View source for Orbital eccentricity

{{About|eccentricity in astrodynamics||Eccentricity (disambiguation)}}
{{short description|Parameter that determines the amount by which an orbit deviates from a perfect circle}}
[[File:Kepler orbits.svg|thumb|An elliptic, parabolic, and hyperbolic [[Kepler orbit]]:<br />
{{legend2|red|border=1px solid #770D18|elliptic&nbsp;(eccentricity &#61; 0.7)}}<br />
{{legend2|limegreen|border=1px solid black|parabolic&nbsp;(eccentricity &#61; 1)}}<br />
{{legend2|blue|border=1px solid #444|hyperbolic orbit&nbsp;(eccentricity &#61; 1.3)}}
]]
{{Astrodynamics}}
The '''orbital eccentricity''' of an [[astronomical object]] is a [[dimensionless quantity|dimensionless parameter]] that determines the amount by which its [[orbit]] around another body deviates from a perfect [[circle]]. A value of 0 is a circular orbit, values between 0 and 1 form an [[elliptic orbit]], 1 is a [[parabola|parabolic]] [[escape orbit]], and greater than 1 is a [[hyperbola]]. The term derives its name from the parameters of [[conic section]]s, as every [[Kepler orbit]] is a conic section. It is normally used for the isolated [[two-body problem]], but extensions exist for objects following a [[Klemperer rosette]] orbit through the galaxy.

==Definition==
 {{multiple image
 | direction = vertical
 | width = 220
 | footer = Orbits in a two-body system for two values of the eccentricity, e. (NB: + is [[barycentre]])
 | image1 = Binary system orbit q=3 e=0.gif
 | alt1 = e=0
 | caption1 = e=0
 | image2 = Binary system orbit q=3 e=0.5.gif
 | alt2 = e=0.5
 | caption2 = e=0.5
 }}

In a [[two-body problem]] with inverse-square-law force, every [[orbit]] is a Kepler orbit. The [[eccentricity (mathematics)|eccentricity]] of this [[Kepler orbit]] is a non-negative number that defines its shape.

The eccentricity may take the following values:
* [[circular orbit]]: ''e'' = 0
* [[elliptic orbit]]: 0 < ''e'' < 1 (see [[ellipse]])
* [[parabolic trajectory]]: ''e'' = 1 (see [[parabola]])
* [[hyperbolic trajectory]]: ''e'' > 1 (see [[hyperbola]])

The eccentricity ''e'' is given by

:<math>e = \sqrt{1 + \frac{2 E L^2}{m_\text{red} \alpha ^2}}</math>

where ''E'' is the total [[orbital energy]], ''L'' is the [[angular momentum]], ''m''<sub>red</sub> is the [[reduced mass]], and ''α'' the coefficient of the inverse-square law [[central force]] such as [[gravity]] or [[electrostatics]] in [[classical physics]]:

:<math>F = \frac{\alpha}{r^2}</math>
::(''α'' is negative for an attractive force, positive for a repulsive one; see also [[Kepler problem]])

or in the case of a gravitational force:
:<math>e = \sqrt{1 + \frac{2 \varepsilon h^{2}}{\mu^2}}</math>

where ''ε'' is the [[specific orbital energy]] (total energy divided by the reduced mass), ''μ'' the [[standard gravitational parameter]] based on the total mass, and ''h'' the [[specific relative angular momentum]] ([[angular momentum]] divided by the reduced mass).

For values of ''e'' from 0 to 1 the orbit's shape is an increasingly elongated (or flatter) ellipse; for values of ''e'' from 1 to infinity the orbit is a [[hyperbola]] branch making a total turn of {{nowrap|2 [[arccsc]] ''e''}}, decreasing from 180 to 0 degrees. The limit case between an ellipse and a hyperbola, when ''e''  equals 1, is parabola.

Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits have zero angular momentum and hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic, parabolic, and hyperbolic orbits each tend to the corresponding type of radial trajectory while ''e'' tends to 1 (or in the parabolic case, remains 1).

For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable.

For elliptical orbits, a simple proof shows that arcsin(<math>e</math>) yields the projection angle of a perfect circle to an [[ellipse]] of eccentricity ''e''.  For example, to view the eccentricity of the planet Mercury (''e'' = 0.2056), one must simply calculate the [[inverse trigonometric functions|inverse sine]] to find the projection angle of 11.86 degrees.  Next, tilt any circular object (such as a coffee mug viewed from the top) by that angle and the apparent [[ellipse]] projected to your eye will be of that same eccentricity.

