Tropical year
Encyclopedia
A tropical year for general purposes, 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. Because of the precession of the equinoxes, the seasonal cycle does not remain exactly synchronised with the position of the Earth in its orbit around the Sun. As a consequence, the tropical year is about 20 minutes shorter than the time it takes Earth to complete one full orbit around the Sun as measured with respect to the fixed stars (the sidereal year
).
Since antiquity, astronomers have progressively refined the definition of the tropical year, and currently define it as the time required for the mean Sun's tropical longitude (longitudinal position along the ecliptic
relative to its position at the vernal equinox) to increase by 360 degree
s (that is, to complete one full seasonal circuit). (Meeus & Savoie, 1992, p. 40)
tropikos meaning "turn". (tropic, 1992) Thus, the tropics of Cancer
and Capricorn
mark the extreme north and south latitude
s 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". The early Chinese, Hindus, Greeks, and others made approximate measures of the tropical year; early astronomers did so by noting the time required between the appearance of the Sun in one of the tropics to the next appearance in the same tropic. (Meeus & Savoie, 1992, p. 40)
introduced a new definition which was still used by some authors in the 20th century, the time required for the Sun to travel from an equinox
to the same equinox again. He measured the length of the year to be 365 solar days, 5 hours, 55 minutes, 12 seconds. A modern computer model gives 365 solar days, 5 hours 49 minutes 19 seconds. He adopted the new definition because the instrument he used, the meridian armillae
, was better able to detect the more rapid motion in declination at the time of the equinoxes, compared to the solstices. (Meeus & Savoie, 1992, p. 40)
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 measured the value as 1° per century, a value that was not improved upon until about 1000 years later, by Arabian astronomers. Since this discovery a distinction has been made between the tropical year and the sidereal year
. (Meeus & Savoie, 1992, p. 40)
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 most recent update in 1978 was by the French National Centre for Scientific Research. The length of the tropical year (using the equinox-based definition) was 365 solar days 5 hours 49 minutes 16 seconds. It was these tables that were used in the reform process that led to the Gregorian calendar
. (Meeus & Savoie, 1992, p. 41)
In the 16th century Copernicus put forward a heliocentric cosmology
. Erasmus Reinhold used Copernicus' theory to compute the Prutenic Tables
in 1551, and found a tropical year length of 365 solar days, 5 hours, 55 minutes, 58 seconds. (Meeus & Savoie, 1992, p. 41)
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. (McCarthy & Seidelmann, 2009, p. 26) 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 tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds. (Meeus & Savoie, 1992, p. 41)
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 Edmund Halley published in 1693 and 1749. (McCarthy & Seidelmann, 2009, pp. 26–28) and provided the underpinnings of all solar system models until Albert Einstein
's theory of General relativity
in the 20th century.
(irregular motions of the axis of rotation of the earth, the main cycle being 18.6 years) and the movement of the Sun caused by the gravitational pull of the planets. 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 either precise observations or an elaborate theory of the motion of the Sun. The necessary theories and mathematical tools came together in the 18th century due to the work of Pierre-Simon Laplace
, Joseph Louis Lagrange
, and other specialists in celestial mechanics
. They were able to express the mean longitude of the Sun as
where T is the time in Julian centuries. The inverse of the derivative of L0, dT/dL0 gives the length of the tropical year as a linear function of T. When this is computed, an expression giving the length of the tropical year as function of T results.
Two equations are given in the table. Both equations estimate the tropical year gets roughly a half second shorter each century.
