Editing Axial tilt

{{Redirect|Obliquity|the book|Obliquity (book)}}
{{short description|Angle between the rotational axis and orbital axis of a body}}
{{Use dmy dates|date=December 2022}}[[Image:Planet axis comparison.png|380px|thumb|The '''positive''' pole of a planet is defined by the [[right-hand rule]]: if the fingers of the right hand are curled in the direction of the rotation then the thumb points to the positive pole. The axial tilt is defined as the angle between the direction of the positive pole and the normal to the orbital plane. The angles for Earth, Uranus, and Venus are approximately 23°, 97°, and 177° respectively.]]In [[astronomy]], '''axial tilt''', also known as '''obliquity''', is the [[angle]] between an object's [[Rotation around a fixed axis|rotational axis]] and its [[orbit]]al axis, which is the line [[perpendicular]] to its [[Orbital plane (astronomy)|orbital plane]]; equivalently, it is the angle between its [[equator]]ial plane and orbital plane.<ref>
{{cite book
 |author=U.S. Naval Observatory Nautical Almanac Office
 |editor=P. Kenneth Seidelmann
 |date=1992
 |title=Explanatory Supplement to the Astronomical Almanac
 |page=733
 |publisher=University Science Books
 |isbn=978-0-935702-68-2
}}</ref> It differs from [[orbital inclination]]. 
At an obliquity of 0 degrees, the two axes point in the same direction; that is, the rotational axis is perpendicular to the orbital plane. 

The rotational axis of [[Earth]], for example, is the imaginary line that passes through both the [[North Pole]] and [[South Pole]], whereas the Earth's orbital axis is the line perpendicular to the imaginary [[Plane (geometry)|plane]] through which the Earth moves as it revolves around the [[Sun]]; the Earth's obliquity or axial tilt is the angle between these two lines. 
 
Over the course of an [[orbital period]], the obliquity usually does not change considerably, and the [[Axial parallelism|orientation of the axis remains the same]] relative to the [[celestial sphere|background]] of [[fixed stars|stars]]. This causes one pole to be pointed more toward the Sun on one side of the orbit, and more away from the Sun on the other side—the cause of the [[season]]s on Earth.

== Standards ==
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 [[Axis–angle representation|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]];<ref>''Explanatory Supplement 1992'', p. 384</ref> under this system, [[Venus]] is tilted 3° and rotates [[retrograde motion|retrograde]], opposite that of most of the other planets.<ref name="CorreiaVenusI">{{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
}}</ref><ref>{{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
}}</ref>
* The IAU also uses the right-hand rule to define a ''positive pole''<ref>{{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}}</ref> for the purpose of determining orientation. Using this convention, Venus is tilted 177° ("upside down") and rotates prograde.

== Earth ==
{{further|Ecliptic#Obliquity of the ecliptic}}
{{See also|Earth's rotation|Earth-centered inertial}}

[[Earth]]'s [[orbit|orbital plane]] is known as the [[ecliptic]] plane, and [[Earth#Axial tilt and seasons|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]].<ref>
{{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
}}</ref> It is denoted by the [[Greek letter]] Epsilon ''[[epsilon|ε]]''.

Earth currently has an axial tilt of about 23.44°.<ref>[https://aa.usno.navy.mil/faq/asa_glossary "Glossary"] in ''Astronomical Almanac Online''. (2023). Washington DC: United States Naval Observatory. s.v. obliquity.</ref> This value remains about the same relative to a stationary orbital plane throughout the cycles of [[axial precession]].<ref>
{{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 & Co.|J. B. Lippincott]]
 |volume=1 |pages=604–605
}}</ref> But the ecliptic (i.e., Earth's orbit) moves due to planetary [[perturbation (astronomy)|perturbations]], and the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about [[second of arc|46.8″]]<ref name=Ray2014>{{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
}}</ref> per [[century]] ''(see details in [[#Short term|Short term]] below)''.

