astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

, 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 point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...

s and . Trojans can share the orbits of

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...

s or of large

moons A natural satellite is, in the most common usage, an astronomical body that orbits a planet, dwarf planet, or small Solar System body (or sometimes another natural satellite). Natural satellites are often colloquially referred to as ''moons'' ...

. Trojans are one type of co-orbital object. In this arrangement, a star and a planet orbit about their common

barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important conc ...

, 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 The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...

as the planet, and the arrangement can remain stable over time. In the Solar System, most known trojans share the orbit of Jupiter. They are divided into the

Greek camp This is a list of Jupiter trojans that lie in the Greek camp, an elongated curved region around the leading Lagrangian point (), 60 ° ahead of Jupiter in its orbit. All the asteroids at Jupiter's point have names corresponding to participants ...

at (ahead of Jupiter) and the

Trojan camp This is a list of Jupiter trojans that lie in the Trojan camp, an elongated curved region around the trailing Lagrangian point, 60° behind Jupiter. All the asteroids at the trailing point have names corresponding to participants on the Troj ...

at (trailing Jupiter). More than a million Jupiter trojans larger than one kilometer are thought to exist, of which more than 7,000 are currently catalogued. In other planetary orbits only nine Mars trojans, 28

Neptune trojan Neptune trojans are bodies that orbit the Sun near one of the stable Lagrangian points of Neptune, similar to the Trojan (astronomy), trojans of other planets. They therefore have approximately the same orbital period as Neptune and follow rough ...

s, two Uranus trojans, and two

Earth trojan An Earth trojan is an asteroid that orbits the Sun in the vicinity of the Earth–Sun Lagrangian points (leading 60°) or (trailing 60°), thus having an orbit similar to Earth's. Only two Earth trojans have so far been discovered. The name " ...

s, have been found to date. A temporary Venus trojan is also known. Numerical orbital dynamics stability simulations indicate that Saturn and Uranus probably do not have any primordial trojans. The same arrangement can appear when the primary object is a planet and the secondary is one of its moons, whereby much smaller

trojan moon 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 ...

s 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

In 1772, the Italian–French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

and

astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiathree-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

. 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

s. 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 In Greek mythology, the Trojan War was waged against the city of Troy by the Achaeans (Greeks) after Paris of Troy took Helen from her husband Menelaus, king of Sparta. The war is one of the most important events in Greek mythology and has ...

Greek mythology A major branch of classical mythology, Greek mythology is the body of myths originally told by the Ancient Greece, ancient Greeks, and a genre of Ancient Greek folklore. These stories concern the Cosmogony, origin and Cosmology#Metaphysical co ...

. By convention, the asteroids orbiting near the point of Jupiter are named for the characters from the Greek side of the war, whereas those orbiting near the of Jupiter are from the Trojan side. There are two exceptions, which were named before the convention was put in place, the Greek

624 Hektor 624 Hektor is the largest Jupiter trojan and the namesake of the Hektor family, with a highly elongated shape equivalent in volume to a sphere of approximately 225 to 250 kilometers diameter. It was discovered on 10 February 1907, by astronomer ...

and the Trojan

617 Patroclus 617 Patroclus ( ) is a large binary Jupiter trojan asteroid. It is a dark D-type asteroid and a slow rotator, due to the 103-hour orbital period of its two components. It is one of five Jovian asteroids targeted by the ''Lucy'' space probe, and ...

. Astronomers estimate that the Jovian trojans are about as numerous as the asteroids of the

asteroid belt The asteroid belt is a torus-shaped region in the Solar System, located roughly between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies, of many sizes, but much smaller than planets, called ...

. Later on, objects were found orbiting near the Lagrangian points of

Neptune Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times ...

Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury (planet), Mercury. In the English language, Mars is named for the Mars (mythology), Roman god of war. Mars is a terr ...

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

Uranus Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus (mythology), Uranus (Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars (mythology), Mars), grandfather ...

, and

Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never fa ...

. Minor planets at the Lagrangian points of planets other than Jupiter may be called Lagrangian minor planets. *Four Martian trojans are known: 5261 Eureka, , , and – the only Trojan body in the leading "cloud" at , There seem to be, also, , , and , but these have not yet been accepted by the

Minor Planet Center The Minor Planet Center (MPC) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory. Function ...

. *There are 28 known Neptunian trojans, but the large Neptunian trojans are expected to outnumber the large Jovian trojans by an

order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...

. * was confirmed to be the first known

in 2011. It is located in the Lagrangian point, which lies ahead of the Earth. was found to be another Earth trojan in 2021. It is also at L4. * was identified as the first Uranus trojan in 2013. It is located at the Lagrangian point. A second one, , was announced in 2017. * is a temporary Venusian trojan, the first one to be identified. *The large asteroids

Ceres Ceres most commonly refers to: * Ceres (dwarf planet), the largest asteroid * Ceres (mythology), the Roman goddess of agriculture Ceres may also refer to: Places Brazil * Ceres, Goiás, Brazil * Ceres Microregion, in north-central Goiás ...

and Vesta have temporary trojans.

Trojans by planet

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 ''m''₁ > 100''m''₂ > 10,000''m''₃ (in which ''m''₁, ''m''₂, and ''m''₃ are the masses of the star, planet, and trojan). More formally, in a three-body system with circular orbits, the stability condition is 27(''m''₁''m''₂ + ''m''₂''m''₃ + ''m''₃''m''₁) < (''m''₁ + ''m''₂ + ''m''₃)². So the trojan being a mote of dust, ''m''₃→0, imposes a lower bound on of ≈ 24.9599. And if the star were hyper-massive, ''m''₁→+∞, then under Newtonian gravity, the system is stable whatever the planet and trojan masses. And if = , 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.

References

{{DEFAULTSORT:Trojan (Astronomy)

Trojan minor planets

Trojans by planet

Stability

See also

References