A lunar calendar is a calendar based on the monthly cycles of the Moon 's phases ( synodic months , lunations ), in contrast to solar calendars , whose annual cycles are based on the solar year . The most widely observed purely lunar calendar is the Islamic calendar . A purely lunar calendar is distinguished from a lunisolar calendar , whose lunar months are brought into alignment with the solar year through some process of intercalation – such as by insertion of a leap month . The details of when months begin vary from calendar to calendar, with some using new , full , or crescent moons and others employing detailed calculations.
86-441: Since each lunation is approximately 29 + 1 ⁄ 2 days, it is common for the months of a lunar calendar to alternate between 29 and 30 days. Since the period of 12 such lunations, a lunar year , is 354 days, 8 hours, 48 minutes, 34 seconds (354.36707 days), purely lunar calendars are 11 to 12 days shorter than the solar year . In purely lunar calendars, which do not make use of intercalation,
172-515: A Fourier series ) is also discouraged for having an error of up to 0.28°. An additional error of up to 0.5° can occur in all equations around the equinoxes if not using a decimal place when selecting N to adjust for the time after UT midnight for the beginning of that day. So the above equation can have up to 2.0° of error, about four times the Sun's angular width, depending on how it is used. The declination can be more accurately calculated by not making
258-477: A year , which resembles a figure-8. An analemma can be pictured by superimposing photographs taken at the same time of day, a few days apart for a year . An analemma can also be considered as a graph of the Sun's declination , usually plotted vertically, against the equation of time , plotted horizontally. Usually, the scales are chosen so that equal distances on the diagram represent equal angles in both directions on
344-405: A "tropical millennium" is decreasing by about 0.06 per millennium (neglecting the oscillatory changes in the real length of the tropical year). This means there should be fewer and fewer leap days as time goes on. A possible reform could 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. but the quantity ΔT
430-533: A complete position of the Sun in the ecliptic coordinate system . This can be converted to the equatorial coordinate system by calculating the obliquity of the ecliptic , ϵ {\displaystyle \epsilon } , and continuing: Right ascension , To get RA at the right quadrant on computer programs use double argument Arctan function such as ATAN2(y,x) and declination , Right-handed rectangular equatorial coordinates in astronomical units are: The Sun appears to move northward during
516-408: A constant speed. Thus, the graph of solar declination, as seen from this highly tilted Earth, would resemble a triangle wave rather than a sine wave, zigzagging between plus and minus 90°, with linear segments between the maxima and minima. If the 90° axial tilt is decreased, then the absolute maximum and minimum values of the declination would decrease, to equal the axial tilt. Also, the shapes of
602-463: 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
688-501: 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: the period of time for the ecliptic longitude of the Sun to increase 360 degrees . Since the Sun's ecliptic longitude
774-411: A number of 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
860-470: A sine wave. Calculating it accurately involves some complexity, as shown below. The declination of the Sun , δ ☉ , is the angle between the rays of the Sun and the plane of the Earth's equator. The Earth's axial tilt (called the obliquity of the ecliptic by astronomers) is the angle between the Earth's axis and a line perpendicular to the Earth's orbit. The Earth's axial tilt changes slowly over thousands of years but its current value of about ε = 23.44°
946-416: A sundial would be ahead of a clock. The equation of time can be positive or negative. An analemma is a diagram that shows the annual variation of the Sun's position on the celestial sphere , relative to its mean position, as seen from a fixed location on Earth. (The word analemma is also occasionally, but rarely, used in other contexts.) It can be considered as an image of the Sun's apparent motion during
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#17327768866381032-599: 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 compute the date of Easter used a conventional date for
1118-402: Is an international standard. It is a solar calendar that is designed to maintain synchrony with the mean 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 mean tropical year of 365.2422 days. The Gregorian calendar
1204-432: Is applied, which depends on the observer's distance away from the center of the Earth. This correction is less than 0.0025°. The error in calculating the position of the center of the Sun can be less than 0.00015°. For comparison, the Sun's width is about 0.5°. The declination calculations described above do not include the effects of the refraction of light in the atmosphere, which causes the apparent angle of elevation of
1290-404: Is given the symbol ♎︎ (because it used to be toward Libra ). Because of the precession of the equinoxes 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
1376-409: Is longer: that tropical year is comparatively short. 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
1462-414: Is measured with respect to the equinox, the tropical year comprises a complete cycle of seasons, and its length is approximated in the long term by the civil (Gregorian) calendar. The mean tropical year is approximately 365 days, 5 hours, 48 minutes, 45 seconds. An equivalent, more descriptive, definition is "The natural basis for computing passing tropical years is the mean longitude of the Sun reckoned from
1548-411: Is nearly constant, so the change in solar declination during one year is nearly the same as during the next year. At the solstices , the angle between the rays of the Sun and the plane of the Earth's equator reaches its maximum value of 23.44°. Therefore, δ ☉ = +23.44° at the northern summer solstice and δ ☉ = −23.44° at the southern summer solstice. At the moment of each equinox , the center of
