Trojan (celestial body)
In 1772, the Italian-French mathematician and astronomer Joseph-Louis Lagrange obtained two constant-pattern solutions of the general three-body problem. These solutions described collinear and equilateral arrangements where a small mass could remain stable relative to two larger bodies. The five possible positions for that negligible mass are now termed Lagrange points. On the 12th of February 1906, Max Wolf discovered the trojan asteroid 588 Achilles. His contemporary Carl Charlier noticed that this asteroid was caught in the L4 point of Jupiter. This discovery marked the first time Lagrange's theoretical calculations applied in practice within our solar system.
Max Wolf found 588 Achilles on the 12th of February 1906 using photographic plates at Heidelberg Observatory. Carl Charlier immediately recognized the object's position near Jupiter's leading Lagrangian point. Astronomers named these objects after figures from the Trojan War of Greek mythology. Asteroids orbiting near the L4 point receive names from the Greek side of the war. Those near the L5 point take names from the Trojan side. Two exceptions exist: 624 Hektor in the L4 group and 617 Patroclus in the L5 group. These were named before the convention was adopted by astronomers.
More than a million Jupiter trojans larger than one kilometer are thought to exist. Over 7,000 of these have been catalogued by astronomers. Only nine Mars trojans have been found to date. Thirty-one Neptune trojans share that planet's orbit. Two Uranus trojans and two Earth trojans remain known. One Saturn trojan has been identified so far. A temporary Venus trojan is also documented. Numerical orbital dynamics stability simulations indicate that Saturn probably does not have any primordial trojans. The large Neptunian trojans are expected to outnumber the large Jovian trojans by an order of magnitude.
All known trojan moons belong to the Saturn system. Telesto and Calypso act as trojans for Tethys. Helene and Polydeuces serve as trojans for Dione. These small satellites share orbits with their much larger parent bodies. They occupy stable positions near Lagrangian points within the planetary system. No other moon in our solar system currently holds this specific relationship. The arrangement creates a unique gravitational dance between four pairs of objects. Each pair maintains its position over long periods despite perturbations from other planets.
Whether a star-planet-trojan system remains stable depends on mass ratios among the three bodies. If the planet equals Earth's mass, a Jupiter-mass object orbiting that star would destabilize the trojan. The system likely stays long-lived if m1 exceeds 100 times m2, which exceeds 10,000 times m3. Here m1, m2, and m3 represent masses of the star, planet, and trojan respectively. A formal stability condition requires 27(m1m2 + m2m3 + m3m1) to stay below (m1 + m2 + m3) squared. When the trojan becomes dust-like, m3 approaches zero, imposing a lower bound on the mass ratio. Once other distant bodies enter the equation, even larger ratios become necessary for survival.
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Common questions
When did Max Wolf discover the first trojan asteroid 588 Achilles?
Max Wolf discovered the trojan asteroid 588 Achilles on the 12th of February 1906. He used photographic plates at Heidelberg Observatory to find this object. This discovery marked the first time Lagrange's theoretical calculations applied in practice within our solar system.
How are Jupiter trojans named according to their position relative to L4 and L5 points?
Asteroids orbiting near the L4 point receive names from the Greek side of the Trojan War. Those near the L5 point take names from the Trojan side. Two exceptions exist: 624 Hektor in the L4 group and 617 Patroclus in the L5 group were named before the convention was adopted by astronomers.
Which planets have known trojan asteroids besides Jupiter?
Only nine Mars trojans have been found to date. Thirty-one Neptune trojans share that planet's orbit. Two Uranus trojans, two Earth trojans, one Saturn trojan, and a temporary Venus trojan remain documented.
What is the relationship between trojan moons and their parent bodies in the Saturn system?
All known trojan moons belong to the Saturn system. Telesto and Calypso act as trojans for Tethys while Helene and Polydeuces serve as trojans for Dione. These small satellites occupy stable positions near Lagrangian points within the planetary system.
What mass ratios determine whether a star-planet-trojan system remains stable?
The system likely stays long-lived if m1 exceeds 100 times m2 which exceeds 10,000 times m3. Here m1, m2, and m3 represent masses of the star, planet, and trojan respectively. A formal stability condition requires 27(m1m2 + m2m3 + m3m1) to stay below (m1 + m2 + m3) squared.