==Etymology==

The word "eccentricity" comes from [[Medieval Latin]] ''eccentricus'', derived from [[Ancient Greek language|Greek]] {{lang|grc|ἔκκεντρος}} ''ekkentros'' "out of the center", from {{lang|grc|ἐκ-}} ''ek-'', "out of" + {{lang|grc|κέντρον}} ''kentron'' "center". "Eccentric" first appeared in English in 1551, with the definition "a circle in which the earth, sun. etc. deviates from its center".{{citation needed|date=November 2017}} By five years later, in 1556, an adjectival form of the word had developed.

==Calculation==

The '''eccentricity''' of an [[orbit]] can be calculated from the [[orbital state vectors]] as the [[magnitude (mathematics)#Euclidean vector space|magnitude]] of the [[eccentricity vector]]:
:<math>e= \left | \mathbf{e} \right |</math>
where:
* '''e''' is the [[eccentricity vector]].

For [[elliptical orbit]]s it can also be calculated from the [[periapsis]] and [[apoapsis]] since {{nowrap|1=''r''<sub>p</sub> = ''a''(1 − ''e'')}} and {{nowrap|1=''r''<sub>a</sub> = ''a''(1 + ''e'')}}, where ''a'' is the [[semimajor axis]].
:<math>\begin{align} e&= {{r_\text{a}-r_\text{p}}\over{r_\text{a}+r_\text{p}}} \\ &= 1-\frac{2}{\frac{r_\text{a}}{r_\text{p}}+1} \end{align}</math>
where:
* ''r''<sub>a</sub> is the radius at [[apoapsis]] (i.e., the farthest distance of the orbit to the [[center of mass]] of the system, which is a [[focus (geometry)|focus]] of the ellipse).
* ''r''<sub>p</sub> is the radius at [[periapsis]] (the closest distance).

The eccentricity of an elliptical orbit can also be used to obtain the ratio of the [[periapsis]] to the [[apoapsis]]:

:<math>{{r_\text{p}}\over{r_\text{a}}}={{1-e}\over{1+e}}</math>

For Earth, orbital eccentricity ≈ 0.0167, [[apoapsis]] is aphelion, and [[periapsis]] is perihelion relative to sun.

For Earth's annual orbit path, r<sub>a</sub>/r<sub>p</sub> ratio = longest_radius / shortest_radius ≈ 1.034 relative to center point of path.

==Examples==
[[File:Eccentricity rocky planets.jpg|500px|thumb|style="right"|[http://www.orbitsimulator.com/gravity/articles/what.html Gravity Simulator] plot of the changing orbital eccentricity of [[Mercury (planet)|Mercury]], [[Venus]], [[Earth]], and [[Mars]] over the next {{gaps|50|000}} years. The arrows indicate the different scales used, as the eccentricities of Mercury and Mars are much greater than those of Venus and Earth. The 0 point on this plot is the year 2007. ]]
{| Class= "wikitable floatright"
|+ Eccentricities of Solar System bodies
! Object || eccentricity
|-
| [[Triton (moon)|Triton]] || {{gaps|0.000|02}}
|-
| [[Venus]] || {{gaps|0.006|8}}
|-
| [[Neptune]] || {{gaps|0.008|6}}
|-
| [[Earth]] || {{gaps|0.016|7}}
|-
| [[Titan (moon)|Titan]] || {{gaps|0.028|8}}
|-
| [[Uranus]] || {{gaps|0.047|2}}
|-
| [[Jupiter]] || {{gaps|0.048|4}}
|-
| [[Saturn]] || {{gaps|0.054|1}}
|-
| [[Moon]] || {{gaps|0.054|9}}
|-
| [[1 Ceres]] || {{gaps|0.075|8}}
|-
| [[4 Vesta]] || {{gaps|0.088|7}}
|-
| [[Mars]] || {{gaps|0.093|4}}
|-
| [[10 Hygiea]] || {{gaps|0.114|6}}
|-
| [[Makemake]] || {{gaps|0.155|9}}
|-
| [[Haumea]] || {{gaps|0.188|7}}
|-
| [[Mercury (planet)|Mercury]] || {{gaps|0.205|6}}
|-
| [[2 Pallas]] || {{gaps|0.231|3}}
|-
| [[Pluto]] || {{gaps|0.248|8}}
|-
| [[3 Juno]] || {{gaps|0.255|5}}
|-
| [[324 Bamberga]] || {{gaps|0.340|0}}
|-
| [[Eris (dwarf planet)|Eris]] || {{gaps|0.440|7}}
|-
| [[Nereid (moon)|Nereid]] || {{gaps|0.750|7}}
|-
| [[90377 Sedna|Sedna]] || {{gaps|0.854|9}}
|-
| [[Halley's Comet]] || {{gaps|0.967|1}}
|-
| [[Comet Hale-Bopp]] || {{gaps|0.995|1}}
|-
| [[Comet Ikeya-Seki]] || {{gaps|0.999|9}}
|-
|[[C/1980 E1 (Bowell)|C/1980 E1]] || 1.057
|-
| [[ʻOumuamua]] || 1.20{{efn|ʻOumuamua was never bound to the Sun.}}
|-
| [[C/2019 Q4 (Borisov)]] || 3.5{{efn|C/2019 Q4 (Borisov) was never bound to the Sun.}}
|}