=== History ===
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.<ref>
{{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]
}}</ref> 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.<ref>
{{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]
}}</ref> In 1437, [[Ulugh Beg]] determined the Earth's axial tilt as 23°30′17″ (23.5047°).<ref>{{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}}</ref>

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 (astronomy)|trepidation]]'' of the equinoxes. Perhaps the first to realize this was incorrect (during historic time) was [[Ibn al-Shatir]] in the fourteenth century<ref>{{cite book 
 |last=Saliba |first=George 
 |date=1994
 |title= A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam 
 |page=235
}}</ref> and the first to realize that the obliquity is decreasing at a relatively constant rate was [[Fracastoro]] in 1538.<ref>
{{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]
}}</ref> The first accurate, modern, western observations of the obliquity were probably those of [[Tycho Brahe]] from [[Denmark]], about 1584,<ref>Dreyer (1890), p. 123</ref> although observations by several others, including [[al-Ma'mun]], [[Sharaf al-Dīn al-Tūsī|al-Tusi]],<ref>
{{cite book 
 |last=Sayili |first=Aydin 
 |date=1981
 |title= The Observatory in Islam 
 |page=78
}}</ref> [[Georg Purbach|Purbach]], [[Regiomontanus]], and [[Bernhard Walther|Walther]], could have provided similar information.

=== Seasons ===
{{main|Season}}
[[File:Earth tilt animation.gif|thumb|An illustration of [[axial parallelism]]. The axis of Earth remains oriented in the same direction with reference to the background stars regardless of where it is in its [[Earth's orbit|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 [[Hemispheres of the Earth|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 [[season]]s. [[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 [[climate change (general concept)|climatic change]] ''(also see [[Milankovitch cycles]])''.

[[File:axial_tilt_vs_tropical_and_polar_circles.svg|thumb|center|420px|Relationship between Earth's axial tilt (ε) to the tropical and polar circles]]

=== Oscillation ===

==== Short term ====
[[File:Obliquity of the ecliptic laskar.PNG|thumb|Obliquity of the ecliptic for 20,000 years, from [[Jacques Laskar|Laskar]] (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 ephemeris|fundamental ephemerides]] as the accuracy of [[Observational astronomy|observation]] improves and as the understanding of the [[Analytical dynamics|dynamics]] increases, and from these ephemerides various astronomical values, including the obliquity, are derived.

Annual [[almanac]]s 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 [[Newcomb's Tables of the Sun|work of Newcomb]], who analyzed positions of the planets until about 1895:

: {{math|''ε'' {{=}} 23°27′8.26″ − 46.845″ ''T'' − 0.0059″ ''T''<sup>2</sup> + {{val|0.00181}}″ ''T''<sup>3</sup>}}

where {{math|''ε''}} is the obliquity and {{math|''T''}} is [[Tropical year|tropical centuries]] from [[Epoch (astronomy)#Besselian years|B1900.0]] to the date in question.<ref>
{{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
}}</ref>

From 1984, the [[Jet Propulsion Laboratory Development Ephemeris|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″ ''T''<sup>2</sup> + {{val|0.001813}}″ ''T''<sup>3</sup>}}

where hereafter {{math|''T''}} is [[Julian year (astronomy)|Julian centuries]] from [[Epoch (astronomy)#Julian years and J2000|J2000.0]].<ref>
{{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
}}</ref>

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:<ref name="ReferenceA">''Astronomical Almanac 2010'', p. B52</ref>

: {{math|''ε'' {{=}} 23°26′21.406″ − {{val|46.836769}}″ ''T'' − {{val|0.0001831}}″ ''T''<sup>2</sup> + {{val|0.00200340}}″ ''T''<sup>3</sup> − 5.76″ × 10<sup>−7</sup> ''T''<sup>4</sup> − 4.34″ × 10<sup>−8</sup> ''T''<sup>5</sup>}}

These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps {{math|±}} several centuries.<ref>
{{cite book
 |last=Newcomb |first=Simon
 |date=1906
 |title=A Compendium of Spherical Astronomy
 |url=https://archive.org/details/acompendiumsphe00newcgoog
 |publisher=[[Macmillan Publishers|MacMillan]]
 |pages=[https://archive.org/details/acompendiumsphe00newcgoog/page/n250 226]–227
}}</ref> [[Jacques Laskar]] computed an expression to order {{math|''T''<sup>10</sup>}} good to 0.02″ over 1000 years and several [[Minute of arc|arcseconds]] over 10,000 years.

