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Trojan asteroids

History

On 1st of January 1801, Guiseppe Piazzi (1746-1826) discovered the first asteroid Ceres (1), which was regarded as a new planet between Mars and Jupiter. In the following years astronomers discovered more and more bodies with the semimajor axes in the range 1.5AU < a < 5.2AU. This belt of asteroids is called the main belt. Today approximately 260000 of these bodies are known and in general we may distinguish 6 main different groups of asteroids:

  1. the asteroids with perihelion distances q < 1.3AU are called near earth

    2. asteroids     (NEAs), some of which even cross the Earth's orbit;
  2. the main belt asteroids, with semimajor axes between those of Mars and Jupiter;

    3. the cloud of Jupiter Trojans following, respectively preceding the gas planet
  3. close to the Lagrangian equilibrium points L4 and L5;

    4. the Centaurs a group of asteroids between Jupiter and Neptune;
  4. the Trans-Neptunian Objects (TNOs) with a mean orbital distance

    5. greater than 30AU. The orbits of the Kuiper Belt Objects extend roughly from
    30AU to 50AU;
  5. the Oort Cloud is a hypothetical cloud of comets with a

    6. mean orbital distance between approximately 50000AU and 100000AU


The discovery of the Trojan asteroids

The Italian-French mathematician Joseph Lagrange (1736-1812) showed that the three-body problem could be solved for a few special cases analytically. In 1772 he discovered five special points in the vicinity of two orbiting masses where a third, much smaller mass can orbit at a fixed distance from the larger masses (Lagrange 1788). More precisely, the Lagrange points mark positions where the gravitational pull of the two large masses equals the centripedal force required to rotate with them. The Lagrangian points are the five stationary solutions of the restricted three-body problem (=RTBP)- in this idealized problem a massless body moves under the gravitational influence of two massive bodies, which themselves are on circular orbits - , as it is demonstrated in Fig.1.

Fig.1: All equilibrium points (L1 up to L5) for the RTBP in the 'Sun-Jupiter' system.


The points L1, L2 and L3 lie on a straight line connecting the other two bodies (Fig. 1) and are points of unstable equilibrium. This means that a small perturbation will cause the third body to drift away. The L4 and L5 points are at the third vertex of an equilateral triangle (Sun - Jupiter - L4 or L5) formed with the other two bodies; they are points of stable equilibrium. Objects orbiting around L4 and L5 are stable because the Coriolis force keeps them spinning around the Lagrange point.
For a long time it was believed that the Lagrange's solutions of the restricted problem are only of theoretical interest. But one century later, in February 1906 Max Wolf (1863-1932) discovered the first Trojan asteroid and named it Achilles (588) after the mythical Achilles, one of the heroes of Homer's Iliad. As of October 1999, 170 had been numbered. By July 2004 that number had grown to 877. There are currently (March 2005) 1759 known Trojan asteroids - 1069 at L4 and 690 at L5.

As the Iliad deals with the events of the Trojan War, the asteroids came to be collectively known as Trojan asteroids. Strictly speaking, the term Trojan applies only to those in the L4 and L5 points of the Sun-Jupiter system, but over time, it has come to be more generally applied to any planetoidal body at the triangular Lagrangian point of any more massive two bodies. Besides Jupiters Trojans, Mars and Neptune have one Trojan each, plus there are Trojan moons around Saturn (Telesto-Thetys-Calypso and Dione-Helene). The moon Helene orbits in Dione's leading Lagrangian point. Calypso is co-orbital with the moon Tethys, and resides in Thety's trailing Lagrangian point L5. The moon Telesto resides in Thetys leading point. Also some Mars Trojans were found, but they could not be confirmed. Now a few comments of the Minor-planet-center (=MPC) to the confirmation of the Mars Trojans (see at MPC):

The list of Martian Trojans are extracted using cuts in osculating element space. This technique works very well for Jupiter Trojans and for the sole Neptunian Trojan, but it has been known to be problematic for Martian Trojans. It has been the intention to replace the Martian Trojans with a list comprising only those that are confirmed (by long-term integration off good orbits) to be Trojans. Therefore the MPC removed some Trojan candidates (5261 Eureka, 1998 VF31, 1999 UJ7, 2001 DH47, 2001 FG24, and 2001 FR127) which are ill-informed speculations.

Earth Trojans have not been found yet, but these objects could someday be of great practical value, though. After the Moon, they would be easily 'accessible objects' in term of the energy required to reach them. The energy required to return materials from an Earth-Trojan orbit to the Earth is much smaller than for the other minor planets in the solar system.