# Tropical year

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Period of time for the ecliptic longitude of the Sun to increase 360°

A **tropical year**, or **solar year** (or **tropical period**), is the time that the Sun takes to return to the same [position](/source/Position_of_the_Sun) in the sky – as viewed from the Earth or another [celestial body](/source/Celestial_body) of the [Solar System](/source/Solar_System) – thus completing a full cycle of [astronomical seasons](/source/Season). For example, it is the time from [vernal equinox](/source/March_equinox) to the next vernal equinox, or from [summer solstice](/source/Summer_solstice) to the next summer solstice. It is the type of year used by [tropical solar calendars](/source/Solar_calendar#Tropical_solar_calendars).

The tropical year is one type of [astronomical year](/source/Astronomical_year) and a particular [orbital period](/source/Orbital_period). Another type is the [sidereal year](/source/Sidereal_year) (or sidereal orbital period), which is the time it takes Earth to complete one full orbit around the Sun as measured with respect to the [fixed stars](/source/Fixed_stars), resulting in a duration of 20 minutes and 24.7 seconds longer than the tropical year due to the [precession of the equinoxes](/source/Axial_precession_(astronomy)).

Since antiquity, astronomers have progressively refined the definition of the tropical year. The entry for "year, tropical" in the *[Astronomical Almanac Online Glossary](/source/Astronomical_Almanac)* states:[1]

the period of time for the [ecliptic longitude](/source/Ecliptic_longitude) of the Sun to increase 360 [degrees](/source/Degree_(angle)). 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](/source/Mean_Sun) crosses the vernal equinox and a new tropical year begins".[2]

The mean tropical year in 2000 was 365.24219 [ephemeris days](/source/Ephemeris_day), each ephemeris day lasting 86,400 SI seconds.[3] This is 365.24217 [mean solar days](/source/Mean_solar_day).[4] For this reason, the calendar year is an approximation of the solar year: the [Gregorian calendar](/source/Gregorian_calendar) (with its rules for catch-up [leap days](/source/Leap_year)) is designed to resynchronize the calendar year with the solar year at regular intervals.

## History

### Origin

The word "tropical" comes from the [Greek](/source/Greek_language) *tropikos* meaning "turn".[5] Thus, the tropics of [Cancer](/source/Tropic_of_Cancer) and [Capricorn](/source/Tropic_of_Capricorn) mark the extreme north and south [latitudes](/source/Latitude) 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" was lent to the period of the seasonal cycle. The early Chinese, Hindus, Greeks, and others made approximate measures of the tropical year.

### Early value, precession discovery

In the 2nd century BC, [Hipparchus](/source/Hipparchus) measured the time required for the Sun to travel from an [equinox](/source/Equinox_(celestial_coordinates)) 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.[6]

Hipparchus also discovered that the equinoctial points moved along the [ecliptic](/source/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](/source/Astronomy_in_medieval_Islam). Since this discovery, a distinction has been made between the tropical year and the [sidereal year](/source/Sidereal_year).[6]

### Middle Ages and the Renaissance

During the Middle Ages and Renaissance, several progressively better tables were published that allowed computation of the positions of the Sun, [Moon](/source/Moon) and [planets](/source/Planets) relative to the fixed stars. An important application of these tables was the [reform of the calendar](/source/Calendar_reform).

The [Alfonsine Tables](/source/Alfonsine_Tables), published in 1252, were based on the theories of [Ptolemy](/source/Ptolemy) 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](/source/Gregorian_calendar) of 1582.[7]

In [Uzbekistan](/source/Ulugh_Beg_Observatory), [Ulugh Beg](/source/Ulugh_Beg)'s [Zij-i Sultani](/source/Zij-i_Sultani) was published in 1437 and gave an estimate of 365 solar days, 5 hours, 49 minutes, 15 seconds (365.242535 days).[8]