1634-460: Is not sufficiently predictable to form more precise proposals. Position of the Sun The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth 's surface. As Earth orbits the Sun over the course of a year , the Sun appears to move with respect to the fixed stars on the celestial sphere , along a circular path called
1720-514: Is the first day of the month. Some are based on the first sighting of the lunar crescent , such as the Hijri calendar observed by most of Islam. Alternatively, in some lunisolar calendars, such as the Hebrew calendar and Chinese calendar , the first day of a month is the day when an astronomical new moon occurs in a particular time zone. In others, such as some Hindu calendars , each month begins on
1806-467: Is to add an additional month every second or third year. Some lunisolar calendars are also calibrated by annual natural events which are affected by lunar cycles as well as the solar cycle. An example of this is the lunisolar calendar of the Banks Islands , which includes three months in which the edible palolo worms mass on the beaches. These events occur at the last quarter of the lunar month, as
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#17327768866381892-557: Is useful in astronomy , navigation , surveying , meteorology , climatology , solar energy , and sundial design. These equations, from the Astronomical Almanac , can be used to calculate the apparent coordinates of the Sun , mean equinox and ecliptic of date , to a precision of about 0°.01 (36″), for dates between 1950 and 2050. Similar equations are coded into a Fortran 90 routine in Ref. and are used to calculate
1978-467: Is usually done even when the analemma is marked on a geographical globe , on which the continents, etc., are shown with west to the left. Some analemmas are marked to show the position of the Sun on the graph on various dates, a few days apart, throughout the year. This enables the analemma to be used to make simple analog computations of quantities such as the times and azimuths of sunrise and sunset . Analemmas without date markings are used to correct
2064-600: 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
2150-432: The ecliptic . Earth's rotation about its axis causes diurnal motion , so that the Sun appears to move across the sky in a Sun path that depends on the observer's geographic latitude . The time when the Sun transits the observer's meridian depends on the geographic longitude . To find the Sun's position for a given location at a given time, one may therefore proceed in three steps as follows: This calculation
2236-513: The ecliptic longitude of the Sun is: The ecliptic latitude of the Sun is nearly: as the ecliptic latitude of the Sun never exceeds 0.00033° (a little over 1″), and the distance of the Sun from the Earth, in astronomical units , is: Where the obliquity of the ecliptic is not obtained elsewhere, it can be approximated: λ {\displaystyle \lambda } , β {\displaystyle \beta } and R {\displaystyle R} form
2322-474: The solar zenith angle and solar azimuth angle as observed from the surface of the Earth. Start by calculating n , the number of days (positive or negative, including fractional days) since Greenwich noon, Terrestrial Time, on 1 January 2000 ( J2000.0 ). If the Julian date for the desired time is known, then The mean longitude of the Sun, corrected for the aberration of light , is: The mean anomaly of
2408-410: 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
2494-420: The Earth's perihelion . The number 0.0167 is the current value of the eccentricity of the Earth's orbit. The eccentricity varies very slowly over time, but for dates fairly close to the present, it can be considered to be constant. The largest errors in this equation are less than ± 0.2°, but are less than ± 0.03° for a given year if the number 10 is adjusted up or down in fractional days as determined by how far
2580-448: 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. The mean tropical year on January 1, 2000,
2666-517: 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. During the Middle Ages and Renaissance
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2752-539: 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
2838-423: 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 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
2924-550: 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 . As of 5 July 2022, 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
3010-484: The Sun (actually, of the Earth in its orbit around the Sun, but it is convenient to pretend the Sun orbits the Earth), is: Put L {\displaystyle L} and g {\displaystyle g} in the range 0° to 360° by adding or subtracting multiples of 360° as needed — which is to say, L {\displaystyle L} and g {\displaystyle g} are really to be evaluated ( mod 360). Finally,
3096-458: The Sun appears to pass through the celestial equator , and δ ☉ is 0°. The Sun's declination at any given moment is calculated by: where EL is the ecliptic longitude (essentially, the Earth's position in its orbit). Since the Earth's orbital eccentricity is small, its orbit can be approximated as a circle which causes up to 1° of error. The circle approximation means the EL would be 90° ahead of
3182-462: 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 solar days . 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 , as used for civil and scientific purposes,
3268-412: The Sun as seen by an observer to be higher than the actual angle of elevation, especially at low Sun elevations. For example, when the Sun is at an elevation of 10°, it appears to be at 10.1°. The Sun's declination can be used, along with its right ascension , to calculate its azimuth and also its true elevation, which can then be corrected for refraction to give its apparent position. In addition to
3354-410: The Sun's declination during a year resembles a sine wave with an amplitude of 23.44°, but one lobe of the wave is several days longer than the other, among other differences. The following phenomena would occur if Earth were a perfect sphere , in a circular orbit around the Sun, and if its axis were tilted 90°, so that the axis itself is on the orbital plane (similar to Uranus ). At one date in
