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) relative to the equinoxes and solstices. 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 the 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).

Subtleties

The motion of the Earth in its orbit (and therefore the apparent motion of the Sun among the stars) is not completely regular. This is 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 elliptical 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 365.24219 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, as measured by clocks, is steadily getting longer (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, 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, Newcomb-style, of 365.24219 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 long-term 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 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

The latest value of the mean tropical year at J2000.0 (1 January 2000, 12:00 TT) according to an incomplete analytical solution by Moisson365.242190419 days = 365.25 days × 1296000" / (6.28307585085 rad × 180°/π × 1296000"/360° + 50.28796195") from X. Moisson, "Solar system planetary motion to third order of the masses", Astronomy and astrophysics 341 (1999) 318-327, p. 324 (N for Earth fitted to DE405) and N. Capitaine et al., "Expressions for IAU 2000 precession quantities" (685 KB pdf file) Astronomy and Astrophysics 412 (2003) 567-586 p. 581 (P03: p_A). was:

365.242 190 419 SI days

An older value from a complete solution described by MeeusDerived from: Jean Meeus (1991), Astronomical Algorithms, Ch.26 p. 166; Willmann-Bell, Richmond, VA. ISBN 0-943396-35-2 ; based on the VSOP-87 planetary ephemeris. was:
(this value is consistent with the linear change and the other ecliptic years that follow)

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 which is based on atomic clocks (formerly, Ephemeris Time was used instead); 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 – January 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 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 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:

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 ^*. This is due to Mars' considerably greater orbital eccentricity.

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. In the distant future, therefore, 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 ten-thousandth 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.

Calendar rules and vernal equinox

The great interest of the tropical year value is to keep the calendar year synchronized with the beginning of seasons. All the progressive solar calendars since Old Egyptian times are arithmetical calendars. This means an easy rule to try to reach the best possible astronomical value.

In the history of solar calendars notably these five rules (approximations) were used, are used or are proposed:

Calendar rule	Mean year in days		Tropical correctness attained
Old Egyptian	365	= 365. 000 000 000	in very far future (several million years)
Julian	365 + 1/4	= 365. 250 000 000	several hundred thousand years ago
Gregorian	365 + 1/4 - 3/400	= 365. 242 500 000	at about 4000 BC
Khayyam	365 + 8/33	= 365. 242 242 242	at about 1000 AD
*Mean tropical year at epoch 2000.0*		= 365. 242 190 419	*astronomical comparsion value*
von Mädler	365 + 1/4 - 1/128	= 365. 242 187 500	expected between 2024 and 2048

Vernal Equinox from AD 2001 to 2048 in Dynamical Time (delta T to UT ≥ 1 min.)
2001	20	13:32	2002	20	19:17	2003	21	01:01	2004	20	06:50
2005	20	12:35	2006	20	18:27	2007	21	00:09	2008	20	05:50
2009	20	11:45	2010	20	17:34	2011	20	23:22	2012	20	05:16
2013	20	11:03	2014	20	16:58	2015	20	22:47	2016	20	04:32
2017	20	10:30	2018	20	16:17	2019	20	22:00	2020	20	03:51
2021	20	09:39	2022	20	15:35	2023	20	21:26	2024	20	03:08
2025	20	09:03	2026	20	14:47	2027	20	20:26	2028	20	02:19
2029	20	08:03	2030	20	13:54	2031	20	19:42	2032	20	01:23
2033	20	07:24	2034	20	13:19	2035	20	19:04	2036	20	01:04
2037	20	06:52	2038	20	12:42	2039	20	18:34	2040	20	00:13
2041	20	06:08	2042	20	11:55	2043	20	17:29	2044	19	23:22
2045	20	05:09	2046	20	11:00	2047	20	16:54	2048	19	22:36
Source: Jean Meeus

Remarks: The current Gregorian rule – with respect to the mean tropical year – was astronomically true about 6000 years ago. However with respect to the vernal equinox year, important for the date of Easter, the Gregorian year is and stays a very good approximation for thousands of years.

Nevertheless, in the Gregorian calendar, the beginning of spring will inevitably shift to 19-20 March, instead of the traditional 20-21 March. Gregorian common year 2100 will temporally replace vernal equinox to 20-21 March, but shift back to 19-20 March in 2176 (=17x128) according to Meeus' equinox tables. The correct von Mädler rule would regulary avoid this shift to 19 March for millennia. The proposed new Universal Time of Florence – which is defined UTC of Greenwich plus 2700 seconds – also avoids the 19 March date of the year 2044.

References

Jean Meeus and Denis Savoie, "The history of the tropical year", Journal of the British Astronomical Association 102 (1992) 40–42.

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