TheInfoList

A tropical year (also known as a solar year or tropical period) is the
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

that the
Sun The Sun is the star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

sky The sky is the panorama obtained from observing the universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang th ...

of a
celestial body In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...
of the
solar system The Solar SystemCapitalization Capitalization ( North American English) or capitalisation ( British English) is writing a word with its first letter as a capital letter (uppercase letter) and the remaining letters in lower case, in writin ...

such as the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

, completing a full cycle of
season A season is a division of the year based on changes in weather Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloud cover, cloudy. On Earth, most ...

s; for example, the time from vernal equinox to vernal equinox, or from
summer solstice The summer solstice, also known as estival solstice or midsummer, occurs when one of Earth's geographical pole, poles has its maximum tilt toward the Sun. It happens twice yearly, once in each Hemispheres of Earth, hemisphere (Northern and Souther ...

to summer solstice. It is the time of a
year A year is the orbital period of a planetary body, for example, the Earth, moving in Earth's orbit, 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 ...
as a
unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
of tropical solar calendars and
calendar era #REDIRECT Calendar era #REDIRECT Calendar era A calendar era is the period of time elapsed since one ''epoch'' of a calendar and, if it exists, before the next one. For example, it is the year as per the Gregorian calendar, which numbers its year ...
s which use tropical solar calendars. The solar year is an astronomical year, like the
sidereal year A sidereal year (, ; from Latin "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 af ...
, which is the time it takes Earth to complete one full orbit around the Sun as measured with respect to the
fixed stars The fixed stars ( la, stellae fixae) compose the background of astronomical object In astronomy, an astronomical object or celestial object is a naturally occurring physical entity, association, or structure that exists in the observable ...
, resulting in a difference in length by 20
minute The minute is a unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation ...
s because of the
precession of the equinoxes In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's Rotation around a fixed axis, rotational axis. In particular, it can refer to the gradual shift in the orientation of E ...
. Since antiquity, astronomers have progressively refined the definition of the tropical year. The entry for "year, tropical" in the '' Astronomical Almanac Online Glossary'' states: 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". The mean tropical year in 2000 was 365.24219 ephemeris days; each ephemeris day lasting 86,400 SI seconds. This is 365.24217
mean solar day Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar times are apparent solar time (sundial tim ...
s. For this reason, the calendar year is an approximation of the solar year: the
Gregorian calendar The Gregorian calendar is the calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, speci ...
(with its rules for catch-up ) is designed so as to resynchronise the calendar year with the solar year at regular intervals.

# History

## Origin

The word "tropical" comes from the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
''tropikos'' meaning "turn". Thus, the tropics of
Cancer Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumor A benign tumor is a mass of cells Cell most often refers to: * Cell (biolo ...

and
Capricorn Capricorn (pl. ''capricorni'' or ''capricorns'') may refer to: Places * Capricorn and Bunker Group, islands of the southern Great Barrier Reef, Australia *Capricorn District Municipality, Limpopo province, South Africa Animals *Capricorn, an anim ...
mark the extreme north and south
latitude In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the ...

s 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 Hipparchus of Nicaea (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the ...
measured the time required for the Sun to travel from an
equinox An equinox is traditionally defined as the time when the plane In mathematics, a plane is a flatness (mathematics), flat, two-dimensional surface (mathematics), surface that extends infinitely far. A plane is the two-dimensional space, two-di ...
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. Hipparchus also discovered that the equinoctial points moved along the
ecliptic The ecliptic is the plane (geometry), plane of Earth's orbit around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the ...

(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.

