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see caption
An animation of theinner Solar Systemplanets' orbit around the Sun. The duration of the year is the time taken to go around the Sun.

Ayearis thetimetaken forastronomical objectsto complete oneorbit.For example, a year onEarthis the time taken for Earth to revolve around theSun.Generally, a year is taken to mean a calendar year, but the word is also used for periods loosely associated with the calendar or astronomical year, such as theseasonal year,thefiscal year,theacademic year,etc. The term can also be used in reference to any long period or cycle, such as theGreat Year.[1]

Due to the Earth'saxial tilt,the course of a year sees the passing of theseasons,marked by changes inweather,the hours ofdaylight,and, consequently,vegetationandsoil fertility.Intemperateandsubpolarregions around the planet, four seasons are generally recognized:spring,summer,autumn,andwinter.Intropicalandsubtropicalregions, several geographical sectors do not present defined seasons; but in theseasonal tropics,the annualwetanddry seasonsare recognized and tracked.

Calendar year

Acalendar yearis an approximation of the number of days of the Earth's orbital period, as counted in a givencalendar.TheGregorian calendar,or modern calendar, presents its calendar year to be either acommon yearof 365 days or aleap yearof 366 days, as do theJulian calendars.For the Gregorian calendar, the average length of the calendar year (the mean year) across the complete leap cycle of 400 years is 365.2425 days (97 out of 400 years are leap years).[2]

Abbreviation

In English, theunit of timefor year is commonly abbreviated as "y" or "yr". The symbol "a" (forLatin:annus,year) is sometimes used in scientific literature, though its exact duration may be inconsistent.

Etymology

Englishyear(viaWest Saxonġēar(/jɛar/),Anglianġēr) continuesProto-Germanic*jǣran(*jē₁ran). Cognates areGermanJahr,Old High Germanjār,Old NorseárandGothicjer,from theProto-Indo-Europeannoun*yeh₁r-om"year, season". Cognates also descended from the same Proto-Indo-European noun (with variation in suffixablaut) areAvestanyārǝ"year",Greekὥρα(hṓra) "year, season, period of time" (whence "hour"),Old Church Slavonicjarŭ,andLatinhornus"of this year".[citation needed]

Latinannus(a2nd declensionmasculine noun;annumis theaccusative singular;annīisgenitivesingular andnominativeplural;annōthedativeandablativesingular) is from aPIEnoun*h₂et-no-,which also yielded Gothicaþn"year" (only the dative pluralaþnamis attested).

Although most languages treat the word as thematic*yeh₁r-o-,there is evidence for an original derivation with an*-r/nsuffix,*yeh₁-ro-.Both Indo-European words for year,*yeh₁-ro-and*h₂et-no-,would then be derived from verbal roots meaning "to go, move",*h₁ey-and*h₂et-,respectively (compareVedic Sanskritéti"goes",atasi"thou goest, wanderest" ). A number of English words are derived from Latinannus,such asannual,annuity,anniversary,etc.;per annummeans "each year",annō Dominīmeans "in the year of the Lord".

The Greek word for "year",ἔτος,is cognate with Latinvetus"old", from the PIE word*wetos-"year", also preserved in this meaning inSanskritvat-sa-ras"year" andvat-sa-"yearling (calf)", the latter also reflected in Latinvitulus"bull calf", Englishwether"ram" (Old Englishweðer,Gothicwiþrus"lamb" ).

In some languages, it is common to count years by referencing to one season, as in "summers", or "winters", or "harvests". Examples include ChineseNăm"year", originallyNhân,an ideographic compound of a person carrying a bundle of wheat denoting "harvest". Slavic besidesgodŭ"time period; year" useslěto"summer; year".

Intercalation

Astronomical years do not have anintegernumber of days or lunar months. Any calendar that follows an astronomical year must have a system ofintercalationsuch as leap years.

Julian calendar

In theJulian calendar,the average (mean) length of a year is 365.25 days. In a non-leap year, there are 365 days, in a leap year there are 366 days. A leap year occurs every fourth year during which a leap day isintercalatedinto the month of February. The name "Leap Day" is applied to the added day.

