Co-orbital configuration

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])

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 $({\lambda }=\varpi +M)$ and the mean longitude is 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 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 $({\Delta }{\lambda },{\Delta }\varpi )$ = (±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, 1 Uranus trojan (2011 QF99) and 1 Earth trojan (2010 TK7) known to exist.

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

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

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.^[6]

More resonant near-Earth objects (NEOs) have since been discovered. These include 54509 YORP, (85770) 1998 UP1, 2002 AA29, 2010 SO16, 2009 BD, and 2015 SO2 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 gravitationally 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.^[7]

See also

• Double planet
• Kordylewski cloud

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:1308.0216. Bibcode:2013MNRAS.436L..30M. doi:10.1093/mnrasl/slt106.
2. Dynamics of two planets in co-orbital motion
3. "Two planets found sharing one orbit". New Scientist. 24 February 2011.
4. Characterization of Kepler-91b and the Investigation of a Potential Trojan Companion Using EXONEST, Ben Placek, Kevin H. Knuth, Daniel Angerhausen, Jon M. Jenkins, (Submitted on 3 Nov 2015)
5. Extrasolar Trojan Planets close to Habitable Zones, R. Dvorak, E. Pilat-Lohinger, R. Schwarz, F. Freistetter
6. 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.
7. Exchange orbits: a possible application to extrasolar planetary systems?, B. Funk, R. Schwarz, R. Dvorak, M. Roth

• Eric B. Ford and Matthew J. Holman (2007). "Using Transit Timing Observations to Search for Trojans of Transiting Extrasolar Planets". The Astrophysical Journal Letters. 664 (1): L51–L54. arXiv:0705.0356. Bibcode:2007ApJ...664L..51F. doi:10.1086/520579.

External links

• QuickTime animation of co-orbital motion from Murray and Dermott
• Cassini Observes the Orbital Dance of Epimetheus and Janus The Planetary Society
• A Search for Trojan Planets Web page of group of astronomers searching for extrasolar trojan planets at Appalachian State University