axial tilt

There are two standard methods of specifying a planet's tilt. One way is based on the planet's north pole, defined in relation to the direction of Earth's north pole, and the other way is based on the planet's positive pole, defined by the right-hand rule:

• The International Astronomical Union (IAU) defines the north pole of a planet as that which lies on Earth's north side of the invariable plane of the Solar System;Explanatory Supplement 1992, p. 384 under this system, Venus is tilted 3° and rotates retrograde, opposite that of most of the other planets.{{cite journal

|author1=Correia, Alexandre C. M.

|author2=Laskar, Jacques

|author3=de Surgy, Olivier Néron

|title=Long-term evolution of the spin of Venus I. theory

|journal=Icarus |volume=163 |issue=1 |pages=1–23

|date=May 2003

|url=http://www.imcce.fr/Equipes/ASD/preprints/prep.2002/venus1.2002.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.imcce.fr/Equipes/ASD/preprints/prep.2002/venus1.2002.pdf |archive-date=9 October 2022 |url-status=live

|doi=10.1016/S0019-1035(03)00042-3

|bibcode=2003Icar..163....1C

}}{{cite journal

|author1=Correia, A. C. M.

|author2=Laskar, J.

|date=2003

|title=Long-term evolution of the spin of Venus: II. numerical simulations

|journal=Icarus

|volume=163 |issue=1 |pages=24–45

|url=http://www.imcce.fr/Equipes/ASD/preprints/prep.2002/venus2.2002.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.imcce.fr/Equipes/ASD/preprints/prep.2002/venus2.2002.pdf |archive-date=9 October 2022 |url-status=live

|doi=10.1016/S0019-1035(03)00043-5

|bibcode=2003Icar..163...24C

}}

• The IAU also uses the right-hand rule to define a positive pole{{cite journal|title=Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006|journal=Celestial Mechanics and Dynamical Astronomy|volume=98|issue=3|pages=155–180|doi=10.1007/s10569-007-9072-y|year=2007|last1=Seidelmann|first1=P. Kenneth|last2=Archinal|first2=B. A.|last3=a'Hearn|first3=M. F.|last4=Conrad|first4=A.|last5=Consolmagno|first5=G. J.|last6=Hestroffer|first6=D.|last7=Hilton|first7=J. L.|last8=Krasinsky|first8=G. A.|last9=Neumann|first9=G.|last10=Oberst|first10=J.|last11=Stooke|first11=P.|last12=Tedesco|first12=E. F.|last13=Tholen|first13=D. J.|last14=Thomas|first14=P. C.|last15=Williams|first15=I. P.|bibcode=2007CeMDA..98..155S|doi-access=free}} for the purpose of determining orientation. Using this convention, Venus is tilted 177° ("upside down") and rotates prograde.

{{further|Ecliptic#Obliquity of the ecliptic}}

{{See also|Earth's rotation|Earth-centered inertial}}

Earth's orbital plane is known as the ecliptic plane, and Earth's tilt is known to astronomers as the obliquity of the ecliptic, being the angle between the ecliptic and the celestial equator on the celestial sphere.

{{cite book

|author1=U.S. Naval Observatory Nautical Almanac Office

|author2=U.K. Hydrographic Office

|author3=H.M. Nautical Almanac Office

|date=2008

|title=The Astronomical Almanac for the Year 2010

|page=M11

|publisher=US Government Printing Office

|isbn=978-0-7077-4082-9

}} It is denoted by the Greek letter Epsilon ε.

Earth currently has an axial tilt of about 23.44°.[https://aa.usno.navy.mil/faq/asa_glossary "Glossary"] in Astronomical Almanac Online. (2023). Washington DC: United States Naval Observatory. s.v. obliquity. This value remains about the same relative to a stationary orbital plane throughout the cycles of axial precession.

{{cite book

|last=Chauvenet |first=William

|date=1906

|title=A Manual of Spherical and Practical Astronomy

|url=https://books.google.com/books?id=yobvAAAAMAAJ

|publisher=J. B. Lippincott

|volume=1 |pages=604–605

}} But the ecliptic (i.e., Earth's orbit) moves due to planetary perturbations, and the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about 46.8″{{cite journal

| title=Long-period tidal variations in the length of day

| journal= Journal of Geophysical Research: Solid Earth

| volume=119 | issue=2 | pages=1498–1509

| first1=Richard D. | last1=Ray | first2=Svetlana Y. | last2=Erofeeva

| doi=10.1002/2013JB010830 | date=4 February 2014

| bibcode=2014JGRB..119.1498R

| doi-access=free

}} per century (see details in Short term below).

