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In astronomy, a trojan is a small celestial body (mostly asteroids) that shares the orbit of a larger body, 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.

Trojans are one type of co-orbital object. In this arrangement, a star and a planet orbit about their common barycenter, which is close to the center of the star because it is usually much more massive than the orbiting planet. In turn, a much smaller mass than both the star and the planet, located at one of the Lagrangian points of the star–planet system, is subject to a combined gravitational force that acts through this barycenter. Hence the smallest object orbits around the barycenter with the same orbital period as the planet, and the arrangement can remain stable over time.^[1]

In the Solar System, most known trojans share the orbit of Jupiter. They are divided into the Greek camp at L₄ (ahead of Jupiter) and the Trojan camp at L₅ (trailing Jupiter). More than a million Jupiter trojans larger than one kilometer are thought to exist,^[2] of which more than 7,000 are currently catalogued. In other planetary orbits only nine Mars trojans, 31 Neptune trojans, two Uranus trojans, and two Earth trojans, have been found to date. A temporary Venus trojan is also known. Numerical orbital dynamics stability simulations indicate that Saturn probably does not have any primordial trojans.^[3]

The same arrangement can appear when the primary object is a planet and the secondary is one of its moons, whereby much smaller trojan moons can share its orbit. All known trojan moons are part of the Saturn system. Telesto and Calypso are trojans of Tethys, and Helene and Polydeuces of Dione.

Trojan minor planets

The Jupiter trojans are seen in this graphic as Greek camp at L₄ ahead of Jupiter and as Trojan camp at L₅ trailing Jupiter along its orbital path. It also shows 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.^[4] 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 Lagrange points.

The term "trojan" originally referred to the "trojan asteroids" (Jovian trojans) that orbit close to the Lagrangian points of Jupiter. These have long been named for figures from the Trojan War of Greek mythology. By convention, the asteroids orbiting near the L₄ point of Jupiter are named for 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, named before the convention was adopted: 624 Hektor in the L4 group, and 617 Patroclus in the L5 group.^[5]

Astronomers estimate that the Jovian trojans are about as numerous as the asteroids of the asteroid belt.^[6]

Later on, objects were found orbiting near the Lagrangian points of Neptune, Mars, Earth,^[7] Uranus, and Venus. Minor planets at the Lagrangian points of planets other than Jupiter may be called Lagrangian minor planets.^[8]

Four Martian trojans are known: 5261 Eureka, (101429) 1998 VF31, (311999) 2007 NS2, and (121514) 1999 UJ7 – the only Trojan body in the leading "cloud" at L₄,^[9]^[10] There seem to be, also, (385250) 2001 DH47, 2011 SC191, and 2011 UN63, but these have not yet been accepted by the Minor Planet Center.
There are 28 known Neptunian trojans,^[11] but the large Neptunian trojans are expected to outnumber the large Jovian trojans by an order of magnitude.^[12]^[13]
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 the Earth.^[14] (614689) 2020 XL5 was found to be another Earth trojan in 2021. It is also at L4.^[15]^[16]
(687170) 2011 QF99 was identified as the first Uranus trojan in 2013. It is located at the L₄ Lagrangian point. A second one, (636872) 2014 YX49, was announced in 2017.^[17]
2013 ND15 is a temporary Venusian trojan, the first one to be identified.
The large asteroids Ceres and Vesta have temporary trojans.^[18]
Saturn has 1 known trojan in the L₄ Lagrangian Point, 2019 UO₁₄.^[19]

Trojans by planet

Planet	Number in L4	Number in L5	List (L4)	List (L5)
Mercury	0	0	—	—
Venus	1	0	2013 ND15	—
Earth	2	0	(706765) 2010 TK7, (614689) 2020 XL5	—
Mars	1	13	(121514) 1999 UJ7, 2023 FW14	many
Jupiter	7508	4044	Greek camp	Trojan camp
Saturn	1	0	2019 UO₁₄	—
Uranus	2	0	(687170) 2011 QF99, (636872) 2014 YX49	—
Neptune	24	4	many	many

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 > 100m₂ > 10,000m₃ (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 ⁠25+√621/2⁠ ≈ 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+√168 ≈ 25.9615. However, this all assumes a three-body system; once other bodies are introduced, even if distant and small, stability of the system requires even larger ratios.