
In
astronomy
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, a trojan is a small
celestial body
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists within the observable universe. In astronomy, the terms ''object'' and ''body'' are of ...
(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 (mechanics), equilibrium for small-mass objects under the gravity, gravitational influence of two massive orbit, orbiting b ...
s and . Trojans can share the orbits of
planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s 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
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 at (ahead of Jupiter) and the
Trojan camp 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, 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.
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

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, mathematical structure, structure, space, Mathematica ...
and
astronomer
An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[three-body problem
In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then calculate their subsequent trajectories using Newton' ...](_blank)
. 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
The Trojan War was a legendary conflict in Greek mythology that took place around the twelfth or thirteenth century BC. The war was waged by the Achaeans (Homer), Achaeans (Ancient Greece, Greeks) against the city of Troy after Paris (mytho ...
of
Greek mythology
Greek mythology is the body of myths originally told by the Ancient Greece, ancient Greeks, and a genre of ancient Greek folklore, today absorbed alongside Roman mythology into the broader designation of classical mythology. These stories conc ...
. 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, named before the convention was adopted:
624 Hektor in the L4 group, and
617 Patroclus in the L5 group.
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, centered on the Sun and roughly spanning the space between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies called asteroids ...
.
Later on, objects were found orbiting near the Lagrangian points of
Neptune
Neptune is the eighth and farthest known planet from the Sun. It is the List of Solar System objects by size, fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 t ...
,
Mars
Mars is the fourth planet from the Sun. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the atmosph ...
,
Earth
Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
,
Uranus
Uranus is the seventh planet from the Sun. It is a gaseous cyan-coloured ice giant. Most of the planet is made of water, ammonia, and methane in a Supercritical fluid, supercritical phase of matter, which astronomy calls "ice" or Volatile ( ...
, and
Venus
Venus is the second planet from the Sun. It is often called Earth's "twin" or "sister" planet for having almost the same size and mass, and the closest orbit to Earth's. While both are rocky planets, Venus has an atmosphere much thicker ...
. 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.
Funct ...
.
*There are 28 known
Neptunian trojans, but the large Neptunian trojans are expected to outnumber the large Jovian trojans by an
order of magnitude.
* was confirmed to be the first known
Earth trojan 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 and
Vesta have temporary trojans.
*
Saturn
Saturn is the sixth planet from the Sun and the second largest in the Solar System, after Jupiter. It is a gas giant, with an average radius of about 9 times that of Earth. It has an eighth the average density of Earth, but is over 95 tim ...
has 1 known trojan in the L
4 Lagrangian Point,
2019 UO14.
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.
See also
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*
*
*
References
{{DEFAULTSORT:Trojan (Astronomy)
Solar System