Trojan (celestial body)
Trojan celestial bodies are small objects sharing the orbit of a larger neighbor, hovering roughly 60 degrees ahead or behind it in a gravitational sweet spot that keeps them locked in place for enormous spans of time. The question this raises is almost philosophical: how can something so small hold its position against the forces pulling at it from every direction? The answer goes back to an 18th-century mathematician who never saw one of these objects with his own eyes, and to a discovery on the 12th of February 1906 that proved him right in a way no one had anticipated. How stable are these arrangements, really? And just how many of these hidden companions are drifting through our solar system right now, waiting to be found?
Joseph-Louis Lagrange, the Italian-French mathematician and astronomer, worked out the mathematics in 1772. He found two constant-pattern solutions to the general three-body problem, one collinear and one equilateral. The equilateral solution is the one that matters here. It describes positions where a third, much smaller mass can sit in balance between two larger bodies, moving with the same orbital period as the larger pair without drifting away.
In the restricted version of this problem, where one mass is negligible, there are five such positions in total. They are now called Lagrange points. The two most relevant to trojans are the equilateral ones, sitting 60 degrees ahead and 60 degrees behind the secondary body along its orbit.
The gravitational logic is elegant. A star and a planet orbit their shared center of mass, the barycenter. Because the star is so much more massive, the barycenter sits close to the star's center. A tiny body at one of the equilateral Lagrange points is pulled by both the star and the planet, and the combined force routes through that same barycenter. The tiny body therefore tracks the same orbital period as the planet itself, and the arrangement holds.
Max Wolf, observing in 1906, spotted asteroid 588 Achilles and placed it squarely at Jupiter's L4 Lagrange point. His contemporary Carl Charlier recognized what had happened: a theoretical prediction made over a century earlier had materialized in the night sky.
The name trojan originally attached only to the asteroids near Jupiter's Lagrange points, and astronomers gave them names drawn from the Trojan War of Greek mythology. The convention that emerged sorted them by loyalty: objects orbiting near Jupiter's L4 point carry names from the Greek side of the war, while those near L5 carry names from the Trojan side.
Two asteroids broke this rule because they were named before the convention was set. Asteroid 624 Hektor, a Trojan warrior's name, sits in the Greek L4 group. Asteroid 617 Patroclus, a Greek hero, sits in the Trojan L5 group. They remain there as permanent diplomatic anomalies.
The population behind these names is staggering. More than seven thousand Jupiter trojans are currently catalogued, and astronomers believe more than a million of them larger than one kilometer actually exist. That makes the Jovian trojans roughly as numerous as all the asteroids in the main asteroid belt. The two camps are named accordingly: the Greek camp at L4 and the Trojan camp at L5.
Objects at the Lagrange points of planets other than Jupiter are sometimes called Lagrangian minor planets rather than trojans, though the term trojan has expanded to cover them all.
Jupiter dominates the trojan count, but it does not hold a monopoly. Nine Mars trojans have been found, including 5261 Eureka, which sits in the leading cloud at L4 and is the only confirmed trojan in that position for Mars. Several more Martian candidates exist but have not yet been accepted by the Minor Planet Center.
Neptune presents a striking case. Only 31 Neptunian trojans are confirmed, but models suggest the large ones may outnumber the large Jovian trojans by an order of magnitude. Distance makes them harder to find, not harder to exist.
Two Earth trojans are known. The first was confirmed in 2011 and sits at Earth's L4 point, ahead of Earth in its orbit. A second was found at L4 as well, confirmed in 2021. Two Uranus trojans have been identified: the first announced in 2013 at a Lagrange point, and a second in 2017. Saturn has one known trojan, 2019 UO14, at the L4 point. Venus has a temporary trojan, the first of its kind identified. Even Ceres and Vesta, the large asteroids, have temporary trojans of their own.
Numerical simulations of orbital dynamics suggest Saturn probably has no primordial trojans at all, making 2019 UO14 a likely transient rather than a long-term resident. As for Mercury, no trojans have been found.
Trojans are not limited to planets. The same gravitational geometry applies when the primary is a planet and the secondary is one of its moons. Much smaller trojan moons can share the orbit of the larger moon, sitting 60 degrees ahead or behind it.
All known trojan moons belong to the Saturn system. Telesto and Calypso share the orbit of Tethys; Telesto leads at L4 and Calypso trails at L5. Helene and Polydeuces share the orbit of Dione in the same arrangement.
These moon trojans are among the tidiest demonstrations of Lagrange's mathematics at work, small bodies locked in formation around a larger companion, which is itself circling Saturn.
Not every trio of bodies can maintain this arrangement. Whether a star, planet, and trojan form a stable system depends on the scale of the perturbations acting on them. A Jupiter-mass object sharing a solar system with a planet no larger than Earth would destabilize that planet's trojan companions far more than a Pluto-mass companion would.