In the 16th century, [Copernicus](/source/Copernicus) put forward a [heliocentric cosmology](/source/Copernican_heliocentrism). Erasmus Reinhold used Copernicus' theory to compute the [Prutenic Tables](/source/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](/source/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](/source/Johannes_Kepler) and [Isaac Newton](/source/Isaac_Newton). In 1609 and 1619, Kepler published his [three laws of planetary motion](/source/Kepler's_laws_of_planetary_motion).[9] In 1627, Kepler used the observations of [Tycho Brahe](/source/Tycho_Brahe) and Waltherus to produce the most accurate tables up to that time, the [Rudolphine Tables](/source/Rudolphine_Tables). He evaluated the mean tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds (365.24219 days).[7]

Newton's three laws of dynamics and theory of gravity were published in his *[Philosophiæ Naturalis Principia Mathematica](/source/Philosophi%C3%A6_Naturalis_Principia_Mathematica)* in 1687. Newton's theoretical and mathematical advances influenced tables by [Edmond Halley](/source/Edmond_Halley) published in 1693 and 1749[10] and provided the underpinnings of all solar system models until [Albert Einstein](/source/Albert_Einstein)'s theory of [general relativity](/source/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) some 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](/source/Nutation) 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](/source/Pierre-Simon_de_Laplace), [Joseph Louis Lagrange](/source/Joseph_Louis_Lagrange), and other specialists in [celestial mechanics](/source/Celestial_mechanics). They were able to compute periodic variations and separate them from the gradual mean motion. They could express the [mean longitude of the Sun](/source/Mean_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.

Tropical year coefficients Name Equation Date on which T = 0 Leverrier[11] Y = 365.24219647 − 6.24×10−6 T January 0.5, 1900, Ephemeris time Newcomb (1898) Y = 365.24219879 − 6.14×10−6 T January 0, 1900, mean time

Newcomb's tables were sufficiently accurate that they were used by the joint American-British *[Astronomical Almanac](/source/Astronomical_Almanac)* for the Sun, [Mercury](/source/Mercury_(planet)), [Venus](/source/Venus), and [Mars](/source/Mars) through 1983.[12]

### 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](/source/Pioneer_4) beginning in 1959[13]

- [radars](/source/Radar_astronomy) able to measure the distance to other planets beginning in 1961[14]

- [lunar laser ranging](/source/Lunar_laser_ranging) since the 1969 [Apollo 11](/source/Apollo_11) left the first of a series of [retroreflectors](/source/Retroreflector) which allow greater accuracy than reflectorless measurements

- artificial satellites such as [LAGEOS](/source/LAGEOS) (1976) and the [Global Positioning System](/source/Global_Positioning_System) (initial operation in 1993)

- [very long baseline interferometry](/source/Very_long_baseline_interferometry) which finds precise directions to [quasars](/source/Quasar) in distant [galaxies](/source/Galaxies), 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.[15]

The complexity of the model used for the Solar System must be limited to the available computational 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](/source/IBM_Selective_Sequence_Electronic_Calculator) was used since 1948. When modern computers became available, it was possible to compute ephemerides using [numerical integration](/source/Numerical_integration) rather than general theories; numerical integration came into use in 1984 for the joint US-UK almanacs.[16]

[Albert Einstein](/source/Albert_Einstein)'s [General Theory of Relativity](/source/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.[17]

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](/source/Solar_day), is not constant. William Ferrel in 1864 and [Charles-Eugène Delaunay](/source/Charles-Eug%C3%A8ne_Delaunay) 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](/source/Shortt%E2%80%93Synchronome_clock) and later in the 1930s when [quartz clocks](/source/Quartz_clock) began to replace pendulum clocks as time standards.[18]

## Time scales and calendar

[Apparent solar time](/source/Solar_time#Apparent_solar_time) is the time indicated by a [sundial](/source/Sundial), 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](/source/Solar_time#Mean_solar_time) 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](/source/Universal_Time), which is the mean solar time at 0° [longitude](/source/Longitude) (the [IERS Reference Meridian](/source/IERS_Reference_Meridian)). [Civil time](/source/Civil_time) is based on UT (actually [UTC](/source/Coordinated_Universal_Time)), 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](/source/Ephemeris_time) (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.[19] 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 [SI](/source/SI) [second](/source/Second#"Atomic"_second), 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.[20] ET as counted by atomic clocks was given a new name, [Terrestrial Time](/source/Terrestrial_Time) (TT), and for most purposes ET = TT = [TAI](/source/International_Atomic_Time) + 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*](/source/%CE%94T_(timekeeping)), or Delta *T*.[21] As of 5 July 2022,[\[update\]](https://en.wikipedia.org/w/index.php?title=Tropical_year&action=edit) TT is ahead of UT1 by 69.28 seconds.[22][23][24]