## 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 The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

and
planets A planet is an astronomical body orbit In physics, an orbit is the gravitationally curved trajectory of an physical body, object, such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit r ...

relative to the fixed stars. An important application of these tables was the reform of the calendar. The
Alfonsine Tables 350px, Alfonsine tables The ''Alfonsine tables'' ( es, Tablas alfonsíes, la, tabulae alphonsinae), sometimes spelled ''Alphonsine tables'', provided data for computing the position of the Sun, Moon The Moon is Earth's only proper natur ...
, published in 1252, were based on the theories of
Ptolemy Claudius Ptolemy (; grc-koi, Κλαύδιος Πτολεμαῖος, , ; la, Claudius Ptolemaeus; AD) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes ...
and were revised and updated after the original publication. 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 The Gregorian calendar is the calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, speci ...
of 1582. In the 16th century
Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, link=no, Niclas Koppernigk, modern: ''Nikolaus Kopernikus''; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic Church, C ...

put forward a . Erasmus Reinhold used Copernicus' theory to compute the 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 A sidereal year (, ; from Latin "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 af ...
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 Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and ...

and
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

. In 1609 and 1619 Kepler published his three laws of planetary motion. In 1627, Kepler used the observations of
Tycho Brahe Tycho Brahe ( ; born Tyge Ottesen Brahe; 14 December 154624 October 1601) was a Danish astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. T ...

and Waltherus to produce the most accurate tables up to that time, the
Rudolphine Tables The ''Rudolphine Tables'' ( la, Tabulae Rudolphinae) consist of a star catalogue and planetary tables published by Johannes Kepler in 1627, using observational data collected by Tycho Brahe (1546–1601). The tables are named in memory of Rudolf ...

. He evaluated the mean tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds (365.24219 days). Newton's three laws of dynamics and theory of gravity were published in his ''
Philosophiæ Naturalis Principia Mathematica (from Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it bec ...
'' in 1687. Newton's theoretical and mathematical advances influenced tables by
Edmond Halley Edmond (or Edmund) Halley (; – ) was an English astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects su ...

published in 1693 and 1749 and provided the underpinnings of all solar system models until
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

's theory of
General relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
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 periodic variations (caused by the gravitational pull of the planets, and the small effect of
nutation Nutation (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman ...
on the equinox). 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 precise observations and 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 Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering Engineering is the use of scientific method, scientific principles ...

,
Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
, and other specialists in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motion (physics), motions of celestial object, objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical obje ...
. They were able to compute periodic variations and separate them from the gradual mean motion. They could express the mean longitude of the Sun in a polynomial such as: :''L''0 = ''A''0 + ''A''1''T'' + ''A''2''T''2 days where ''T'' is the time in Julian centuries. The derivative of this formula is an expression of the mean angular velocity, and the inverse of this gives an expression for the length of the tropical year as a linear function of ''T''. Two equations are given in the table. Both equations estimate that the tropical year gets roughly a half second shorter each century. Newcomb's tables were sufficiently accurate that they were used by the joint American-British ''
Astronomical Almanac ''The Astronomical Almanac''The ''Astronomical Almanac'' for the Year 2015, (United States Naval Observatory/Nautical Almanac Office, 2014) . is an almanac An almanac (also spelled ''almanack'' and ''almanach'') is an listing a set of current, ...
'' for the Sun,
Mercury Mercury usually refers to: * Mercury (planet) Mercury is the smallest planet in the Solar System and the closest to the Sun. Its orbit around the Sun takes 87.97 Earth days, the shortest of all the Sun's planets. It is named after the Roman g ...

,
Venus Venus is the second planet from the Sun. It is named after the Venus (mythology), Roman goddess of love and beauty. As List of brightest natural objects in the sky, the brightest natural object in Earth's night sky after the Moon, Venus can ...

, and
Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, being larger than only Mercury (planet), Mercury. In English, Mars carries the name of the Mars (mythology), Roman god of war and is often referred to ...

through 1983.