Newcomb's tables were successful enough that they were used by the joint American-British Astronomical Almanac
for the Sun, Mecury
, Venus
, and Mars
through 1983. (Seidelmann, 1992, p. 317)
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. At 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. (McCarthy & Seidelmann, 2009, p. 32)
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. (McCarthy & Seidelmann, 2009, p. 37)
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 indicated the rotation of the Earth was being retarded by tides. In 1921 William H Shortt invented the Shortt-Synchronome clock
, the most accurate commercially produced pendulum clock; it was the first clock capable of measuring variations in the Earth's rotation. The next major time-keeping advance was the quartz clock
, first built by Warren Marrison and J. W. Horton in 1927; in the late 1930s quartz clocks began to replace pendulum clocks as time standards. (McCarthy and Seidelmann, 2009, ch. 9)
A series of experiments beginning in the late 1930s led to the development of the first atomic clock
by Louis Essen
and J. V. L. Parry in 1955. Their clock was based on a transition in the cesium atom. (McCarthy & Seidelmann, 2009, pp. 157–9) Due to the accuracy the General Conference on Weights and Measures
in 1960 redefined the second in terms of the cesium transition. The atomic second, often called the SI
second, 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. (McCarthy & Seidelman, 2009, pp. 81–2, 191–7)
(especially the UT1 variant), which is the mean solar time at 0 degrees longitude
(the Greenwich meridian). One second of UT is 1/86,400 of a mean solar day. This time scale is known to be somewhat variable. Since all civil calendars count actual solar days, all civil calendars are based on UT.
The other time scale has two parts. Ephemeris time (ET) is the independent variable in the equations of motion of the solar system, in particular, the equations in use from 1960 to 1984. (McCarthy & Seidelmann, 2009, p. 378) That is, the length of the second used in the solar system calculations could be adjusted until the length that gives the best agreement with observations is found. With the introduction of atomic clocks in the 1950s, it was found that ET could be better realized as atomic time. This also means that ET is a uniform time scale, as is atomic time. ET was given a new name, Terrestrial Time
(TT), and for most purposes ET = TT = International Atomic Time
+ 32.184 SI seconds. As of January 2010, TT is ahead of UT1 by about 66 seconds. (International Earth Rotation Service, 2010; McCarthy & Seidelman, 2009, pp. 86–7).
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
. Of course the 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 amount that TT is ahead of UT1 is known as ΔT, or Delta T. The table below gives Morrison and Stephenson's (S & M) 2004 estimates and standard error
s (σ) for dates significant in the process of developing the Gregorian calendar.
The low precision extrapolations are computed with an expression provided by Morrison and Stephenson
where t is measured in Julian centuries from 1820. The extrapolation is provided only to show ΔT is not negligible when evaluating the calendar for long periods; Borkowski (1991, p. 126) cautions that "many researchers have attempted to fit a parabola to the measured ΔT values in order to determine the magnitude of the deceleration of the Earth's rotation. The results, when taken together, are rather discouraging."
(the Earth's equator projected into space). These two planes intersect in a line. The direction along the line from the Earth in the general direction of the zodiac sign
Aries
(Ram) is the March equinox, and is given the symbol ♈ (the symbol looks like the horns of a ram
).
The opposite direction, along the line in the general direction of the sign Libra
, is the September equinox and is given the symbol ♎. Because of precession and nutation these directions change, compared to the direction of distant stars and galaxies, whose directions have no measurable motion due to their great distance (see International Celestial Reference Frame
).
The ecliptic longitude of the Sun is the angle between ♈ and the Sun, measured eastward along the ecliptic. This creates a complicated 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.
Using the over-simplified definition, 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 a duration of 365 d 5 h 49 m 30s. (Astronomical Applications Dept., 2009) 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). (Seidelmann, 1992, p. 104, expression for pA)
If a different starting longitude for the Sun is chosen, the duration for the Sun to return to the same longitude will be different. This is because although ♈ changes at a nearly steady rate there is considerable variation in the angular speed of the Sun. Thus, the 50 or so arcseconds that the Sun does not have to move to complete the tropical year "saves" varying amounts of time depending on the position in the orbit.
, and to the planetary perturbations
acting on the Sun. Meeus and Savoie (1992, p. 41) provided the following examples of intervals between northward equinoxes:
Until the beginning of the 19th century, the length of the tropical year was found by comparing equinox dates that were separated by many years; this approach yielded the mean tropical year. (Meeus & Savoie, 1992, p. 42)
Values of mean time intervals between equinoxes and solstices were provided by Meeus and Savoie (1992, p. 42) for the years 0 and 2000.
