[ from Greek "tropos" : turn ]
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 perturbations 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.
Current mean value
At the epoch J2000 (1 January 2000, 12h TT), the mean 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. y is measured in Julian years (365.25 days) from the epoch (2000). The time scale is ephemeris time (more precisely TT) 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 Delta-T) 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.
However, 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 regression of the equinox 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 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 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 21000 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 tropical year for the cardinal ecliptic points are :
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. Several different ways of computing the day of Easter were used in early christian times, but eventually the unified rule was accepted that Easter would be celebrated on the Sunday on or after the first full moon 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 1000 A.D. the mean tropical year has become increasingly shorter than this mean interval between vernal equinoxes.
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 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 millenia and should remain 365.2424 days (to the nearest ten-thousandth of a calendar day) for some millenia 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.
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:
365 365 1/4 (Julian year) 365 7/29 365 8/33 (Khayyam year) 365 143/590