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](/source/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 table below gives Morrison and Stephenson's estimates and [standard errors](/source/Standard_error) (*σ*) for ΔT at dates significant in the process of developing the Gregorian calendar.[25]

Event Year Nearest S & M Year ΔT σ Julian calendar begins −44[26] 0 2h 56m 20s 4m 20s First Council of Nicaea 325 300 2h 8m 2m Gregorian calendar begins 1582 1600 2m 20s Low-precision extrapolation 4000 4h 13m 10,000 2d 11h

The low-precision extrapolations are computed with an expression provided by Morrison and Stephenson:[25]

- Δ*T* in seconds = −20 + 32*t*2

where *t* is measured in Julian centuries from 1820. The extrapolation is provided only to show that Δ*T* is not negligible when evaluating the calendar for long periods;[27] Borkowski cautions that "many researchers have attempted to fit a parabola to the measured Δ*T* values to determine the magnitude of the deceleration of the Earth's rotation. The results, when taken together, are rather discouraging."[27]

## Length of tropical year

**This section contains uncommon [Unicode](/source/Unicode) characters.** Without proper [rendering support](https://en.wikipedia.org/wiki/Help:Multilingual_support), you may see [question marks, boxes, or other symbols](/source/Specials_(Unicode_block)#Replacement_character) instead of the intended characters.

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

♈︎ ♎︎o Equinox symbols In Unicode U+2648 ♈ ARIES U+264E ♎ LIBRA

Before considering an example, the [equinox](/source/Equinox_(celestial_coordinates)) must be examined. There are two important planes in solar system calculations: the plane of the [ecliptic](/source/Ecliptic) (the Earth's orbit around the Sun), and the plane of the [celestial equator](/source/Celestial_equator) (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](/source/March_equinox) which is given the symbol ♈︎ (the symbol looks like the horns of a [ram](/source/Bighorn_sheep) because it used to be toward the constellation [Aries](/source/Aries_(astrology))). The opposite *direction* is given the symbol ♎︎ (because it used to be toward [Libra](/source/Libra_(astrology))). Because of the [precession of the equinoxes](/source/Axial_precession) and [nutation](/source/Astronomical_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](/source/International_Celestial_Reference_Frame)).

The [ecliptic longitude](/source/Ecliptic_coordinate_system) 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 from which the angle is measured 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.[28] 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[29]). This is why the tropical year is 20 minutes shorter than the sidereal year.

When tropical year measurements from several successive years are compared, variations are found which are due to the [perturbations](/source/Perturbation_(astronomy)) 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:[7]

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.[11]

### 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](/source/Perihelion), slower in the [aphelion](/source/Aphelion). 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](/source/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](/source/Astronomical_year_numbering#Usage_of_the_year_zero) and 2000.[11] These are smoothed values which take account of the Earth's orbit being elliptical, using well-known procedures (including solving [Kepler's equation](/source/Kepler's_equation)). 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.[30]

Year 0 Year 2000 Between two March equinoxes 365.242137 days 365.242374 days Between two June solstices 365.241726 365.241626 Between two September equinoxes 365.242496 365.242018 Between two December solstices 365.242883 365.242740 Mean tropical year (Laskar's expression) 365.242310 365.242189

### Mean tropical year current value

The mean tropical year on January 1, 2000, was 365.2421897 or 365 [ephemeris days](/source/Ephemeris_day), 5 hours, 48 minutes, 45.19 seconds. This changes slowly; an expression suitable for calculating the length of a tropical year in ephemeris days, is

