Orbital eccentricity

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:

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

e = 1 + 2 E L 2 m red α 2

where E is the total orbital energy, L is the angular momentum, m_red is the reduced mass, and α the coefficient of the inverse-square law central force such as gravity or electrostatics in classical physics:

F = α r 2 (α is negative for an attractive force, positive for a repulsive one; see also Kepler problem)

or in the case of a gravitational force:

e = 1 + 2 ε h 2 μ 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).

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. 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( 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. 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

From Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros "out of the center", from ἐκ- ek-, "out of" + 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." Five years later, in 1556, an adjective form of the word was added.

Calculation

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

e = | e |

where:

e is the eccentricity vector.

For elliptical orbits it can also be calculated from the periapsis and apoapsis since r_p = a(1 − e) and r_a = a(1 + e), where a is the semimajor axis.

e = r a − r p r a + r p = 1 − 2 r a r p + 1

where:

r_a 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 of the ellipse).

r_p 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:

r p r a = 1 − e 1 + e

For Earth, orbital eccentricity ≈ 0.0167, apoapsis= aphelion = apogee and periapsis= perihelion = perigee relative to sun.

For Earth's annual orbit path, r_a/r_p ratio = longest_radius / shortest_radius ≈ 1.034 relative to center point of path.

Examples

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 eccentricity. 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 graph).

The table lists the values for all planets and dwarf planets, and selected asteroid, 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 6999855000000000000♠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. 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 has an eccentricity of 6995160000000000000♠1.6×10⁻⁵ (6995160000000000000♠0.000016), the smallest eccentricity of any known body in the Solar System; its orbit is as close to a perfect circle as can be currently 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, 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 and comet C/2006 P1 (McNaught) with a value of 7000100001900000000♠1.000019. As Hale–Bopp's value is less than 1, its orbit is elliptical and it will in fact return. 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 10⁵ years. As of a 2010 Epoch, Comet C/1980 E1 has the largest eccentricity of any known hyperbolic comet with an eccentricity of 1.057, and will leave the Solar System indefinitely.

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, but from 1800 to 2050 has a mean eccentricity of 6997859000000000000♠0.00859.

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 days longer than fall due to the Milankovitch cycles.

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 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. 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 exoplanets have a higher orbital eccentricity than our solar system. Exoplanets found with low orbital eccentricity, near circular orbits, are very close to its star and are tidal locked to the star, like planet Mercury. 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 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 solar systems discovered have either no planetesimal systems or one very large planetesimal system. Low eccentricity is needed for habitability, especially advanced life. High multiplicity planet systems are much more likely to have habitable exoplanets. The Grand tack hypothesis of the solar system also helps understand the near-circular orbits of the solar system and the other unique features of the solar system.