− T − T2 + T3
where T is in Julian centuries of 36,525 days measured from noon January 1, 2000 TT (in negative numbers for dates in the past). (McCarthy & Seidelmann, 2009, p. 18.; Laskar, 1986)
Modern astronomers define the tropical year as time for the Sun's mean longitude 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 (1986, p. 64). When viewed over a 1 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°. (Meeus & Savoie, 1992, p. 42).
, as used for civil purposes, is an international standard. It is a solar calendar, meaning that it is designed to maintain synchrony with the tropical year. It has a cycle of 400 years (146,097 days). Each cycle repeats the months, dates, and weekdays. The average year length is 146,097/400 = 365+97/400 = 365.2425 days per year, a close approximation to the tropical year. (Seidelmann, 1992, pp. 576–81)
The Gregorian calendar is a reformed version of the Julian calendar
. By the time of the reform in 1582, 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. According to North, the real motivation for reform was not primarily a matter of getting agricultural cycles back to where they had once been in the seasonal cycle; the primary concern of Christians was the correct observance of Easter. The rules used to compute the date of Easter
used a conventional date for the vernal equinox (March 21), and it was considered important to keep March 21 close to the actual equinox. (North, 1983, pp. 75–76)
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 100 tropical years. Also, the mean solar day is getting longer at a rate of about 1.5 ms per 100 tropical years. These effects will cause the calendar to be nearly a day behind in 3200. A possible reform would be to omit the leap day in 3200, keep 3600 and 4000 as leap years, and thereafter make all centennial years common except 4500, 5000, 5500, 6000, etc. The effects are not sufficiently predictable to form more precise proposals. (Blackburn & Holford-Strevens, 2003, p. 692)
Borkowski (1991, p. 121) states "because of high uncertainty in the Earth's rotation it is premature at present to suggest any reform that would reach further than a few thousand years into the future." He estimates that in 4000 the Gregorian year (which counts actual solar days) will be behind the tropical year by 0.8 to 1.1 days. (p. 126)
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
takes to return to the same position in the cycle of seasons, as seen from Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
; for example, the time from vernal equinox to vernal equinox, or from summer solstice
Summer solstice
The summer solstice occurs exactly when the axial tilt of a planet's semi-axis in a given hemisphere is most inclined towards the star that it orbits. Earth's maximum axial tilt to our star, the Sun, during a solstice is 23° 26'. Though the summer solstice is an instant in time, the term is also...
to summer solstice. Because of the precession of the equinoxes, the seasonal cycle does not remain exactly synchronised with the position of the Earth in its orbit around the Sun. As a consequence, the tropical year is about 20 minutes shorter than the time it takes Earth to complete one full orbit around the Sun as measured with respect to the fixed stars (the sidereal year
Sidereal year
A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...
).
Since antiquity, astronomers have progressively refined the definition of the tropical year, and currently define it as the time required for the mean Sun's tropical longitude (longitudinal position along the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
relative to its position at the vernal equinox) to increase by 360 degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s (that is, to complete one full seasonal circuit). (Meeus & Savoie, 1992, p. 40)
Origin
The word "tropical" comes from the GreekGreek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
tropikos meaning "turn". (tropic, 1992) Thus, the tropics of Cancer
Tropic of Cancer
The Tropic of Cancer, also referred to as the Northern tropic, is the circle of latitude on the Earth that marks the most northerly position at which the Sun may appear directly overhead at its zenith...
and Capricorn
Tropic of Capricorn
The Tropic of Capricorn, or Southern tropic, marks the most southerly latitude on the Earth at which the Sun can be directly overhead. This event occurs at the December solstice, when the southern hemisphere is tilted towards the Sun to its maximum extent.Tropic of Capricorn is one of the five...
mark the extreme north and south latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
s 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". The early Chinese, Hindus, Greeks, and others made approximate measures of the tropical year; early astronomers did so by noting the time required between the appearance of the Sun in one of the tropics to the next appearance in the same tropic. (Meeus & Savoie, 1992, p. 40)
Early value, precession discovery
In the 2nd century BC HipparchusHipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
introduced a new definition which was still used by some authors in the 20th century, the time required for the Sun to travel from an equinox
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...
to the same equinox again. He measured the length of the year to be 365 solar days, 5 hours, 55 minutes, 12 seconds. A modern computer model gives 365 solar days, 5 hours 49 minutes 19 seconds. He adopted the new definition because the instrument he used, the meridian armillae
Armillary sphere
An armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...