## 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 *artificial satellites *tracking of deep space probes such as
Pioneer 4 Pioneer 4 was an American spin-stabilized Spin-stabilisation is the method of stabilizing a satellite or launch vehicle by means of spin. For most satellite applications this approach has been superseded by three-axis stabilisation. It is also u ...
beginning in 1959 *
radars Radar (radio detection and ranging) is a detection system that uses radio waves to determine the distance (''ranging''), angle, or velocity of objects. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
able to measure the distance to other planets beginning in 1961 *
lunar laser ranging Lunar Laser Ranging (LLR) is the practice of measuring the distance between the surfaces of the Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's surface is land con ...
since the 1969
Apollo 11 Apollo 11 was the spaceflight Spaceflight (or space flight) is an application of astronautics to fly spacecraft into or through outer space, either human spaceflight, with or uncrewed spaceflight, without humans on board. Most spacefli ...

left the first of a series of
retroreflector A retroreflector (sometimes called a retroflector or cataphote) is a device or surface that radiation (usually light) back to its source with minimum . This works at a wide range of , unlike a planar , which does this only if the mirror is exact ...
s which allow greater accuracy than reflectorless measurements *artificial satellites such as
LAGEOS LAGEOS, Laser Geodynamics Satellite or Laser Geometric Environmental Observation Survey, are a series of two scientific research satellite In the context of spaceflight Spaceflight (or space flight) is an application of astronautics to ...
(1976) and the
Global Positioning System The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government The federal government of the United States (U.S. federal government or U.S. governme ...
(initial operation in 1993) *
Very Long Baseline Interferometry Very-long-baseline interferometry (VLBI) is a type of astronomical interferometer, astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescop ...
which finds precise directions to
quasar A quasar (; also known as a quasi-stellar object, abbreviated QSO) is an extremely luminous active galactic nucleus An active galactic nucleus (AGN) is a compact region at the center of a galaxy A galaxy is a gravitation Gravity () ...

s in distant
galaxies A galaxy is a gravitationally bound system of star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to ...

, and allows determination of the Earth's orientation with respect to these objects whose distance is so great they can be considered to show minimal space motion. 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 In analysis, numerical integration comprises a broad family of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm procee ...
rather than general theories; numerical integration came into use in 1984 for the joint US-UK almanacs.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

's
General Theory of Relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
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. 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 Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar times are apparent solar time (sundial tim ...
, is not constant. William Ferrel in 1864 and
Charles-Eugène Delaunay Charles-Eugène Delaunay (9 April 1816 – 5 August 1872) was a French astronomer and mathematician. His lunar theory, lunar motion studies were important in advancing both the theory of planetary motion and mathematics. Life Born in Lusigny- ...
in 1865 predicted that the rotation of the Earth is being retarded by tides. This could be verified by observation only in the 1920s with the very accurate Shortt-Synchronome clock and later in the 1930s when
quartz clock Quartz clocks and quartz watches are timepieces that use an electronic oscillator An electronic oscillator is an electronic circuit 200px, A circuit built on a printed circuit board (PCB). An electronic circuit is composed of individual el ...

s began to replace pendulum clocks as time standards.

# Time scales and calendar

Apparent solar time planet like the Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's surface is land consisting of continent A continent is one of several large landmasses. ...
is the time indicated by a
sundial A sundial is a horological device that tells the time of day (in modern usage referred to as civil time In modern usage, civil time refers to statutory time scales designated by civilian authorities, or to local time indicated by clocks. M ...

, and is determined by the apparent motion of the Sun caused by the rotation of the Earth around its axis as well as the revolution of the Earth around the Sun.
Mean solar time planet like the Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's surface is land consisting of continent A continent is one of several large landmasses. ...
is corrected for the periodic variations in the apparent velocity of the Sun as the Earth revolves in its orbit. The most important such time scale is
Universal Time#REDIRECT Universal Time Universal Time (UT) is a time standard based on Earth's rotation. There are several versions of Universal Time, which differ by up to a few seconds. The most commonly used are Coordinated Universal Time (UTC) and UT1 (see ...

