Tropical year
A tropical year (also known as a solar year), for general purposes, is the length of 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, and currently define it as the time required for the mean Sun's tropical longitude (longitudinal position along the ecliptic relative to its position at the vernal equinox) to increase by 360 degrees (that is, to complete one full seasonal circuit). (Meeus & Savoie, 1992, p. 40)
Contents
History
Origin
The word "tropical" comes from the Greek tropikos meaning "turn". (tropic, 1992) Thus, the tropics of Cancer and Capricorn mark the extreme north and south latitudes where the Sun can appear directly overhead, and where it appears to "turn" in its annual seasonal motion. Because of this connection between the tropics and the seasonal cycle of the apparent position of the Sun, the word "tropical" also lent its name to the "tropical year". The early Chinese, Hindus, Greeks, and others made approximate measures of the tropical year; early astronomers did so by noting the time required between the appearance of the Sun in one of the tropics to the next appearance in the same tropic. (Meeus & Savoie, 1992, p. 40)
Early value, precession discovery
In the 2nd century BC Hipparchus introduced a new definition which was still used by some authors in the 20th century, 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 365 solar days, 5 hours, 55 minutes, 12 seconds (365.24667 days). A modern computer model gives 365 solar days, 5 hours 49 minutes 19 seconds (365.24258 days). He adopted the new definition because the instrument he used, the meridian armillae, was better able to detect the more rapid motion in declination at the time of the equinoxes, compared to the solstices. (Meeus & Savoie, 1992, p. 40)
Hipparchus also discovered that the equinoctial points moved along the ecliptic (plane of the Earth's orbit, or what Hipparchus would have thought of as the plane of the Sun's orbit about the Earth) in a direction opposite that of the movement of the Sun, a phenomenon that came to be named "precession of the equinoxes". He reckoned the value as 1° per century, a value that was not improved upon until about 1000 years later, by Islamic astronomers. Since this discovery a distinction has been made between the tropical year and the sidereal year. (Meeus & Savoie, 1992, p. 40)
Middle Ages and the Renaissance
During the Middle Ages and Renaissance a number of progressively better tables were published that allowed computation of the positions of the Sun, Moon and planets relative to the fixed stars. An important application of these tables was the reform of the calendar.
The Alfonsine Tables, published in 1252, were based on the theories of Ptolemy and were revised and updated after the original publication; the most recent update in 1978 was by the French National Centre for Scientific Research. The length of the tropical year (using the equinoxbased definition) was 365 solar days 5 hours 49 minutes 16 seconds (365.24255 days). It was these tables that were used in the reform process that led to the Gregorian calendar. (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 found a tropical year length of 365 solar days, 5 hours, 55 minutes, 58 seconds (365.24720 days). (Meeus & Savoie, 1992, p. 41)
Major advances in the 17th century were made by Johannes Kepler and Isaac Newton. In 1609 and 1619 Kepler published his three laws of planetary motion. (McCarthy & Seidelmann, 2009, p. 26) In 1627, Kepler used the observations of Tycho Brahe and Waltherus to produce the most accurate tables up to that time, the Rudolphine Tables. He evaluated the tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds (365.24219 days). (Meeus & Savoie, 1992, p. 41)
Newton's three laws of dynamics and theory of gravity were published in his Philosophiæ Naturalis Principia Mathematica in 1687. Newton's theoretical and mathematical advances influenced tables by Edmund Halley published in 1693 and 1749 (McCarthy & Seidelmann, 2009, pp. 26–28) and provided the underpinnings of all solar system models until Albert Einstein's theory of General relativity in the 20th century.
18th and 19th century
From the time of Hipparchus and Ptolemy, the year was based on two equinoxes (or two solstices) a number of years apart, to average out both observational errors and the effects of nutation (irregular 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 motion of the Sun. The necessary theories and mathematical tools came together in the 18th century due to the work of PierreSimon 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
where 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 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.
