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tropical year

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**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*(USNO|2015}}|2015) 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" {{harv|Borkowski|1991|p=122}}.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 days {{harv|Richards|2013|p=587}}.

## History

### Origin

The word "tropical" comes from the Greek*tropikos*meaning "turn" {{harv|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 {{harv|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 1Â° 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 {{harv|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 (â‰ˆ 365.24255 days). This length was used in devising the Gregorian calendar of 1582 {{harv|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 {{harv|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; {{harvnb|Meeus|Savoie|1992|p=41}}).Newton's three laws of dynamics and theory of gravity were published in his*PhilosophiÃ¦ Naturalis Principia Mathematica*in 1687. Newton's theoretical and mathematical advances influenced tables by Edmund Halley published in 1693 and 1749 {{harv|McCarthy|Seidelmann|2009|pp=26â€“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*L*0 =

*A*0 +

*A*1

*T*+

*A*2

*T*2 days

*T*is the time in Julian centuries. The inverse of the derivative of

*L*0,

*dT*/

*dL*0 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.{| class="wikitable" border="1"|+Tropical year coefficients! Name! Equation! Date on which

*T*= 0

Urbain Le Verrier>Leverrier {{harv | Savoie | p=42}} | Y = {{gaps>365.242 | 47}} − 6.24{{e|-6}} T|January 0.5, 1900, Ephemeris time |

last=Newcomb | author-link=Simon Newcomb | pp=9-10}} | Y = {{gaps>365.242 | 79}} − 6.14{{e|-6}} T|January 0, 1900, mean time |

*Astronomical Almanac*for the Sun, Mercury, Venus, and Mars through 1983 {{harv|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- artificial satellites
- tracking of deep space probes such as Pioneer 4 beginning in 1959 (Jet Propulsion Laboratory JPL|2005}}|2005)
- radars able to measure the distance to other planets beginning in 1961 {{harv|Butrica|1996}}
- lunar laser ranging since the 1969 Apollo 11 left the first of a series of retroreflectors which allow greater accuracy than reflectorless measurements
- artificial satellites such as LAGEOS (1976) and the Global Positioning System (initial operation in 1993)
- Very Long Baseline Interferometry which finds precise directions to quasars in distant 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 {{harv|McCarthy|Seidelmann|2009|p=265}}.

*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 {{harv|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 {{harv|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-EugÃ¨ne 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 {{harv|McCarthy|Seidelmann|2009|loc=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 {{harv|McCarthy|Seidelmann|2009|pp=157â€“9}}. Due to the accuracy the General Conference on Weights and Measures in 1960 redefined the second in terms of the cesium transition."The second is the duration of {{gaps|9|192|631|770}} 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 BIPM|2006}}|2006, 133) 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 {{harv|McCarthy|Seidelmann|2009|pp=81â€“2, 191â€“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 {{harv|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|2017|01|post=,}} TT is ahead of UT1 by 69.184 seconds (International Earth Rotation Service IERS|2017}}|2017; {{harvnb|McCarthy|Seidelmann|2009|pp=86â€“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 Î”*T*, or Delta

*T*. The table below gives Morrison|Stephenson|2004}}|Morrison and Stephenson's (S & M) 2004 estimates and standard errors (

*Ïƒ*) for dates significant in the process of developing the Gregorian calendar.{| class=wikitable! Event || Year || Nearest S & M Year || Î”

*T*|| Ïƒ

| 4m20s |

First Council of Nicaea > | | 2m |

| 20s |

| |

| |

Î”

where *T*in seconds = −20 + 32*t*2*t*is measured in Julian centuries from 1820. The extrapolation is provided only to show Î”

*T*is not negligible when evaluating the calendar for long periods; {{harvtxt|Borkowski|1991|p=126}} cautions that "many researchers have attempted to fit a parabola to the measured Î”