In astronomy, theJulian yearis a unit of time defined as 365.25 days, each of exactly 86,400seconds(SI base unit), totaling exactly 31,557,600 seconds in the Julian astronomical year.[3][4]

Revised Julian calendar

TheRevised Julian calendar,proposed in 1923 and used in someEastern Orthodox Churches,has 218 leap years every 900 years, for the average (mean) year length of365.2422222days, close to the length of the mean tropical year,365.24219days (relative error of 9·10). In the year 2800 CE, the Gregorian and Revised Julian calendars will begin to differ by one calendar day.[5]

Gregorian calendar

TheGregorian calendaraims to ensure that thenorthward equinoxfalls on or shortly before March 21 and hence it follows thenorthward equinox year,ortropical year.[6]Because 97 out of 400 years are leap years, the mean length of the Gregorian calendar year is365.2425days; with a relative error below oneppm(8·10) relative to the current length of the meantropical year(365.24219days) and even closer to the currentMarch equinox yearof365.242374days that it aims to match.

Other calendars

Historically, lunisolar calendars intercalated entireleap monthson an observational basis. Lunisolar calendars have mostly fallen out of use except for liturgical reasons (Hebrew calendar,variousHindu calendars).

A modern adaptation of the historicalJalali calendar,known as theSolar Hijri calendar(1925), is a purelysolar calendarwith an irregular pattern of leap days based on observation (or astronomical computation), aiming to place new year (Nowruz) on the day ofvernal equinox(for the time zone ofTehran), as opposed to using an algorithmic system of leap years.

Year numbering

Acalendar eraassigns acardinal numberto each sequential year, using a reference event in the past (called theepoch) as the beginning of the era.

The Gregorian calendar era is the world's most widely usedcivil calendar.[7]Its epoch is a6th century estimateof the date of birth ofJesus of Nazareth.Two notations are used to indicate year numbering in the Gregorian calendar: the Christian "Anno Domini"(meaning" in the year of the Lord "), abbreviated AD; and"Common Era",abbreviated CE, preferred by many of other faiths and none. Year numbers are based oninclusive counting,so that there is no "year zero". Years before the epoch are abbreviated BC forBefore Christor BCE forBefore the Common Era.InAstronomical year numbering,positive numbers indicate years AD/CE, the number0designates 1 BC/BCE, −1 designates 2 BC/BCE, and so on.

Other eras include that ofAncient Rome,Ab Urbe Condita( "from the foundation ofthe city), abbreviated AUC;Anno Mundi( "year of the world" ), used for theHebrew calendarand abbreviated AM; and theJapanese imperial eras.The IslamicHijri year,(year of theHijrah,Anno Hegiraeabbreviated AH), is alunar calendarof twelvelunar monthsand thus is shorter than a solar year.

Pragmatic divisions

Financial and scientific calculations often use a365-day calendarto simplify daily rates.

Fiscal year

Afiscal yearor financial year is a 12-month period used for calculating annual financial statements in businesses and other organizations. In many jurisdictions, regulations regarding accounting require such reports once per twelve months, but do not require that the twelve months constitute a calendar year.

For example, inCanadaandIndiathe fiscal year runs from April 1; in theUnited Kingdomit runs from April 1 for purposes of corporation tax and government financial statements, but from April 6 for purposes of personal taxation and payment of state benefits; inAustraliait runs from July 1; while in theUnited Statesthe fiscal year of thefederal governmentruns from October 1.

Academic year

An academic year is the annual period during which a student attends aneducational institution.The academic year may be divided intoacademic terms,such as semesters or quarters. The school year in many countries starts in August or September and ends in May, June or July. In Israel the academic year begins around October or November, aligned with the second month of the Hebrew calendar.

Some schools in the UK, Canada and the United States divide the academic year intothreeroughly equal-length terms (calledtrimestersorquartersin the United States), roughly coinciding with autumn, winter, and spring. At some, a shortened summer session, sometimes considered part of the regular academic year, is attended by students on a voluntary or elective basis. Other schools break the year intotwomain semesters, a first (typically August through December) and a second semester (January through May). Each of these main semesters may be split in half by mid-term exams, and each of the halves is referred to as aquarter(ortermin some countries). There may also be a voluntary summer session or a short January session.