Name  Equation  Date on which T = 0 

Leverrier (Meeus & Savoie, 1992, p. 42)  Y = 365.24219647 − 6.24×10^{−6} T  January 0.5, 1900, Ephemeris Time 
Newcomb (1898, p. 9–10)  Y = 365.24219879 − 6.14×10^{−6} T  January 0, 1900, mean time 
Newcomb's tables were successful enough that they were used by the joint AmericanBritish Astronomical Almanac for the Sun, Mercury, Venus, and Mars through 1983. (Seidelmann, 1992, p. 317)
20th and 21st centuries
The length of the mean tropical year is derived from a model of the solar system, so any advance that improves the solar system model potentially improves the accuracy of the mean tropical year. Many new observing instruments became available, including
 artificial satellites
 tracking of deep space probes such as Pioneer 4 beginning in 1959 (Jet Propulsion Laboratory 2005)
 radars able to measure other planets beginning in 1961 (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 (McCarthy & Seidelmann, 2009, p. 265)
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. At the American Ephemeris an electromagnetic computer, the IBM Selective Sequence Electronic Calculator was used since 1948. When modern computers became available, it was possible to compute ephemerides using numerical integration rather than general theories; numerical integration came into use in 1984 for the joint USUK almanacs. (McCarthy & Seidelmann, 2009, p. 32)
Einstein's General Theory of Relativity provided a more accurate theory, but the accuracy of theories and observations did not require the refinement provided by this theory (except for the advance of the perihelion of Mercury) until 1984. Time scales incorporated general relativity beginning in the 1970s. (McCarthy & Seidelmann, 2009, p. 37)
A key development in understanding the tropical year over long periods of time is the discovery that the rate of rotation of the earth, or equivalently, the length of the mean solar day, is not constant. William Ferrel in 1864 and CharlesEugène Delaunay in 1865 indicated the rotation of the Earth was being retarded by tides. In 1921 William H Shortt invented the ShorttSynchronome 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 timekeeping advance was the quartz clock, first built by Warren Marrison and J. W. Horton in 1927; in the late 1930s quartz clocks began to replace pendulum clocks as time standards. (McCarthy and Seidelmann, 2009, ch. 9)
A series of experiments beginning in the late 1930s led to the development of the first atomic clock by Louis Essen and J. V. L. Parry in 1955. Their clock was based on a transition in the cesium atom. (McCarthy & Seidelmann, 2009, pp. 157–9) Due to the accuracy the General Conference on Weights and Measures in 1960 redefined the second in terms of the cesium transition.^{1} The atomic second, often called the SI second, was intended to agree with the ephemeris second based on Newcomb's work, which in turn makes it agree with the mean solar second of the mid19th century. (McCarthy & Seidelman, 2009, pp. 81–2, 191–7)
Time scales
As mentioned in History, advances in timekeeping 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, all civil calendars are based on UT.
The other time scale has two parts. Ephemeris time (ET) is the independent variable in the equations of motion of the solar system, in particular, the equations in use from 1960 to 1984. (McCarthy & Seidelmann, 2009, p. 378) That is, the length of the second used in the solar system calculations could be adjusted until the length that gives the best agreement with observations is found. With the introduction of atomic clocks in the 1950s, it was found that ET could be better realized as atomic time. This also means that ET is a uniform time scale, as is atomic time. ET was given a new name, Terrestrial Time (TT), and for most purposes ET = TT = International Atomic Time + 32.184 SI seconds. As of January 2010, TT is ahead of UT1 by about 66 seconds. (International Earth Rotation Service, 2010; McCarthy & Seidelman, 2009, pp. 86–7).
As explained below, long term estimates of the length of the tropical year were used in connection with the reform of the Julian calendar, which resulted in the Gregorian calendar. Of course the participants in that reform were unaware of the nonuniform rotation of the earth, but now this can be taken into account to some degree. The amount that TT is ahead of UT1 is known as ΔT, or Delta T. The table below gives Morrison and Stephenson's (S & M) 2004 estimates and standard errors (σ) for dates significant in the process of developing the Gregorian calendar.
Event  Year  Nearest S & M Year  ΔT  σ 

Julian calendar begins  −44  0  2h56m20s  4m20s 
First Council of Nicaea  325  300  2h8m  2m 
Gregorian calendar begins  1583  1600  2m  20s 
low precision extrapolation  4000  4h13m  
low precision extrapolation  10,000  2d11h 
The low precision extrapolations are computed with an expression provided by Morrison and Stephenson
 ΔT = −20 + 32t^{2}
where t is measured in Julian centuries from 1820. The extrapolation is provided only to show ΔT is not negligible when evaluating the calendar for long periods; Borkowski (1991, p. 126) cautions that "many researchers have attempted to fit a parabola to the measured ΔT values in order to determine the magnitude of the deceleration of the Earth's rotation. The results, when taken together, are rather discouraging."
