Orbital eccentricity

This article is about eccentricity in astrodynamics. For other uses, seeEccentricity (disambiguation).

Inastrodynamics,theorbital eccentricityof anastronomical objectis adimensionless parameterthat determines the amount by which itsorbitaround another body deviates from a perfectcircle.A value of 0 is acircular orbit,values between 0 and 1 form anelliptic orbit,1 is aparabolicescape orbit(or capture orbit), and greater than 1 is ahyperbola.The term derives its name from the parameters ofconic sections,as everyKepler orbitis a conic section. It is normally used for the isolatedtwo-body problem,but extensions exist for objects following arosetteorbit through the Galaxy.

An elliptic, parabolic, and hyperbolicKepler orbit:
Elliptic (eccentricity = 0.7)
Parabolic (eccentricity = 1)
Hyperbolic orbit (eccentricity = 1.3)

Elliptic orbit by eccentricity
0·0.2·0.4·0.6·0.8

Definition

In atwo-body problemwith inverse-square-law force, everyorbitis aKepler orbit.Theeccentricityof this Kepler orbit is anon-negative numberthat defines its shape.

The eccentricity may take the following values:

• Circular orbit:e= 0
• Elliptic orbit:0 <e< 1
• Parabolic trajectory:e= 1
• Hyperbolic trajectory:e> 1

The eccentricityeis given by

$$e={\sqrt {1+{\frac {2EL^{2}}{m_{\text{red}}\,\ Alpha ^{2}}}}}$$

whereEis the totalorbital energy,Lis theangular momentum,m_redis thereduced mass,and$\ Alpha$the coefficient of the inverse-square lawcentral forcesuch as in the theory ofgravityorelectrostaticsinclassical physics: $$F={\frac {\ Alpha }{r^{2}}}$$ ($\ Alpha$is negative for an attractive force, positive for a repulsive one; related to theKepler problem)

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

whereεis thespecific orbital energy(total energy divided by the reduced mass),μthestandard gravitational parameterbased on the total mass, andhthespecific relative angular momentum(angular momentumdivided by the reduced mass).

For values ofefrom 0 to 1 the orbit's shape is an increasingly elongated (or flatter) ellipse; for values ofefrom 1 to infinity the orbit is ahyperbolabranch making a total turn of2arccsc(e),decreasing from 180 to 0 degrees. Here, the total turn is analogous toturning number,but for open curves (an angle covered by velocity vector). The limit case between an ellipse and a hyperbola, wheneequals 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 whileetends 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$\arcsin(e)$yields the projection angle of a perfect circle to anellipseof eccentricitye.For example, to view the eccentricity of the planet Mercury (e= 0.2056), one must simply calculate theinverse sineto 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.

Etymology

The word "eccentricity" comes fromMedieval Latineccentricus,derived fromGreekἔκκεντροςekkentros"out of the center", fromἐκ-ek-,"out of" +κέντρονkentron"center". "Eccentric" first appeared in English in 1551, with the definition "...a circle in which the earth, sun. etc. deviates from its center".In 1556, five years later, an adjectival form of the word had developed.

Calculation

The eccentricity of an orbit can be calculated from theorbital state vectorsas themagnitudeof theeccentricity vector: $$e=\left|\mathbf {e} \right|$$ where:

• eis the eccentricity vector ("Hamilton's vector").

Forelliptical orbitsit can also be calculated from theperiapsisandapoapsissince$r_{\text{p}}=a\,(1-e)$and$r_{\text{a}}=a\,(1+e)\,,$whereais the length of thesemi-major axis.$${\begin{aligned}e&={\frac {r_{\text{a}}-r_{\text{p}}}{r_{\text{a}}+r_{\text{p}}}}\\\,\\&={\frac {r_{\text{a}}/r_{\text{p}}-1}{r_{\text{a}}/r_{\text{p}}+1}}\\\,\\&=1-{\frac {2}{\;{\frac {r_{\text{a}}}{r_{\text{p}}}}+1\;}}\end{aligned}}$$ where:

• r_ais the radius atapoapsis(also "apofocus", "aphelion", "apogee" ), i.e., the farthest distance of the orbit to thecenter of massof the system, which is afocusof the ellipse.
• r_pis the radius atperiapsis(or "perifocus" etc.), the closest distance.

The semi-major axis, a, is also the path-averaged distance to the centre of mass,while the time-averaged distance is a(1 + e e / 2).[1]

The eccentricity of an elliptical orbit can be used to obtain the ratio of theapoapsisradius to theperiapsisradius: $${\frac {r_{\text{a}}}{r_{\text{p}}}}={\frac {\,a\,(1+e)\,}{\,a\,(1-e)\,}}={\frac {1+e}{1-e}}$$

For Earth, orbital eccentricitye≈0.01671,apoapsisis aphelion andperiapsisis perihelion, relative to the Sun.

For Earth's annual orbit path, the ratio of longest radius (r_a) / shortest radius (r_p) is${\frac {\,r_{\text{a}}\,}{r_{\text{p}}}}={\frac {\,1+e\,}{1-e}}{\text{ ≈ 1.03399.}}$

Examples

Plot of the changing orbital eccentricities ofMercury,Venus,EarthandMarsover 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.