Some other schools, including some in the United States, havefourmarking periods. Some schools in the United States, notablyBoston Latin School,may divide the year intofive or moremarking periods. Some state in defense of this that there is perhaps apositive correlationbetween report frequency and academic achievement.

There are typically 180 days of teaching each year in schools in the US, excluding weekends and breaks, while there are 190 days for pupils in state schools in Canada, New Zealand and the United Kingdom, and 200 for pupils in Australia.

In India the academic year normally starts from June 1 and ends on May 31. Though schools start closing from mid-March, the actual academic closure is on May 31 and in Nepal it starts from July 15.[citation needed]

Schools and universities in Australia typically have academic years that roughly align with the calendar year (i.e., starting in February or March and ending in October to December), as the southern hemisphere experiences summer from December to February.

Astronomical years

Julian year

The Julian year, as used in astronomy and other sciences, is a time unit defined as exactly 365.25 days of 86,400SI secondseach ( "ephemeris days"). This is the normal meaning of the unit" year "used in various scientific contexts. The Julian century of36525ephemeris days and the Julian millennium of365250ephemeris days are used in astronomical calculations. Fundamentally, expressing a time interval in Julian years is a way to precisely specify an amount of time (not how many "real" years), for long time intervals where stating the number of ephemeris days would be unwieldy and unintuitive. By convention, the Julian year is used in the computation of the distance covered by alight-year.

In theUnified Code for Units of Measure(but not according to theInternational Union of Pure and Applied Physicsor theInternational Union of Geological Sciences,see below), the symbola(without subscript) always refers to the Julian year,aj,of exactly31557600seconds.

365.25 d ×86400s = 1 a = 1 aj=31.5576Ms

TheSI multiplier prefixesmay be applied to it to form "ka", "Ma", etc.[8]

Sidereal, tropical, and anomalistic years

Each of these three years can be loosely called anastronomical year.

The sidereal year is the time taken for the Earth to complete one revolution of itsorbit,as measured against a fixed frame of reference (such as the fixed stars, Latinsidera,singularsidus). Its average duration is365.256363004days (365 d 6 h 9 min 9.76 s) (at the epochJ2000.0= January 1, 2000, 12:00:00TT).[9]

Today the mean tropical year is defined as the period of time for the meanecliptic longitudeof the Sun to increase by 360 degrees.[10]Since the Sun's ecliptic longitude is measured with respect to the equinox,[11]the tropical year comprises a complete cycle of the seasons and is the basis ofsolar calendarssuch as the internationally usedGregorian calendar.The modern definition of mean tropical year differs from the actual time between passages of, e.g., the northward equinox, by a minute or two, for several reasons explained below. Because of the Earth'saxial precession,this year is about 20 minutes shorter than the sidereal year. The mean tropical year is approximately 365 days, 5 hours, 48 minutes, 45 seconds, using the modern definition[12]( = 365.24219 d × 86 400 s). The length of the tropical year varies a bit over thousands of years because the rate of axial precession is not constant.

The anomalistic year is the time taken for the Earth to complete one revolution with respect to itsapsides.The orbit of the Earth is elliptical; the extreme points, called apsides, are theperihelion,where the Earth is closest to the Sun, and theaphelion,where the Earth is farthest from the Sun. The anomalistic year is usually defined as the time between perihelion passages. Its average duration is 365.259636 days (365 d 6 h 13 min 52.6 s) (at the epoch J2011.0).[13]

Draconic year

The draconic year, draconitic year, eclipse year, or ecliptic year is the time taken for the Sun (as seen from the Earth) to complete one revolution with respect to the samelunar node(a point where the Moon's orbit intersects the ecliptic). The year is associated witheclipses:these occur only when both the Sun and the Moon are near these nodes; so eclipses occur within about a month of every half eclipse year. Hence there are twoeclipse seasonsevery eclipse year. The average duration of the eclipse year is

346.620075883days (346 d 14 h 52 min 54 s) (at the epoch J2000.0).

This term is sometimes erroneously used for the draconic or nodal period oflunar precession,that is the period of a complete revolution of the Moon's ascending node around the ecliptic:18.612815932Julian years (6798.331019days; at the epoch J2000.0).