:{{math|''ε'' {{=}} 23°26′21.448″ − 4680.93″ ''t'' − 1.55″ ''t''<sup>2</sup> + 1999.25″ ''t''<sup>3</sup> − 51.38″ ''t''<sup>4</sup> − 249.67″ ''t''<sup>5</sup> − 39.05″ ''t''<sup>6</sup> + 7.12″ ''t''<sup>7</sup> + 27.87″ ''t''<sup>8</sup> + 5.79″ ''t''<sup>9</sup> + 2.45″ ''t''<sup>10</sup>}}

where here {{math|''t''}} is multiples of 10,000 [[Julian day|Julian years]] from [[Epoch (astronomy)#Julian years and J2000|J2000.0]].<ref name="laskar">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.</ref>

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 [[minute of arc|arcseconds]]) short-period (about 18.6 years) oscillations of the rotation axis of Earth, known as [[astronomical nutation|nutation]], which add a periodic component to Earth's obliquity.<ref>''Explanatory Supplement'' (1961), sec. 2C</ref><ref>
{{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
}}</ref> The ''true'' or instantaneous obliquity includes this nutation.<ref>
{{cite book
 |last=Meeus |first=Jean
 |date=1991
 |chapter=Chapter 21
 |title=Astronomical Algorithms
 |publisher=Willmann-Bell
 |isbn=978-0-943396-35-4
}}</ref>

==== Long term ====
{{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 [[Addend|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″}}.<ref>
{{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
}}</ref>

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 resonance]]s and [[Stability of the Solar System|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]]'').<ref name="LaskarRobutel">
{{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 (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
}}</ref><ref>
{{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
}}</ref> However, more recent numerical simulations<ref>
{{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 (journal)|Icarus]]
 |volume=217 |issue= 1 |pages=77–87
 |doi=10.1016/j.icarus.2011.10.013
 |bibcode = 2012Icar..217...77L
}}</ref> 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°.<ref>{{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}}</ref> 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.<ref>
{{cite journal
 |author1=Ward, W.R.
 |date=1982
 |title=Comments on the Long-Term Stability of the Earth's Obliquity
 |journal=[[Icarus (journal)|Icarus]]
 |volume=50 |issue= 2–3 |pages=444–448
 |bibcode=1982Icar...50..444W
 |doi=10.1016/0019-1035(82)90134-8
}}</ref>

{{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&nbsp;million years; note the approx. 41,000-year period of variation. In both graphs, the red point represents the year 1850.<ref>Berger, 1976.</ref>
}}

== Solar System bodies ==
{{see also|Poles of astronomical bodies#Poles of rotation}}[[File:Planets and dwarf planets' tilt and rotation speed.webm|thumb|upright=1.5|Axial tilt of eight [[Planet|planets]] and two dwarf planets, [[Ceres (dwarf planet)|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.<ref name="Ward">{{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}}</ref> The rate varies due to [[Tidal acceleration|tidal dissipation]] and [[Planetary core|core]]-[[Mantle (geology)|mantle]] interaction, among other things. When a planet's precession rate approaches certain values, [[orbital resonance]]s 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.<ref name="Ward" />

[[Mercury (planet)|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 [[Moon#Formation|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 [[Perturbation (astronomy)|perturbations]] of the planets.<ref name="LaskarRobutel" /><ref>
{{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 (journal)|Science]]
 |volume=259 |issue= 5099|pages=1294–1297
 |bibcode=1993Sci...259.1294T
 |doi=10.1126/science.259.5099.1294
 |pmid=17732249
 |s2cid=42933021
}}</ref> 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.<ref name="CorreiaVenusI" /><ref name=Correia2009>{{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
}}</ref>

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.<ref>{{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}}</ref><ref>{{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}}</ref>

The obliquities of the outer planets are considered relatively stable.

