Mobile Suit Gundam: High FrontierLife In The Universal CenturyJoseph-Louis Lagrange (1736–1813) was a mathematician who, until the advent of the space program, was most famous for developing Lagrangian Mechanics, a branch of physics that uses energy principles instead of forces to describe how the universe evolves. Lagrange’s hobby was the so-called “n-body” problem of Newtonian physics. Newton’s laws define the motion of two bodies due to their mutual interaction. When two objects, such as the Earth and the Moon, rotate around each other, one need only set up the differential equations of motion and solve them using calculus to determine what the system will look like at any point in time—a “two-body” problem. The n-body problem seeks to derive a similar solution for the interaction of “n” number of bodies, where n is all numbers greater than two. There are ten integrals of the Newtonian equations for motion for an arbitrary system of n objects, six describing the motions of center-of-mass, three the angular momentum and one for conservation of total energy. The “problem” was that the solution to these equations “proved” that many-body systems were unstable and couldn’t exist. How, then, does one explain the solar system and other demonstrably stable multiple-body systems? In 1772, Lagrange published his “Essai sur le Problème des Trois Corps” (L’Academe Royale de Sciences de Paris, Volume 9), an elegant approximation to the “three-body” problem. By assuming that the third mass was negligible in comparison to the other two, he proved that there were certain equilibrium or “libration” solutions. These became important two centuries later because the mass of an artificial satellite is negligible in comparison to the Earth and the Moon and Lagrange’s solutions find practical application. One of these solutions showed that the third body could come to rest at one of five libration points, which are now called Lagrange points and designated L1 to L5. 
L1 was the easiest to find—the point between the Earth and the Moon where the gravitational attractions negate each other (325,000 kilometers from the Earth and 56,000 kilometers from the Moon). L2 was a point on the other side of the Moon, where the component forces combine (447,000 kilometers from the Earth and 67,000 kilometers from the Moon). L3 was a point in the Moon’s orbit directly opposite the Moon (380,000 kilometers from the Earth and 760,000 kilometers from the Moon). L1, L2 and L3 are all saddle-shaped gravitational “valleys” in which a body displaced perpendicular to the Earth-Moon axis is drawn back toward the axis. Since displacement along the axis can continue indefinitely, these as known as points of unstable equilibrium. (Circular “halo” orbits around these points, where the plane of the orbit is perpendicular to the line joining Earth and Moon, are just as stable as orbits around a true celestial body) 
L4 and L5, on the other hand, are bowl-shaped valleys in which a body displaced in any direction returns to the center and are therefore known as points of stable equilibrium. They are located in the Moon’s orbit, equidistant from both the Earth and the Moon, with which form congruent equilateral triangles. L4 is the point 60° east (i.e., the direction of the Earth’s rotation or counterclockwise) of the Moon, leading it in orbit around the Earth. L5 is the point 60° west (clockwise) of the Moon, trailing it in orbit around the Earth. Lagrange’s theories were confirmed a century later with the discovery of the Trojan asteroids in the orbit of Jupiter in 1906, exactly where L4 and L5 would’ve been had Jupiter been the Moon and the Sun been the Earth. The discovery was so profound at the time that, to this day, astronomers call the L4 and L5 points in any orbit the leading or trailing Trojan points. The theories were further refined in 1970, when A. A. Kamel published his doctoral dissertation “Perturbation Theory Based On Lie Transforms and its Application to the Stability of Motion Near Sun-Perturbed Earth-Moon Triangular Libration Points” with Stanford professor John Bleakwell. 
The Lagrangian libration points are critical to the building of orbital space habitats because they give us a place in which to build. Massive objects placed in the vicinity of the Trojan points in particular will orbit those points once every 29½ days while accompanying the Earth and the Moon around the Sun, without the need to expend propellant mass and energy, which might be needed elsewhere. In addition to offering zones of stability in which a space habitat can orbit without frequent and expensive course corrections, the five Lagrange points have another advantage: they occupy the same orbit as the Moon. This puts them out of the vicinity of the Earth and its orbital collection of man-made debris, but between 9.6 and 156 hours transit time using a minimum-energy trajectory such as the classic doubly tangent Hohmann orbit. They are, in effect, equidistant from the Earth in terms of both metric distance and velocity change (ΔV). Since the majority of velocity change (about 8.6 kilometers per second or two-thirds of a total 12.7 kilometers per second ΔV) is expended in reaching low Terrestrial orbit, one can travel to or between any of the colonies with equal facility.
|
||||||||||||||||||||||||||
Return to Top of Page