Orbital eccentricity

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In astrodynamics, the orbital eccentricity of an astronomical object is a 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 parabolic escape orbit (or capture orbit), and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, 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 rosette orbit through the Galaxy.

Definition

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:

The eccentricity e is given by

${\displaystyle e={\sqrt {1+{\frac {2EL^{2}}{m_{\text{red}}\,\alpha ^{2}}}}}}$ [1]

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

${\displaystyle F={\frac {\alpha }{r^{2}}}}$

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

or in the case of a gravitational force:

${\displaystyle e={\sqrt {1+{\frac {2\varepsilon h^{2}}{\mu ^{2}}}}}}$ [2]

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). [3]

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 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 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 ${\displaystyle \arcsin(e)}$ 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 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.

Etymology

The word "eccentricity" comes from Medieval Latin eccentricus, derived from Greek ἔκκεντροςekkentros "out of the center", from ἐκ-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".[ citation needed ] In 1556, five years later, 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 of the eccentricity vector:

${\displaystyle e=\left|\mathbf {e} \right|}$

where:

• e is the eccentricity vector ("Hamilton's vector"). [4]

For elliptical orbits it can also be calculated from the periapsis and apoapsis since ${\displaystyle \,r_{\text{p}}=a\,(1-e)\,}$ and ${\displaystyle \,r_{\text{a}}=a\,(1+e)\,,}$ where a is the length of the semi-major axis, the geometric-average and time-average distance. [5] [ failed verification ]

{\displaystyle {\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:

• ra 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.
• rp is the radius at periapsis (or "perifocus" etc.), the closest distance.

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

${\displaystyle {\frac {r_{\text{a}}}{r_{\text{p}}}}={\frac {\,a\,(1+e)\,}{\,a\,(1-e)\,}}={\frac {1+e}{1-e}}}$

For Earth, orbital eccentricity e0.01671 , apoapsis is aphelion and periapsis is perihelion, relative to the Sun.

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

Examples

Eccentricities of Solar System bodies
Objecteccentricity
Triton 0.00002
Venus 0.0068
Neptune 0.0086
Earth 0.0167
Titan 0.0288
Uranus 0.0472
Jupiter 0.0484
Saturn 0.0541
Moon 0.0549
1 Ceres 0.0758
4 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
C/1980 E1 1.057
ʻOumuamua 1.20 [lower-alpha 1]
C/2019 Q4 (Borisov) 3.5 [lower-alpha 2]

The eccentricity of Earth's orbit is currently about 0.0167; its 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. [6]

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 = 0.2056). 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 = 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 0.855 due to its estimated aphelion of 937 AU and perihelion of about 76 AU.

Most of the Solar System's asteroids have orbital eccentricities between 0 and 0.35 with an average value of 0.17. [7] 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 an eccentricity of less than 0.01. Neptune's largest moon Triton has an eccentricity of 1.6×10−5 (0.000016), [8] the smallest eccentricity of any known moon in the Solar System;[ citation needed ] its orbit is as close to a perfect circle as can be currently[ when? ] measured. However, smaller moons, particularly irregular moons, can have significant eccentricity, such as Neptune's third largest moon Nereid (0.75).

Comets have very different values of eccentricity. Periodic comets have eccentricities mostly between 0.2 and 0.7, [9] 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 0.995 [10] and comet C/2006 P1 (McNaught) with a value of 1.000019. [11] As Hale–Bopp's value is less than 1, its orbit is elliptical and it will return. [10] Comet McNaught has a hyperbolic orbit while within the influence of the planets, [11] but is still bound to the Sun with an orbital period of about 105 years. [12] Comet C/1980 E1 has the largest eccentricity of any known hyperbolic comet of solar origin with an eccentricity of 1.057, [13] and will eventually leave the Solar System.

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 the Sun. It was discovered 0.2 AU (30000000 km; 19000000 mi) from Earth and is roughly 200 meters in diameter. It has an interstellar speed (velocity at infinity) of 26.33 km/s (58900 mph).