, was better able to detect the more rapid motion in declination at the time of the equinoxes, compared to the solstices. (Meeus & Savoie, 1992, p. 40)
Hipparchus also discovered that the equinoctial points moved along the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
(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 measured the value as 1° per century, a value that was not improved upon until about 1000 years later, by Arabian astronomers. Since this discovery a distinction has been made between the tropical year and the sidereal year
Sidereal year
A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...
. (Meeus & Savoie, 1992, p. 40)
Middle Ages and the Renaissance
During the Middle Ages and Renaissance a number of progressively better tables were published that allowed computation of the positions of the Sun, MoonMoon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
and planets relative to the fixed stars. An important application of these tables was the reform of the calendar
Calendar reform
A calendar reform is any significant revision of a calendar system. The term sometimes is used instead for a proposal to switch to a different calendar.Most calendars have several rules which could be altered by reform:...
.
The Alfonsine Tables
Alfonsine tables
The Alfonsine tables provided data for computing the position of the Sun, Moon and planets relative to the fixed stars....
, published in 1252, were based on the theories of Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
and were revised and updated after the original publication; the most recent update in 1978 was by the French National Centre for Scientific Research. The length of the tropical year (using the equinox-based definition) was 365 solar days 5 hours 49 minutes 16 seconds. It was these tables that were used in the reform process that led to the Gregorian calendar
Gregorian calendar
The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...
. (Meeus & Savoie, 1992, p. 41)
In the 16th century Copernicus put forward a heliocentric cosmology
Copernican heliocentrism
Copernican heliocentrism is the name given to the astronomical model developed by Nicolaus Copernicus and published in 1543. It positioned the Sun near the center of the Universe, motionless, with Earth and the other planets rotating around it in circular paths modified by epicycles and at uniform...
. Erasmus Reinhold used Copernicus' theory to compute the Prutenic Tables
Prutenic Tables
The Prutenic Tables , were an ephemeris by the astronomer Erasmus Reinhold published in 1551. They are sometimes called the Prussian Tables after Albert I, Duke of Prussia, who supported Reinhold and financed the printing...
in 1551, and found a tropical year length of 365 solar days, 5 hours, 55 minutes, 58 seconds. (Meeus & Savoie, 1992, p. 41)
Major advances in the 17th century were made by Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
and Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
. In 1609 and 1619 Kepler published his three laws of planetary motion. (McCarthy & Seidelmann, 2009, p. 26) In 1627, Kepler used the observations of Tycho Brahe
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...
and Waltherus to produce the most accurate tables up to that time, the Rudolphine Tables. He evaluated the tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds. (Meeus & Savoie, 1992, p. 41)
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 Edmund Halley published in 1693 and 1749. (McCarthy & Seidelmann, 2009, pp. 26–28) and provided the underpinnings of all solar system models until Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theory of General relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
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) a number of years apart, to average out both observational errors and the effects of nutationNutation
Nutation is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behavior of a mechanism...
(irregular motions of the axis of rotation of the earth, the main cycle being 18.6 years) and the movement of the Sun caused by the gravitational pull of the planets. 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 either precise observations or an elaborate theory of the motion of the Sun. The necessary theories and mathematical tools came together in the 18th century due to the work of Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
, Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
, and other specialists in celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
. They were able to express the mean longitude of the Sun as
- L0 = A0 + A1T + A2T2 days
where T is the time in Julian centuries. The inverse of the derivative of L0, dT/dL0 gives the length of the tropical year as a linear function of T. When this is computed, an expression giving the length of the tropical year as function of T results.
