Orbital eccentricity

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

The eccentricity may take the following values:

• Circular orbit: {{math|e {{=}} 0}}
• Elliptic orbit: {{math|0 < e < 1}}
• Parabolic trajectory: {{math|e {{=}} 1}}
• Hyperbolic trajectory: {{math|e > 1}}

The eccentricity {{mvar|e}} is given by{{cite book| last=Abraham |first=Ralph | year=2008 | title=Foundations of Mechanics | edition=2nd | publisher=AMS Chelsea Pub./American Mathematical Society | others=Marsden, Jerrold E. | isbn=978-0-8218-4438-0 | location=Providence, RI | oclc=191847156}}

$$e\; =\; \backslash sqrt\{1\; +\; \backslash frac\{\backslash \; 2\backslash \; E\backslash \; L^2\backslash \; \}\{\backslash \; m\_\backslash text\{rdc\}\backslash \; \backslash alpha^2\backslash \; \}\}$$

where {{math|E}} is the total orbital energy, {{math|L}} is the angular momentum, {{math|m_rdc}} is the reduced mass, and $\backslash alpha$ the coefficient of the inverse-square law central force such as in the theory of gravity or electrostatics in classical physics:

$$F\; =\; \backslash frac\{\backslash alpha\}\{r^2\}$$

($\backslash alpha$ is negative for an attractive force, positive for a repulsive one; related to the Kepler problem)

or in the case of a gravitational force:{{cite book|url=https://books.google.com/books?id=UEC9DwAAQBAJ | title=Fundamentals of Astrodynamics | last1=Bate|first1=Roger R. | last2=Mueller | first2=Donald D. | last3=White|first3=Jerry E. | last4=Saylor|first4=William W. | publisher=Courier Dover | date=2020 | access-date=4 March 2022 | isbn=978-0-486-49704-4}}{{rp|p=24}}

$$e\; =\; \backslash sqrt\{1\; +\; \backslash frac\{2\; \backslash varepsilon\; h^\{2\}\}\{\backslash mu^2\}\}$$

where {{math|ε}} is the specific orbital energy (total energy divided by the reduced mass), {{math|μ}} the standard gravitational parameter based on the total mass, and {{math|h}} the specific relative angular momentum (angular momentum divided by the reduced mass).{{rp|pp=12–17}}

For values of {{mvar|e}} from {{math|0}} to just under {{math|1}} the orbit's shape is an increasingly elongated (or flatter) ellipse; for values of {{mvar|e}} just over {{math|1}} to infinity the orbit is a hyperbola branch making a total turn of {{nobr|{{math|  2 arccsc(e)}} ,}} decreasing from 180 to 0 degrees. Here, the total turn is analogous to turning number, but for open curves (an angle covered by velocity vector). The limit case between an ellipse and a hyperbola, when {{mvar|e}} equals {{math|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 {{mvar|e}} tends to {{math|1}} (or in the parabolic case, remains {{math|1}}).

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

For elliptical orbits, a simple proof shows that $\backslash \; \backslash arcsin(e)\backslash$ gives the projection angle of a perfect circle to an ellipse of eccentricity {{mvar|e}}. For example, to view the eccentricity of the planet Mercury ({{math|e {{=}} 0.2056}}), one must simply calculate the inverse sine to find the projection angle of 11.86 degrees. Then, tilting any circular object by that angle, the apparent ellipse of that object projected to the viewer's eye will be of the same eccentricity.

The word "eccentricity" comes from Medieval Latin eccentricus, derived from 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}} In 1556, five years later, an adjectival form of the word had developed.

The eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector:

$$e\; =\; \backslash left\; |\; \backslash mathbf\{e\}\; \backslash right\; |$$

where:

• {{math|e}} is the eccentricity vector ("Hamilton's vector").{{rp|pp=25, 62–63}}

For elliptical orbits it can also be calculated from the periapsis and apoapsis since $r\_\backslash text\{p\}\; =\; a\; \backslash ,\; (1\; -\; e\; )$ and $r\_\backslash text\{a\}\; =\; a\; \backslash ,\; (1\; +\; e\; )\backslash ,,$ where {{mvar|a}} is the length of the semi-major axis.$$$$

\begin{align}

e &= \frac{ r_\text{a} - r_\text{p} }{ r_\text{a} + r_\text{p} } \\ \, \\

&= \frac{ r_\text{a} / r_\text{p} - 1 }{ r_\text{a} / r_\text{p} + 1 } \\ \, \\

&= 1 - \frac{2}{\; \frac{ r_\text{a} }{ r_\text{p} } + 1 \;}

\end{align}

where:

• {{mvar|r}}{{sub|a}} is the radius at apoapsis (also "apofocus", "aphelion", "apogee"), i.e., the farthest distance of the orbit to the center of mass of the system, which is a focus of the ellipse.
• {{mvar|r}}{{sub|p}} is the radius at periapsis (or "perifocus" etc.), the closest distance.

The semi-major axis, a, is also the path-averaged distance to the centre of mass, {{rp|pp=24–25}} while the time-averaged distance is a(1 + e e / 2).[https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/09%3A_The_Two_Body_Problem_in_Two_Dimensions/9.10%3A_Mean_Distance_in_an_Elliptic_Orbit]

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

$$\backslash frac\{r\_\backslash text\{a\}\; \}\{\; r\_\backslash text\{p\}\; \}\; =\; \backslash frac\{\backslash ,a\backslash ,(1\; +\; e)\backslash ,\}\{\backslash ,a\backslash ,(1\; -\; e)\backslash ,\; \}\; =\; \backslash frac\{1\; +\; e\; \}\{\; 1\; -\; e\; \}$$

For Earth, orbital eccentricity {{math|e ≈ {{val|0.01671}}}}, apoapsis is aphelion and periapsis is perihelion, relative to the Sun.

For Earth's annual orbit path, the ratio of longest radius ({{mvar|r}}{{sub|a}}) / shortest radius ({{mvar|r}}{{sub|p}}) is $\backslash frac\{\backslash ,\; r\_\backslash text\{a\}\; \backslash ,\}\{\; r\_\backslash text\{p\}\; \}\; =\; \backslash frac\{\backslash ,\; 1\; +\; e\; \backslash ,\}\{\; 1\; -\; e\; \}\; \backslash text\{\; \approx \; 1.03399\; .\}$

File:Eccentricity rocky planets.jpg, Venus, Earth and Mars over the next 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 x-axis in this plot is the year 2007.]]

The table lists the values for all planets and dwarf planets, and selected asteroids, comets, and moons. Mercury has the greatest orbital eccentricity of any planet in the Solar System (e = {{val|0.2056}}), followed by Mars of {{gaps|0.093|4}}. Such eccentricity is sufficient for Mercury to receive twice as much solar irradiation at perihelion compared to aphelion. Before its demotion from planet status in 2006, Pluto was considered to be the planet with the most eccentric orbit (e = {{val|0.248}}). Other Trans-Neptunian objects have significant eccentricity, notably the dwarf planet Eris (0.44). Even further out, Sedna has an extremely-high eccentricity of {{val|0.855}} due to its estimated aphelion of 937 AU and perihelion of about 76 AU, possibly under influence of unknown object(s).

The eccentricity of Earth's orbit is currently about {{gaps|0.016|7}}; its orbit is nearly circular. Neptune's and Venus's have even lower eccentricities of {{gaps|0.008|6}} and {{gaps|0.006|8}} respectively, the latter being the least orbital eccentricity of any planet in the Solar System. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly {{gaps|0.003|4}} to almost 0.058 as a result of gravitational attractions among the planets.{{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

|name-list-style=amp

|url=http://www.museum.state.il.us/exhibits/ice_ages/eccentricity_graph.html

|archive-url=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}}

Luna's value is {{gaps|0.054|9}}, the most eccentric of the large moons in the Solar System. The four Galilean moons (Io, Europa, Ganymede and Callisto) have their eccentricities of less than 0.01. Neptune's largest moon Triton has an eccentricity of {{val|1.6e-5}} ({{gaps|0.000|016}}),{{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

}} 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. Smaller moons, particularly irregular moons, can have significant eccentricities, such as Neptune's third largest moon, Nereid, of {{val|0.75}}.