, which is the mean solar time at 0 degrees
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

(the
Greenwich meridian The prime meridian is a geographical reference line that passes through the Royal Observatory, Greenwich Greenwich ( , , , or ) is a town in South London, south-east London, England, centred east-southeast of Charing Cross and located ...
).
Civil time In modern usage, civil time refers to statutory time scales designated by civilian authorities, or to local time indicated by clocks. Modern civil time is generally standard time in a time zone at a UTC offset, fixed offset from Coordinated Univers ...
is based on UT (actually UTC), and civil calendars count mean solar days. However the rotation of the Earth itself is irregular and is slowing down, with respect to more stable time indicators: specifically, the motion of planets, and atomic clocks.
Ephemeris time The term ephemeris time (often abbreviated ET) can in principle refer to time in association with any ephemeris In and , an ephemeris (plural: ephemerides) is a book with tables that gives the of naturally occurring as well as in the , i.e. ...
(ET) is the independent variable in the equations of motion of the solar system, in particular, the equations from Newcomb's work, and this ET was in use from 1960 to 1984. These ephemerides were based on observations made in solar time over a period of several centuries, and as a consequence represent the mean solar second over that period. The
second The second (symbol: s, also abbreviated: sec) is the base unit of time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, th ...
, defined in atomic time, 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. ET as counted by atomic clocks was given a new name,
Terrestrial Time Terrestrial Time (TT) is a modern astronomical time standard defined by the International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) exists to promote and safeguard t ...
(TT), and for most purposes ET = TT =
International Atomic Time International Atomic Time (TAI, from the French name ) is a high-precision atomic coordinate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with a ...
+ 32.184 SI seconds. Since the era of the observations, the rotation of the Earth has slowed down and the mean solar second has grown somewhat longer than the SI second. As a result, the time scales of TT and UT1 build up a growing difference: the amount that TT is ahead of UT1 is known as Δ''T'', or Delta ''T''. TT is ahead of UT1 by 69.184 seconds. As a consequence, the tropical year following the seasons on Earth as counted in solar days of UT is increasingly out of sync with expressions for equinoxes in ephemerides in TT. As explained below, long-term estimates of the length of the tropical year were used in connection with the reform of the
Julian calendar The Julian calendar, proposed by Julius Caesar Gaius Julius Caesar (; 12 July 100 BC – 15 March 44 BC) was a Roman Roman or Romans usually refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century B ...
, 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 table below gives Morrison and Stephenson's estimates and
standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpret ...
s (''σ'') for ΔT at dates significant in the process of developing the Gregorian calendar. The low-precision extrapolations are computed with an expression provided by Morrison and Stephenson: :Δ''T'' in seconds = −20 + 32''t''2 where ''t'' is measured in Julian centuries from 1820. The extrapolation is provided only to show Δ''T'' is not negligible when evaluating the calendar for long periods; Borkowski cautions that "many researchers have attempted to fit a parabola to the measured Δ''T'' 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

One 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.

## Mean time interval between equinoxes

Before considering an example, the
equinox An equinox is traditionally defined as the time when the plane In mathematics, a plane is a flatness (mathematics), flat, two-dimensional surface (mathematics), surface that extends infinitely far. A plane is the two-dimensional space, two-di ...
must be examined. There are two important planes in solar system calculations: the plane of the
ecliptic The ecliptic is the plane (geometry), plane of Earth's orbit around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the ...

(the Earth's orbit around the Sun), and the plane of the
celestial equator The celestial equator is the great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensiona ...
(the Earth's equator projected into space). These two planes intersect in a line. One ''direction'' points to the so-called vernal, northward, or March equinox which is given the symbol (the symbol looks like the horns of a
ram Random-access memory (RAM; ) is a form of computer memory In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic proces ...