Mean eccentricity

The mean eccentricity of an object is the average eccentricity as a result of perturbations over a given time period. Neptune currently has an instant (current epoch) eccentricity of 0.0113, [14] but from 1800 to 2050 has a mean eccentricity of 0.00859. [15]

Climatic effect

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 the Milankovitch cycles. [16] [17]

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. Over the next 10000 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. [18] This will reduce the mean orbital radius and raise temperatures in both hemispheres closer to the mid-interglacial peak.

Exoplanets

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. [19] 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 one very large one. Low eccentricity is needed for habitability, especially advanced life. [20] High multiplicity planet systems are much more likely to have habitable exoplanets. [21] [22] The grand tack hypothesis of the Solar System also helps understand its near-circular orbits and other unique features. [23] [24] [25] [26] [27] [28] [29] [30]

Footnotes

1. ʻOumuamua was never bound to the Sun, so its orbit is hyperbolic: e ≈ 1.20 > 1 .
2. C/2019 Q4 (Borisov) was never bound to the Sun, so its orbit is hyperbolic: e ≈ 3.5 >> 1 .

Related Research Articles

In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.

An apsis is the farthest or nearest point in the orbit of a planetary body about its primary body. The apsides of Earth's orbit of the Sun are two: the aphelion, where Earth is farthest from the sun, and the perihelion, where it is nearest. "Apsides" can also refer to the distance of the extreme range of an object orbiting a host body.

The orbital period is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars.

In astronautics, the Hohmann transfer orbit is an orbital maneuver used to transfer a spacecraft between two circular orbits of different altitudes around a central body. Examples would be used for travel between low Earth orbit and the Moon, or another solar planet or asteroid. It is accomplished by placing the craft into an elliptical orbit that is tangential to both the initial and target orbits in the same plane. The maneuver uses two engine impulses: the first prograde impulse places it on the transfer orbit by raising the craft's apoapsis to the target orbit's altitude; and the second raises the craft's periapsis to match the target orbit. The Hohmann maneuver often uses the lowest possible amount of impulse to accomplish the transfer, but requires a relatively longer travel time than higher-impulse transfers. In some cases where one orbit is much larger than the other, a bi-elliptic transfer can use even less impulse, but with a greater penalty in travel time.

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

In astronomy, the barycenter is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important concept in fields such as astronomy and astrophysics. The distance from a body's center of mass to the barycenter can be calculated as a two-body problem.

In gravitationally bound systems, the orbital speed of an astronomical body or object is the speed at which it orbits around either the barycenter or, if one body is much more massive than the other bodies of the system combined, its speed relative to the center of mass of the most massive body.

Earth orbits the Sun at an average distance of 149.60 million km in a counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes 365.256 days, during which time Earth has traveled 940 million km. Ignoring the influence of other Solar System bodies, Earth's orbit is an ellipse with the Earth-Sun barycenter as one focus and a current eccentricity of 0.0167. Since this value is close to zero, the center of the orbit is relatively close to the center of the Sun.

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).

In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In astrodynamics an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or the focus.

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

Tidal heating occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either the surface ocean or interior of a planet or satellite. When an object is in an elliptical orbit, the tidal forces acting on it are stronger near periapsis than near apoapsis. Thus the deformation of the body due to tidal forces varies over the course of its orbit, generating internal friction which heats its interior. This energy gained by the object comes from its gravitational energy, so over time in a two-body system, the initial elliptical orbit decays into a circular orbit. Sustained tidal heating occurs when the elliptical orbit is prevented from circularizing due to additional gravitational forces from other bodies that keep tugging the object back into an elliptical orbit. In this more complex system, gravitational energy still is being converted to thermal energy; however, now the orbit's semimajor axis would shrink rather than its eccentricity.