3440-469: 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
3526-749: The accuracy of the mean tropical year. Many new observing instruments became available, including 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
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3612-450: The annual north–south oscillation of the Sun's apparent position, corresponding to the variation of its declination described above, there is also a smaller but more complex oscillation in the east–west direction. This is caused by the tilt of the Earth's axis, and also by changes in the speed of its orbital motion around the Sun produced by the elliptical shape of the orbit. The principal effects of this east–west oscillation are variations in
3698-400: The apparent position of the Sun, compared with its mean position. A westward shift causes the sundial to be ahead of the clock. Since the main effect of this oscillation concerns time, it is called the equation of time , using the word "equation" in a somewhat archaic sense meaning "correction". The oscillation is measured in units of time, minutes and seconds, corresponding to the amount that
3784-448: The apparent speed of the Sun) varies in its elliptical orbit: faster in the perihelion , slower in the aphelion . The equinox moves with respect to the perihelion (and both move with respect to the fixed sidereal frame). From one equinox passage to the next, or from one solstice passage to the next, the Sun completes not quite a full elliptic orbit. The time saved depends on where it starts in
3870-511: 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 degrees 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,
3956-401: The celestial sphere. Thus 4 minutes (more precisely 3 minutes, 56 seconds), in the equation of time, are represented by the same distance as 1° in the declination , since Earth rotates at a mean speed of 1° every 4 minutes, relative to the Sun. An analemma is drawn as it would be seen in the sky by an observer looking upward. If north is shown at the top, then west is to the right . This
4042-534: The day after the full moon. The length of each lunar cycle varies slightly from the average value. In addition, observations are subject to uncertainty and weather conditions. Thus, to minimise uncertainty, there have been attempts to create fixed arithmetical rules to determine the start of each calendar month. The best known of these is the Tabular Islamic calendar : in brief, it has a 30-year cycle with 11 leap years of 355 days and 19 years of 354 days. In
4128-522: The days part of the ordinal date −1). The number 10, in (N+10), is the approximate number of days after the December solstice to January 1. This equation overestimates the declination near the September equinox by up to +1.5°. The sine function approximation by itself leads to an error of up to 0.26° and has been discouraged for use in solar energy applications. The 1971 Spencer formula (based on
4214-420: The ecliptic longitude by using terms in addition to the 1st-order eccentricity correction above. They also correct the 23.44° obliquity which changes very slightly with time. Corrections may also include the effects of the moon in offsetting the Earth's position from the center of the pair's orbit around the Sun. After obtaining the declination relative to the center of the Earth, a further correction for parallax
4300-401: The gradual mean motion. They could express the mean longitude of the Sun in a polynomial such as: 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
4386-404: 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
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#17327768866384472-413: 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. If a different starting longitude for the Sun is chosen than 0° ( i.e. ♈︎), then the duration for the Sun to return to the same longitude will be different. This is a second-order effect of the circumstance that the speed of the Earth (and conversely
4558-421: 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
4644-572: The long term, it is accurate to one day in about 2,500 solar years or 2,570 lunar years. It also deviates from observation by up to about one or two days in the short term. The algorithm was introduced by Muslim astronomers in the 8th century to predict the approximate date of the first crescent moon, which is used to determine the first day of each month in the Islamic lunar calendar . Most calendars referred to as "lunar" calendars are in fact lunisolar calendars . Their months are based on observations of
4730-532: The lunar cycle, with periodic intercalation being used to restore them into general agreement with the solar year. The solar " civic calendar " that was used in ancient Egypt showed traces of its origin in the earlier lunar calendar, which continued to be used alongside it for religious and agricultural purposes. Present-day lunisolar calendars include the Chinese , Korean , Vietnamese , Hindu , Hebrew and Thai calendars. The most common form of intercalation
4816-559: The lunar months cycle through all the seasons of a solar year over the course of a 33–34 lunar-year cycle (see, e.g., list of Islamic years ). A lunisolar calendar was found at Warren Field in Scotland and has been dated to c. 8000 BC , during the Mesolithic period . Some scholars argue for lunar calendars still earlier— Rappenglück in the marks on a c. 17,000 year-old cave painting at Lascaux and Marshack in
4902-450: The magnitude of the deceleration of the Earth's rotation. The results, when taken together, are rather discouraging." One 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:
4988-520: The marks on a c. 27,000 year-old bone baton—but their findings remain controversial. Scholars have argued that ancient hunters conducted regular astronomical observations of the Moon back in the Upper Palaeolithic . Samuel L. Macey dates the earliest uses of the Moon as a time-measuring device back to 28,000–30,000 years ago. Lunar and lunisolar calendars differ as to which day
5074-444: The maxima and minima on the graph would become less acute, being curved to resemble the maxima and minima of a sine wave. However, even when the axial tilt equals that of the actual Earth, the maxima and minima remain more acute than those of a sine wave. In reality, Earth's orbit is elliptical . Earth moves more rapidly around the Sun near perihelion , in early January, than near aphelion , in early July. This makes processes like