The ancient Greeks had good measurements of the obliquity since about 350 BCE, when Pytheas of Marseilles measured the shadow of a gnomon at the summer solstice.

{{cite book

|last=Gore |first=J. E.

|date=1907

|title=Astronomical Essays Historical and Descriptive

|publisher=Chatto & Windus |url=https://archive.org/details/astronomicaless00goregoog

|page=[https://archive.org/details/astronomicaless00goregoog/page/n78 61]

}} About 830 CE, the Caliph Al-Mamun of Baghdad directed his astronomers to measure the obliquity, and the result was used in the Arab world for many years.

{{cite book

|last=Marmery |first=J. V.

|date=1895

|title=Progress of Science

|publisher=Chapman and Hall, ld. |url=https://archive.org/details/in.ernet.dli.2015.45033

|page=[https://archive.org/details/in.ernet.dli.2015.45033/page/n67 33]

}} In 1437, Ulugh Beg determined the Earth's axial tilt as 23°30′17″ (23.5047°).{{cite book |first=L.P.E.A. |last=Sédillot |title=Prolégomènes des tables astronomiques d'OlougBeg: Traduction et commentaire |location=Paris |publisher=Firmin Didot Frères |year=1853 |pages=87 & 253}}

During the Middle Ages, it was widely believed that both precession and Earth's obliquity oscillated around a mean value, with a period of 672 years, an idea known as trepidation of the equinoxes. Perhaps the first to realize this was incorrect (during historic time) was Ibn al-Shatir in the fourteenth century{{cite book

|last=Saliba |first=George

|date=1994

|title= A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam

|page=235

}} and the first to realize that the obliquity is decreasing at a relatively constant rate was Fracastoro in 1538.

{{cite book

|last=Dreyer |first=J. L. E.

|date=1890

|url=https://archive.org/details/tychobraheapict00dreygoog

|title=Tycho Brahe

|publisher=A. & C. Black |page=[https://archive.org/details/tychobraheapict00dreygoog/page/n387 355]

}} The first accurate, modern, western observations of the obliquity were probably those of Tycho Brahe from Denmark, about 1584,Dreyer (1890), p. 123 although observations by several others, including al-Ma'mun, al-Tusi,

{{cite book

|last=Sayili |first=Aydin

|date=1981

|title= The Observatory in Islam

|page=78

}} Purbach, Regiomontanus, and Walther, could have provided similar information.

{{main|Season}}

File:Earth tilt animation.gif. The axis of Earth remains oriented in the same direction with reference to the background stars regardless of where it is in its orbit. Northern hemisphere summer occurs at the right side of this diagram, where the north pole (red) is directed toward the Sun, winter at the left.]]

Earth's axis remains tilted in the same direction with reference to the background stars throughout a year (regardless of where it is in its orbit) – this is known as axial parallelism. This means that one pole (and the associated hemisphere of Earth) will be directed away from the Sun at one side of the orbit, and half an orbit later (half a year later) this pole will be directed towards the Sun. This is the cause of Earth's seasons. Summer occurs in the Northern hemisphere when the north pole is directed toward and the south pole away from the Sun. Variations in Earth's axial tilt can influence the seasons and is likely a factor in long-term climatic change (also see Milankovitch cycles).

File:axial_tilt_vs_tropical_and_polar_circles.svg

File:Obliquity of the ecliptic laskar.PNG (1986). The red point represents the year 2000.]]

The exact angular value of the obliquity is found by observation of the motions of Earth and planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, and from these ephemerides various astronomical values, including the obliquity, are derived.

Annual almanacs are published listing the derived values and methods of use. Until 1983, the Astronomical Almanac's angular value of the mean obliquity for any date was calculated based on the work of Newcomb, who analyzed positions of the planets until about 1895:

: {{math|ε {{=}} 23°27′8.26″ − 46.845″ T − 0.0059″ T² + {{val|0.00181}}″ T³}}

where {{math|ε}} is the obliquity and {{math|T}} is tropical centuries from B1900.0 to the date in question.

