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A tropical year or solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky – as viewed from the Earth or another celestial body of the Solar System – thus completing a full cycle of season. For example, it is the time from March equinox to the next vernal equinox, or from summer solstice to the next summer solstice. It is the type of year used by tropical solar calendars.
The tropical year is one type of astronomical year and particular orbital period. Another type is the sidereal year (or sidereal orbital period), which is the time it takes Earth to complete one full orbit around the Sun as measured with respect to the fixed stars, resulting in a duration of 20 minutes longer than the tropical year, because of the precession of the equinoxes.
Since antiquity, astronomers have progressively refined the definition of the tropical year. The entry for "year, tropical" in the Astronomical Almanac Online Glossary states:
An equivalent, more descriptive, definition is "The natural basis for computing passing tropical years is the mean longitude of the Sun reckoned from the precessionally moving equinox (the dynamical equinox or equinox of date). Whenever the longitude reaches a multiple of 360 degrees the mean Sun crosses the vernal equinox and a new tropical year begins".
The mean tropical year in 2000 was 365.24219 , each ephemeris day lasting 86,400 SI seconds. This is 365.24217 mean solar days. For this reason, the calendar year is an approximation of the solar year: the Gregorian calendar (with its rules for catch-up leap year) is designed so as to resynchronize the calendar year with the solar year at regular intervals.
Hipparchus also discovered that the equinoctial points moved along the ecliptic (plane of the Earth's orbit, or what Hipparchus would have thought of as the plane of the Sun's orbit about the Earth) in a direction opposite that of the movement of the Sun, a phenomenon that came to be named "precession of the equinoxes". He reckoned the value as 1° per century, a value that was not improved upon until about 1000 years later, by Islamic astronomers. Since this discovery a distinction has been made between the tropical year and the sidereal year.
The Alfonsine Tables, published in 1252, were based on the theories of Ptolemy and were revised and updated after the original publication. The length of the tropical year was given as 365 solar days 5 hours 49 minutes 16 seconds (≈ 365.24255 days). This length was used in devising the Gregorian calendar of 1582.
In Uzbekistan, Ulugh Beg's Zij-i Sultani was published in 1437 and gave an estimate of 365 solar days 5 hours 49 minutes 15 seconds (365.242535 days).
In the 16th century Copernicus put forward a heliocentric cosmology. Erasmus Reinhold used Copernicus' theory to compute the Prutenic Tables in 1551, and gave a tropical year length of 365 solar days, 5 hours, 55 minutes, 58 seconds (365.24720 days), based on the length of a sidereal year and the presumed rate of precession. This was actually less accurate than the earlier value of the Alfonsine Tables.
Major advances in the 17th century were made by Johannes Kepler and Isaac Newton. In 1609 and 1619 Kepler published his three laws of planetary motion. In 1627, Kepler used the observations of Tycho Brahe and Waltherus to produce the most accurate tables up to that time, the Rudolphine Tables. He evaluated the mean tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds (365.24219 days).
Newton's three laws of dynamics and theory of gravity were published in his Philosophiæ Naturalis Principia Mathematica in 1687. Newton's theoretical and mathematical advances influenced tables by Edmond Halley published in 1693 and 1749 and provided the underpinnings of all solar system models until Albert Einstein's theory of General relativity in the 20th century.
where T is the time in Julian centuries. The derivative of this formula is an expression of the mean angular velocity, and the inverse of this gives an expression for the length of the tropical year as a linear function of T.
Two equations are given in the table. Both equations estimate that the tropical year gets roughly a half second shorter each century.
| +Tropical year coefficients | ||
| Leverrier | Y = − 6.24 T | January 0.5, 1900, Ephemeris time |
| Y = − 6.14 T | January 0, 1900, mean time |
Newcomb's tables were sufficiently accurate that they were used by the joint American-British Astronomical Almanac for the Sun, Mercury, Venus, and Mars through 1983.
The complexity of the model used for the Solar System must be limited to the available computation facilities. In the 1920s punched card equipment came into use by L. J. Comrie in Britain. For the American Ephemeris an electromagnetic computer, the IBM Selective Sequence Electronic Calculator was used since 1948. When modern computers became available, it was possible to compute ephemerides using numerical integration rather than general theories; numerical integration came into use in 1984 for the joint US-UK almanacs.
Albert Einstein's General Theory of Relativity provided a more accurate theory, but the accuracy of theories and observations did not require the refinement provided by this theory (except for the advance of the perihelion of Mercury) until 1984. Time scales incorporated general relativity beginning in the 1970s.
A key development in understanding the tropical year over long periods of time is the discovery that the rate of rotation of the earth, or equivalently, the length of the Solar day, is not constant. William Ferrel in 1864 and Charles-Eugène Delaunay in 1865 predicted that the rotation of the Earth is being retarded by tides. This could be verified by observation only in the 1920s with the very accurate Shortt-Synchronome clock and later in the 1930s when began to replace pendulum clocks as time standards.
However the rotation of the Earth itself is irregular and is slowing down, with respect to more stable time indicators: specifically, the motion of planets, and atomic clocks.
