Tropical year Totally Explained

Everything about Solar Year totally explained

A tropical year (also known as a solar year) 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. A tropical year can equivalently be defined as the time taken for the 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).
   For the reasons explained below, the length of a tropical year varies slightly, by up to a minute or two, depending on the seasonal starting point. The tropical year measured between (northern) vernal equinoxes (one of the four cardinal points along the ecliptic), is called the vernal equinox tropical year, or just vernal equinox year. The mean tropical year is calculated by averaging the (slightly differing) tropical years over all possible starting points through the four seasons. When used without qualification, the term "tropical year" often refers to the mean tropical year.
   Because of a phenomenon known as the precession of the equinoxes, the tropical year, which is based on the seasonal cycle, is slightly shorter than the sidereal year, which is the time it takes for the Sun to return to the same apparent position relative to the backdrop of stars. This difference was 20.400 minutes in AD 1900 and 20.409 minutes in AD 2000. Because it's desirable for everyday-use calendars to keep in synchronisation with the seasons, it's the tropical year that, in principle, these calendars track. Although the yearly differences are small, they're cumulative, and after many years amount to a very noticeable discrepancy.
   The word "tropical" comes from the Greek tropos meaning "turn". 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".

Vernal equinox and mean tropical year

Tropical years have been defined for specific points on the ecliptic, as well as an average over all solstices and equinoxes on the ecliptic, with a length of about 365.24219 SI days.
   Time can be measured 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 solar days, defined by the rotation of the Earth with respect to the Sun. The duration of the mean solar day, as measured by clocks, is getting longer (or clock days are getting shorter, as measured by a sundial). With the mean solar day, the length of each solar day varies regularly during the year, as the equation of time shows.
   The motion of the Earth in its orbit (and therefore the apparent motion of the Sun among the stars) isn't completely regular, caused by gravitational perturbations by the Moon and planets.
   The time between successive passages of a specific point on the ecliptic, and the speed of the Earth in its orbit vary (because the orbit is elliptical rather than circular). The position of the equinox on the orbit changes because of precession. The length of a tropical year (explained below) depends on the specific point selected on the ecliptic (as measured from, and moving together with, the equinox) that the Sun should return to. Nevertheless, the vernal equinox year that begins and ends when the Sun is at the vernal equinox isn't an astronomer's tropical year.
   The tropical year isn't equal to the time interval between two successive spring equinoxes for two reasons: the vernal equinox varies from year to year because of the nutation of the earth and the tidal influence of other planets; and on average the vernal equinoxes come slightly further apart because the vernal equinox is close to the Earth's perihelion. As the perihelion precesses (in about 21,000 years), this effect will also average out.
   Vernal equinox years are of chief interest to the Christian, Jewish and Iranian calendars, and the mean length of that year in the second and third millennia will be about 365.2424 days. Error in Statement of Tropical Year

explains that isn't correct to use the value of the "mean tropical year" to refer to the vernal equinox year defined above. The words "tropical year" in astronomical jargon refer only to the mean tropical year, Newcomb-style, of 365.24219 SI days.
The number of mean solar 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 equinox of rotation and 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. About AD 200, it was 365.2423 SI days, and was near 365.2422 SI days by AD 2000.

Specific equinox and solstice tropical year current values

As already mentioned, there's 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.
The northern solstice point is now near the aphelion, where the Sun moves slower than average. The time gained because this point approached the Sun (by the same angular arc distance as happens at the southern solstice point) is greater. 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 was: ť 365.242 190 419 SI days

An older value from a complete solution described by Meeus was:
(this value is consistent with the linear change and the other ecliptic years that follow) ť 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:
ť difference (days) = −0.000 000 061 62×a days,

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.

Calendar year

The distinction between tropical years is relevant for calendar studies.
   The established Hebrew calendar created a mathematical resolution for the differences that arise between the solar and lunar years so that all Jewish holidays occur at the same season each year.
   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 (ecclesiastical) full moon on or after the day of the (ecclesiastical, not actual) vernal equinox, which was established to fall on 21 March.
   The Catholic Church made it, therefore, an objective to keep the day of the (actual) vernal 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.
   The currently widely-used Gregorian calendar has an average year of:
ť 365 + 97/400 = 365.2425 days.

Modern calculations show that the vernal equinox year has remained between 365.2423 and 365.2424 calendar days (for example 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 main 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) shown below were used, are used or are proposed.

Calendar rule	Mean year in days		calendar year minus mean tropical year in hh:mm:ss
Old Egyptian	365	= 365. 000 000 000	-05:48:45.25
Julian	365 + ¼	= 365. 250 000 000	00:11:14.75
Gregorian	365 +	= 365. 242 500 000	00:00:26.75
Khayyam	365 +	= 365. 24 24 24 24	00:00:20.20
Revised Julian	365 +	= 365. 24 22 22 22	00:00:02.75
von Mädler	365 +	= 365. 242 187 500	-00:00:00.25
*Mean tropical year at epoch J2000.0*		= 365. 242 190 419	N/A

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 matched the mean tropical year measured in SI seconds about 6000 years ago. With respect to the vernal equinox year measured in mean solar days, important for the calendar date of Easter, the Gregorian year is and stays a very good approximation for thousands of years.
When using the Gregorian calendar in constant time scales (TT or TAI), so when ignoring DeltaT, the vernal equinox 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 von Mädler rule would regularly avoid this shift to 19 March for millennia.

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