Most of the Solar System's asteroids have orbital eccentricities between 0 and 0.35 with an average value of 0.17.[http://filer.case.edu/sjr16/advanced/asteroid.html Asteroids]{{webarchive

|url=https://web.archive.org/web/20070304161718/http://filer.case.edu/sjr16/advanced/asteroid.html

|date=4 March 2007}} Their comparatively high eccentricities are probably due to under influence of Jupiter and to past collisions.

Comets have very different values of eccentricities. Periodic comets have eccentricities mostly between 0.2 and 0.7,

{{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

}} but some of them have highly eccentric elliptical orbits with eccentricities just below 1; for example, Halley's Comet has a value of 0.967. Non-periodic comets follow near-parabolic orbits and thus have eccentricities even closer to 1. Examples include Comet Hale–Bopp with a value of {{gaps|0.995|1}}, Comet Ikeya-Seki with a value of {{gaps|0.999|9}} and Comet McNaught (C/2006 P1) with a value of {{gaps|1.000|019}}.{{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

|access-date=17 December 2009}} As first two's values are less than 1, their orbit are elliptical and they will return.{{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

|access-date=5 December 2008}}

McNaught has a hyperbolic orbit but within the influence of the inner planets, is still bound to the Sun with an orbital period of about 10⁵ years. Comet C/1980 E1 has the largest eccentricity of any known hyperbolic comet of solar origin with an eccentricity of 1.057,{{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

|access-date=22 March 2010}} and will eventually leave the Solar System.

{{okina}}Oumuamua is the first interstellar object to be found passing through the Solar System. Its orbital eccentricity of 1.20 indicates that {{okina}}Oumuamua has never been gravitationally bound to the Sun. It was discovered 0.2 AU ({{gaps|30|000|000}} km; {{gaps|19|000|000}} mi) from Earth and is roughly 200 meters in diameter. It has an interstellar speed (velocity at infinity) of 26.33 km/s ({{gaps|58|900}} mph).

The exoplanet HD 20782 b has the most eccentric orbit known of 0.97 ± 0.01,{{cite journal |journal=MNRAS |arxiv=0810.1589 |title=Selection Functions in Doppler Planet Searches |author=S. J. O'Toole |author2=C. G. Tinney |author3=H. R. A. Jones |author4=R. P. Butler |author5=G. W. Marcy |author6=B. Carter |author7=J. Bailey |volume=392, 641 |date=2009 |issue=2 |pages=641–654 |bibcode = 2009MNRAS.392..641O |doi = 10.1111/j.1365-2966.2008.14051.x |doi-access=free |s2cid=7248338}} followed by HD 80606 b of 0.93226 {{±|0.00064|0.00069}}.{{Cite web |date=March 28, 2016 |title=Investigating the Mystery of Migrating 'Hot Jupiters' |url=https://www.jpl.nasa.gov/news/investigating-the-mystery-of-migrating-hot-jupiters/ |url-status=live |archive-url=https://web.archive.org/web/20230324065920/https://www.nasa.gov/feature/jpl/investigating-the-mystery-of-migrating-hot-jupiters |archive-date=24 March 2023 |publisher=Jet Propulsion Laboratory}}

The mean eccentricity of an object is the average eccentricity as a result of perturbations over a given time period. For example: Neptune currently has an instant (current epoch) eccentricity of {{gaps|0.011|3}},{{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.

