In 1772 the French mathematician Joseph Louis Lagrange (1736-1813) calculated that in the orbit of Jupiter around the Sun there would be two stable positions, one 60° ahead of the planet, the other 60° behind, where a comparatively tiny mass would remain in stable orbit around the Sun rather than being swept up by Jupiter's gravitational field. (More than a century later two groups of Asteroids, the Trojans, were found at these positions in Jupiter's orbit – hence the frequently used term Trojan points.) This is a general principle, part of what is sometimes called the three-body problem, although usually more than three bodies must be considered; for example, if planning to site a Space Habitat at one of the Lagrange Points (or Lagrangian 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 at the intuitively obvious position between these bodies where their gravitational attractions balance; L2 on the far side of the smaller mass; and L3 on the far side on 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 of Counter-Earth.
The Princeton physicist Gerard K O'Neill, an important propagandist for space colonies, argued in The 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 sf Cliché as the site for fictional space cities consisting of clusters of habitats. The numerous examples include Dean Ing's "Down & Out on Ellfive Prime" (March 1979 Omni) and Mack Reynolds's Lagrange Five (1979) and its sequels.
The Trojan points (L4 and L5) of planet-Star systems are often referenced in Hard SF dealing with orbital mechanics: the gimmick of Hal Clement's "'Trojan Fall'" (June 1944 Astounding) is that no stable Trojan-point orbit is available in the special case of the three-body problem depicted in this story. L1 in the Earth-Sun system is the location of a huge artificial shield constructed to protect Earth from Sun-induced Disaster in Sunstorm (2005) by Arthur C Clarke and Stephen Baxter. [PN/DRL]
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