The eccentricity of the [[Earth]]'s orbit is currently about 0.0167; the Earth's orbit is nearly circular. [[Venus]] and [[Neptune]] have even lower eccentricities. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly 0.0034 to almost 0.058 as a result of gravitational attractions among the planets (see [https://web.archive.org/web/20180106070653/http://www.museum.state.il.us/exhibits/ice_ages/eccentricity_graph.html graph]).<ref name="Berger1991">{{cite web
  |date=1991
  |title=Graph of the eccentricity of the Earth's orbit
  |publisher=Illinois State Museum (Insolation values for the climate of the last 10 million years)
  |author=A. Berger
  |author2=M.F. Loutre
  |last-author-amp=yes
  |url=http://www.museum.state.il.us/exhibits/ice_ages/eccentricity_graph.html
  |archiveurl=https://web.archive.org/web/20180106070653/http://www.museum.state.il.us/exhibits/ice_ages/eccentricity_graph.html
  |archive-date=6 January 2018
  |url-status=dead}}</ref>

The table lists the values for all planets and dwarf planets, and selected asteroids, comets, and moons. [[Mercury (planet)|Mercury]] has the greatest orbital eccentricity of any planet in the [[Solar System]] (''e'' = 0.2056). Such eccentricity is sufficient for Mercury to receive twice as much solar irradiation at perihelion compared to aphelion. [[2006 definition of planet|Before its demotion from planet status in 2006]], [[Pluto]] was considered to be the planet with the most eccentric orbit (''e'' = 0.248). Other Trans-Neptunian objects have significant eccentricity, notably the dwarf planet Eris (0.44). Even further out, [[90377 Sedna|Sedna]], has an extremely high eccentricity of {{val|0.855}} due to its estimated aphelion of 937&nbsp;AU and perihelion of about 76&nbsp;AU.

Most of the Solar System's [[asteroid]]s have orbital eccentricities between 0 and 0.35 with an average value of 0.17.<ref>[http://filer.case.edu/sjr16/advanced/asteroid.html Asteroids<!-- Bot generated title -->] {{webarchive|url=https://web.archive.org/web/20070304161718/http://filer.case.edu/sjr16/advanced/asteroid.html|date=4 March 2007}}</ref> Their comparatively high eccentricities are probably due to the influence of [[Jupiter]] and to past collisions.

The [[Moon]]'s value is 0.0549, the most eccentric of the large moons of the Solar System. The four [[Galilean moons]] have eccentricity < 0.01. [[Neptune]]'s largest moon [[Triton (moon)|Triton]] has an eccentricity of {{val|1.6e-5}} ({{val|0.000016}}),<ref name=Triton>{{cite web
 |title=Neptunian Satellite Fact Sheet
 |publisher=NASA
 |author=David R. Williams
 |date=22 January 2008
 |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/neptuniansatfact.html
}}</ref> the smallest eccentricity of any known moon in the Solar System;{{Citation needed|date=September 2016}} its orbit is as close to a perfect circle as can be currently{{when|date=September 2016}} measured. However, smaller moons, particularly [[irregular moons]], can have significant eccentricity, such as Neptune's third largest moon [[Nereid (moon)|Nereid]] (0.75).

