number of days of the Earth's orbital period as counted in a given calendar. The Gregorian calendar, or modern calendar, presents its calendar year to be either a common year of 365 days or a leap year of 366 days, as do the Julian calendars; see below. For the Gregorian calendar, the average length of the calendar year (the mean year) across the complete leap cycle of 400 years is 365.2425 days. The ISO standard ISO 80000-3, Annex C, supports the symbol a (for Latin annus) to represent a year of either 365 or 366 days. In English, the abbreviations y and yr are commonly used.

In astronomy, the Julian year is a unit of time; it is defined as 365.25 days of exactly 86,400 seconds (SI base unit), totalling exactly 31,557,600 seconds in the Julian astronomical year.The word year is also used for periods loosely associated with, but not identical to, the calendar or astronomical year, such as the seasonal year, the fiscal year, the academic year, etc. Similarly, year can mean the orbital period of any planet; for example, a Martian year and a Venusian year are examples of the time a planet takes to transit one complete orbit. The term can also be used in reference to any long period or cycle, such as the Great Year.

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