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Trojan (celestial body)
Trojan (celestial body)
Trojan (celestial body)
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The trojan points are located on the L4 and L5 Lagrange points, on the orbital path of the secondary object (blue), around the primary object (yellow). All of the Lagrange points are highlighted in red.

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, two Earth trojans, and one Saturn trojan 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

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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. On February 12th, 1906, Max Wolf discovered the trojan asteroid 588 Achilles; His contemporary Carl Charlier noticed that the asteroid was caught in the L4 point of Jupiter, and thus 588 Achilles was the first discovered instance of Lagrange's theoretical calculations applying in practice.[5][6]

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.[7]

Astronomers estimate that the Jovian trojans are about as numerous as the asteroids of the asteroid belt.[8]

Later on, objects were found orbiting near the Lagrangian points of Neptune, Mars, Earth,[9] Uranus, and Venus. Minor planets at the Lagrangian points of planets other than Jupiter may be called Lagrangian minor planets.[10]

Trojans by planet

[edit]
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 2 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

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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.

See also

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

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Trojan (celestial body)

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from Grokipedia
Trojan asteroids, commonly referred to as Trojans, are small Solar System bodies that share the orbit of a larger planet around the Sun while remaining dynamically stable at one of the two equilateral Lagrangian points, L4 or L5, located approximately 60 degrees ahead of or behind the planet in its orbit. These points arise from the balanced gravitational influences of the Sun and the planet, allowing the asteroids to librate without colliding with the planet. The largest known population orbits Jupiter, with over 15,000 cataloged as of 2025, making up two distinct swarms: the Greek camp at L4 (leading Jupiter) and the Trojan camp at L5 (trailing Jupiter). Smaller Trojan populations have been identified for other planets, including Neptune, Mars, Uranus, and Earth. As primitive remnants of the early Solar System, Trojans are believed to preserve material from the planet-forming region beyond the water ice line, providing crucial insights into the origins and dynamical evolution of the outer Solar System. The first Trojan asteroid, (588) Achilles, was discovered on February 22, 1906, by German astronomer Max Wolf using photographic techniques at Heidelberg Observatory; its unusual orbit, which remained consistently ahead of , prompted theoretical predictions by from 1772 to be confirmed observationally. Subsequent discoveries, such as (617) Patroclus in 1906, expanded the known population, with naming conventions drawing from figures of the —Greek heroes for the L4 swarm and Trojan heroes for the L5 swarm—to reflect their positional "camps." By the mid-20th century, photographic and telescopic surveys had identified hundreds, but modern wide-field surveys like those from the and Catalina Sky Survey have dramatically increased the tally to thousands. Trojan asteroids exhibit a range of sizes, from sub-kilometer objects to large bodies like (624) Hektor, which spans about 200 kilometers in diameter and is potentially a . Spectrally, they display uniformly low geometric albedos averaging around 5.3%, reddish slopes across to near-infrared wavelengths, and surfaces dominated by fine-grained anhydrous silicates such as and , with possible traces of ammonia-bearing materials in some cases. These dark, primitive compositions suggest origins in the outer Solar System, akin to cometary nuclei but lacking prominent ice signatures on their surfaces, likely due to or eolian processes. NASA's mission, launched in 2021, represents the first dedicated exploration of Trojans, aiming to fly by multiple targets to analyze their , composition, and densities for deeper understanding of Solar System formation models.

Definition and Orbital Mechanics

Lagrangian Points

In the circular restricted (CRTBP), which models the motion of a negligible-mass third body under the gravitational influence of two primary masses m1≥m2m_1 \geq m_2 orbiting their common in circular orbits, there exist five equilibrium points known as Lagrangian points where the third body can remain stationary relative to the primaries in the corotating frame. These points, labeled L1 through L5, arise from the balance of gravitational and centrifugal forces. The collinear points L1, L2, and L3 lie along the line connecting the two primaries, while L4 and L5 form the vertices of equilateral triangles with the primaries and are thus termed triangular points; L4 is located 60 degrees ahead of the secondary mass m2m_2 in its orbit, and L5 is 60 degrees behind. The derivation of these points relies on the in the rotating frame, where the for the third body are derived from the J=12v2+Ω(x,y)J = \frac{1}{2} v^2 + \Omega(x, y), with Ω(x,y)=1−μr1+μr2+12(x2+y2)\Omega(x, y) = \frac{1 - \mu}{r_1} + \frac{\mu}{r_2} + \frac{1}{2} (x^2 + y^2) as the (in normalized units where the distance between primaries is 1, G(m1+m2)=1G(m_1 + m_2) = 1, and rotation rate ω=1\omega = 1); here, μ=m2/(m1+m2)\mu = m_2 / (m_1 + m_2) is the mass parameter, r1r_1 is the distance from the third body to m1m_1 at position (−μ,0)(- \mu, 0), and r2r_2 is the distance to m2m_2 at (1−μ,0)(1 - \mu, 0). Equilibrium occurs where ∇Ω=0\nabla \Omega = 0, so the partial derivatives vanish: ∂Ω∂x=−(1−μ)x+μr13−μx+μ−1r23+x=0\frac{\partial \Omega}{\partial x} = -(1 - \mu) \frac{x + \mu}{r_1^3} - \mu \frac{x + \mu - 1}{r_2^3} + x = 0 and ∂Ω∂y=−(1−μ)yr13−μyr23+y=0\frac{\partial \Omega}{\partial y} = -(1 - \mu) \frac{y}{r_1^3} - \mu \frac{y}{r_2^3} + y = 0. For the collinear points (L1–L3), set y=0y = 0, reducing the problem to solving the single equation ∂Ω∂x=0\frac{\partial \Omega}{\partial x} = 0 along the x-axis, which yields a quintic equation in the distance γ\gamma from the secondary: (1−μ)γ5+(3−2μ)γ4+(3−5μ+2μ2)γ3+μ(1−μ)(1−2μ)γ2+2μ2(1−μ)γ+μ3=0(1 - \mu) \gamma^5 + (3 - 2\mu) \gamma^4 + (3 - 5\mu + 2\mu^2) \gamma^3 + \mu(1 - \mu)(1 - 2\mu) \gamma^2 + 2\mu^2 (1 - \mu) \gamma + \mu^3 = 0; the three real roots correspond to L1 (γ>0\gamma > 0, between primaries), L2 (γ<0\gamma < 0, beyond m2m_2), and L3 (far beyond m1m_1). For the equilateral points L4 and L5, the positions are x=μ−12x = \mu - \frac{1}{2}, y=±32y = \pm \frac{\sqrt{3}}{2}
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