{{cite book

|author=U.S. Naval Observatory Nautical Almanac Office

|author2=H.M. Nautical Almanac Office

|date=1961

|title=Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac

|publisher=H.M. Stationery Office

|at=Section 2B

}}

From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated:

: {{math|ε {{=}} 23°26′21.448″ − 46.8150″ T − 0.00059″ T² + {{val|0.001813}}″ T³}}

where hereafter {{math|T}} is Julian centuries from J2000.0.

{{cite book

|last=U.S. Naval Observatory

|author2=H.M. Nautical Almanac Office

|date=1989

|title=The Astronomical Almanac for the Year 1990

|page=B18

|publisher=US Government Printing Office

|isbn=978-0-11-886934-8

}}

JPL's fundamental ephemerides have been continually updated. For instance, according to IAU resolution in 2006 in favor of the P03 astronomical model, the Astronomical Almanac for 2010 specifies:Astronomical Almanac 2010, p. B52

: {{math|ε {{=}} 23°26′21.406″ − {{val|46.836769}}″ T − {{val|0.0001831}}″ T² + {{val|0.00200340}}″ T³ − 5.76″ × 10⁻⁷ T⁴ − 4.34″ × 10⁻⁸ T⁵}}

These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps {{math|±}} several centuries.

{{cite book

|last=Newcomb |first=Simon

|date=1906

|title=A Compendium of Spherical Astronomy

|url=https://archive.org/details/acompendiumsphe00newcgoog

|publisher=MacMillan

|pages=[https://archive.org/details/acompendiumsphe00newcgoog/page/n250 226]–227

}} Jacques Laskar computed an expression to order {{math|T¹⁰}} good to 0.02″ over 1000 years and several arcseconds over 10,000 years.

:{{math|ε {{=}} 23°26′21.448″ − 4680.93″ t − 1.55″ t² + 1999.25″ t³ − 51.38″ t⁴ − 249.67″ t⁵ − 39.05″ t⁶ + 7.12″ t⁷ + 27.87″ t⁸ + 5.79″ t⁹ + 2.45″ t¹⁰}}

where here {{math|t}} is multiples of 10,000 Julian years from J2000.0.See table 8 and eq. 35 in {{cite journal

|last=Laskar |first=J.

|date=1986

|title=Secular terms of classical planetary theories using the results of general theory

|journal=Astronomy and Astrophysics

|volume=157 |issue=1

|pages=59–70

|bibcode = 1986A&A...157...59L

}} and erratum to article

{{cite journal

|last = Laskar |first = J.

|title = Erratum: Secular terms of classical planetary theories using the results of general theory

|journal = Astronomy and Astrophysics

|volume = 164

|date=1986

|page=437

|bibcode = 1986A&A...164..437L

}} Units in article are arcseconds, which may be more convenient.

These expressions are for the so-called mean obliquity, that is, the obliquity free from short-term variations. Periodic motions of the Moon and of Earth in its orbit cause much smaller (9.2 arcseconds) short-period (about 18.6 years) oscillations of the rotation axis of Earth, known as nutation, which add a periodic component to Earth's obliquity.Explanatory Supplement (1961), sec. 2C

{{Cite web

|url=http://www2.jpl.nasa.gov/basics/bsf2-1.php#nutation

|title=Basics of Space Flight, Chapter 2

|date=29 October 2013

|access-date=26 March 2015

|work=Jet Propulsion Laboratory/NASA

}} The true or instantaneous obliquity includes this nutation.

{{cite book

|last=Meeus |first=Jean

|date=1991

|chapter=Chapter 21

|title=Astronomical Algorithms

|publisher=Willmann-Bell

|isbn=978-0-943396-35-4

}}

{{main|Formation and evolution of the Solar System}}

{{main|Milankovitch cycles}}

Using numerical methods to simulate Solar System behavior over a period of several million years, long-term changes in Earth's orbit, and hence its obliquity, have been investigated. For the past 5 million years, Earth's obliquity has varied between {{nowrap|22°2′33″}} and {{nowrap|24°30′16″}}, with a mean period of 41,040 years. This cycle is a combination of precession and the largest term in the motion of the ecliptic. For the next 1 million years, the cycle will carry the obliquity between {{nowrap|22°13′44″}} and {{nowrap|24°20′50″}}.

