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 astronomical seasons. For example, it is the time from vernal 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 a 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 and 24.7 seconds longer than the tropical year due to 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 ephemeris days, 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 days) is designed to resynchronize the calendar year with the solar year at regular intervals.
History Origin The word "tropical" comes from the Greek tropikos 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" was lent to the period of the seasonal cycle. The early Chinese, Hindus, Greeks, and others made approximate measures of the tropical year.
Early value, precession discovery In the 2nd century BC, Hipparchus measured the time required for the Sun to travel from an equinox to the same equinox again. He reckoned the length of the year to be 1/300 of a day less than 365.25 days (365 days, 5 hours, 55 minutes, 12 seconds, or 365.24667 days). Hipparchus used this method because he was better able to detect the time of the equinoxes, compared to that of the solstices.
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.
Middle Ages and the Renaissance During the Middle Ages and Renaissance, several progressively better tables were published that allowed computation of the positions of the Sun, Moon and planets relative to the fixed stars. An important application of these tables was the reform of the calendar.
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.
18th and 19th century From the time of Hipparchus and Ptolemy, the year was based on two equinoxes (or two solstices) some years apart, to average out both observational errors and periodic variations (caused by the gravitational pull of the planets, and the small effect of nutation on the equinox). These effects did not begin to be understood until Newton's time. To model short-term variations of the time between equinoxes (and prevent them from confounding efforts to measure long-term variations) requires precise observations and an elaborate theory of the apparent motion of the Sun. The necessary theories and mathematical tools came together in the 18th century due to the work of Pierre-Simon de Laplace, Joseph Louis Lagrange, and other specialists in celestial mechanics. They were able to compute periodic variations and separate them from the gradual mean motion. They could express the mean longitude of the Sun in a polynomial such as:
L0 = A0 + A1T + A2T2 days
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 Name Equation Date on which T = 0 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.
20th and 21st centuries The length of the mean tropical year is derived from a model of the Solar System, so any advance that improves the solar system model potentially improves the accuracy of the mean tropical year. Many new observing instruments became available, including artificial satellites tracking of deep space probes such as Pioneer 4 beginning in 1959 radars able to measure the distance to other planets beginning in 1961 lunar laser ranging since the 1969 Apollo 11 left the first of a series of retroreflectors which allow greater accuracy than reflectorless measurements artificial satellites such as LAGEOS (1976) and the Global Positioning System (initial operation in 1993) very long baseline interferometry which finds precise directions to quasars in distant galaxies, and allows determination of the Earth's orientation with respect to these objects whose distance is so great they can be considered to show minimal space motion.
The complexity of the model used for the Solar System must be limited to the available computational 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 mean 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 quartz clocks began to replace pendulum clocks as time standards.
Time scales and calendar Apparent solar time is the time indicated by a sundial, and is determined by the apparent motion of the Sun caused by the rotation of the Earth around its axis as well as the revolution of the Earth around the Sun. Mean solar time is corrected for the periodic variations in the apparent velocity of the Sun as the Earth revolves in its orbit. The most important such time scale is Universal Time, which is the mean solar time at 0° longitude (the IERS Reference Meridian). Civil time is based on UT (actually UTC), and civil calendars count mean solar days.
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 = . 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 ΔT, or Delta T. TT is ahead of UT1 by approximately 69.110 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 standard errors (σ) for ΔT at dates significant in the process of developing the Gregorian calendar.
Event Year Nearest S & M Year ΔT σ Julian calendar begins −44 0 2h 56m 20s 4m 20s First Council of Nicaea 325 300 2h 8m 2m Gregorian calendar begins 1582 1600 2m 20s Low-precision extrapolation 4000 4h 13m 10,000 2d 11h The low-precision extrapolations are computed with an expression provided by Morrison and Stephenson: