# Co-orbital configuration

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In astronomy, a co-orbital configuration is a configuration of two or more astronomical objects (such as asteroids, moons, or planets) orbiting at the same, or very similar, distance from their primary, i.e. they are in a 1:1 mean-motion resonance. (or 1:−1 if orbiting in opposite directions). [1]

## Contents

There are several classes of co-orbital objects, depending on their point of libration. The most common and best-known class is the trojan, which librates around one of the two stable Lagrangian points (Trojan points), L4 and L5, 60° ahead of and behind the larger body respectively. Another class is the horseshoe orbit, in which objects librate around 180° from the larger body. Objects librating around 0° are called quasi-satellites. [2]

An exchange orbit occurs when two co-orbital objects are of similar masses and thus exert a non-negligible influence on each other. The objects can exchange semi-major axes or eccentricities when they approach each other.

## Parameters

Orbital parameters that are used to describe the relation of co-orbital objects are the longitude of the periapsis difference and the mean longitude difference. The longitude of the periapsis is the sum of the mean longitude and the mean anomaly ${\displaystyle ({\lambda }=\varpi +M)}$ and the mean longitude is the sum of the longitude of the ascending node and the argument of periapsis ${\displaystyle (\varpi =\Omega +\omega )}$.

## Trojans

Trojan objects orbit 60° ahead of (L4) or behind (L5) a more massive object, both in orbit around an even more massive central object. The best known example are the asteroids that orbit ahead of or behind Jupiter around the Sun. Trojan objects do not orbit exactly at one of either Lagrangian points, but do remain relatively close to it, appearing to slowly orbit it. In technical terms, they librate around ${\displaystyle ({\Delta }{\lambda },{\Delta }\varpi )}$ = (±60°, ±60°). The point around which they librate is the same, irrespective of their mass or orbital eccentricity. [2]

### Trojan minor planets

There are several thousand known trojan minor planets orbiting the Sun. Most of these orbit near Jupiter's Lagrangian points, the traditional Jupiter trojans. As of 2015 there are also 13 Neptune trojans, 7 Mars trojans, 2 Uranus trojans ( and ) and 1 Earth trojan () known to exist.

### Trojan moons

The Saturnian system contains two sets of trojan moons. Both Tethys and Dione have two trojan moons each, Telesto and Calypso in Tethys's L4 and L5 respectively, and Helene and Polydeuces in Dione's L4 and L5 respectively.

Polydeuces is noticeable for its wide libration: it wanders as far as ±30° from its Lagrangian point and ±2% from its mean orbital radius, along a tadpole orbit in 790 days (288 times its orbital period around Saturn, the same as Dione's).

### Trojan planets

A pair of co-orbital exoplanets was proposed to be orbiting the star Kepler-223, but this was later retracted. [3]

The possibility of a trojan planet to Kepler-91b was studied but the conclusion was that the transit-signal was a false-positive. [4]

One possibility for the habitable zone is a trojan planet of a giant planet close to its star. [5]

#### Formation of the Earth–Moon system

According to the giant impact hypothesis, the Moon formed after a collision between two co-orbital objects: Theia, thought to have had about 10% of the mass of Earth (about as massive as Mars), and the proto-Earth—whose orbits were perturbed by other planets, bringing Theia out of its trojan position and causing the collision.

## Horseshoe orbits

Objects in a horseshoe orbit librate around 180° from the primary. Their orbits encompass both equilateral Lagrangian points, i.e. L4 and L5. [2]

### Co-orbital moons

The Saturnian moons Janus and Epimetheus share their orbits, the difference in semi-major axes being less than either's mean diameter. This means the moon with the smaller semi-major axis will slowly catch up with the other. As it does this, the moons gravitationally tug at each other, increasing the semi-major axis of the moon that has caught up and decreasing that of the other. This reverses their relative positions proportionally to their masses and causes this process to begin anew with the moons' roles reversed. In other words, they effectively swap orbits, ultimately oscillating both about their mass-weighted mean orbit.

### Earth co-orbital asteroids

A small number of asteroids have been found which are co-orbital with Earth. The first of these to be discovered, asteroid 3753 Cruithne, orbits the Sun with a period slightly less than one Earth year, resulting in an orbit that (from the point of view of Earth) appears as a bean-shaped orbit centered on a position ahead of the position of Earth. This orbit slowly moves further ahead of Earth's orbital position. When Cruithne's orbit moves to a position where it trails Earth's position, rather than leading it, the gravitational effect of Earth increases the orbital period, and hence the orbit then begins to lag, returning to the original location. The full cycle from leading to trailing Earth takes 770 years, leading to a horseshoe-shaped movement with respect to Earth. [6]

More resonant near-Earth objects (NEOs) have since been discovered. These include 54509 YORP, , , , , and which exist in resonant orbits similar to Cruithne's. is the first and so far only identified Earth trojan.