Eccentricities of Solar System bodies

Object	Eccentricity
Triton	0.00002
Venus	0.0068
Neptune	0.0086
Earth	0.0167
Titan	0.0288
Uranus	0.0472
Jupiter	0.0484
Saturn	0.0541
Luna (Moon)	0.0549
Ceres	0.0758
Vesta	0.0887
Mars	0.0934
10 Hygiea	0.1146
Makemake	0.1559
Haumea	0.1887
Mercury	0.2056
2 Pallas	0.2313
Pluto	0.2488
3 Juno	0.2555
324 Bamberga	0.3400
Eris	0.4407
Nereid	0.7507
Sedna	0.8549
Halley's Comet	0.9671
Comet Hale-Bopp	0.9951
Comet Ikeya-Seki	0.9999
Comet McNaught	1.0002
C/1980 E1	1.057
ʻOumuamua	1.20
2I/Borisov	3.5

The table lists the values for all planets and dwarf planets, and selected asteroids, comets, and moons.Mercuryhas the greatest orbital eccentricity of any planet in theSolar System(e=0.2056), followed byMarsof0.0934.Such eccentricity is sufficient for Mercury to receive twice as muchsolar irradiationat perihelion compared to aphelion. Before its demotion fromplanet statusin 2006,Plutowas considered to be the planet with the most eccentric orbit (e=0.248). OtherTrans-Neptunian objectshave significant eccentricity, notably the dwarf planetEris(0.44). Even further out,Sednahas an extremely-high eccentricity of0.855due to its estimated aphelion of 937 AU and perihelion of about 76 AU, possibly under influence ofunknown object(s).

The eccentricity ofEarth's orbitis currently about0.0167;its orbit is nearly circular.Neptune'sandVenus'shave even lower eccentricities of0.0086and0.0068respectively, 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 nearly0.0034to almost 0.058 as a result of gravitational attractions among the planets.

Luna's value is0.0549,the most eccentric of the large moons in the Solar System. The fourGalilean moons(Io,Europa,GanymedeandCallisto) have their eccentricities of less than 0.01.Neptune's largest moonTritonhas an eccentricity of1.6×10⁻⁵(0.000016),the smallest eccentricity of any known moon in the Solar System;its orbit is as close to a perfect circle as can be currentlymeasured. Smaller moons, particularlyirregular moons,can have significant eccentricities, such as Neptune's third largest moon,Nereid,of0.75.

Most of the Solar System'sasteroidshave orbital eccentricities between 0 and 0.35 with an average value of 0.17.Their comparatively high eccentricities are probably due to under influence ofJupiterand to past collisions.

Cometshave very different values of eccentricities.Periodic cometshave eccentricities mostly between 0.2 and 0.7,but some of them have highly eccentric elliptical orbits with eccentricities just below 1; for example,Halley's Comethas a value of 0.967. Non-periodic comets follow near-parabolic orbitsand thus have eccentricities even closer to 1. Examples includeComet Hale–Boppwith a value of0.9951,Comet Ikeya-Sekiwith a value of0.9999andComet McNaught(C/2006 P1) with a value of1.000019.As first two's values are less than 1, their orbit are elliptical and they will return. McNaught has ahyperbolic orbitbut within the influence of the planets,is still bound to the Sun with an orbital period of about 10⁵years.Comet C/1980 E1has the largest eccentricity of any known hyperbolic comet of solar origin with an eccentricity of 1.057,and will eventually leave the Solar System.

ʻOumuamuais the firstinterstellar objectto be found passing through the Solar System. Its orbital eccentricity of 1.20 indicates that ʻOumuamua has never been gravitationally bound to the Sun. It was discovered 0.2 AU (30000000km;19000000mi) from Earth and is roughly 200 meters in diameter. It has an interstellar speed (velocity at infinity) of 26.33 km/s (58900mph).

Mean average

The mean eccentricity of an object is the average eccentricity as a result ofperturbationsover a given time period. Neptune currently has an instant (currentepoch) eccentricity of0.0113,but from 1800 to 2050 has a mean eccentricity of0.00859.

Climatic effect

Orbital mechanics require that the duration of the seasons be proportional to the area of Earth's orbit swept between thesolsticesandequinoxes,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.

Apsidal precessionalso 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 asaxial precession.The climatic effects of this change are part of theMilankovitch cycles.Over the next10000years, 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.This will reduce the mean orbital radius and raise temperatures in both hemispheres closer to the mid-interglacial peak.

Exoplanets

Of the manyexoplanetsdiscovered, 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 aretidally-lockedto 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.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 othermultiplanetary systemshave been found, but none resemble the Solar System. The Solar System has uniqueplanetesimalsystems, which led the planets to have near-circular orbits. Solar planetesimal systems include theasteroid belt,Hilda family,Kuiper belt,Hills cloud,and theOort cloud.The exoplanet systems discovered have either no planetesimal systems or a very large one. Low eccentricity is needed for habitability, especially advanced life.High multiplicity planet systems are much more likely to have habitable exoplanets.Thegrand tack hypothesisof the Solar System also helps understand its near-circular orbits and other unique features.

See also

• Equation of time

Further reading

• Prussing, John E.; Conway, Bruce A. (1993).Orbital Mechanics.New York: Oxford University Press.ISBN0-19-507834-9.

External links

• World of Physics: Eccentricity
• The NOAA page on Climate Forcing Dataincludes (calculated) data fromBerger (1978), Berger and Loutre (1991).Laskar et al. (2004)on Earth orbital variations, Includes eccentricity over the last 50 million years and for the coming 20 million years.
• The orbital simulations by Varadi, Ghil and Runnegar (2003)provides series for Earth orbital eccentricity and orbital inclination.
• Kepler's Second law's simulation