- 365.2421896698 − 6.15359 × 10 − 6 T − 7.29 × 10 − 10 T 2 + 2.64 × 10 − 10 T 3 {\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.[31]

Modern astronomers define the tropical year as the time for the [Sun's mean longitude](/source/Mean_sun) 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.[32] When viewed over one year, 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°.[11][33]

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

## Calendar year

The [Gregorian calendar](/source/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.[34] 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+97⁄400 = 365.2425 days per year, a close approximation to the mean tropical year of 365.2422 days.[35]

The Gregorian calendar is a reformed version of the Julian calendar organized by the Catholic Church and enacted in 1582. By the time of the reform, the date of the vernal equinox had shifted about 10 days, from about March 21 at the time of the [First Council of Nicaea](/source/First_Council_of_Nicaea) in 325, to about March 11. The motivation for the change was the correct observance of Easter. The rules used to [compute the date of Easter](/source/Computus) used a conventional date for the vernal equinox (March 21), and it was considered important to keep March 21 close to the actual equinox.[36]

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 between three hours and four days 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 seconds per century, the mean solar day is getting longer at a rate of about 1.5 ms per century, and length of the "tropical millennium" is decreasing by about 0.06 solar days per millennium (neglecting the oscillatory changes in the real length of the tropical year).[37] These effects will cause the calendar to be as much as one day behind the Sun in 3200. As a result, many have suggested that the number of leap days should decrease as time goes on. One possible reform that has been proposed involves omitting the leap day in 3200, keeping 3600 and 4000 as leap years, and making all centennial years common except 4500, 5000, 5500, 6000, etc. (i.e. making centennial leap years occur once every 500 years instead of 400 starting from the year 4000), but the quantity [ΔT](/source/%CE%94T_(timekeeping)) is not sufficiently predictable to form more precise proposals.[38]

## See also

- [Anomalistic year](/source/Year#Sidereal,_tropical,_and_anomalistic_years)

- [Gregorian calendar](/source/Gregorian_calendar)

- [Sidereal and tropical astrology](/source/Sidereal_and_tropical_astrology)

## Notes

1. **[^](#cite_ref-AAOG2020_1-0)** ["Astronomical almanac online glossary"](https://asa.hmnao.com/SecM/Glossary.html#_Y). US Naval Observatory. 2020.

1. **[^](#cite_ref-FOOTNOTEBorkowski1991122_2-0)** [Borkowski 1991](#CITEREFBorkowski1991), p. 122.

1. **[^](#cite_ref-SIsecond_3-0)** "13th CGPM (1967/68, Resolution 1; CR, 103 and *Metrologia*, 1968, 4, 43)". [The International System of Units](https://web.archive.org/web/20081216221824/http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf) (PDF) (Report). [Bureau International des Poids et Mesures](/source/Bureau_International_des_Poids_et_Mesures). 2006. p. 113. Archived from [the original](http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf) (PDF) on December 16, 2008. 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. Via ["The SI brochure"](https://web.archive.org/web/20091001065750/http://www.bipm.org/en/si/si_brochure/general.html). *BIMP*. Archived from [the original](http://www.bipm.org/en/si/si_brochure/general.html) on October 1, 2009.

1. **[^](#cite_ref-Richards2013_587_4-0)** Richards, E.G. "Calendars". In [Urban & Seidelmann (2013)](#CITEREFUrbanSeidelmann2013), p. 587.

1. **[^](#cite_ref-tropicAHD1992_5-0)** "tropic". *American Heritage Dictionary* (3rd ed.). Boston: Houghton-Mifflin. 1992.

1. ^ [***a***](#cite_ref-FOOTNOTEMeeusSavoie199240_6-0) [***b***](#cite_ref-FOOTNOTEMeeusSavoie199240_6-1) [Meeus & Savoie 1992](#CITEREFMeeusSavoie1992), p. 40.

1. ^ [***a***](#cite_ref-FOOTNOTEMeeusSavoie199241_7-0) [***b***](#cite_ref-FOOTNOTEMeeusSavoie199241_7-1) [***c***](#cite_ref-FOOTNOTEMeeusSavoie199241_7-2) [Meeus & Savoie 1992](#CITEREFMeeusSavoie1992), p. 41.

