are seen in this graphic as
Greek camp at ahead of Jupiter and as
Trojan camp at trailing Jupiter along its orbital path. It also shows the
asteroid belt between
Mars and
Jupiter and the
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. 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 point of Jupiter, and thus 588 Achilles was the first discovered instance of Lagrange's theoretical calculations applying in practice. 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 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. Later on, objects were found orbiting near the Lagrangian points of
Neptune,
Mars,
Earth,
Uranus, and
Venus. 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. • 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. •
Saturn has 1 known trojan in the L4 Lagrangian Point,
2019 UO14.
Trojans by planet ==Stability==