Full moon cycle

Thefull moon cycleis the time for the Sun (as seen from the Earth) to complete one revolution with respect to theperigeeof the Moon's orbit. This period is associated with the apparent size of thefull moon,and also with the varying duration of thesynodic month.The duration of one full moon cycle is:

411.78443029days (411 days 18 hours 49 minutes 35 seconds) (at the epoch J2000.0).

Lunar year

Thelunar yearcomprises twelve full cycles of the phases of the Moon, as seen from Earth. It has a duration of approximately 354.37 days.Muslimsuse this for celebrating theirEidsand for marking the start of the fasting month ofRamadan.A Muslim calendar year is based on the lunar cycle. TheJewish calendaris also essentially lunar, except that an intercalary lunar month is added once every two or three years, in order to keep the calendar synchronized with the solar cycle as well. Thus, a lunar year on the Jewish (Hebrew) calendar consists of either twelve or thirteen lunar months.

Vague year

The vague year, fromannus vagusor wandering year, is an integral approximation to the year equaling 365 days, which wanders in relation to more exact years. Typically the vague year is divided into 12schematicmonths of 30 days each plus 5epagomenaldays. The vague year was used in the calendars ofEthiopia,Ancient Egypt,Iran,Armeniaand inMesoamericaamong theAztecsandMaya.[14]It is still used by many Zoroastrian communities.

Heliacal year

A heliacal year is the interval between theheliacal risingsof a star. It differs from thesidereal yearfor stars away from theeclipticdue mainly to theprecession of the equinoxes.

Sothic year

TheSothic yearis the heliacal year, the interval between heliacal risings, of the starSirius.It is currently less than thesidereal yearand its duration is very close to the Julian year of 365.25 days.

Gaussian year

TheGaussian yearis the sidereal year for a planet of negligible mass (relative to the Sun) and unperturbed by other planets that is governed by theGaussian gravitational constant.Such a planet would be slightly closer to the Sun than Earth's mean distance. Its length is:

365.2568983days (365 d 6 h 9 min 56 s).

Besselian year

TheBesselian yearis a tropical year that starts when the (fictitious) mean Sun reaches an ecliptic longitude of 280°. This is currently on or close to January 1. It is named after the 19th-century German astronomer and mathematicianFriedrich Bessel.The following equation can be used to compute the current Besselian epoch (in years):[15]

B = 1900.0 + (Julian dateTT2415020.31352) /365.242198781

The TT subscript indicates that for this formula, the Julian date should use theTerrestrial Timescale, or its predecessor,ephemeris time.

Variation in the length of the year and the day

The exact length of an astronomical year changes over time.

  • The positions of the equinox and solstice points with respect to the apsides of Earth's orbit change: the equinoxes and solstices move westward relative to the stars because ofprecession,and the apsides move in the other direction because of the long-term effects of gravitational pull by the other planets. Since the speed of the Earth varies according to its position in its orbit as measured from its perihelion, Earth's speed when in a solstice or equinox point changes over time: if such a point moves toward perihelion, the interval between two passages decreases a little from year to year; if the point moves towards aphelion, that period increases a little from year to year. So a "tropical year" measured from one passage of the northward ( "vernal" ) equinox to the next, differs from the one measured between passages of the southward ( "autumnal" ) equinox. The average over the full orbit does not change because of this, so the length of the average tropical year does not change because of this second-order effect.
  • Each planet's movement is perturbed by the gravity of every other planet. This leads to short-term fluctuations in its speed, and therefore its period from year to year. Moreover, it causes long-term changes in its orbit, and therefore also long-term changes in these periods.
  • Tidal dragbetween the Earth and the Moon and Sun increases the length of the day and of the month (by transferring angular momentum from the rotation of the Earth to the revolution of the Moon); since the apparent mean solar day is the unit with which we measure the length of the year in civil life, the length of the year appears to decrease. The rotation rate of the Earth is also changed by factors such aspost-glacial reboundandsea level rise.