because it used to be toward the constellation
Aries Aries may refer to: *Aries (astrology) *Aries (constellation) Vehicles *Dodge Aries, an automobile *Ariès, a French automobile *Aries Motorsport, a British kit car manufacturer *EP-3E Aries, an reconnaissance aircraft Other uses *Aries (rocket) * ...
). The opposite ''direction'' is given the symbol (because it used to be toward
Libra The Libra Party or ''Equal List'' ( sq, Lista e Barabartë) is a list of political parties in Albania, political party in Albania. It was founded in 2016. History The party was founded by Ben Blushi, a prominent politician/writer, and Mimoza H ...
). Because of the
precession of the equinoxes In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's Rotation around a fixed axis, rotational axis. In particular, it can refer to the gradual shift in the orientation of E ...
and
nutation Nutation (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman ...
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 International Celestial Reference System (ICRS) is the current standard celestial reference system adopted by the International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internati ...
). The
ecliptic longitude The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent place, apparent positions and orbits of Solar System objects. Because most planets (except Mercury (planet), Mercury) and many Small Solar S ...
of the Sun is the angle between and the Sun, measured eastward along the ecliptic. This creates a relative and not an absolute 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. 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 an interval - and a duration of the tropical year - of 365 days 5 hours 48 minutes 34.5 seconds. 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). This is why the tropical year is 20 min. shorter than the sidereal year. When tropical year measurements from several successive years are compared, variations are found which are due to the perturbations by the Moon and planets acting on the Earth, and to nutation. Meeus and Savoie provided the following examples of intervals between March (northward) equinoxes: 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.

## Different tropical year definitions

If a different starting longitude for the Sun is chosen than 0° (''i.e.'' ), then the duration for the Sun to return to the same longitude will be different. This is a second-order effect of the circumstance that the speed of the Earth (and conversely the apparent speed of the Sun) varies in its elliptical orbit: faster in the
perihelion upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

, slower in the
aphelion upright=1.15, The two-body system of interacting primary body A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravity, gravitationally bound, multi-object system. This object consti ...

. The equinox moves with respect to the perihelion (and both move with respect to the fixed sidereal frame). From one equinox passage to the next, or from one solstice passage to the next, the Sun completes not quite a full elliptic orbit. The time saved depends on where it starts in the orbit. If the starting point is close to the perihelion (such as the December solstice), then the speed is higher than average, and the apparent Sun saves little time for not having to cover a full circle: the "tropical year" is comparatively long. If the starting point is near aphelion, then the speed is lower and the time saved for not having to run the same small arc that the equinox has precessed is longer: that tropical year is comparatively short. The "mean tropical year" is based on the mean sun, and is not exactly equal to any of the times taken to go from an equinox to the next or from a solstice to the next. The following values of time intervals between equinoxes and solstices were provided by Meeus and Savoie 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 In orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculate ...
). They do not take into account 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.

## Mean tropical year current value

The mean tropical year on January 1, 2000, was 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 : $365.242 189 669 8-6.153 59\times 10^T-7.29\times 10^T^2 + 2.64\times 10^T^3$ where T is in Julian centuries of 36,525 days of 86,400 SI seconds measured from noon January 1, 2000 TT. 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. 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°. The above formulae give the length of the tropical year in ephemeris days (equal to 86,400 SI seconds), not
solar day A synodic day (or synodic rotation period or solar day) is the rotation period, period for a celestial object to rotate once in relation to the star it is orbiting, and is the basis of solar time. The synodic day is distinguished from the sidereal ...
s. 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 The Gregorian calendar is the calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, speci ...
, 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. 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 = = 365.2425 days per year, a close approximation to the mean tropical year of 365.2422 days. 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 The First Council of Nicaea (; grc, Νίκαια ) was a council of Christian bishops convened in the Bithynia Bithynia (; Koine Greek Koine Greek (, , Greek approximately ;. , , , lit. "Common Greek"), also known as Alexandrian dialec ...
in 325, to about March 11. According to North, 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. 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 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).365242×1.5/8640000. 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.

*
Anomalistic year A year is the orbital period The orbital period is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy Astronomy (from el, ἀστρονομία, literally meaning the ...
*
Gregorian calendar The Gregorian calendar is the calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, speci ...
*
Sidereal and tropical astrology ''Sidereal'' and ''tropical The tropics are the region of Earth surrounding the Equator. They are delimited in latitude by the Tropic of Cancer in the Northern Hemisphere at N and the Tropic of Capricorn in the Southern Hemisphere at ...

# References

* * * * * * Note: In the article at this URL page 68 should be put before page 66. * * * Contains updates to Meeus & Savoie 1992. * * *