[[Comet]]s have very different values of eccentricity. [[Periodic comet]]s have eccentricities mostly between 0.2 and 0.7,<ref name=Comets>
{{cite book
 |title=Physics and Chemistry of the Solar System
 |publisher=Academic Press
 |last=Lewis
 |first=John
 |date=2 December 2012
 |url=https://books.google.com/books?id=35uwarLgVLsC&pg=PA281
|isbn=9780323145848
 }}</ref> but some of them have highly eccentric [[elliptical orbit]]s with eccentricities just below 1, for example, [[Halley's Comet]] has a value of 0.967. Non-periodic comets follow near-[[parabolic orbit]]s and thus have eccentricities even closer to 1. Examples include [[Comet Hale–Bopp]] with a value of 0.995<ref name=Hale-Bopp-jpl/> and comet [[C/2006 P1|C/2006&nbsp;P1]] (McNaught) with a value of {{val|1.000019}}.<ref name=McNaught-jpl>{{cite web
  |type=2007-07-11 last obs
  |title=JPL Small-Body Database Browser: C/2006 P1 (McNaught)
  |url=http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=C/2006+P1
  |accessdate=17 December 2009}}</ref> As Hale–Bopp's value is less than 1, its orbit is elliptical and it will return.<ref name=Hale-Bopp-jpl>{{cite web
  |type=2007-10-22 last obs
  |title=JPL Small-Body Database Browser: C/1995 O1 (Hale-Bopp)
  |url=http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=Hale-Bopp
  |accessdate=5 December 2008}}</ref> [[C/2006 P1|Comet McNaught]] has a [[hyperbolic orbit]] while within the influence of the planets, but is still bound to the Sun with an orbital period of about 10<sup>5</sup> years.<ref name="Perth">{{cite web |date=22 January 2007 |title=Comet C/2006 P1 (McNaught) – facts and figures |publisher=Perth Observatory in Australia |url=http://www.perthobservatory.wa.gov.au/information/comet_mcnaught_info.html|url-status=dead |archiveurl=https://web.archive.org/web/20110218054618/http://www.perthobservatory.wa.gov.au/information/comet_mcnaught_info.html |archivedate=18 February 2011  }}</ref> As of a 2010 [[Epoch (astronomy)|Epoch]], Comet [[C/1980 E1]] has the largest eccentricity of any known hyperbolic comet with an eccentricity of 1.057,<ref name="C/1980E1-jpl">{{cite web
  |type=1986-12-02 last obs
  |title=JPL Small-Body Database Browser: C/1980 E1 (Bowell)
  |url=http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=1980E1
  |accessdate=22 March 2010}}</ref> and will leave the [[Solar System]] eventually.

[[ʻOumuamua]] is the first [[interstellar object]] found passing through the Solar System. Its orbital eccentricity of 1.20 indicates that ʻOumuamua has never been gravitationally bound to our sun. It was discovered 0.2 AU (30,000,000&nbsp;km; 19,000,000&nbsp;mi) from Earth and is roughly 200 meters in diameter. It has an interstellar speed (velocity at infinity) of 26.33&nbsp;km/s (58,900&nbsp;mph).

==Mean eccentricity==

The mean eccentricity of an object is the average eccentricity as a result of [[Perturbation (astronomy)|perturbations]] over a given time period.  Neptune currently has an instant (current [[Epoch (astronomy)|epoch]]) eccentricity of 0.0113,<ref name="nssdc-Neptune">{{cite web
  |title=Neptune Fact Sheet
  |date=29 November 2007
  |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/neptunefact.html
  |publisher=NASA
  |last=Williams |first=David R.
}}</ref> but from 1800 to 2050 has a ''mean'' eccentricity of {{val|0.00859}}.<ref name=ssd-mean>{{cite web
 |title=Keplerian elements for 1800 A.D. to 2050 A.D.
 |publisher=JPL Solar System Dynamics
 |url=http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt
 |accessdate=17 December 2009}}</ref>