Numerical value of year variation
Mean year lengths in this section are calculated for 2000, and differences in year lengths, compared to 2000, are given for past and future years. In the tables a day is 86,400 SI seconds long.[16][17][18][19]

Mean year lengths for 2000
Type of year Days Hours Minutes Seconds
Tropical 365 5 48 45
Sidereal 365 6 9 10
Anomalistic 365 6 13 53
Eclipse 346 14 52 55
Year length difference from 2000
(seconds; positive when length for tabulated year is greater than length in 2000)
Year Tropical Sidereal Anomalistic Eclipse
−4000 −8 −45 −15 −174
−2000 4 −19 −11 −116
0 7 −4 −5 −57
2000 0 0 0 0
4000 −14 −3 5 54
6000 −35 −12 10 104

Summary

Some of the year lengths in this table are in averagesolar days,which are slowly getting longer (at a rate that cannot be exactly predicted in advance) and are now around 86,400.002SI seconds.

Days Year type
346.62 Draconic, also called eclipse
354.37 Lunar
365 Solar days: vague, and acommon yearin manysolar calendars
365.24219 Tropical, also called solar, averaged and then rounded for epochJ2000.0
365.2425 Gregorian, solar days averaged over the 400-year cycle
365.25 Julian, solar days averaged over the four-year cycle
365.25636 Sidereal, for epochJ2000.0
365.259636 Anomalistic, averaged and then rounded for epoch J2011.0
366 Leap yearin manysolar calendars

An average Gregorian year may be said to be 365.2425days(52.1775weeks,and if an hour is defined as one twenty-fourth of a day,8765.82hours,525949.2minutesor31556952seconds). Note however that in absolute time the average Gregorian year is not adequately defined unless the period of the averaging (start and end dates) is stated, because each period of 400 years is longer (by more than 1000 seconds) than the preceding one as the rotation of the Earth slows. In this calendar, a common year is 365 days (8760hours,525600minutes or31536000seconds), and a leap year is 366 days (8784hours,527040minutes or31622400seconds). The 400-year civil cycle of the Gregorian calendar has146097days and hence exactly20871weeks.

Greater astronomical years

Equinoctial cycle

TheGreat Year,or equinoctial cycle, corresponds to a complete revolution of the equinoxes around the ecliptic. Its length is about 25,700 years.[20][21]

Galactic year

TheGalactic yearis the time it takes Earth'sSolar Systemto revolve once around theGalactic Center.It comprises roughly 230 million Earth years.[22]

Seasonal year

A seasonal year is the time between successive recurrences of a seasonal event such as the flooding of a river, the migration of a species of bird, the flowering of a species of plant, the first frost, or the first scheduled game of a certain sport. All of these events can have wide variations of more than amonthfrom year to year.

Symbols and abbreviations

A common symbol for the year as aunit of timeis "a", taken from the Latin wordannus. For example, the U.S.National Institute of Standards and Technology(NIST)Guide for the Use of the International System of Units (SI)supports the symbol "a" as the unit of time for a year.[23]

In English, the abbreviations "y" or "yr" are more commonly used in non-scientific literature.[24]In someEarth sciencesbranches (geologyandpaleontology), "kyr,myr,byr"(thousands, millions, and billions of years, respectively) and similar abbreviations are used to denote intervals of time remote from the present.[25][26]Inastronomythe abbreviations kyr, Myr and Gyr are in common use for kiloyears, megayears and gigayears.[27][28]

TheUnified Code for Units of Measure(UCUM) disambiguates the varying symbologies of ISO 1000, ISO 2955 and ANSI X3.50 by using:[8]

at=365.24219days for the mean tropical year;
aj= 365.25 days for the mean Julian year;
ag=365.2425days for the meanGregorian year;

In the UCUM, the symbol "a", without any qualifier, equals 1 aj. The UCUM also minimizes confusion withare,a unit of area, by using the abbreviation "ar".