Hungaria asteroids were found to be one of the possible sources for co-orbital objects of the Earth with a lifetime up to ~58 kyrs [7]

## Quasi-satellite

Quasi-satellites are co-orbital objects that librate around 0° from the primary. Low-eccentricity quasi-satellite orbits are highly unstable, but for moderate to high eccentricities such orbits can be stable. [2] From a co-rotating perspective the quasi-satellite appears to orbit the primary like a retrograde satellite, although at distances so large that it is not gravitationally bound to it. [2] Two examples of quasi-satellites of the Earth are [8] and 469219 Kamoʻoalewa. [9] [10]

## Exchange orbits

In addition to swapping semi-major axes like Saturn's moons Epimetheus and Janus, another possibility is to share the same axis, but swap eccentricities instead. [11]

## Related Research Articles

In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationship is found between a pair of objects. The physical principle behind orbital resonance is similar in concept to pushing a child on a swing, whereby the orbit and the swing both have a natural frequency, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance between bodies with similar orbital radii causes large solar system bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.

The Jupiter trojans, commonly called trojan asteroids or simply trojans, are a large group of asteroids that share the planet Jupiter's orbit around the Sun. Relative to Jupiter, each trojan librates around one of Jupiter's stable Lagrange points: either L4, existing 60° ahead of the planet in its orbit, or L5, 60° behind. Jupiter trojans are distributed in two elongated, curved regions around these Lagrangian points with an average semi-major axis of about 5.2 AU.

3753 Cruithne is a Q-type, Aten asteroid in orbit around the Sun in 1:1 orbital resonance with Earth, making it a co-orbital object. It is an asteroid that, relative to Earth, orbits the Sun in a bean-shaped orbit that effectively describes a horseshoe, and that can change into a quasi-satellite orbit. Cruithne does not orbit Earth and at times it is on the other side of the Sun, placing Cruithne well outside of Earth's Hill sphere. Its orbit takes it near the orbit of Mercury and outside the orbit of Mars. Cruithne orbits the Sun in about one year, but it takes 770 years for the series to complete a horseshoe-shaped movement around Earth.

The Hilda asteroids are a dynamical group of more than 5,000 asteroids located beyond the asteroid belt but within Jupiter's orbit, in a 3:2 orbital resonance with Jupiter. The namesake is the asteroid 153 Hilda.

2003 YN107 is a tiny asteroid, classified as a near-Earth object of the Aten group moving in a 1:1 mean-motion resonance with Earth. Because of that, it is in a co-orbital configuration relative to Earth.

A quasi-satellite is an object in a specific type of co-orbital configuration with a planet where the object stays close to that planet over many orbital periods.

In celestial mechanics, a horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The osculating (instantaneous) orbital period of the smaller body remains very near that of the larger body, and if its orbit is a little more eccentric than that of the larger body, during every period it appears to trace an ellipse around a point on the larger object's orbit. However, the loop is not closed but drifts forward or backward so that the point it circles will appear to move smoothly along the larger body's orbit over a long period of time. When the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, with the larger body between the 'horns'.

In astronomy, a resonant trans-Neptunian object is a trans-Neptunian object (TNO) in mean-motion orbital resonance with Neptune. The orbital periods of the resonant objects are in a simple integer relations with the period of Neptune, e.g. 1:2, 2:3, etc. Resonant TNOs can be either part of the main Kuiper belt population, or the more distant scattered disc population.

In celestial mechanics, the Kozai mechanism or Lidov–Kozai mechanism or Kozai–Lidov mechanism, also known as the Kozai, Lidov–Kozai or Kozai–Lidoveffect, oscillations, cycles or resonance, is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions, causing the orbit's argument of pericenter to oscillate about a constant value, which in turn leads to a periodic exchange between its eccentricity and inclination. The process occurs on timescales much longer than the orbital periods. It can drive an initially near-circular orbit to arbitrarily high eccentricity, and flip an initially moderately inclined orbit between a prograde and a retrograde motion.

(524522) 2002 VE68, provisional designation 2002 VE68, is a sub-kilometer sized asteroid and temporary quasi-satellite of Venus. It was the first such object to be discovered around a major planet in the Solar System. In a frame of reference rotating with Venus, it appears to travel around it during one Venerean year but it actually orbits the Sun, not Venus.

