Tropical year

[ from Greek "tropos" : turn or return ]

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 position is measured from the vernal equinox, one of the 4 cardinal points along the ecliptic.

Because the vernal equinox moves back along the ecliptic due to precession, a tropical year is shorter than a [sidereal year]? (the time for the Sun to return to the same position among the stars as measured in a fixed frame of reference).

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 perturbation?s by the Moon and planets. Therefore the time between successive passages of a specific point on the ecliptic will vary. Conventionally, this is ignored when talking about the tropical year, and some average value is used as if the Sun is having a completely regular circular motion.

At the epoch J2000 (1 January 2000, 12h TDT?), the average tropical year was:

	365.242189670 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:

	-0.00000006162*y days (y in Julian years from 2000),

or about 5 ms/year.

Note: these and following formulae use days of exactly 86400 SI seconds. The time scale is ephemeris time (more precisely TDT?) which is based on atomic clocks; this is different from UT?, which follows the somewhat unpredictable rotation of the Earth. The (small but accumulating) difference 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.

Repeated fractions of the decimal value quoted above give successive approaches for the average length of the tropical year 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 tropical year:

	365
	365  1/4	(Julian year)
	365  7/29
	365  8/33
	365 31/128

However, there is some choice in the length of the tropical year depending on the point that one selects. The reason is, that while the regression of the equinox is fairly steady, the apparent speed of the Sun during the year is not. When the Earth is near the perihelium? of its orbit (around 2 January), 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 22 December), which is close to the perihelium?. Conversely, the northern solstice point is near the aphelium?, where the Sun moves slower than average. Hence the time gained because this point has approached by the same angular arc distance as 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 tropical years measured for these are closer to the value of the average tropical year as quoted above.

Values of the tropical year for the cardinal ecliptic points are [1]:

	vernal equinox: 	365.24237404 + 0.00000010338*y days
	northern solstice:	365.24162603 + 0.00000000650*y days
	autumn equinox: 	365.24201767 - 0.00000023150*y days
	southern solstice:	365.24274049 - 0.00000012446*y days

This distinction is relevant for calendar studies. The main christian moving feast has been Easter. At the consilium of Nicea in 325 it was decided that Easter would be celebrated on the Sunday on or after the first full moon after 21 March, the day of the vernal equinox. The church therefore always had an objective to keep the day of the vernal (spring) equinox on 21 March, and the calendar year has to be synchronized with the tropical year as measured for the vernal equinox: this is slightly longer than the average tropical year.

Now our current Gregorian calendar has an average year of:

	365 97/400 = 365.2425 days.

This is slightly too long, and not an optimal approximation when considering the repeated fractions listed above. However, it is closer to the tropical year as measured for the vernal equinox. Indeed, the approximation of 365 8/33 is even better, and it has been considered in Elizabethan England as a protestant alternative for the catholic Gregorian calendar reform of 1582.

References: [1] Derived from: Jean Meeus (1991), "Astronomical Algorithms" Ch.26 p. 166; Willmann-Bell, Richmond, VA. ISBN 0-943396-35-2 (amazon.com, search) ; based on the VSOP-87 planetary ephemeris?.