{{cite journal

|last=Berger |first=A.L.

|date=1976

|title=Obliquity and Precession for the Last 5000000 Years

|journal=Astronomy and Astrophysics

|volume=51 |issue= 1|pages=127–135

|bibcode=1976A&A....51..127B

}}

The Moon has a stabilizing effect on Earth's obliquity. Frequency map analysis conducted in 1993 suggested that, in the absence of the Moon, the obliquity could change rapidly due to orbital resonances and chaotic behavior of the Solar System, reaching as high as 90° in as little as a few million years (also see Orbit of the Moon).

{{cite journal

|author1=Laskar, J.

|author2=Robutel, P.

|date=1993

|title=The Chaotic Obliquity of the Planets

|url=http://bugle.imcce.fr/fr/presentation/equipes/ASD/person/Laskar/misc_files/Laskar_Robutel_1993.pdf

|journal=Nature

|volume=361 |issue=6413 |pages=608–612

|bibcode=1993Natur.361..608L

|doi=10.1038/361608a0

|s2cid=4372237

|url-status=dead

|archive-url=https://web.archive.org/web/20121123093109/http://bugle.imcce.fr/fr/presentation/equipes/ASD/person/Laskar/misc_files/Laskar_Robutel_1993.pdf

|archive-date=23 November 2012

}}

{{cite journal

|author1=Laskar, J.

|author2=Joutel, F.

|author3=Robutel, P.

|date=1993

|title=Stabilization of the Earth's Obliquity by the Moon

|url=http://www.imcce.fr/Equipes/ASD/person/Laskar/misc_files/Laskar_Joutel_Robutel_1993.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.imcce.fr/Equipes/ASD/person/Laskar/misc_files/Laskar_Joutel_Robutel_1993.pdf |archive-date=9 October 2022 |url-status=live

|journal=Nature

|volume=361 |issue= 6413 |pages=615–617

|bibcode=1993Natur.361..615L

|doi=10.1038/361615a0

|s2cid=4233758

}} However, more recent numerical simulations

{{cite journal

|author1=Lissauer, J.J.

|author2=Barnes, J.W.

|author3=Chambers, J.E.

|date=2011

|title=Obliquity variations of a moonless Earth

|url=http://barnesos.net/publications/papers/2012.01.Icarus.Barnes.Moonless.Earth.pdf |archive-url=https://web.archive.org/web/20130608154841/http://barnesos.net/publications/papers/2012.01.Icarus.Barnes.Moonless.Earth.pdf |archive-date=8 June 2013 |url-status=live

|journal=Icarus

|volume=217 |issue= 1 |pages=77–87

|doi=10.1016/j.icarus.2011.10.013

|bibcode = 2012Icar..217...77L

}} made in 2011 indicated that even in the absence of the Moon, Earth's obliquity might not be quite so unstable; varying only by about 20–25°. To resolve this contradiction, diffusion rate of obliquity has been calculated, and it was found that it takes more than billions of years for Earth's obliquity to reach near 90°.{{Cite journal|last1=Li|first1=Gongjie|last2=Batygin|first2=Konstantin|date=20 July 2014|title=On the Spin-axis Dynamics of a Moonless Earth|journal=Astrophysical Journal|volume=790|issue=1|pages=69–76|arxiv=1404.7505|bibcode=2014ApJ...790...69L|doi=10.1088/0004-637X/790/1/69|s2cid=119295403}} The Moon's stabilizing effect will continue for less than two billion years. As the Moon continues to recede from Earth due to tidal acceleration, resonances may occur which will cause large oscillations of the obliquity.