Ephemeris time (ET) is the independent variable in the equations of motion of the Solar System, in particular, the equations from Newcomb's work, and this ET was in use from 1960 to 1984. These ephemerides were based on observations made in solar time over a period of several centuries, and as a consequence represent the mean solar second over that period. The SI second, defined in atomic time, was intended to agree with the ephemeris second based on Newcomb's work, which in turn makes it agree with the mean solar second of the mid-19th century. ET as counted by atomic clocks was given a new name, Terrestrial Time (TT), and for most purposes ET = TT = International Atomic Time + 32.184 SI seconds. Since the era of the observations, the rotation of the Earth has slowed down and the mean solar second has grown somewhat longer than the SI second. As a result, the time scales of TT and UT1 build up a growing difference: the amount that TT is ahead of UT1 is known as Δ T, or Delta T. TT is ahead of UT1 by 69.28 seconds.
As a consequence, the tropical year following the seasons on Earth as counted in solar days of UT is increasingly out of sync with expressions for equinoxes in ephemerides in TT.
As explained below, long-term estimates of the length of the tropical year were used in connection with the reform of the Julian calendar, which resulted in the Gregorian calendar. Participants in that reform were unaware of the non-uniform rotation of the Earth, but now this can be taken into account to some degree. The table below gives Morrison and Stephenson's estimates and ( σ) for ΔT at dates significant in the process of developing the Gregorian calendar.
| 4m20s |
| 2m |
| 20s |
where t is measured in Julian centuries from 1820. The extrapolation is provided only to show Δ T is not negligible when evaluating the calendar for long periods; Borkowski cautions that "many researchers have attempted to fit a parabola to the measured Δ T values in order to determine the magnitude of the deceleration of the Earth's rotation. The results, when taken together, are rather discouraging."
The ecliptic longitude of the Sun is the angle between and the Sun, measured eastward along the ecliptic. This creates a relative and not an absolute measurement, because as the Sun is moving, the direction the angle is measured from is also moving. It is convenient to have a fixed (with respect to distant stars) direction to measure from; the direction of at noon January 1, 2000, fills this role and is given the symbol 0.
There was an equinox on March 20, 2009, 11:44:43.6 TT. The 2010 March equinox was March 20, 17:33:18.1 TT, which gives an interval - and a duration of the tropical year - of 365 days 5 hours 48 minutes 34.5 seconds. While the Sun moves, moves in the opposite direction. When the Sun and met at the 2010 March equinox, the Sun had moved east 359°59'09" while had moved west 51" for a total of 360° (all with respect to 0). This is why the tropical year is 20 min. shorter than the sidereal year.
When tropical year measurements from several successive years are compared, variations are found which are due to the perturbations by the Moon and planets acting on the Earth, and to nutation. Meeus and Savoie provided the following examples of intervals between March (northward) equinoxes:
| 58 |
| 15 |
| 38 |
| 42 |
| 06 |
The "mean tropical year" is based on the mean sun, and is not exactly equal to any of the times taken to go from an equinox to the next or from a solstice to the next.
The following values of time intervals between equinoxes and solstices were provided by Meeus and Savoie for the years 0 and 2000. These are smoothed values which take account of the Earth's orbit being elliptical, using well-known procedures (including solving Kepler's equation). They do not take into account periodic variations due to factors such as the gravitational force of the orbiting Moon and gravitational forces from the other planets. Such perturbations are minor compared to the positional difference resulting from the orbit being elliptical rather than circular.
| days |
where T is in Julian centuries of 36,525 days of 86,400 SI seconds measured from noon January 1, 2000, TT.In negative numbers for dates in the past; , calculated from planetary model of .
Modern astronomers define the tropical year as time for the Mean sun to increase by 360°. The process for finding an expression for the length of the tropical year is to first find an expression for the Sun's mean longitude (with respect to ), such as Newcomb's expression given above, or Laskar's expression. When viewed over a one-year period, the mean longitude is very nearly a linear function of Terrestrial Time. To find the length of the tropical year, the mean longitude is differentiated, to give the angular speed of the Sun as a function of Terrestrial Time, and this angular speed is used to compute how long it would take for the Sun to move 360°.
The above formulae give the length of the tropical year in ephemeris days (equal to 86,400 SI seconds), not . It is the number of solar days in a tropical year that is important for keeping the calendar in synch with the seasons (see below).
The Gregorian calendar is a reformed version of the Julian calendar organized by the Catholic Church and enacted in 1582. By the time of the reform, the date of the vernal equinox had shifted about 10 days, from about March 21 at the time of the First Council of Nicaea in 325, to about March 11. The motivation for the change was the correct observance of Easter. The rules used to Computus used a conventional date for the vernal equinox (March 21), and it was considered important to keep March 21 close to the actual equinox.
If society in the future still attaches importance to the synchronization between the civil calendar and the seasons, another reform of the calendar will eventually be necessary. According to Blackburn and Holford-Strevens (who used Newcomb's value for the tropical year) if the tropical year remained at its 1900 value of days the Gregorian calendar would be 3 days, 17 min, 33 s behind the Sun after 10,000 years. Aggravating this error, the length of the tropical year (measured in Terrestrial Time) is decreasing at a rate of approximately 0.53 s per century and the mean solar day is getting longer at a rate of about 1.5 ms per century. These effects will cause the calendar to be nearly a day behind in 3200. The number of solar days in a "tropical millennium" is decreasing by about 0.06 per millennium (neglecting the oscillatory changes in the real length of the tropical year).365242×1.5/8640000. This means there should be fewer and fewer leap days as time goes on. One possible reform that has been proposed involves omitting the leap day in 3200, keeping 3600 and 4000 as leap years, and making all centennial years common except 4500, 5000, 5500, 6000, etc. (i.e. making centennial leap years occur once every 500 years instead of 400 starting from the year 4000), but the quantity ΔT is not sufficiently predictable to form more precise proposals.
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