Editing Axial tilt (section)

=== 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>
}}