}} but from 1800 to 2050 has a mean eccentricity of {{val|0.00859}}.{{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

|access-date=17 December 2009}}

{{expand section|date=January 2025}}

Orbital mechanics require that the duration of the seasons be proportional to the area of 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. Northern hemisphere autumn and winter occur at closest approach (perihelion), when Earth is moving at its maximum velocity—while the opposite occurs in the southern hemisphere. As a result, in the northern hemisphere, autumn 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 days longer than autumn due to orbital eccentricity.Data from [http://aa.usno.navy.mil/data/docs/EarthSeasons.php United States Naval Observatory] {{Webarchive|url=https://web.archive.org/web/20071013000301/http://aa.usno.navy.mil/data/docs/EarthSeasons.php |date=13 October 2007 }}{{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|doi-access=free}}

Apsidal precession also slowly changes the place in Earth's orbit where the solstices and equinoxes occur. This is a slow change in the orbit of Earth, not the axis of rotation, which is referred to as axial precession. The climatic effects of this change are part of the Milankovitch cycles. Over the next {{gaps|10|000}} years, the northern hemisphere winters will become gradually longer and summers will become shorter. 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.{{Cite web|url=http://ircamera.as.arizona.edu/NatSci102/NatSci102/lectures/climate.htm|title=Long Term Climate|website=ircamera.as.arizona.edu|access-date=1 September 2016|archive-date=2 June 2015|archive-url=https://web.archive.org/web/20150602033750/http://ircamera.as.arizona.edu/NatSci102/NatSci102/lectures/climate.htm|url-status=dead}} This will reduce the mean orbital radius and raise temperatures in both hemispheres closer to the mid-interglacial peak.

{{See also|Eccentric Jupiter}}

Of the many exoplanets discovered, most have a higher orbital eccentricity than planets in the 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.{{Cite web|url=http://exoplanets.org/ecc.html|title=ECCENTRICITY|website=exoplanets.org}} 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 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 a very large one. Low eccentricity is needed for habitability, especially advanced life.{{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}} High multiplicity planet systems are much more likely to have habitable exoplanets.{{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 | doi-access=free }}{{Cite journal|title=Planetesimals in Debris Disks|first1=Andrew N.|last1=Youdin|first2=George H.|last2=Rieke|date=15 December 2015|arxiv=1512.04996}} The grand tack hypothesis of the Solar System also helps understand its near-circular orbits and other unique features.{{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| access-date=4 November 2015| archive-date=9 June 2011| archive-url=https://web.archive.org/web/20110609234138/http://www.nasa.gov/topics/solarsystem/features/young-jupiter.html| url-status=dead}}{{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| date=23 August 2011| access-date=4 November 2015}}{{cite web| last1=Choi| first1=Charles Q.| title=Jupiter's 'Smashing' Migration May Explain Our Oddball Solar System| date=23 March 2015| url=http://www.space.com/28901-wandering-jupiter-oddball-solar-system.html| publisher=Space.com| access-date=4 November 2015}}{{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| date=9 March 2014| access-date=7 November 2015}}{{cite web| last1=Raymond| first1=Sean| title=The Grand Tack| url=http://planetplanet.net/2013/08/02/the-grand-tack/| website=PlanetPlanet| date=2 August 2013| access-date=7 November 2015}}{{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|year=2014| volume=239| pages=74–84| doi=10.1016/j.icarus.2014.05.009|arxiv = 1407.3290 |bibcode = 2014Icar..239...74O | s2cid=51737711}}{{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 |first1=Abraham |last1=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 |s2cid=118489638 }}{{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 }}

• Equation of time

{{notelist}}

{{Reflist}}

• {{cite book |last1=Prussing |first1=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 }}

• [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)]{{dead link|date=May 2025|bot=medic}}{{cbignore|bot=medic}}. [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.
• [https://web.archive.org/web/20170424171326/http://www.walter-fendt.de/ph14e/keplerlaw2.htm Kepler's Second law's simulation]

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