*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

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 {{Emoji presentation|â™ˆ|text}} (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 {{Emoji presentation|â™Ž|text}} (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 {{Emoji presentation|â™ˆ|text}} 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 {{Emoji presentation|â™ˆ|text}} at noon January 1, 2000 fills this role and is given the symbol {{Emoji presentation|â™ˆ|text}}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., USNO|2009}}|2009). While the Sun moves, {{Emoji presentation|â™ˆ|text}} moves in the opposite direction . When the Sun and {{Emoji presentation|â™ˆ|text}} met at the 2010 March equinox, the Sun had moved east 359Â°59'09" while {{Emoji presentation|â™ˆ|text}} had moved west 51" for a total of 360Â° (all with respect to {{Emoji presentation|â™ˆ|text}}0; {{harvnb|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 {{Emoji presentation|â™ˆ|text}} changes at a nearly steady rateThe expression for pA in {{harvnb|Seidelmann|1992|p=104}} shows the progression per 365 days of TT increases steadily from 50.25865 arcseconds (January 1, 2009 â€“ January 1, 2010) to 50.25889 arcseconds (December 1, 2009 â€“ December 1, 2010). 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. {{harvtxt|Meeus|Savoie|1992|p=41}} provided the following examples of intervals between northward equinoxes:{| class=wikitable! !!days!!hours!!min!!s|58 |

|15 |

|38 |

|42 |

|06 |

*mean*tropical year {{harv|Meeus|Savoie|1992|p=42}}.The following values of time intervals between equinoxes and solstices were provided by {{harvtxt|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.{{harv|Meeus|2002|p=362}}{| class=wikitable! !!Year 0 !!Year 2000

Northward equinoxes > | 365.242 | 365.242|374}} days |

Northern solstices > | 365.241 | 365.241|626}} |

Southward equinoxes > | 365.242 | 365.242|018}} |

Southern solstices > | 365.242 | 365.242|740}} |

365.242 | 365.242|189}} |

### Mean tropical year current value

The mean tropical year on January 1, 2000 was {{gaps|365.242|189|7}} 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 59times 10^{-6}T-7.29times 10^{-10}T^2 + 2.64times 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; {{harvnb|McCarthy|Seidelmann|2009|p=18}}, calculated from planetary model of {{harvnb|Laskar|1986}}).Modern astronomers define the tropical year as time for the Sun's mean longitude to increase by 360Â°. 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 {{Emoji presentation|â™ˆ|text}}), such as Newcomb's expression given above, or Laskar's expression (Laskar|1986}}|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 360Â° ({{harvnb|Meeus|Savoie|1992|p=42}}; USNO|2010}}|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 {{harv|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 = {{frac|365|97|400}} = 365.2425 days per year, a close approximation to the mean tropical year {{harv|Seidelmann|1992|pp=576â€“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 {{harvtxt|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 {{harv|North|1983|pp=75â€“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 {{gaps|365.242|198|781|25}} 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).365242Ã—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 Î”T is not sufficiently predictable to form more precise proposals {{harv|Blackburn|Holford-Strevens|2003|p=692}}.## See also