1. **[^](#cite_ref-FOOTNOTEO'ConnorRobertson1999_8-0)** [O'Connor & Robertson 1999](#CITEREFO'ConnorRobertson1999).

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann200926_9-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), p. 26.

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann200926–28_10-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), pp. 26–28.

1. ^ [***a***](#cite_ref-FOOTNOTEMeeusSavoie199242_11-0) [***b***](#cite_ref-FOOTNOTEMeeusSavoie199242_11-1) [***c***](#cite_ref-FOOTNOTEMeeusSavoie199242_11-2) [***d***](#cite_ref-FOOTNOTEMeeusSavoie199242_11-3) [Meeus & Savoie 1992](#CITEREFMeeusSavoie1992), p. 42.

1. **[^](#cite_ref-FOOTNOTESeidelmann1992317_12-0)** [Seidelmann 1992](#CITEREFSeidelmann1992), p. 317.

1. **[^](#cite_ref-JPL2005_13-0)** [Jet Propulsion Laboratory](/source/Jet_Propulsion_Laboratory) (2005). [*DSN: History*](https://deepspace.jpl.nasa.gov/about/history/). NASA.

1. **[^](#cite_ref-FOOTNOTEButrica1996[[Category:Wikipedia_articles_needing_page_number_citations_from_February_2022]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">&#91;<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears.&#32;(February_2022)">page&nbsp;needed</span>]]</i>&#93;</sup>_14-0)** [Butrica 1996](#CITEREFButrica1996), p. [*[page needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources)*].

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann2009265_15-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), p. 265.

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann200932_16-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), p. 32.

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann200937_17-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), p. 37.

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann2009ch._9_18-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), ch. 9.

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann2009378_19-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), p. 378.

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann200981–82,_191–197_20-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), pp. 81–82, 191–197.

1. **[^](#cite_ref-FOOTNOTEMcCarthySeidelmann200986–67_21-0)** [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), pp. 86–67.

1. **[^](#cite_ref-22)** [International Earth Rotation Service](/source/International_Earth_Rotation_and_Reference_Systems_Service) (July 1, 2022). ["Bulletin B 413"](https://hpiers.obspm.fr/iers/bul/bulb_new/bulletinb.dat). *IERS Bulletin B*.

1. **[^](#cite_ref-23)** ["Bulletin C"](https://hpiers.obspm.fr/iers/bul/bulc/bulletinc.dat). *Earth Orientation Center*. July 5, 2022.

1. **[^](#cite_ref-24)** ["Common Units and Conversions in Earth Orientation"](https://maia.usno.navy.mil/information/eo-values). *United States Naval Observatory*.

1. ^ [***a***](#cite_ref-FOOTNOTEMorrisonStephenson2004_25-0) [***b***](#cite_ref-FOOTNOTEMorrisonStephenson2004_25-1) [Morrison & Stephenson 2004](#CITEREFMorrisonStephenson2004).

1. **[^](#cite_ref-FOOTNOTEUrbanSeidelmann2013595_26-0)** [Urban & Seidelmann 2013](#CITEREFUrbanSeidelmann2013), p. 595.

1. ^ [***a***](#cite_ref-FOOTNOTEBorkowski1991126_27-0) [***b***](#cite_ref-FOOTNOTEBorkowski1991126_27-1) [Borkowski 1991](#CITEREFBorkowski1991), p. 126.

1. **[^](#cite_ref-AADUNNO2009_28-0)** Astronomical Applications Dept. of [United States Naval Observatory](/source/United_States_Naval_Observatory) (2009). *Multiyear interactive computer almanac*. 2.2. Richmond VA: Willman-Bell.

1. **[^](#cite_ref-FOOTNOTESeidelmann1992p._104,_expression_for_p<sub>A</sub>_29-0)** [Seidelmann 1992](#CITEREFSeidelmann1992), p. 104, expression for pA.

