Item of scientificTerminology denoting each of the five locations in a two-body orbital arrangement where under the influence ofGravity alone a third, smaller body can remain in orbit with respect to the first two. These Lagrange or Lagrangian Points are named for the French mathematician Joseph Louis Lagrange (1736-1813), whose 1772 essay on what is generally called the three-body problem extended the earlier work of his mentor Leonhard Euler (1707-1783), who had found three solutions in which the bodies are arranged in a straight line. Lagrange, considering the Earth-Moon system, added two further solutions in which the three bodies form an equilateral triangle.

Thus in the orbit ofJupiter around theSun there are two equilateral positions, one 60° ahead of the planet, the other 60° behind, where a comparatively tiny mass can remain in stable orbit around the Sun rather than being swept up by Jupiter's gravitational field. More than a century after Lagrange's work, two groups ofAsteroids, now known as the Trojans, were found at these positions in Jupiter's orbit – hence the frequently used term Trojan points. The three-body problem is an idealizedThought Experiment: usually more than three bodies must be considered. For example, if one plans to site aSpace Habitat at one of the Lagrange Points of the Earth-Moon system, one must take into account also the gravitational presence of the Sun (the mass of the habitat itself can be discounted as trivially small). There are five Lagrange Points in the Earth-Moon system; they are not absolutely fixed in relation to the Earth and Moon but, because of the Sun's influence, slowly circle "Lagrange Regions". They are numbered L1 to L5. Speaking generally, the first three L-points all lie on an extended line joining the two principal masses: L1 between these two bodies, L2 on the far side of the smaller mass, and L3 on the far side of the larger. (In the Earth-Sun system, L3 is close to the point of Earth's orbit diametrically opposite its current position – the traditional sf location ofCounter-Earth.) The equilateral-triangle points are L4 and L5. The "Trojan" term comes from the emerging convention of naming Jovian-orbit asteroids for characters in the legendary Trojan War – Greeks in the L4 region and Trojans at L5.

The Trojan points or regions, L4 and L5, of planet-Star systems are often referenced inHard SF dealing with orbital mechanics. In George OSmith'sVenus Equilateral stories, published inAstounding from October 1942, the titularSpace Station occupies a Trojan position in the orbit ofVenus. The gimmick of HalClement's "'Trojan Fall'" (June 1944Astounding) is that no stable Trojan-point orbit is available in the special case of the three-body problem depicted in this story. In Poul Anderson's "Admiralty" (June 1965F&SF; inThe Star Fox fixup1965), a warship orbiting at the L2 point of an extrasolar planet-moon system is concealed by the moon from planet-based foes. L1 in the Earth-Sun system is the location of a huge artificial shield constructed to protect Earth fromSun-inducedDisaster inSunstorm (2005) by Arthur CClarke and StephenBaxter.

The Princeton physicist Gerard KO'Neill, an important propagandist for space colonies, argued inThe High Frontier (1977) that good sites for such colonies would be L4 and L5 of the Earth-Moon system, 60° ahead of and behind the Moon in its orbit – also known as the system's Trojan points. He particularly liked L5, and this region soon became something of an sfCliché as the site for fictional spaceCities consisting of clusters of self-supporting habitats. The numerous examples include DeanIng's "Down & Out on Ellfive Prime" (March 1979Omni) and MackReynolds'sLagrange Five (1979) and its sequels. L5 colonies feature in the background of many further stories, such as Robert AHeinlein'sFriday (1982) – which opens with the protagonist's return to Earth from "Ell-Five" – and JohnMcLoughlin'sToolmaker Koan (1987). [DRL/PN]

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