Length of tropical year
An oversimplified definition of the tropical year would be the time required for the Sun, beginning at a chosen ecliptic longitude, to make one complete cycle of the seasons and return to the same ecliptic longitude. Before considering an example, the equinox must be examined. There are two important planes in solar system calculations, the plane of the ecliptic (the Earth's orbit around the Sun), and the plane of the celestial equator (the Earth's equator projected into space). These two planes intersect in a line. The direction along the line from the Earth in the general direction of the zodiac sign Aries (Ram) is the Northward equinox, and is given the symbol ♈ (the symbol looks like the horns of a ram).
The opposite direction, along the line in the general direction of the sign Libra, is the Southward equinox and is given the symbol ♎. Because of precession and nutation these directions change, compared to the direction of distant stars and galaxies, whose directions have no measurable motion due to their great distance (see International Celestial Reference Frame).
The ecliptic longitude of the Sun is the angle between ♈ and the Sun, measured eastward along the ecliptic. This creates a complicated measurement, because as the Sun is moving, the direction the angle is measured from is also moving. It is convenient to have a fixed (with respect to distant stars) direction to measure from; the direction of ♈ at noon January 1, 2000 fills this role and is given the symbol ♈_{0}.
Using the oversimplified definition, there was an equinox on March 20, 2009, 11:44:43.6 TT. The 2010 March equinox was March 20, 17:33:18.1 TT, which gives a duration of 365 d 5 h 49 m 30s. (Astronomical Applications Dept., 2009) While the Sun moves, ♈ moves in the opposite direction . When the Sun and ♈ met at the 2010 March equinox, the Sun had moved east 359°59'09" while ♈ had moved west 51" for a total of 360° (all with respect to ♈_{0}). (Seidelmann, 1992, p. 104, expression for p_{A})
If a different starting longitude for the Sun is chosen, the duration for the Sun to return to the same longitude will be different. This is because although ♈ changes at a nearly steady rate^{2} there is considerable variation in the angular speed of the Sun. Thus, the 50 or so arcseconds that the Sun does not have to move to complete the tropical year "saves" varying amounts of time depending on the position in the orbit.
Mean equinox tropical year
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 the instruments were most sensitive there. When tropical year measurements from several successive years are compared, variations are found which are due to nutation, and to the planetary perturbations acting on the Sun. Meeus and Savoie (1992, p. 41) provided the following examples of intervals between northward equinoxes:
days  hours  min  s  

1985–1986  365  5  48  58 
1986–1987  365  5  49  15 
1987–1988  365  5  46  38 
1988–1989  365  5  49  42 
1989–1990  365  5  51  06 
Until the beginning of the 19th century, the length of the tropical year was found by comparing equinox dates that were separated by many years; this approach yielded the mean tropical year. (Meeus & Savoie, 1992, p. 42)
Values of mean time intervals between equinoxes and solstices were provided by Meeus and Savoie (1992, p. 42) for the years 0 and 2000.
Year 0  Year 2000  

Between two Northward equinoxes  365.242137 days  365.242374 days 
Between two Northern solstices  365.241726  365.241626 
Between two Southward equinoxes  365.242496  365.242018 
Between two Southern solstices  365.242883  365.242740 
Mean tropical year (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 days, 5 hours, 48 minutes, 45.19 seconds. This changes slowly; an expression suitable for calculating the length in days for the distant past is
365.2421896698 − 6.15359×10 ^{−6}T − 7.29×10 ^{−10}T^{2} + 2.64×10 ^{−10}T^{3}
where T is in Julian centuries of 36,525 days measured from noon January 1, 2000 TT (in negative numbers for dates in the past). (McCarthy & Seidelmann, 2009, p. 18.; Laskar, 1986)
Modern astronomers define the tropical year as time for the Sun's mean longitude to increase by 360°. The process for finding an expression for the length of the tropical year is to first find an expression for the Sun's mean longitude (with respect to ♈), such as Newcomb's expression given above, or Laskar's expression (1986, p. 64). When viewed over a 1 year period, the mean longitude is very nearly a linear function of Terrestrial Time. To find the length of the tropical year, the mean longitude is differentiated, to give the angular speed of the Sun as a function of Terrestrial Time, and this angular speed is used to compute how long it would take for the Sun to move 360°. (Meeus & Savoie, 1992, p. 42).