==Climatic effect==

Orbital mechanics require that the duration of the seasons be proportional to the area of the Earth's orbit swept between the [[solstices]] and [[equinoxes]], so when the orbital eccentricity is extreme, the seasons that occur on the far side of the orbit ([[aphelion]]) can be substantially longer in duration.  Today, northern hemisphere fall and winter occur at closest approach ([[perihelion]]), when the earth is moving at its maximum velocity—while the opposite occurs in the southern hemisphere.  As a result, in the northern hemisphere, fall and winter are slightly shorter than spring and summer—but in global terms this is balanced with them being longer below the equator.  In 2006, the northern hemisphere summer was 4.66 days longer than winter, and spring was 2.9&nbsp;days longer than fall due to the [[Milankovitch cycles]].<ref>Data from [http://aa.usno.navy.mil/data/docs/EarthSeasons.php United States Naval Observatory]</ref><ref>{{cite journal |author1=Berger A. |author2=Loutre M.F. |author3=Mélice J.L. |title=Equatorial insolation: from precession harmonics to eccentricity frequencies|journal=Clim. Past Discuss.|volume=2 |pages=519–533|year=2006|doi=10.5194/cpd-2-519-2006|url=http://www.clim-past-discuss.net/2/519/2006/cpd-2-519-2006.pdf |issue=4}}</ref>

[[Apsidal precession]] also slowly changes the place in the Earth's orbit where the solstices and equinoxes occur.  Note that this is a slow change in the ''orbit'' of the Earth, not the axis of rotation, which is referred to as [[axial precession]] (see {{section link|Precession|Astronomy}}).  Over the next 10,000 years, the northern hemisphere winters will become gradually longer and summers will become shorter. However, any cooling effect in one hemisphere is balanced by warming in the other, and any overall change will be counteracted by the fact that the eccentricity of Earth's orbit will be almost halved.<ref>[http://ircamera.as.arizona.edu/NatSci102/NatSci102/lectures/climate.htm Arizona U., Long Term Climate]</ref> This will reduce the mean orbital radius and raise temperatures in both hemispheres closer to the mid-interglacial peak.

