Tropical year

A tropical year is the length of time that the Sun, as viewed from the Earth, takes to return to the same position along the ecliptic (its path among the stars on the celestial sphere). The precise length of time depends on which point of the ecliptic one chooses: starting from the (northern) vernal equinox, one of the four cardinal points along the ecliptic, yields the vernal equinox year; averaging over all starting points on the ecliptic yields the mean tropical year.
On Earth, we notice the progress of the tropical year from the slow motion of the Sun from south to north and back; the word "tropical" is derived from Greek tropos meaning "turn". The tropics of Cancer and Capricorn mark the extreme north and south latitudes the Sun reaches during this cycle. The position of the Sun can be measured by the variation from day to day of the length of the shadow at noon of a gnomon (a vertical pillar or stick). This is the most "natural" way of measuring the year in the sense that the variations of insolation drive the seasons.
Because the vernal equinox moves back along the ecliptic due to precession, a tropical year is shorter than a sidereal year (in 2000, the difference was 20.409 minutes; it was 20.400 min in 1900).
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Subtleties
The motion of the Earth in its orbit (and therefore the apparent motion of the Sun among the stars) is not completely regular due to gravitational perturbations by the Moon and planets. Therefore the time between successive passages of a specific point on the ecliptic will vary.
Moreover, the speed of the Earth in its orbit varies (because the orbit is elliptic rather than circular). Furthermore, the position of the equinox on the orbit changes due to precession. As a consequence (explained below) the length of a tropical year depends on the specific point that you select on the ecliptic (as measured from, and moving together with, the equinox) that the Sun should return to.
Therefore astronomers defined a mean tropical year, that is an average over all points on the ecliptic; it has a length of about 365.2422 SI days. Besides this, tropical years have been defined for specific points on the ecliptic: in particular the vernal equinox year, that start and ends when the Sun is at the vernal equinox. Its length is about 365.2424 days.
An additional complication: We can measure time either in "days of fixed length": SI days of 86,400 SI seconds, defined by atomic clocks, or dynamical days defined by the motion of the Moon and planets; or in mean "natural" days, defined by the rotation of the Earth with respect to the Sun. The duration of the mean natural day is steadily getting longer as measured by clocks (or conversely, clock days are steadily getting shorter, as measured by a sundial). One must use the mean natural day because the "instantaneous" natural day varies regularly over time, as the equation of time shows.
As explained at Error in Statement of Tropical Year (http://www.hermetic.ch/cal_stud/cassidy/err_trop.htm), using the value of the "mean tropical year" to refer to the vernal equinox year defined above, is strictly speaking an error. The words "tropical year" in astronomical jargon refer only to the mean tropical year Newcombstyle of 365.2422 SI days. The vernal equinox year of 365.2424 natural days is also important, because it is the basis of most solar calendars, but it is not the "tropical year" of modern astronomers.
The number of natural days in a vernal equinox year has been oscillating between 365.2424 and 365.2423 for several millennia and will likely remain near 365.2424 for a few more. This longterm stability is pure chance, because in our era the slowdown of the rotation, the acceleration of the mean orbital motion, and the effect at the vernal point of shape changes in the Earth's orbit's happen to almost cancel out.
In contrast, the mean tropical year, measured in SI days, is getting shorter. It was 365.2423 SI days at about AD 200, and is currently near 365.2422 SI days.
Current mean value
At the epoch J2000 (1 January 2000, 12h TT), the mean tropical year was:
 365.242 189 670 SI days.
Due to changes in the precession rate and in the orbit of the Earth, there exists a steady change in the length of the tropical year. This can be expressed with a polynomial in time; the linear term is:
 −0.000 000 061 62×a days (a in Julian years from 2000),
or about 5 ms/year, which means that 2000 years ago the tropical year was 10 seconds longer.
Note: these and following formulae use days of exactly 86400 SI seconds. a is measured in Julian years (365.25 days) from the epoch (2000). The time scale is Terrestrial Time (formerly Ephemeris Time) which is based on atomic clocks; this is different from Universal Time, which follows the somewhat unpredictable rotation of the Earth. The (small but accumulating) difference (called ΔT) is relevant for applications that refer to time and days as observed from Earth, like calendars and the study of historical astronomical observations such as eclipses.
Different lengths
As already mentioned, there is some choice in the length of the tropical year depending on the point of reference that one selects. The reason is that, while the precession of the equinoxes is fairly steady, the apparent speed of the Sun during the year is not. When the Earth is near the perihelion of its orbit (presently, around January 3–4), it (and therefore the Sun as seen from Earth) moves faster than average; hence the time gained when reaching the approaching point on the ecliptic is comparatively small, and the "tropical year" as measured for this point will be longer than average. This is the case if one measures the time for the Sun to come back to the southern solstice point (around December 21–22), which is close to the perihelion.