1. **[^](#cite_ref-FOOTNOTEMeeusSavoie1992362_30-0)** [Meeus & Savoie 1992](#CITEREFMeeusSavoie1992), p. 362.

1. **[^](#cite_ref-31)** In negative numbers for dates in the past; [McCarthy & Seidelmann 2009](#CITEREFMcCarthySeidelmann2009), p. 18, calculated from planetary model of [Laskar 1986](#CITEREFLaskar1986).

1. **[^](#cite_ref-FOOTNOTELaskar198664_32-0)** [Laskar 1986](#CITEREFLaskar1986), p. 64.

1. **[^](#cite_ref-33)** *Astronomical almanac for the year 2011*. Washington: Astronomical Almanac Office US Naval Observatory. 2010. p. L8.

1. **[^](#cite_ref-Dobrzycki1983_123_34-0)** Dobrzycki, J. "Astronomical aspects of the calendar reform". In [Coyne, Hoskin & Pedersen (1983)](#CITEREFCoyneHoskinPedersen1983), p. 123.

1. **[^](#cite_ref-FOOTNOTESeidelmann1992576–581_35-0)** [Seidelmann 1992](#CITEREFSeidelmann1992), pp. 576–581.

1. **[^](#cite_ref-North1983_756_36-0)** North, J.D. "The Western calendar – 'Intolerabilis, horribilis, et derisibilis'; four centuries of discontent". In [Coyne, Hoskin & Pedersen (1983)](#CITEREFCoyneHoskinPedersen1983), pp. 75–76.

1. **[^](#cite_ref-37)** 365242×1.5/8640000.

1. **[^](#cite_ref-BlackburnHS2003_692_38-0)** Blackburn, B.; Holford-Strevens, L. (2003). *The Oxford companion to the year*. Corrected reprint of 1999. Oxford University Press. p. 692.

## References

- Borkowski, K.M. (1991). "The tropical year and the solar calendar". *Journal of the Royal Astronomical Society of Canada*. **85** (3): 121–130. [Bibcode](/source/Bibcode_(identifier)):[1991JRASC..85..121B](https://ui.adsabs.harvard.edu/abs/1991JRASC..85..121B).

- Butrica, A.J. (1996). [*SP-4218: To See the Unseen*](https://web.archive.org/web/20080310133443/http://history.nasa.gov/SP-4218/contents.htm). The NASA History Series. NASA History Office. Archived from [the original](https://history.nasa.gov/SP-4218/contents.htm) on March 10, 2008. Via ["To See the Unseen – A History of Planetary Radar Astronomy"](https://web.archive.org/web/20070823124845/https://history.nasa.gov/SP-4218/sp4218.htm). *NASA History Division*. Archived from [the original](https://history.nasa.gov/SP-4218/sp4218.htm) on August 23, 2007.

- [Coyne, G.V.](/source/George_Coyne); Hoskin, M.A.; Pedersen, O., eds. (1983). [*Gregorian reform of the calendar*](https://www.pas.va/en/publications/extra-series/es3pas.html). Vatican Observatory.

- Laskar, J. (1986). "Secular terms of classical planetary theories using the results of general theory". *Astronomy and Astrophysics*. **157** (1): 59–70. [Bibcode](/source/Bibcode_(identifier)):[1986A&A...157...59L](https://ui.adsabs.harvard.edu/abs/1986A&A...157...59L). [ISSN](/source/ISSN_(identifier)) [0004-6361](https://search.worldcat.org/issn/0004-6361). Note: In the article at this URL, page 68 should be put before page 66.

- [McCarthy, D.D.](/source/Dennis_McCarthy_(scientist)); Seidelmann, P.K. (2009). *Time from Earth rotation to atomic physics*. Weinhein: Wiley-VCH Verlag GmbH & Co. KGaA.

- [Meeus, J.](/source/Jean_Meeus); Savoie, D. (1992). "The history of the tropical year". *Journal of the British Astronomical Association*. **102** (1): 40–42. [Bibcode](/source/Bibcode_(identifier)):[1992JBAA..102...40M](https://ui.adsabs.harvard.edu/abs/1992JBAA..102...40M).