Since 1989, theInternational Astronomical Union(IAU) recognizes the symbol "a" rather than "yr" for a year, notes the different kinds of year, and recommends adopting the Julian year of 365.25 days, unless otherwise specified (IAUStyle Manual).[29][30]

Since 1987, theInternational Union of Pure and Applied Physics(IUPAP) notes "a" as the general symbol for the time unit year (IUPAPRed Book).[31] Since 1993, theInternational Union of Pure and Applied Chemistry(IUPAC)Green Bookalso uses the same symbol "a", notes the difference between Gregorian year and Julian year, and adopts the former (a=365.2425 days),[32]also noted in theIUPACGold Book.[33]

In 2011, the IUPAC and theInternational Union of Geological Sciencesjointly recommended defining the "annus", with symbol "a", as the length of the tropical year in the year 2000:[34]

a =31556925.445seconds (approximately365.24219265ephemeris days)

This differs from the above definition of 365.25 days by about 20parts per million.The joint document says that definitions such as the Julian year "bear an inherent, pre-programmed obsolescence because of the variability of Earth's orbital movement", but then proposes using the length of the tropical year as of 2000 AD (specified down to the millisecond), which suffers from the same problem.[35](The tropical year oscillates with time by more than a minute.)

The notation has proved controversial as it conflicts with an earlier convention among geoscientists to use "a" specifically for "years ago" (e.g. 1 Ma for 1 million years ago), and "y" or "yr" for a one-year time period.[35][36] However, this historical practice does not comply with the NISTGuide,[23]considering the unacceptability of mi xing information concerning thephysical quantitybeing measured (in this case, time intervals or points in time) with the units and also the unnaceptability of using abbreviations for units. Furthermore, according to theUK Metric Association(UKMA), language-independent symbols are more universally understood (UKMAStyle guide).[37]

SI prefix multipliers

For the following, there are alternative forms that elide the consecutive vowels, such askilannus,megannus,etc. The exponents and exponential notations are typically used for calculating and in displaying calculations, and for conserving space, as in tables of data.

Units of time with SI prefixes
Symbol Definition Common scientific uses and notes
ka (forkiloannus) Onethousandor 103years, also known as amillenniuminanthropologyand calendar uses. Geology, paleontology, andarchaeologyfor theHoloceneandPleistoceneperiods,where a non−radiocarbon datingtechnique such asice coredating,dendrochronology,uranium-thorium datingorvarveanalysis is used as the primary method for age determination. If age is determined primarily by radiocarbon dating, then the age should be expressed in either radiocarbon or calendar (calibrated) yearsBefore Present.
Ma (formegaannus) Onemillionor 106years. Geology, paleontology, andcelestial mechanics.In astronomical applications, the year used is the Julian year of precisely 365.25 days. In geology and paleontology, the year is not so precise and varies depending on the author.
Ga (forgigaannus) Onebillionor 109years. Cosmologyand geology[38]For example,the formation of the Earthoccurred approximately 4.54 Ga (4.54 billion years) ago and theage of the universeis approximately 13.8 Ga.
Ta (forteraannus) Onetrillionor 1012years An extremely long unit of time, about 70 times as long as the age of the universe. It is the same order of magnitude as the expected life span of a smallred dwarf.
Pa (forpetaannus) Onequadrillionor 1015years. Thehalf-lifeof thenuclidecadmium-113is about 8 Pa.[39]This symbol coincides with that for thepascalwithout a multiplier prefix, but context will normally be sufficient to distinguish long time periods from pressure values.
Ea (forexaannus) Onequintillionor 1018years. The half-life oftungsten-180is 1.8 Ea.[40]

Abbreviations for "years ago"

In geology and paleontology, a distinction sometimes is made between abbreviation "yr" foryearsand "ya" foryears ago,combined with prefixes for thousand, million, or billion.[25][41]In archaeology, dealing with more recent periods, normally expressed dates, e.g. "10,000 BC", may be used as a more traditional form thanBefore Present( "BP" ).

These abbreviations include:

Non-SI
abbreviations
Short for SI-prefixed
equivalent
Definition Examples
Event Time
kilo years ka Thousand years
myr
Myr
million years
Mega years
Ma Million years
byr
Gyr
billion years
Giga years
Ga Billion years
(thousand millionyears)
kya
kilo years ago Time ago in ka

Around 200 kya
Around 60 kya
Around 20 kya
Around 10 kya

mya
Mya
million years ago
Mega years ago
Time ago in Ma 5.3 to 2.6 mya
0.78 mya
0.13 mya
0.01 mya
bya
Gya
billion years ago
giga years ago
Time ago in Ga 2 bya
4.5 bya
13.8 bya