A working rule of thumb holds that the system is likely to be long-lived when the star's mass exceeds the planet's mass by a factor of at least 100, and the planet's mass exceeds the trojan's mass by a factor of at least 100 in turn. Written compactly: m1 must exceed 100 times m2, and m2 must exceed 100 times m3.
The formal stability condition for circular orbits is more precise. It states that 27 times the sum of the three pairwise mass products must be less than the square of the total mass. When the trojan's mass approaches zero, this sets a lower bound on the ratio of the star's mass to the planet's mass of approximately 24.9599. When the masses of the star and planet are equal, both must individually exceed roughly 25.9615 times the trojan's mass to preserve stability.
Once other bodies enter the picture, even distant and small ones, the required mass ratios grow larger still. The three-body math is clean; the real solar system is not, which is why long-term stability for most trojan populations depends on their neighborhood staying relatively uncrowded.
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Common questions
What is a trojan celestial body in astronomy?
A trojan is a small celestial body, mostly an asteroid, that shares the orbit of a larger body while remaining roughly 60 degrees ahead of or behind it near one of its Lagrange points. The gravitational balance at these points allows the trojan to orbit with the same period as the larger body without drifting away.
When was the first trojan asteroid discovered?
The first trojan asteroid, 588 Achilles, was discovered on the 12th of February 1906 by Max Wolf. Carl Charlier recognized that the asteroid was caught at Jupiter's Lagrange point, confirming in practice the theoretical calculations Joseph-Louis Lagrange had worked out in 1772.
How many Jupiter trojans are there?
More than 7,000 Jupiter trojans are currently catalogued, but astronomers estimate that more than a million Jupiter trojans larger than one kilometer actually exist. They are divided into the Greek camp at the L4 Lagrange point and the Trojan camp at L5, and their total number is thought to rival that of the main asteroid belt.
Why are trojan asteroids named after Trojan War figures?
By convention, Jupiter trojans are named for figures from the Trojan War of Greek mythology. Asteroids at the L4 point carry names of Greek warriors, while those at L5 carry Trojan names. Two exceptions predate the convention: 624 Hektor sits in the Greek L4 group, and 617 Patroclus sits in the Trojan L5 group.
What planets in the solar system have known trojan objects?
Jupiter, Mars, Neptune, Uranus, Earth, Saturn, and Venus all have known trojan objects. Jupiter has the most with more than 7,000 catalogued; Mars has nine, Neptune has 31, Uranus and Earth each have two, and Saturn has one known trojan, 2019 UO14. Venus has a temporary trojan. Mercury has none.
What is the mass ratio needed for a trojan system to be stable?
As a rule of thumb, a trojan system is likely to be long-lived when the star's mass is at least 100 times the planet's mass, and the planet's mass is at least 100 times the trojan's mass. The formal stability condition for circular orbits requires that 27 times the sum of the three pairwise mass products be less than the square of the total system mass.
All sources
21 references cited across the entry
- 1bookDynamics of Small Solar System Bodies and ExoplanetsPhilippe Robutel et al. — Springer — 2010
- 2journalSize Distribution of Faint Jovian L4 Trojan AsteroidsF. Yoshida et al. — Dec 2005
- 3journalA Thick Cloud of Neptune Trojans and their ColorsScott S. Sheppard et al. — June 2006
- 4journalEssai sur le Problème des Trois CorpsJoseph-Louis Lagrange — 1772
- 6webHow Were the Trojan Asteroids Discovered and Named?William Steigerwald — 22 February 2021
- 7journalPlanetary science: The Trojan is out thereAlison Wright — 1 August 2011
- 8journalSize distribution of faint L4 Trojan asteroidsFumi Yoshida et al. — 2005
- 9journalEarth's Trojan asteroidMartin Connors et al. — 27 July 2011
- 10journalA CCD Search for Lagrangian Asteroids of the Earth–Sun SystemRobert J. Whiteley et al. — November 1998
- 12journalThree new stable L5 Mars TrojansC. de la Fuente Marcos et al. — 15 May 2013
- 13webList of Neptune Trojans28 October 2018
- 14journalNeptune Trojans as a Testbed for Planet FormationEugene I. Chiang et al. — 20 July 2005
- 15newsNeptune May Have Thousands of EscortsDavid Powell — 30 January 2007
- 16newsFirst Asteroid Companion of Earth Discovered at LastCharles Q. Choi — 27 July 2011
- 17journalThe Second Earth Trojan 2020 XL5Man-To Hui — Nov 2021
- 18journalTrojan asteroid: Another object found that shares Earth's orbitLeah Crane — Nov 22, 2021
- 19journalAsteroid 2014 YX49: a large transient Trojan of UranusCarlos de la Fuente Marcos et al. — 21 May 2017
- 20journalA population of main belt asteroids co-orbiting with Ceres and VestaApostolos A. Christou et al. — January 2012
- 21webSaturn gets its 1st confirmed Trojan asteroid — but it might be stolenRobert Lea — 2024-10-24