Two equations are given in the table. Both equations estimate the tropical year gets roughly a half second shorter each century.
| Name | Equation | Date on which T = 0 |
|---|---|---|
| Leverrier (Meeus & Savoie, 1992, p. 42) | Y = − 6.24 T | January 0.5, 1900, Ephemeris Time Ephemeris time The term ephemeris time can in principle refer to time in connection with any astronomical ephemeris. In practice it has been used more specifically to refer to:... |
| Newcomb Simon Newcomb Simon Newcomb was a Canadian-American astronomer and mathematician. Though he had little conventional schooling, he made important contributions to timekeeping as well as writing on economics and statistics and authoring a science fiction novel.-Early life:Simon Newcomb was born in the town of... (1898, p. 9–10) |
Y = − 6.14 T | January 0, 1900, mean time |
Newcomb's tables were successful enough that they were used by the joint American-British Astronomical Almanac
Astronomical Almanac
The Astronomical Almanac is an almanac published by the United States Naval Observatory and Her Majesty's Nautical Almanac Office, containing solar system ephemeris and catalogs of selected stellar and extragalactic objects....
for the Sun, Mecury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
, Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
, and Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
through 1983. (Seidelmann, 1992, p. 317)
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 4Pioneer 4Pioneer 4 was a spin-stabilized spacecraft launched as part of the Pioneer program on a lunar flyby trajectory and into a heliocentric orbit making it the first U.S. probe to escape from the Earth's gravity. It carried a payload similar to Pioneer 3: a lunar radiation environment experiment using a...
beginning in 1959 (Jet Propulsion Laboratory 2005) - radarsRadar astronomyRadar astronomy is a technique of observing nearby astronomical objects by reflecting microwaves off target objects and analyzing the echoes. This research has been conducted for six decades. Radar astronomy differs from radio astronomy in that the latter is a passive observation and the former an...
able to measure other planets beginning in 1961 (Butrica, 1996) - lunar laser ranging since the 1969 Apollo 11Apollo 11In early 1969, Bill Anders accepted a job with the National Space Council effective in August 1969 and announced his retirement as an astronaut. At that point Ken Mattingly was moved from the support crew into parallel training with Anders as backup Command Module Pilot in case Apollo 11 was...
left the first of a series of retroreflectorRetroreflectorA retroreflector is a device or surface that reflects light back to its source with a minimum scattering of light. An electromagnetic wave front is reflected back along a vector that is parallel to but opposite in direction from the wave's source. The device or surface's angle of incidence is...
s which allow greater accuracy than reflectorless measurements - artificial satellites such as LAGEOSLAGEOSLAGEOS, or Laser Geodynamics Satellites, are a series of scientific research satellites designed to provide an orbiting laser ranging benchmark for geodynamical studies of the Earth...
(1976) and the Global Positioning SystemGlobal Positioning SystemThe Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...
(initial operation in 1993) - Very Long Baseline InterferometryVery Long Baseline InterferometryVery Long Baseline Interferometry is a type of astronomical interferometry used in radio astronomy. It allows observations of an object that are made simultaneously by many telescopes to be combined, emulating a telescope with a size equal to the maximum separation between the telescopes.Data...
which finds precise directions to quasarQuasarA quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...
s 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 (McCarthy & Seidelmann, 2009, p. 265)
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. At 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
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
rather than general theories; numerical integration came into use in 1984 for the joint US-UK almanacs. (McCarthy & Seidelmann, 2009, p. 32)
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. (McCarthy & Seidelmann, 2009, p. 37)
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
Charles-Eugène Delaunay
Charles-Eugène Delaunay was a French astronomer and mathematician. His lunar motion studies were important in advancing both the theory of planetary motion and mathematics.-Life:...
in 1865 indicated the rotation of the Earth was being retarded by tides. In 1921 William H Shortt invented the Shortt-Synchronome clock
Shortt-synchronome clock
The Shortt-Synchronome free pendulum clock was a complex precision electromechanical pendulum clock invented in 1921 by British railway engineer William Hamilton Shortt in collaboration with horologist Frank Hope-Jones, and manufactured by the Synchronome Co., Ltd. of London, UK...