5160-448: 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,
5246-428: 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
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#17327768866385332-560: The northern spring , crossing the celestial equator on the March equinox . Its declination reaches a maximum equal to the angle of Earth's axial tilt (23.44° or 23°26') on the June solstice , then decreases until reaching its minimum (−23.44° or -23°26') on the December solstice , when its value is the negative of the axial tilt. This variation produces the seasons . A line graph of
5418-558: 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
5504-407: The orbit. If the starting point is close to the perihelion (such as the December solstice), then the speed is higher than average, and the apparent Sun saves little time for not having to cover a full circle: the "tropical year" is comparatively long. If the starting point is near aphelion, then the speed is lower and the time saved for not having to run the same small arc that the equinox has precessed
5590-425: The plane of the ecliptic (the Earth's orbit around the Sun), and the plane of the celestial equator (the Earth's equator projected into space). These two planes intersect in a line. One direction points to the so-called vernal, northward, or March equinox which is given the symbol ♈︎ (the symbol looks like the horns of a ram because it used to be toward the constellation Aries ). The opposite direction
5676-420: 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
5762-477: The previous year's December solstice occurred before or after noon on December 22. These accuracies are compared to NOAA's advanced calculations which are based on the 1999 Jean Meeus algorithm that is accurate to within 0.01°. (The above formula is related to a reasonably simple and accurate calculation of the Equation of Time , which is described here .) More complicated algorithms correct for changes to
5848-475: The reproductive cycle of the palolos is synchronized with the moon. Tropical year 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
5934-589: The solar year: the Gregorian calendar (with its rules for catch-up leap days ) is designed so as to resynchronise the calendar year with the solar year at regular intervals. 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
6020-414: The solstices in Earth's orbit (at the equinoxes), so that sin(EL) can be written as sin(90+NDS)=cos(NDS) where NDS is the number of days after the December solstice. By also using the approximation that arcsin[sin(d)·cos(NDS)] is close to d·cos(NDS), the following frequently used formula is obtained: where N is the day of the year beginning with N=0 at midnight Universal Time (UT) as January 1 begins (i.e.
6106-459: 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
6192-422: The timing of events such as sunrise and sunset, and in the reading of a sundial compared with a clock showing local mean time . As the graph shows, a sundial can be up to about 16 minutes fast or slow, compared with a clock. Since the Earth rotates at a mean speed of one degree every four minutes, relative to the Sun, this 16-minute displacement corresponds to a shift eastward or westward of about four degrees in
6278-403: The tropical year gets roughly a half second shorter each century. 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 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
6364-401: 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: Until the beginning of the 19th century,
6450-413: 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. 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
6536-420: The two approximations, using the parameters of the Earth's orbit to more accurately estimate EL: which can be simplified by evaluating constants to: N is the number of days since midnight UT as January 1 begins (i.e. the days part of the ordinal date −1) and can include decimals to adjust for local times later or earlier in the day. The number 2, in (N-2), is the approximate number of days after January 1 to
6622-512: The underpinnings of all solar system models until Albert Einstein 's theory of General relativity in the 20th 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 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
6708-527: The variation of the solar declination happen faster in January than in July. On the graph, this makes the minima more acute than the maxima. Also, since perihelion and aphelion do not happen on the exact dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of change before and after are not quite equal. The graph of apparent solar declination is therefore different in several ways from
6794-463: 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 365.242 198 781 25 days
6880-424: The very accurate Shortt-Synchronome clock and later in the 1930s when quartz clocks began to replace pendulum clocks as time standards. 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
6966-627: The year, the Sun would be directly overhead at the North Pole , so its declination would be +90°. For the next few months, the subsolar point would move toward the South Pole at constant speed, crossing the circles of latitude at a constant rate, so that the solar declination would decrease linearly with time. Eventually, the Sun would be directly above the South Pole, with a declination of −90°; then it would start to move northward at
7052-480: Was 365.242 189 7 or 365 ephemeris days , 5 hours, 48 minutes, 45.19 seconds. This changes slowly; an expression suitable for calculating the length of a tropical year in ephemeris days, between 8000 BC and 12000 AD is where T is in Julian centuries of 36,525 days of 86,400 SI seconds measured from noon January 1, 2000 TT. 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
7138-401: 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
7224-437: 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
7310-447: 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
7396-480: 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
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