588 Achilles (ə-kil'-eez), provisional designation 1906 TG, is a large and dark asteroid, classified as Jupiter trojan, the first and 6th-largest of its kind ever confirmed by astronomers (A904 RD was discovered 2 years previous, but was not observed long enough to calculate an orbit). It was discovered on 22 February 1906, by the German astronomer Max Wolf at Heidelberg Observatory in southern Germany. It measures about 135 kilometers in diameter and was named after Achilles from Greek mythology.
The D-type asteroid, classified as a DU-subtype in the Tholen taxonomic scheme, orbits the Sun at a distance of 4.4–6.0 AU in the L4 Lagrangian point of the Sun–Jupiter System once every 11 years and 10 months (4,337 days). Its orbit shows an eccentricity of 0.15 and an inclination of 10 degrees from the plane of the ecliptic. The asteroid is the first known example of the stable solution of the three-body problem worked out by French mathematician Joseph Lagrange in 1772, after whom the minor planet 1006 Lagrangea is named. After the discovery of other asteroids with similar orbital characteristics, which were also named after heroes from the Trojan War (see below), the term "Trojan asteroids" or "Jupiter trojans" became commonly used. In addition, a rule was established that the L4 point was the "Greek camp", whereas the L5 point was the "Trojan camp", though not before each camp had acquired a "spy" (624 Hektor in the Greek camp and 617 Patroclus in the Trojan camp).
Photometric observations of this asteroid during 1994 were used to build a light-curve showing a rotation period of 7000732000000000000♠7.32±0.02 hours with a brightness variation of 6999310000000000000♠0.31±0.01 magnitude. This result is in good agreement with prior studies. According to the surveys carried out by the Infrared Astronomical Satellite, IRAS, the Japanese Akari satellite, and the NEOWISE mission of NASA's Wide-field Infrared Survey Explorer, the body's surface has a very low albedo in the range between 0.033 and 0.043, and a corresponding diameter of 130.1 to 135.4 kilometers.
The minor planet's name was suggested by Austrian astronomer Johann Palisa. It was named after Achilles, the legendary hero from Greek mythology and central figure in Homer's Iliad which tells the accounts of the Trojan War (also see 5700 Homerus and 6604 Ilias). As an infant, Archilles was plunged in the River Styx by his mother Thetis (also see 17 Thetis), thus rendering his body invulnerable excepting the heel by which he was held. He slew Hector (see also 624 Hektor), the greatest Trojan warrior. He was eventually killed by an arrow in the heel by Paris (see 3317 Paris).