Use of "mya" and "bya" is deprecated in modern geophysics, the recommended usage being "Ma" and "Ga" for datesBefore Present,but "m.y." for the durations of epochs.[25][26]Thisad hocdistinction between "absolute" time and time intervals is somewhat controversial amongst members of the Geological Society of America.[43]

See also

References

Notes

  1. ^OED,s.v. "year", entry 2.b.: "transf.Applied to a very long period or cycle (in chronology or mythology, or vaguely in poetic use). "
  2. ^"Calendar Calculations".Glenn Learning Technologies Project.National Aeronautics Space Administration.RetrievedNovember 11,2023.
  3. ^"SI units".International Astronomical Union IAU.RetrievedFebruary 18,2010.
  4. ^Wilkins, George A. (1989)."The IAU Style Manual"(PDF).IAU Transactions.XXB.
  5. ^Shields, Miriam Nancy (1924). "The new calendar of the eastern churches".Popular Astronomy.32:407.Bibcode:1924PA.....32..407S.
  6. ^Ziggelaar, A. (1983)."The Papal Bull of 1582 Promulgating a Reform of the Calendar".In G. V. Coyne; M. A. Hoskin; O. Pedersen (eds.).Gregorian Reform of the Calendar: Proceedings of the Vatican Conference to Commemorate its 400th Anniversary.Vatican City: Pontifical Academy of Sciences. p. 223.
  7. ^Richards, E.G. (2013). "Calendars". In Urban, S.E.; Seidelmann, P.K. (eds.).Explanatory Supplement to the Astronomical Almanac(PDF)(3rd ed.). Mill Valley, CA: University Science Books. pp. 585, 590.ISBN978-1-891389-85-6.Archived fromthe original(PDF)on April 30, 2019.RetrievedMay 9,2018.
  8. ^ab"The Unified Code for Units of Measure".UCUM.November 21, 2017.RetrievedJuly 27,2022.
  9. ^International Earth Rotation and Reference System Service. (2010).IERS EOP PC Useful constants.ArchivedOctober 29, 2012, at theWayback Machine
  10. ^Richards, E.G. (2013). Calendars. In S.E. Urban & P.K. Seidelmann (Eds.),Explanatory Supplement to the Astronomical Almanac(3rd ed.). Mill Valley, CA: University Science Books. p. 586.
  11. ^"longitude, ecliptic".Archivedfrom the original on September 8, 2023,and"dynamical equinox".Archivedfrom the original on September 8, 2023,(c. 2022). In "Glossary",The Astronomical Almanac Online.United States Naval Observatory.
  12. ^ "Glossary".Astronomical Applications Department.United States Naval Observatory. c. 2022. s.v. year, tropical.Archivedfrom the original on September 8, 2023.RetrievedNovember 6,2023.
  13. ^ Astronomical Almanac for the Year 2011.Washington and Taunton:US Government Printing Officeand theUK Hydrographic Office.2009. pp. A1, C2.
  14. ^Calendar Description and CoordinationArchivedApril 26, 2012, at theWayback MachineMaya World Studies Center
  15. ^ Astronomical Almanac for the Year 2010.Washington and Taunton: U.S. Government Printing Office and the U.K. Hydrographic Office. 2008. p. B3.
  16. ^ U.S. Naval Observatory Nautical Almanac Office and Her Majesty's Nautical Almanac Office (2010).Astronomical Almanac for the year 2011.Washington: U.S. Government Printing Office. pp. C2, L8.
  17. ^ Simon, J.L.; Bretagnon, P.; Chapront, J.; Chapront-Touzé, M.; Francou, G.; Laskar, J. (February 1994). "Numerical expressions for precession formulae and mean elements for the Moon and planets".Astronomy and Astrophysics.282(2): 663–683.Bibcode:1994A&A...282..663S.
  18. ^ Taff, Lawrence G. (1985).Celestial Mechanics: A Computational Guide for the Practitioner.New York: John Wiley & Sons. p. 103.ISBN978-0-471-89316-5.Values in tables agree closely for 2000, and depart by as much as 44 seconds for the years furthest in the past or future; the expressions are simpler than those recommended in theAstronomical Almanac for the Year 2011.
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