==Exoplanets==

Of the many [[exoplanet]]s discovered, most have a higher orbital eccentricity than planets in our solar system.  Exoplanets found with low orbital eccentricity, near circular orbits, are very close to their star and are [[tidally locked]] to the star. All eight planets in the Solar System have near-circular orbits.  The exoplanets discovered show that the solar system, with its unusually low eccentricity, is rare and unique.<ref>[http://exoplanets.org/ecc.html exoplanets.org, ORBITAL ECCENTRICITES, by G.Marcy, P.Butler, D.Fischer, S.Vogt, 20 Sept 2003]</ref> One theory attributes this low eccentricity to the high number of planets in the Solar System; another suggests it arose because of its unique asteroid belts. A few other [[List of multiplanetary systems|multiplanetary systems]] have been found, but none resemble the Solar System. The Solar System has unique [[planetesimal]] systems, which led the planets to have near-circular orbits. Solar planetesimal systems include the [[asteroid belt]], [[Hilda family]], [[Kuiper belt]], [[Hills cloud]], and the [[Oort cloud]]. The exoplanet systems discovered have either no planetesimal systems or one very large one. Low eccentricity is needed for habitability, especially advanced life.<ref>{{cite book|ref=Ward|author= Ward, Peter |author2=Brownlee, Donald |title=Rare Earth: Why Complex Life is Uncommon in the Universe|publisher= Springer|year= 2000|isbn=0-387-98701-0| pages=122–123}}</ref> High multiplicity planet systems are much more likely to have habitable exoplanets.<ref>{{cite journal | pmc = 4291657 | pmid=25512527 | doi=10.1073/pnas.1406545111 | volume=112 | issue=1 | title=Exoplanet orbital eccentricity: multiplicity relation and the Solar System | journal=Proc Natl Acad Sci U S A | pages=20–4 | last1 = Limbach | first1 = MA | last2 = Turner | first2 = EL|arxiv = 1404.2552 |bibcode = 2015PNAS..112...20L | year=2015 }}</ref><ref>[https://arxiv.org/pdf/1512.04996.pdf Steward Observatory, University of Arizona, Tucson, Planetesimals in Debris Disks, by Andrew N. Youdin and George H. Rieke, 2015]</ref> The [[grand tack hypothesis]] of the Solar System also helps understand its near-circular orbits and other unique features.<ref name=Zubritsky_2011>{{cite web| last1=Zubritsky| first1=Elizabeth| title=Jupiter's Youthful Travels Redefined Solar System| url=http://www.nasa.gov/topics/solarsystem/features/young-jupiter.html| publisher=[[NASA]]| accessdate=4 November 2015}}</ref><ref name=Sanders_2011>{{cite web| last1=Sanders| first1=Ray| title=How Did Jupiter Shape Our Solar System?| url=http://www.universetoday.com/88374/how-did-jupiter-shape-our-solar-system/| work=[[Universe Today]]| accessdate=4 November 2015}}</ref><ref name=Choi_2015>{{cite web| last1=Choi| first1=Charles Q.| title=Jupiter's 'Smashing' Migration May Explain Our Oddball Solar System| url=http://www.space.com/28901-wandering-jupiter-oddball-solar-system.html| publisher=Space.com| accessdate=4 November 2015}}</ref><ref name=Davidsson_2014>{{cite web| last1=Davidsson| first1=Dr. Björn J. R.| title=Mysteries of the asteroid belt| url=https://thehistoryofthesolarsystem.wordpress.com/tag/the-grand-tack/| website=The History of the Solar System| accessdate=7 November 2015}}</ref><ref name=Raymond_2013>{{cite web| last1=Raymond| first1=Sean| title=The Grand Tack| url=http://planetplanet.net/2013/08/02/the-grand-tack/| website=PlanetPlanet| accessdate=7 November 2015}}</ref><ref name="O'Brien_2014">{{cite journal| last1=O'Brien| first1=David P.| last2=Walsh| first2=Kevin J.| last3=Morbidelli| first3=Alessandro| last4=Raymond| first4=Sean N.| last5=Mandell| first5=Avi M.| title=Water delivery and giant impacts in the 'Grand Tack' scenario| journal=Icarus| date=2014| volume=239| pages=74–84| doi=10.1016/j.icarus.2014.05.009|arxiv = 1407.3290 |bibcode = 2014Icar..239...74O }}</ref><ref>{{cite journal |title=Relative Likelihood for Life as a Function of Cosmic Time |journal=Journal of Cosmology and Astroparticle Physics |volume=2016 |issue=8 |pages=040 |first=Abraham |last=Loeb |first2=Rafael |last2=Batista |first3=David |last3=Sloan |date=August 2016 |doi=10.1088/1475-7516/2016/08/040 |arxiv=1606.08448 |bibcode=2016JCAP...08..040L }}</ref><ref>{{cite web |url=https://www.cfa.harvard.edu/news/2016–17 |publisher=Harvard-Smithsonian Center for Astrophysics |title=Is Earthly Life Premature from a Cosmic Perspective? |date=1 August 2016 }}</ref>

==See also==
* [[Eccentricity (mathematics)]]
* [[Eccentricity vector]]
* [[Equation of time]]
* [[Milankovitch cycles]]
* [[Orbit]]s

==Footnotes==
{{notelist}}

==References==
{{Reflist}}

==Further reading==
*{{cite book |last=Prussing |first=John E. |first2=Bruce A. |last2=Conway |title=Orbital Mechanics |location=New York |publisher=Oxford University Press |year=1993 |isbn=0-19-507834-9 |url-access=registration |url=https://archive.org/details/orbitalmechanics00prus }}

==External links==
* [http://scienceworld.wolfram.com/physics/Eccentricity.html World of Physics: Eccentricity]
* [http://www.ncdc.noaa.gov/paleo/forcing.html The NOAA page on Climate Forcing Data] includes (calculated) data from [ftp://ftp.ncdc.noaa.gov/pub/data/paleo/insolation/ Berger (1978), Berger and Loutre (1991)]. [http://www.imcce.fr/Equipes/ASD/insola/earth/earth.html Laskar et al. (2004)] on Earth orbital variations, Includes eccentricity over the last 50 million years and for the coming 20 million years.
* [https://web.archive.org/web/20060210012245/http://www.astrobiology.ucla.edu/OTHER/SSO/ The orbital simulations by Varadi, Ghil and Runnegar (2003)] provides series for Earth orbital eccentricity and orbital inclination.
* [http://www.walter-fendt.de/ph14e/keplerlaw2.htm Kepler's Second law's simulation]
{{orbits}}
{{Use dmy dates|date=September 2019}}

[[Category:Orbits|Eccentricity]]