, the most accurate commercially produced pendulum clock; it was the first clock capable of measuring variations in the Earth's rotation. The next major time-keeping advance was the quartz clock
Quartz clock
A quartz clock is a clock that uses an electronic oscillator that is regulated by a quartz crystal to keep time. This crystal oscillator creates a signal with very precise frequency, so that quartz clocks are at least an order of magnitude more accurate than good mechanical clocks...
, first built by Warren Marrison and J. W. Horton in 1927; in the late 1930s quartz clocks began to replace pendulum clocks as time standards. (McCarthy and Seidelmann, 2009, ch. 9)
A series of experiments beginning in the late 1930s led to the development of the first atomic clock
Atomic clock
An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...
by Louis Essen
Louis Essen
Louis Essen FRS O.B.E. was an English physicist whose most notable achievements were in the precise measurement of time and the determination of the speed of light...
and J. V. L. Parry in 1955. Their clock was based on a transition in the cesium atom. (McCarthy & Seidelmann, 2009, pp. 157–9) Due to the accuracy the General Conference on Weights and Measures
General Conference on Weights and Measures
The General Conference on Weights and Measures is the English name of the Conférence générale des poids et mesures . It is one of the three organizations established to maintain the International System of Units under the terms of the Convention du Mètre of 1875...
in 1960 redefined the second in terms of the cesium transition. The atomic second, often called the SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
second, 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. (McCarthy & Seidelman, 2009, pp. 81–2, 191–7)
Time scales
As mentioned in History, advances in time-keeping have resulted in various time scales. One useful time scale is Universal TimeUniversal Time
Universal Time is a time scale based on the rotation of the Earth. It is a modern continuation of Greenwich Mean Time , i.e., the mean solar time on the Prime Meridian at Greenwich, and GMT is sometimes used loosely as a synonym for UTC...
(especially the UT1 variant), which is the mean solar time at 0 degrees longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
(the Greenwich meridian). One second of UT is 1/86,400 of a mean solar day. This time scale is known to be somewhat variable. Since all civil calendars count actual solar days, all civil calendars are based on UT.
The other time scale has two parts. Ephemeris time (ET) is the independent variable in the equations of motion of the solar system, in particular, the equations in use from 1960 to 1984. (McCarthy & Seidelmann, 2009, p. 378) That is, the length of the second used in the solar system calculations could be adjusted until the length that gives the best agreement with observations is found. With the introduction of atomic clocks in the 1950s, it was found that ET could be better realized as atomic time. This also means that ET is a uniform time scale, as is atomic time. ET was given a new name, Terrestrial Time
Terrestrial Time
Terrestrial Time is a modern astronomical time standard defined by the International Astronomical Union, primarily for time-measurements of astronomical observations made from the surface of the Earth....
(TT), and for most purposes ET = TT = International Atomic Time
International Atomic Time
International Atomic Time is a high-precision atomic coordinate time standard based on the notional passage of proper time on Earth's geoid...
+ 32.184 SI seconds. As of January 2010, TT is ahead of UT1 by about 66 seconds. (International Earth Rotation Service, 2010; McCarthy & Seidelman, 2009, pp. 86–7).
As explained below, long term estimates of the length of the tropical year were used in connection with the
reform of the Julian calendar
Julian calendar
The Julian calendar began in 45 BC as a reform of the Roman calendar by Julius Caesar. It was chosen after consultation with the astronomer Sosigenes of Alexandria and was probably designed to approximate the tropical year .The Julian calendar has a regular year of 365 days divided into 12 months...
, which resulted in the Gregorian calendar
Gregorian calendar
The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...