{| class="wikitable" style="margin: 0.5em auto; text-align:right;"
|+ Axis and rotation of selected Solar System bodies
|-
! rowspan=3 | Body
! colspan=4 style="background:#F2FEEC;" | [[NASA]], [[J2000]].0<ref name="NASA">[http://nssdc.gsfc.nasa.gov/planetary/planetfact.html Planetary Fact Sheets], at http://nssdc.gsfc.nasa.gov</ref> epoch
! colspan=4 style="background:#edf3fe;" | [[IAU]], 0h 0 January 2010 [[terrestrial time|TT]]<ref>''Astronomical Almanac 2010'', pp. B52, C3, D2, E3, E55</ref> epoch
|-
! rowspan=2 style="background: #F2FEEC;" | Axial tilt <br>(degrees)
! colspan=2 style="background: #F2FEEC;" | North Pole
! rowspan=2 style="background: #F2FEEC;" | Rotational <br>period <br>(hours)
! rowspan=2 style="background: #edf3fe;" | Axial tilt <br>(degrees)
! colspan=2 style="background: #edf3fe;" | North Pole
! rowspan=2 style="background: #edf3fe;" | Rotation <br>(deg./day)
|-
! style="background: #F2FEEC;" | [[Right ascension|R.A.]] (degrees)
! style="background: #F2FEEC;" | [[Declination|Dec.]] (degrees)
! style="background: #edf3fe;" | [[Right ascension|R.A.]] (degrees)
! style="background: #edf3fe;" | [[Declination|Dec.]] (degrees)
|-
| style="text-align:left;" | [[Sun]]
| 7.25 || 286.13 || 63.87 || 609.12{{efn|group=upper-alpha|At 16° latitude; the Sun's rotation varies with latitude.}} || 7.25{{efn|group=upper-alpha|With respect to the [[ecliptic]] of 1850.}} || 286.15 ||63.89 || 14.18
|-
| style="text-align:left;" |  [[Mercury (planet)|Mercury]]
| 0.03 || 281.01 || 61.41 || 1407.6 || 0.01 || 281.01 || 61.45 || 6.14
|-
| style="text-align:left;" |  [[Venus]]
| 2.64 || 272.76 || 67.16 || −5832.6 || 2.64 || 272.76 || 67.16 || −1.48
|-
| style="text-align:left;" |  [[Earth]]
| 23.44 || 0.00 || 90.00 || 23.93 || 23.44 || {{n/a|Undefined}} || 90.00 || 360.99
|-
| style="text-align:left;" |  [[Moon]]
| 6.68 || – || – || 655.73 || 1.54{{efn|group=upper-alpha|With respect to the ecliptic; the Moon's orbit is inclined 5.16° to the ecliptic.}} || 270.00 || 66.54 || 13.18
|-
| style="text-align:left;" |  [[Mars]]
| 25.19 || 317.68 || 52.89 || 24.62 || 25.19 || 317.67 || 52.88 || 350.89
|-
| style="text-align:left;" | [[Jupiter]]
| 3.13 || 268.06 || 64.50 || 9.93{{efn|group=upper-alpha|name=clouds|From the origin of the radio emissions; the visible clouds generally rotate at different rate.}} || 3.12 || 268.06 || 64.50 || 870.54{{efn|group=upper-alpha|name=clouds}}
|-
| style="text-align:left;" | [[Saturn]]
| 26.73 || 40.59 || 83.54 || 10.66{{efn|group=upper-alpha|name=clouds}} || 26.73 || 40.59 || 83.54 || 810.79{{efn|group=upper-alpha|name=clouds}}
|-
| style="text-align:left;" | [[Uranus]]
| 82.23 || 257.31 || −15.18 || −17.24{{efn|group=upper-alpha|name=clouds}} || 82.23 || 257.31 || −15.18 || −501.16{{efn|group=upper-alpha|name=clouds}}
|-
| style="text-align:left;" | [[Neptune]]
| 28.32 || 299.33 || 42.95 || 16.11{{efn|group=upper-alpha|name=clouds}} || 28.33 || 299.40 || 42.95 || 536.31{{efn|group=upper-alpha|name=clouds}}
|-
| style="text-align:left;" | [[Pluto]]{{efn|group=upper-alpha|name=plutopole|NASA lists the coordinates of Pluto's positive pole; noted values have been reinterpreted to correspond to the north/negative pole.}}
| 57.47 || 312.99{{efn|group=upper-alpha|name=plutopole}}
| 6.16{{efn|group=upper-alpha|name=plutopole}} || −153.29 || 60.41 || 312.99 || 6.16 || −56.36
|-
| colspan=9 style="padding:8px 16px; font-size:0.85em;" | {{noteslist|group=upper-alpha}}
|}

== Extrasolar planets ==
The stellar obliquity {{math|''ψ''<sub>s</sub>}}, 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,<ref name=HRM>
{{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
}}</ref> which serves as a lower limit to {{math|''ψ''<sub>s</sub>}}. 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.<ref>[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</ref>

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,<ref name=Heller_2011>
{{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
}}</ref><ref name=Heller_2011b>
{{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
}}</ref> which means that they would not have tilt-induced seasons as Earth has.

== See also ==
* [[Axial parallelism]]
* [[Milankovitch cycles]]
* [[Polar motion]]
* [[Pole shift]]
* [[Rotation around a fixed axis]]
* [[True polar wander]]

== References ==
{{Reflist|30em}}

== External links ==
* [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<!-- written A'hearn here, mostly A'Hearn elsewhere -->| 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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