# Co-orbital configuration

(Redirected from Trojan moon)

In astronomy, a co-orbital configuration refers to two or more celestial objects (such as asteroids, moons, or planets) that orbit at the same, or very similar, distance from their parent object as each other, i.e. they are in a 1:1 mean motion resonance. (or 1:−1 if orbiting in opposite directions1)

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 librate 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 ( ${\lambda}=M + \varpi$ ) and the mean longitude of the sum of the longitude of the ascending node and the argument of periapsis ( $\varpi = \Omega + \omega\,$ ).

## Trojans

Trojan points are the points labelled L4 and L5, highlighted in red, on the orbital path of the secondary object (blue), around the primary object (yellow).

Trojan objects orbit 60° ahead (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 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 $({\Delta}{\lambda}, {\Delta}\varpi)$ = (±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. Neptune has 9 known trojan objects, Mars 7 known ones (and a strong candidate), and Earth one, 2010 TK7.

### Trojan moons

The Saturnian system contains two sets of trojan moons. Both Tethys and Dione have two trojan moons, 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

The discovery of a pair of co-orbital exoplanets has been reported but later retracted.3

One possibility for the habitable zone is a trojan planet of a gas giant close to its star.4

#### Formation of the Earth–Moon system

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

## Horseshoe orbits

Rotating-frame depiction of the horseshoe exchange orbits of Janus and Epimetheus

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.5

More resonant near-Earth objects (NEOs) have since been discovered. These include 54509 YORP, (85770) 1998 UP1, 2002 AA29, and 2009 BD, which exist in resonant orbits similar to Cruithne's. 2010 TK7 is the first and so far only identified Earth trojan.

## 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 gravitionally bound to it.2

## 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.6

## References

1. ^ Morais, M.H.M.; F. Namouni. "Asteroids in retrograde resonance with Jupiter and Saturn". Monthly Notices of the Royal Astronomical Society Letters (in press). arXiv:arXiv:1308.0216.
2. Dynamics of two planets in co-orbital motion
3. ^ Two planets found sharing one orbit, New Scientist, 24 February 2011
4. ^ Extrasolar Trojan Planets close to Habitable Zones, R. Dvorak, E. Pilat-Lohinger, R. Schwarz, F. Freistetter
5. ^ Christou, A. A.; Asher, D. J. (2011). "A long-lived horseshoe companion to the Earth". Monthly Notices of the Royal Astronomical Society 414 (4): 2965. arXiv:1104.0036. Bibcode:2011MNRAS.414.2965C. doi:10.1111/j.1365-2966.2011.18595.x.
6. ^ Exchange orbits: a possible application to extrasolar planetary systems?, B. Funk, R. Schwarz, R. Dvorak, M. Roth

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