{{cite journal

|author1=Ward, W.R.

|date=1982

|title=Comments on the Long-Term Stability of the Earth's Obliquity

|journal=Icarus

|volume=50 |issue= 2–3 |pages=444–448

|bibcode=1982Icar...50..444W

|doi=10.1016/0019-1035(82)90134-8

}}

{{multiple image

|direction = horizontal

|align= center

|width1= 264

|width2= 272

|image1=Obliquity berger -5000000 to 0.png

|image2=Obliquity berger 0 to 1000000.png

|footer=Long-term obliquity of the ecliptic. Left: for the past 5 million years; the obliquity varies only from about 22.0° to 24.5°. Right: for the next 1 million years; note the approx. 41,000-year period of variation. In both graphs, the red point represents the year 1850.Berger, 1976.

}}

{{see also|Poles of astronomical bodies#Poles of rotation}}File:Planets and dwarf planets' tilt and rotation speed.webm and two dwarf planets, Ceres and Pluto]]

All four of the innermost, rocky planets of the Solar System may have had large variations of their obliquity in the past. Since obliquity is the angle between the axis of rotation and the direction perpendicular to the orbital plane, it changes as the orbital plane changes due to the influence of other planets. But the axis of rotation can also move (axial precession), due to torque exerted by the Sun on a planet's equatorial bulge. Like Earth, all of the rocky planets show axial precession. If the precession rate were very fast the obliquity would actually remain fairly constant even as the orbital plane changes.{{cite journal|last1=William Ward|title=Large-Scale Variations in the Obliquity of Mars|journal=Science|volume=181|issue=4096|pages=260–262|date=20 July 1973|doi=10.1126/science.181.4096.260|pmid=17730940|bibcode=1973Sci...181..260W|s2cid=41231503}} The rate varies due to tidal dissipation and core-mantle interaction, among other things. When a planet's precession rate approaches certain values, orbital resonances may cause large changes in obliquity. The amplitude of the contribution having one of the resonant rates is divided by the difference between the resonant rate and the precession rate, so it becomes large when the two are similar.

Mercury and Venus have most likely been stabilized by the tidal dissipation of the Sun. Earth was stabilized by the Moon, as mentioned above, but before its formation, Earth, too, could have passed through times of instability. Mars's obliquity is quite variable over millions of years and may be in a chaotic state; it varies as much as 0° to 60° over some millions of years, depending on perturbations of the planets.

{{cite journal

|last1=Touma |first1=J.

|last2=Wisdom |first2=J.

|date=1993

|title=The Chaotic Obliquity of Mars

|url=http://groups.csail.mit.edu/mac/users/wisdom/mars-obliquity.pdf |archive-url=https://web.archive.org/web/20100625103119/http://groups.csail.mit.edu/mac/users/wisdom/mars-obliquity.pdf |archive-date=25 June 2010 |url-status=live

|journal=Science

|volume=259 |issue= 5099|pages=1294–1297

|bibcode=1993Sci...259.1294T

|doi=10.1126/science.259.5099.1294

|pmid=17732249

|s2cid=42933021

}} Some authors dispute that Mars's obliquity is chaotic, and show that tidal dissipation and viscous core-mantle coupling are adequate for it to have reached a fully damped state, similar to Mercury and Venus.{{cite journal

|last1=Correia |first1=Alexandre C.M

|last2=Laskar |first2=Jacques

|title=Mercury's capture into the 3/2 spin-orbit resonance including the effect of core-mantle friction

|journal=Icarus |date=2009

|doi=10.1016/j.icarus.2008.12.034

|arxiv=0901.1843

|volume=201

|issue=1

|pages=1–11

|bibcode=2009Icar..201....1C

|s2cid=14778204

}}

The occasional shifts in the axial tilt of Mars have been suggested as an explanation for the appearance and disappearance of rivers and lakes over the course of the existence of Mars. A shift could cause a burst of methane into the atmosphere, causing warming, but then the methane would be destroyed and the climate would become arid again.{{cite journal|last1=Rebecca Boyle|title=Methane burps on young Mars helped it keep its liquid water|journal=New Scientist|date=7 October 2017|url=https://www.newscientist.com/article/mg23631464-100}}{{cite journal|last1=Edwin Kite|display-authors=et al|title=Methane bursts as a trigger for intermittent lake-forming climates on post-Noachian Mars|journal=Nature Geoscience|volume=10|issue=10|pages=737–740|date=2 October 2017|doi=10.1038/ngeo3033|arxiv=1611.01717|bibcode=2017NatGe..10..737K|s2cid=102484593|url=https://authors.library.caltech.edu/80639/4/ngeo3033-s1.pdf |archive-url=https://web.archive.org/web/20180723193849/https://authors.library.caltech.edu/80639/4/ngeo3033-s1.pdf |archive-date=23 July 2018 |url-status=live}}

The obliquities of the outer planets are considered relatively stable.