In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

References

1. Abraham, Ralph (2008). Foundations of mechanics. Jerrold E. Marsden (2nd ed.). Providence, R.I.: AMS Chelsea Pub./American Mathematical Society. ISBN   978-0-8218-4438-0. OCLC   191847156.
2. Bate et al. 2020, p. 24.
3. Bate et al. 2020, pp. 12–17.
4. Bate et al. 2020, p. 25, 62–63.
5. Bate et al. 2020, p. 24–25.
6. A. Berger & M.F. Loutre (1991). "Graph of the eccentricity of the Earth's orbit". Illinois State Museum (Insolation values for the climate of the last 10 million years). Archived from the original on 6 January 2018.
7. Asteroids Archived 4 March 2007 at the Wayback Machine
8. David R. Williams (22 January 2008). "Neptunian Satellite Fact Sheet". NASA.
9. Lewis, John (2 December 2012). Physics and Chemistry of the Solar System. Academic Press. ISBN   9780323145848.
10. "JPL Small-Body Database Browser: C/1995 O1 (Hale-Bopp)" (2007-10-22 last obs). Retrieved 5 December 2008.
11. "JPL Small-Body Database Browser: C/2006 P1 (McNaught)" (2007-07-11 last obs). Retrieved 17 December 2009.
12. "Comet C/2006 P1 (McNaught) – facts and figures". Perth Observatory in Australia. 22 January 2007. Archived from the original on 18 February 2011.
13. "JPL Small-Body Database Browser: C/1980 E1 (Bowell)" (1986-12-02 last obs). Retrieved 22 March 2010.
14. Williams, David R. (29 November 2007). "Neptune Fact Sheet". NASA.
15. "Keplerian elements for 1800 A.D. to 2050 A.D." JPL Solar System Dynamics. Retrieved 17 December 2009.
16. Data from United States Naval Observatory Archived 13 October 2007 at the Wayback Machine
17. Berger A.; Loutre M.F.; Mélice J.L. (2006). "Equatorial insolation: from precession harmonics to eccentricity frequencies" (PDF). Clim. Past Discuss. 2 (4): 519–533. doi:.
18. Ward, Peter; Brownlee, Donald (2000). Rare Earth: Why Complex Life is Uncommon in the Universe. Springer. pp. 122–123. ISBN   0-387-98701-0.
19. Limbach, MA; Turner, EL (2015). "Exoplanet orbital eccentricity: multiplicity relation and the Solar System". Proc Natl Acad Sci U S A. 112 (1): 20–4. arXiv:. Bibcode:2015PNAS..112...20L. doi:. PMC  . PMID   25512527.
20. Zubritsky, Elizabeth. "Jupiter's Youthful Travels Redefined Solar System". NASA . Retrieved 4 November 2015.
21. Sanders, Ray (23 August 2011). "How Did Jupiter Shape Our Solar System?". Universe Today . Retrieved 4 November 2015.
22. Choi, Charles Q. (23 March 2015). "Jupiter's 'Smashing' Migration May Explain Our Oddball Solar System". Space.com. Retrieved 4 November 2015.
23. Davidsson, Dr. Björn J. R. "Mysteries of the asteroid belt". The History of the Solar System. Retrieved 7 November 2015.
24. Raymond, Sean (2 August 2013). "The Grand Tack". PlanetPlanet. Retrieved 7 November 2015.
25. O'Brien, David P.; Walsh, Kevin J.; Morbidelli, Alessandro; Raymond, Sean N.; Mandell, Avi M. (2014). "Water delivery and giant impacts in the 'Grand Tack' scenario". Icarus. 239: 74–84. arXiv:. Bibcode:2014Icar..239...74O. doi:10.1016/j.icarus.2014.05.009. S2CID   51737711.
26. Loeb, Abraham; Batista, Rafael; Sloan, David (August 2016). "Relative Likelihood for Life as a Function of Cosmic Time". Journal of Cosmology and Astroparticle Physics. 2016 (8): 040. arXiv:. Bibcode:2016JCAP...08..040L. doi:10.1088/1475-7516/2016/08/040. S2CID   118489638.
27. "Is Earthly Life Premature from a Cosmic Perspective?". Harvard-Smithsonian Center for Astrophysics. 1 August 2016.