- 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** (3): 327–336. [Bibcode](/source/Bibcode_(identifier)):[2004JHA....35..327M](https://ui.adsabs.harvard.edu/abs/2004JHA....35..327M). [doi](/source/Doi_(identifier)):[10.1177/002182860403500305](https://doi.org/10.1177%2F002182860403500305). [S2CID](/source/S2CID_(identifier)) [119021116](https://api.semanticscholar.org/CorpusID:119021116).

- [Newcomb, S.](/source/Simon_Newcomb) (1898). [*Tables of the four inner planets*](https://archive.org/details/06AstronomicalPapersPreparedForTheUse). Astronomical papers prepared for the use of the American ephemeris and nautical almanac. Vol. 6 (2nd ed.). Washington: Bureau of Equipment, Navy Department.

- O'Connor, J. J.; Robertson, E. F. (November 1999). ["Biography of Ulugh Beg"](https://mathshistory.st-andrews.ac.uk/Biographies/Ulugh_Beg/). *MacTutor*. Retrieved March 8, 2025.

- Seidelmann, P. K., ed. (1992). [*Explanatory Supplement to the Astronomical Almanac*](https://archive.org/stream/131123ExplanatorySupplementAstronomicalAlmanac/131123-explanatory-supplement-astronomical-almanac#page/n306/mode/1up) (2nd ed.). Sausalito, CA: University Science Books. [ISBN](/source/ISBN_(identifier)) [0-935702-68-7](https://en.wikipedia.org/wiki/Special:BookSources/0-935702-68-7).

- Urban, S.E.; Seidelmann, P. K., eds. (2013). [*Explanatory supplement to the astronomical almanac*](https://web.archive.org/web/20190430134555/http://aa.usno.navy.mil/publications/docs/c15_usb_online.pdf) (PDF) (3rd ed.). Mill Valley, CA: University Science Books. [ISBN](/source/ISBN_(identifier)) [978-1-891389-85-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-891389-85-6). Archived from [the original](http://aa.usno.navy.mil/publications/docs/c15_usb_online.pdf) (PDF) on April 30, 2019. Retrieved May 6, 2018.

## Further reading

- Dershowitz, N.; [Reingold, E.M.](/source/Edward_M._Reingold) (2008). [*Calendrical calculations*](/source/Calendrical_Calculations) (3rd ed.). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-0-521-70238-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-70238-6).

- [Meeus, Jean](/source/Jean_Meeus) (August 10, 2009) [1998]. *Astronomical Algorithms* (2nd, with corrections as of August 10, 2009 ed.). Richmond, VA: Willmann-Bell. [ISBN](/source/ISBN_(identifier)) [978-0-943396-61-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-943396-61-3).

- [Meeus, Jean](/source/Jean_Meeus) (2002). *More astronomical astronomy morsels*. Richmond, VA: Willmann-Bell. [ISBN](/source/ISBN_(identifier)) [0-943396-74-3](https://en.wikipedia.org/wiki/Special:BookSources/0-943396-74-3). Contains updates to [Meeus & Savoie 1992](#CITEREFMeeusSavoie1992).

- Simon, J. L.; Bretagnon, P.; Chapront, J.; Chapront-Touze, M.; Francou, G.; Laskar, J. (February 1994). ["Numerical expressions for precession formulae and mean elements for the Moon and the planets"](https://ui.adsabs.harvard.edu/link_gateway/1994A&A...282..663S/ADS_PDF). *Astronomy and Astrophysics*. **282**: 663–683. [Bibcode](/source/Bibcode_(identifier)):[1994A&A...282..663S](https://ui.adsabs.harvard.edu/abs/1994A&A...282..663S). [ISSN](/source/ISSN_(identifier)) [0004-6361](https://search.worldcat.org/issn/0004-6361). Referenced in *Astronomical almanac for the year 2011* and contains expressions used to derive the length of the tropical year.

## External links

- Media related to [Tropical year](https://commons.wikimedia.org/wiki/Category:Tropical_year) at Wikimedia Commons

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