The stellar obliquity {{math|ψ_s}}, i.e. the axial tilt of a star with respect to the orbital plane of one of its planets, has been determined for only a few systems. By 2012, 49 stars have had sky-projected spin-orbit misalignment {{math|λ}} has been observed,

{{cite web

|last=Heller |first=R.

|title=Holt-Rossiter-McLaughlin Encyclopaedia

|url=http://www.aip.de/People/RHeller

|publisher=René Heller

|access-date=24 February 2012

}} which serves as a lower limit to {{math|ψ_s}}. Most of these measurements rely on the Rossiter–McLaughlin effect.

As of 2024 the axial tilt of 4 exoplanets have been measured with one of them VHS 1256 b having a Uranus like tilt of 90 degrees ± 25 degrees.[https://arxiv.org/abs/2410.02672 Leaning Sideways: VHS 1256-1257 b is a Super-Jupiter with a Uranus-like Obliquity], Michael Poon, Marta L. Bryan, Hanno Rein, Caroline V. Morley, Gregory Mace, Yifan Zhou, Brendan P. Bowler, 3 Oct 2024

Astrophysicists have applied tidal theories to predict the obliquity of extrasolar planets. It has been shown that the obliquities of exoplanets in the habitable zone around low-mass stars tend to be eroded in less than a billion years,

{{cite journal

|last1=Heller |first1=R.

|last2=Leconte |first2=J.

|last3=Barnes |first3=R.

|title=Tidal obliquity evolution of potentially habitable planets

|journal=Astronomy and Astrophysics

|date=2011

|volume=528 |pages=A27

|bibcode=2011A&A...528A..27H

|doi=10.1051/0004-6361/201015809

|arxiv=1101.2156

|s2cid=118784209

}}

{{cite journal

|last1=Heller |first1=R.

|last2=Leconte |first2=J.

|last3=Barnes |first3=R.

|date=2011

|title=Habitability of Extrasolar Planets and Tidal Spin Evolution

|journal=Origins of Life and Evolution of Biospheres

|volume= 41 |issue= 6 |pages=539–43

|bibcode=2011OLEB...41..539H

|doi=10.1007/s11084-011-9252-3

|pmid=22139513

|arxiv=1108.4347

|s2cid=10154158

}} which means that they would not have tilt-induced seasons as Earth has.

• Axial parallelism
• Milankovitch cycles
• Polar motion
• Pole shift
• Rotation around a fixed axis
• True polar wander

{{Reflist|30em}}

• [http://nssdc.gsfc.nasa.gov/planetary/ National Space Science Data Center]
• {{cite journal| doi = 10.1007/s10569-007-9072-y| last1 = Seidelmann| first1 = P. Kenneth| last2 = Archinal| first2 = Brent A.| last3 = A'Hearn| first3 = Michael F.| display-authors = 3| last4 = Conrad| first4 = Albert R.| last5 = Consolmagno| first5 = Guy J.| last6 = Hestroffer| first6 = Daniel| last7 = Hilton| first7 = James L.| last8 = Krasinsky| first8 = Georgij A.| last9 = Neumann| first9 = Gregory A.| last10=Oberst | first10=Jürgen | last11=Stooke | first11=Philip J. | last12=Tedesco | first12=Edward F. | last13=Tholen | first13=David J. | last14=Thomas | first14=Peter C. | last15=Williams | first15=Iwan P. | year = 2007| title = Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006| journal = Celestial Mechanics and Dynamical Astronomy| volume = 98| issue = 3| pages = 155–180| bibcode = 2007CeMDA..98..155S| ref = {{sfnRef|Seidelmann Archinal A'hearn et al.|2007}}| doi-access = free}}
• [http://neoprogrammics.com/obliquity_of_the_ecliptic/ Obliquity of the Ecliptic Calculator]

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Category:Precession

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