In astronomy, a trojan is a small celestial body (mostly asteroids) that shares the orbit of a larger one, remaining in a stable orbit approximately 60° ahead of or behind the main body near one of its Lagrangian points L4 and L5. Trojans can share the orbits of planets or of large moons.

Claims of the existence of other moons of Earth—that is, of one or more natural satellites with relatively stable orbits of Earth, other than the Moon—have existed for some time. Several candidates have been proposed, but none has been confirmed. Since the 19th century, scientists have made genuine searches for more moons, but the possibility has also been the subject of a number of dubious non-scientific speculations as well as a number of likely hoaxes.

The Mars trojans are a group of Trojan objects that share the orbit of the planet Mars around the Sun. They can be found around the two Lagrangian points 60° ahead of and behind Mars. The origin of the Mars trojans is not well understood. One theory suggests that they were primordial objects left over from the formation of Mars that were captured in its Lagrangian points as the Solar System was forming. However, spectral studies of the Mars trojans indicate this may not be the case. Another explanation involves asteroids chaotically wandering into the Mars Lagrangian points later in the Solar System's formation. This is also questionable considering the short dynamical lifetimes of these objects. The spectra of Eureka and two other Mars trojans indicates an olivine-rich composition. Since olivine-rich objects are rare in the asteroid belt it has been suggested that some of the Mars trojans are captured debris from a large orbit-altering impact on Mars when it encountered a planetary embryo.

An Earth trojan is an asteroid that orbits the Sun in the vicinity of the Earth–Sun Lagrangian points L4 (leading 60°) or L5 (trailing 60°), thus having an orbit similar to Earth's. Only two Earth trojans have so far been discovered. The name "trojan" was first used in 1906 for the Jupiter trojans, the asteroids that were observed near the Lagrangian points of Jupiter's orbit.

The Nicemodel is a scenario for the dynamical evolution of the Solar System. It is named for the location of the Observatoire de la Côte d'Azur — where it was initially developed in 2005 — in Nice, France. It proposes the migration of the giant planets from an initial compact configuration into their present positions, long after the dissipation of the initial protoplanetary disk. In this way, it differs from earlier models of the Solar System's formation. This planetary migration is used in dynamical simulations of the Solar System to explain historical events including the Late Heavy Bombardment of the inner Solar System, the formation of the Oort cloud, and the existence of populations of small Solar System bodies such as the Kuiper belt, the Neptune and Jupiter trojans, and the numerous resonant trans-Neptunian objects dominated by Neptune.

Asteroid 2011 QF99 is a minor planet from the outer Solar System and the first known Uranus trojan to be discovered. It measures approximately 60 kilometers (37 miles) in diameter, assuming an albedo of 0.05. It was first observed 29 August 2011 during a deep survey of trans-Neptunian objects conducted with the Canada–France–Hawaii Telescope, but its identification as Uranian trojan was not announced until 2013.

2014 OL339 (also written 2014 OL339) is an Aten asteroid that is a temporary quasi-satellite of Earth, the fourth known Earth quasi-satellite.

The hypothetical Planet Nine would modify the orbits of extreme trans-Neptunian objects via a combination of effects. On very long timescales exchanges of angular momentum with Planet Nine cause the perihelia of anti-aligned objects to rise until their precession reverses direction, maintaining their anti-alignment, and later fall, returning them to their original orbits. On shorter timescales mean-motion resonances with Planet Nine provides phase protection, which stabilizes their orbits by slightly altering the objects' semi-major axes, keeping their orbits synchronized with Planet Nine's and preventing close approaches. The inclination of Planet Nine's orbit weakens this protection, resulting in a chaotic variation of semi-major axes as objects hop between resonances. The orbital poles of the objects circle that of the Solar System's Laplace plane, which at large semi-major axes is warped toward the plane of Planet Nine's orbit, causing their poles to be clustered toward one side.

2020 XL5 is a near-Earth asteroid and Earth trojan discovered by the Pan-STARRS 1 survey at Haleakala Observatory, Hawaii on 12 December 2020. It oscillates around the Sun–Earth L4 Lagrangian point (leading 60°), one of the dynamically stable locations where the combined gravitational force acts through the Sun's and Earth's barycenter. Analysis of 2020 XL5's trojan orbit stability suggests it will remain around Earth's L4 point for about 10,000 years until gravitational peturbations from repeated close encounters with Venus destabilize its trojan configuration. It is the second Earth trojan discovered, after 2010 TK7 in 2010.

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