Trojan (astronomy)

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

In astronomy, a trojan is a minor planet or moon that shares the orbit of a planet or larger moon, wherein the trojan remains in the same, stable position relative to the larger object. In particular, a trojan remains near one of the two trojan points of stability – designated L₄ and L₅ – which lie approximately 60° ahead of and behind the larger body, respectively. Trojan points are two of five types of Lagrangian points, and a trojan is a type of Lagrangian object.

They are one type of co-orbital object. In this arrangement, the massive star and the smaller planet orbit about their common barycenter. A much smaller mass located at one of the Lagrangian points is subject to a combined gravitational force that acts through this barycenter. Hence the object can orbit around the barycenter with the same orbital period as the planet, and the arrangement can remain stable over time.^[1]

The Jupiter trojans account for most known trojans in the Solar System. They are divided into the Greek camp (L₄) in front of and the Trojan camp (L₅) trailing behind Jupiter in their orbit. More than 6,000 have been found so far and more than a million Jupiter trojans larger than one kilometer are thought to exist, whereas only a few Mars trojans (7) and Neptune trojans (13) have been found to date. Numerical calculations of the orbital dynamics involved indicate that Saturn and Uranus probably do not have any primordial trojans.^[2] The discovery of the first Earth trojan, 2010 TK7, was announced by NASA in 2011.^[3]

Unlike trojan minor planets that share the orbit with a planet, a trojan moon is a moon orbiting near the trojan point of another, larger moon. All known trojan moons are part of the Saturn system. Telesto and Calypso are trojans of Tethys, and Helene and Polydeuces of Dione. There is also a theoretical concept of a trojan planet, a planet that orbits at the trojan point of another, larger planet. Such a pair of co-orbital exoplanets was already thought to exist in another star system, but this claim was later retracted.^[4]

Trojan minor planets

The Jupiter trojans in front of and behind the planet along its orbital path, the asteroid belt between Mars and Jupiter, and the Hilda asteroids.

Jupiter trojans

Asteroid belt

Hilda asteroids

In 1772, the Italian–French mathematician and astronomer Joseph-Louis Lagrange obtained two constant-pattern solutions (collinear and equilateral) of the general three-body problem.^[5] In the restricted three-body problem, with one mass negligible (which Lagrange did not consider), the five possible positions of that mass are now termed Lagrangian points.

The term "trojan" originally referred to the "trojan asteroids" (Jupiter trojans) that orbit close to the Lagrangian points of Jupiter. These have long been named after characters from the Trojan War of Greek mythology. By convention, the asteroids orbiting near the L₄ point of Jupiter are named after the characters from the Greek side of the war, whereas those orbiting near the L₅ of Jupiter are from the Trojan side. There are two exceptions, which were named before the convention was put in place, the Greek-themed 617 Patroclus and the Trojan-themed 624 Hektor, which were assigned to the wrong sides.^[6]

Astronomers estimate that the Jupiter trojans are about as numerous as the asteroids of the asteroid belt.^[7]

Later on, objects were found orbiting near the Lagrangian points of Neptune, Mars, Earth,^[8] Uranus, and Venus. Minor planets at the Lagrangian points of planets other than Jupiter may be called Lagrangian minor planets.^[9]

• Seven Mars trojans are known: 5261 Eureka, (101429) 1998 VF31, (121514) 1999 UJ7 – the only trojan in the leading cloud at L₄, and (311999) 2007 NS2,^[10][11] as well as 2001 DH47, 2011 SC191 and 2011 UN63, which have not yet been accepted by the Minor Planet Center.
• Nine Neptune trojans^[12] are known, but large Neptune trojans are expected to outnumber the large Jupiter trojans by an order of magnitude.^[13][14]
• 2010 TK7 was confirmed to be the first known Earth trojan in 2011. It is located in the L₄ Lagrangian point, which lies ahead of Earth.^[15]
• 2011 QF99 was identified as the first Uranus trojan in 2013; it is located in the L₄ Lagrangian point.
• 2013 ND15 is a temporary Venus trojan, the first to be identified.
• Asteroids Ceres and Vesta have temporary trojans.^[16]

Stability

Whether or not a system of star, planet, and trojan is stable depends on how large the perturbations are to which it is subject. If, for example, the planet is the mass of Earth, and there is also a Jupiter-mass object orbiting that star, the trojan's orbit would be much less stable than if the second planet had the mass of Pluto.

As a rule of thumb, the system is likely to be long-lived if m₁ > 100×m₂ > 10000×m₃ (in which m₁, m₂, and m₃ are the masses of the star, planet, and trojan).

More formally, in a three-body system with circular orbits, the stability condition is 27(m₁m₂ + m₂m₃ + m₃m₁) < (m₁ + m₂ + m₃)². So the trojan being a mote of dust, m₃→0, imposes a lower bound on m₁÷m₂ of ½(25+3√69) ≈ 24.9599. And if the star were hyper-massive, m₁→+∞, then under Newtonian gravity, the system is stable whatever the planet and trojan masses. And if m₁÷m₂ = m₂÷m₃, then both must exceed 13+2√42 ≈ 25.9615. But all this assumes a three-body system: once other bodies are introduced, even if distant and small, stability of the system requires larger ratios.

See also

Look up Trojan, Trojan asteroid, Trojan moon, or Trojan planet in Wiktionary, the free dictionary.

• Lissajous orbit
• List of objects at Lagrangian points
• Tadpole orbit

References

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