Conversely, the northern solstice point presently is near the aphelion, where the Sun moves slower than average. Hence the time gained because this point has approached the Sun (by the same angular arc distance as happens at the southern solstice point) is notably greater: so the tropical year as measured for this point is shorter than average. The equinoctial points are in between, and at present the tropical years measured for these are closer to the value of the mean tropical year as quoted above. As the equinox completes a full circle with respect to the perihelion (in about 21,000 years), the length of the tropical year as defined with reference to a specific point on the ecliptic oscillates around the mean tropical year.
Current values and their annual change of the time of return to the cardinal ecliptic points are [1]:
 vernal equinox: 365.24237404 + 0.00000010338×a days
 northern solstice: 365.24162603 + 0.00000000650×a days
 autumn equinox: 365.24201767 − 0.00000023150×a days
 southern solstice: 365.24274049 − 0.00000012446×a days
Notice that the average of these four is 365.2422 SI days (the mean tropical year). This figure is currently getting smaller, which means years get shorter, when measured in seconds. Now, actual days get slowly and steadily longer, as measured in seconds. So the number of actual days in a year is decreasing too.
The differences between the various types of year are relatively minor for the present configuration of Earth's orbit. On Mars, however, the differences between the different types of years are an order of magnitude greater: vernal equinox year = 668.5907 Martian days (sols), summer solstice year = 668.5880 sols, autumn equinox year = 668.5940 sols, winter solstice year = 668.5958 sols, with the tropical year being 668.5921 sols [1] (http://pweb.jps.net/~tgangale/mars/facts/factsi.htm). This is due to Mars' considerably greater orbital eccentricity. It should be noted that Earth's orbit goes through cycles of increasing and decreasing eccentricity over a timescale of about 100,000 years (Milankovitch cycles), and its eccentricity can reach as high as about 0.06, so in the distant future, Earth will also have much more divergent values of the various equinox and solstice years.
Calendar year
This distinction is relevant for calendar studies. The main Christian moving feast has been Easter. Several different ways of computing the date of Easter were used in early christian times, but eventually the unified rule was accepted that Easter would be celebrated on the Sunday after the first full moon on or after the day of the vernal equinox, which was established to fall on 21 March. The church therefore made it an objective to keep the day of the vernal (spring) equinox on or near 21 March, and the calendar year has to be synchronized with the tropical year as measured by the mean interval between vernal equinoxes. From about AD 1000 the mean tropical year (measured in SI days) has become increasingly shorter than this mean interval between vernal equinoxes (measured in actual days), though the interval between successive vernal equinoxes measured in SI days has become increasingly longer.
Now our current Gregorian calendar has an average year of:
 365 + 97/400 = 365.2425 days.
Although it is close to the vernal equinox year (in line with the intention of the Gregorian calendar reform of 1582), it is slightly too long, and not an optimal approximation when considering the continued fractions listed below. Note that the approximation of 365 + 8/33 used in the Iranian calendar is even better, and 365 + 8/33 was considered in Rome and England as an alternative for the Catholic Gregorian calendar reform of 1582.
Moreover, modern calculations show that the vernal equinox year has remained between 365.2423 and 365.2424 calendar days (i.e. mean solar days as measured in Universal Time) for the last four millennia and should remain 365.2424 days (to the nearest tenthousandth of a calendar day) for some millennia to come. This is due to the fortuitous mutual cancellation of most of the factors affecting the length of this particular measure of the tropical year during the current era.
Approximations
Continued fractions of the decimal value for the vernal equinox year quoted above give successive approaches to the average interval between vernal equinoxes, in terms of fractions of a day. These can be used to intercalate years of 365 days with leap years of 366 days to keep the calendar year synchronized with the vernal equinox:
 365 (No intercalated days)
 365 + 1/4 (Julian intercalation cycle; 1in4)
 365 + 7/29 (6 × Julian cycle + 1in5; 7in29)
 365 + 8/33 (Khayyam cycle; 7 × 1in4 + 1in5)
 365 + 143/590 (17 × Khayyam cycle + 7in29) etc.
Note that 590 years hence, the year length will have changed, postponing the need for any 7in29 subcycle.
See also
References
[1] Derived from: Jean Meeus (1991), Astronomical Algorithms, Ch.26 p. 166; WillmannBell, Richmond, VA. ISBN 0943396352 ; based on the VSOP87 planetary ephemeris.
 Jean Meeus and Denis Savoie, "The history of the tropical year", Journal of the British Astronomical Association 102 (1992) 40–42.[2] (http://adsbit.harvard.edu/cgibin/nphiarticle_query?1992JBAA..102...40M)ca:Any tròpic
de:Tropisches Jahr es:Año tropical it:Anno tropico ja:太陽年 nl:Tropisch jaar pl:Rok zwrotnikowy pt:Ano trópico sl:tropsko leto vi:Năm chí tuyến zh:回归年