. Of course the 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 amount that TT is ahead of UT1 is known as ΔT, or Delta T. The table below gives Morrison and Stephenson's (S & M) 2004 estimates and standard error
Standard error
Standard error can refer to:* Standard error , the estimated standard deviation or error of a series of measurements* Standard error stream, one of the standard streams in Unix-like operating systems...
s (σ) for dates significant in the process of developing the Gregorian calendar.
| Event | Year | Nearest S & M Year | ΔT | σ |
|---|---|---|---|---|
| Julian calendar begins | −44 | 0 | 2h56m20s | 4m20s |
| First Council of Nicaea First Council of Nicaea The First Council of Nicaea was a council of Christian bishops convened in Nicaea in Bithynia by the Roman Emperor Constantine I in AD 325... |
325 | 300 | 2h8m | 2m |
| Gregorian calendar begins | 1583 | 1600 | 2m | 20s |
| low precision extrapolation | 4000 | 4h13m | ||
| low precision extrapolation | 10,000 | 2d11h |
The low precision extrapolations are computed with an expression provided by Morrison and Stephenson
- ΔT = −20 + 32t2
where t is measured in Julian centuries from 1820. The extrapolation is provided only to show ΔT is not negligible when evaluating the calendar for long periods; Borkowski (1991, p. 126) cautions that "many researchers have attempted to fit a parabola to the measured ΔT values in order to determine the magnitude of the deceleration of the Earth's rotation. The results, when taken together, are rather discouraging."
Length of tropical year
An over-simplified definition of the tropical year would be the time required for the Sun, beginning at a chosen ecliptic longitude, to make one complete cycle of the seasons and return to the same ecliptic longitude. Before considering an example, the equinox must be examined. There are two important planes in solar system calculations, the plane of the ecliptic (the Earth's orbit around the Sun), and the plane of the celestial equatorCelestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...
(the Earth's equator projected into space). These two planes intersect in a line. The direction along the line from the Earth in the general direction of the zodiac sign
Astrological sign
Astrological signs represent twelve equal segments or divisions of the zodiac. According to astrology, celestial phenomena reflect or govern human activity on the principle of "as above, so below", so that the twelve signs are held to represent twelve basic personality types or characteristic modes...
Aries
Aries (astrology)
Aries is the first astrological sign in the Zodiac, which spans the zodiac between the zero degree and the 29th degree of celestial longitude. The Sun enters Aries when it reaches the northern vernal equinox, which is usually on March 21 each year, and remains in this sign until around April 20...
(Ram) is the March equinox, and is given the symbol ♈ (the symbol looks like the horns of a ram
Bighorn Sheep
The bighorn sheep is a species of sheep in North America named for its large horns. These horns can weigh up to , while the sheep themselves weigh up to . Recent genetic testing indicates that there are three distinct subspecies of Ovis canadensis, one of which is endangered: Ovis canadensis sierrae...
).
The opposite direction, along the line in the general direction of the sign Libra
Libra (astrology)
Libra is the seventh astrological sign in the Zodiac, originating from the constellation of Libra. In astrology, Libra is considered a "masculine", positive sign. It is also considered an air sign and is one of four cardinal signs...
, is the September equinox and is given the symbol ♎. Because of precession and nutation these directions change, compared to the direction of distant stars and galaxies, whose directions have no measurable motion due to their great distance (see International Celestial Reference Frame
International Celestial Reference Frame
The International Celestial Reference Frame is a quasi-inertial reference frame centered at the barycenter of the Solar System, defined by the measured positions of 212 extragalactic sources . Although relativity implies that there is no true inertial frame, the extragalactic sources used to...
).
The ecliptic longitude of the Sun is the angle between ♈ and the Sun, measured eastward along the ecliptic. This creates a complicated 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.
Using the over-simplified definition, 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 a duration of 365 d 5 h 49 m 30s. (Astronomical Applications Dept., 2009) 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). (Seidelmann, 1992, p. 104, expression for pA)
If a different starting longitude for the Sun is chosen, the duration for the Sun to return to the same longitude will be different. This is because although ♈ changes at a nearly steady rate there is considerable variation in the angular speed of the Sun. Thus, the 50 or so arcseconds that the Sun does not have to move to complete the tropical year "saves" varying amounts of time depending on the position in the orbit.
Mean equinox tropical year
As already mentioned, there is some choice in the length of the tropical year depending on the point of reference that one selects. But during the period when return of the Sun to a chosen longitude was the method in use by astronomers, one of the equinoxes was usually chosen because the instruments were most sensitive there. When tropical year measurements from several successive years are compared, variations are found which are due to nutationNutation
Nutation is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behavior of a mechanism...