Calendar year
The Gregorian calendar, as used for civil purposes, is an international standard. It is a solar calendar that is designed to maintain synchrony with the vernal equinox tropical year.^{3} 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 tropical year. (Seidelmann, 1992, pp. 576–81)
The Gregorian calendar is a reformed version of the Julian calendar. By the time of the reform in 1582, the date of the vernal equinox had shifted about 10 days, from about March 21 at the time of the First Council of Nicaea in 325, to about March 11. According to North, the real motivation for reform was not primarily a matter of getting agricultural cycles back to where they had once been in the seasonal cycle; the primary concern of Christians was the correct observance of Easter. The rules used to compute the date of Easter used a conventional date for the vernal equinox (March 21), and it was considered important to keep March 21 close to the actual equinox. (North, 1983, pp. 75–76)
If society in the future still attaches importance to the synchronization between the civil calendar and the seasons, another reform of the calendar will eventually be necessary. According to Blackburn and HolfordStrevens (who used Newcomb's value for the tropical year) if the tropical year remained at its 1900 value of 365.24219878125 days the Gregorian calendar would be 3 days, 17 min, 33 s behind the Sun after 10,000 years. Aggravating this error, the length of the tropical year (measured in Terrestrial Time) is decreasing at a rate of approximately 0.53 s per 100 tropical years. Also, the mean solar day is getting longer at a rate of about 1.5 ms per 100 tropical years. These effects will cause the calendar to be nearly a day behind in 3200. 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. The effects are not sufficiently predictable to form more precise proposals. (Blackburn & HolfordStrevens, 2003, p. 692)
Borkowski (1991, p. 121) states "because of high uncertainty in the Earth's rotation it is premature at present to suggest any reform that would reach further than a few thousand years into the future." He estimates that in 4000 the Gregorian year (which counts actual solar days) will be behind the tropical year by 0.8 to 1.1 days. (p. 126)
See also
Notes
 ^ "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." CGPM (1960, Resolution 9, as quoted in Bureau International des Poids et Mesures 2006, 133)
 ^ The expression for p_{A} in Seidelmann, 1992, p. 104 shows the progression per 365 days of TT increases steadily from 50.25865 arcseconds (Jan. 1, 2009–Jan. 1, 2010) to 50.25889 arcseconds (Dec. 1, 2009–Dec. 1, 2010).
 ^ (Dobrzycki 1983, p. 123)
References
 Astronomical Applications Dept. of United States Naval Observatory. (2009). Multiyear Interactive Computer Almanac (ver. 2.2). Richmond VA: WillmanBell.
 Blackburn, B. & HolfordStrevens, L. (2003, corrected reprint of 1999). The Oxford companion to the year. Oxford University Press.
 Borkowski, K. M. The tropical year and the solar calendar. (1991). Journal of the Royal Astronomical Society of Canada, 85(3), 121–130.
 Bureau International des Poids et Mesures. (2006). The International System of Units. author.
 Butrica, A. J. (1996) NASA SP4218: To See the Unseen  A History of Planetary Radar Astronomy. NASA.
 Dershowitz, N. & Reingold, E.M. (2008). Calendrical Calculations (3rd ed.). Cambridge University Press. ISBN 9780521702386
 Dobrzycki, J (1983). Astronomical aspects of the calendar reform. In G. V. Coyne, M. A. Hoskin, & O. Pedersen (Eds.). Gregorian reform of the calendar. Vatican Observatory.
 International Earth Rotation Service (2010, January 28). Bulletin A 23(4). author.
 Jet Propulsion Laboratory. (2005). DSN: History. author.
 Laskar, J. (1986). Secular terms of the classical planetary theories using the results of general theory. Astronomy and Astrophysics, 157, 59–70. ISSN 00046361.
 McCarthy, D. D. & Seidelmann, P. K. (2009). Time from Earth rotation to atomic physics. Weinhein: WileyVCH Verlag GmbH & Co. KGaA.
 Meeus, J. & Savoie, D. (1992). The history of the tropical year. Journal of the British Astronomical Association, 102(1), 40–42.
 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 Vol. 35, Part 3, No. 120, p. 327–336
 Newcomb, S. (1898). Tables of the four inner planets (Vol. 6) in Astronomical papers prepared for the use of the American ephemeris and nautical almanac (2nd ed.). Washington: Bureau of Equipment, Navy Dept.
 North, J. D. (1983). The Western calendar  "Intolerabilis, horribilis, et derisibilis"; four centuries of discontent. In G. V. Coyne, M. A. Hoskin, & O. Pedersen (Eds.). Gregorian reform of the calendar. Vatican Observatory.
 Seidelmann, P. K. (Ed.). (1992). Explanatory supplement to the Astronomical almanac. Sausalito, CA: University Science Books. ISBN 0935702687
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