## Notes

{{reflist}}## References

- WEB,weblink Astronomical almanac online glossary, 2015, US Naval Observatory, {{harvid, USNO, 2015, }}
- BOOK, Astronomical almanac for the year 2011, 2010, Washington, Astronomical Almanac Office US Naval Observatory, {{harvid, USNO, 2010, }}
- BOOK, Astronomical Applications Dept. of United States Naval Observatory, 2009, Multiyear interactive computer almanac, 2.2, Richmond VA, Willman-Bell, {{harvid, USNO, 2009, }}
- BOOK, Blackburn, B., Holford-Strevens, L., 2003, The Oxford companion to the year, Oxford University Press, harv, Corrected reprint of 1999.
- JOURNAL, Borkowski, K.M.,weblink PDF, The tropical year and the solar calendar, 1991, Journal of the Royal Astronomical Society of Canada, 85, 3, 121â€“130, harv, 1991JRASC..85..121B,
- WEB, Bureau International des Poids et Mesures, 2006,weblink The International System of Units, {{harvid, BIPM, 2006, |deadurl=yes |archiveurl=https://web.archive.org/web/20091001065750weblink |archivedate=October 1, 2009 }}
- WEB, Butrica, A.J., 1996,weblink NASA SP-4218: To see the unseenâ€”A history of planetary radar astronomy, NASA, harv, yes,weblink August 23, 2007,
- BOOK, Dershowitz, N., Edward M. Reingold, Reingold, E.M., 2008, Calendrical calculations, 3rd, Cambridge University Press, 978-0-521-70238-6, harv,
- BOOK, Dobrzycki, J., 1983, Astronomical aspects of the calendar reform, George Coyne, G.V., Coyne, M.A., Hoskin, O., Pedersen,weblink Gregorian reform of the calendar, Vatican Observatory, harv,
- WEB, International Earth Rotation Service, International Earth Rotation Service, February 2, 2017,weblink
*Bulletin A 30*(5), US Naval Observatory, {{harvid, IERS, 2017, }} - BOOK, Jet Propulsion Laboratory, Jet Propulsion Laboratory, 2005,weblink DSN: History, NASA, {{harvid, JPL, 2005, }}
- JOURNAL, Laskar, J., 1986, Secular terms of classical planetary theories using the results of general theory, Astronomy and Astrophysics, 157, 59â€“70, 0004-6361, harv, 1986A&A...157...59L, Note: In the article at this URL page 68 should be put before page 66.
- BOOK, Dennis McCarthy (scientist), McCarthy, D.D., Seidelmann, P.K., 2009, Time from Earth rotation to atomic physics, Weinhein, Wiley-VCH Verlag GmbH & Co. KGaA, harv,
- JOURNAL, Jean Meeus, Meeus, J., Savoie, D., 1992, The history of the tropical year, Journal of the British Astronomical Association, 102, 1, 40â€“42, harv, 1992JBAA..102...40M,
- BOOK, Meeus, Jean, More astronomical astronomy morsels, 2002, Richmond, VA, Willmann-Bell, 0-943396-74-3, Jean Meeus, harv, Contains updates to Meeus & Savoie 1992.
- JOURNAL, Morrison, L.V., Stephenson, F.R., 2004, Historical values of the Earth's clock error Î”T and the calculation of eclipses, Journal for the History of Astronomy, 35, Part 3, 120, 327â€“336, harv, 2004JHA....35..327M,
- BOOK, Newcomb, S., Simon Newcomb, 1898,weblink Tables of the four inner planets, 6, Astronomical papers prepared for the use of the American ephemeris and nautical almanac, 2nd, Washington, Bureau of Equipment, Navy Department, harv,
- BOOK, North, J.D., 1983, The Western calendar - 'Intolerabilis, horribilis, et derisibilis'; four centuries of discontent, George Coyne, G.V., Coyne, M.A., Hoskin,, O., Pedersen,weblink Gregorian reform of the calendar, Vatican Observatory, harv,
- BOOK, Richards, E.G., 2013, Calendars, S.E., Urban, P. K., Seidelmann,weblink Explanatory Supplement to the Astronomical Almanac, 3rd, 585â€“624, Mill Valley, CA, University Science Books, 978-1-891389-85-6, harv,
- BOOK, Seidelmann, P. K., 1992, Explanatory Supplement to the Astronomical Almanac,weblink Sausalito, CA, University Science Books, 2nd, 0-935702-68-7, harv,
- DICTIONARY, tropic, 1992, American Heritage Dictionary, 3rd, Boston, Houghton-Mifflin, {{harvid, tropic, 1992, }}
- BOOK, Urban, S.E., Seidelmann, P. K., 2013, Explanatory supplement to the astronomical almanac, 3rd, Mill Valley, CA, University Science Books, 978-1-891389-85-6, harv,

## Further reading

- BOOK, Meeus, Jean, Astronomical Algorithms, 2nd, with corrections as of August 10, 2009, August 10, 2009, 1998, Richmond, VA, Willmann-Bell, 0-943396-61-1, Jean Meeus,
- JOURNAL, 0004-6361, 282, 663âˆ’683, Simon, J. L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J., Numerical expressions for precession formulae and mean elements for the Moon and the planets., Astronomy and Astrophysics, February 1994,weblink Referenced in
*Astronomical almanac for the year 2011*and contains expressions used to derive the length of the tropical year.

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