, and to the planetary perturbations
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....
acting on the Sun. Meeus and Savoie (1992, p. 41) provided the following examples of intervals between northward equinoxes:
| days | hours | min | s | |
|---|---|---|---|---|
| 1985–1986 | 365 | 5 | 48 | 58 |
| 1986–1987 | 365 | 5 | 49 | 15 |
| 1987–1988 | 365 | 5 | 46 | 38 |
| 1988–1989 | 365 | 5 | 49 | 42 |
| 1989–1990 | 365 | 5 | 51 | 06 |
Until the beginning of the 19th century, the length of the tropical year was found by comparing equinox dates that were separated by many years; this approach yielded the mean tropical year. (Meeus & Savoie, 1992, p. 42)
Values of mean time intervals between equinoxes and solstices were provided by Meeus and Savoie (1992, p. 42) for the years 0 and 2000.
| Year 0 | Year 2000 | |
|---|---|---|
| Between two March equinoxes | days | days |
| Between two June solstices | ||
| Between two September equinoxes | ||
| Between two December solsticies | ||
| Mean tropical year (Laskar's expression) |
Mean tropical year current value
The mean tropical year, as of January 1, 2000 was or 365 days, 5 hours, 48 minutes, 45.19 seconds. This changes slowly; an expression suitable for calculating the length in days for the distant past is− T − T2 + T3
where T is in Julian centuries of 36,525 days measured from noon January 1, 2000 TT (in negative numbers for dates in the past). (McCarthy & Seidelmann, 2009, p. 18.; Laskar, 1986)
Modern astronomers define the tropical year as time for the Sun's mean longitude 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 (1986, p. 64). When viewed over a 1 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°. (Meeus & Savoie, 1992, p. 42).
Calendar year
The Gregorian calendarGregorian calendar
The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...
, as used for civil purposes, is an international standard. It is a solar calendar, meaning that it is designed to maintain synchrony with the tropical year. It has a cycle of 400 years (146,097 days). Each cycle repeats the months, dates, and weekdays. The average year length is 146,097/400 = 365+97/400 = 365.2425 days per year, a close approximation to the tropical year. (Seidelmann, 1992, pp. 576–81)
The Gregorian calendar is a reformed version of the Julian calendar
Julian calendar
The Julian calendar began in 45 BC as a reform of the Roman calendar by Julius Caesar. It was chosen after consultation with the astronomer Sosigenes of Alexandria and was probably designed to approximate the tropical year .The Julian calendar has a regular year of 365 days divided into 12 months...
. By the time of the reform in 1582, the date of the vernal equinox had shifted about 10 days, from about March 21 at the time of the First Council of Nicaea
First Council of Nicaea
The First Council of Nicaea was a council of Christian bishops convened in Nicaea in Bithynia by the Roman Emperor Constantine I in AD 325...
in 325, to about March 11. According to North, the real motivation for reform was not primarily a matter of getting agricultural cycles back to where they had once been in the seasonal cycle; the primary concern of Christians was the correct observance of Easter. The rules used to compute the date of Easter
Computus
Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....
used a conventional date for the vernal equinox (March 21), and it was considered important to keep March 21 close to the actual equinox. (North, 1983, pp. 75–76)
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 100 tropical years. Also, the mean solar day is getting longer at a rate of about 1.5 ms per 100 tropical years. These effects will cause the calendar to be nearly a day behind in 3200. A possible reform would be to omit the leap day in 3200, keep 3600 and 4000 as leap years, and thereafter make all centennial years common except 4500, 5000, 5500, 6000, etc. The effects are not sufficiently predictable to form more precise proposals. (Blackburn & Holford-Strevens, 2003, p. 692)
Borkowski (1991, p. 121) states "because of high uncertainty in the Earth's rotation it is premature at present to suggest any reform that would reach further than a few thousand years into the future." He estimates that in 4000 the Gregorian year (which counts actual solar days) will be behind the tropical year by 0.8 to 1.1 days. (p. 126)
See also
- Anomalistic year
- Tropical astrology