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A tropical year (also known as a solar year) is the length of time the Sun,
as seen 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 length of time depends on the point of the ecliptic. 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, the progress of the tropical year seems to slow the Sun from south to
north and back. The word "tropical" comes from the Greek tropos meaning "turn".
The tropics of Cancer and Capricorn mark the extreme north and south latitudes
where the Sun can appear directly overhead. 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 to measure
the year in the sense that the variations of insolation drive the seasons.
The vernal equinox moves back along the ecliptic caused by 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, 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.
Astronomers defined a mean tropical year, an average over all points on the
ecliptic, with a length of about 365.24219 SI days. Tropical years have been
defined for specific points on the ecliptic. The vernal equinox year begins and
ends when the Sun is at the vernal equinox. Its length is about 365.2424 days.
Time can be measure 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.
Error in Statement of Tropical Year explains using the value of the "mean
tropical year" to refer to the vernal equinox year defined above is 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
mean solar days is the basis of most solar calendars, but not the "tropical
year" of modern astronomers.
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. 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 Moisson was:
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)
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:
difference (days) = ?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.
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 are:
vernal equinox 365.242 374 04 + 0.000 000 103 38×a days
northern solstice 365.241 626 03 + 0.000 000 006 50×a days
autumn equinox 365.242 017 67 ? 0.000 000 231 50×a days
southern solstice 365.242 740 49 ? 0.000 000 124 46×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, 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 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 church therefore made it 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.
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 Match mean tropical year in SI time
Old Egyptian 365 = 365. 000 000 000 in very far future (several million years)
Julian 365 + ? = 365. 250 000 000 several hundred thousand years ago
Gregorian 365 + ? - 3/400 = 365. 242 500 000 at about 4000 BC
Khayyam 365 + 8/33 = 365. 24 24 24 24 at about 1000 AD
Mean tropical year at epoch 2000.0 = 365. 242 190 419 astronomical comparsion
value
von M?dler 365 + 31/128 = 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 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 M?dler rule would
regularly avoid this shift to 19 March for millennia.
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