Joseph-Louis Lagrange (1736–1813) was a professional teaching and research astronomer and 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 produced his “Essai sur le Problème des Trois Corps” (Recueil des pièces qui ont remporté les prix de l’Académie royale des sciences, Volume 9), an interesting treatment of the “three-body” problem. He proved that there were two types of equilibrium or “libration” solutions: collinear and equilateral. Natural instances of one type were discovered over a century later. These solutions became important following the 4 October 1957 launch of Sputnik-1, because these solutions showed that the “third body” (i.e., the artificial satellite, the other two bodies in this particular three-body system being the Earth and Moon) could orbit at one of five co-rotating libration points, which are now called Lagrange Points and designated L1 to L5.
At each Lagrange Point, the combination of the attractions of the primary and secondary bodies provides the inwards force necessary for the orbit of the tertiary body to have the same period.
The Earth–Moon System
L1 is between the Earth and the Moon, 325,000 kilometers (202,000 miles) from the Earth and 56,000 kilometers (35,000 miles) from the Moon.
L2 is on the far side of the Moon, 447,000 kilometers (278,000 miles) from the Earth and 67,000 kilometers (42,000 miles) from the Moon.
L3 is almost in the Moon’s orbit directly opposite the Moon, 380,000 kilometers (236,000 miles) from the Earth and 760,000 kilometers (472,000 miles) 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, have proven to be very useful indeed since a spacecraft can be made to execute a small orbit about one of these otherwise unstable Lagrange Points with a very small expenditure of energy. They have provided useful places to “park” a spacecraft for observations and may be equally viable for space habitats.)
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 almost in the Moon’s orbit, forming equilateral triangles with both the Earth and the Moon. 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 “Trojan” asteroids in the orbit of Jupiter in 1906, in the region 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 L4 and L5 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 89 days (three times the 29½-day period of the Lunar orbit) 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 L4 and L5 libration points have another advantage: they occupy much 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 transfer 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.
For more information on the orbital dynamics of the Lagrange libration points and other gravity-related topics, including a link to an English translation of Lagrange’s original essay, browse Dr. J. R. Stockton’s Web page Gravity 4 : The Lagrange Points and its links.
It’s worth noting that L4 and L5 in the Earth–Moon System may not actually be stable due to the influence of the Sun, but that large stable orbits around these libration points have been determined analytically in 1968 by Hans B. Schechter and 1968 by Ahmed A. Kamel, working independently, and numerically in 1968 by Ronald Kolenkiewicz and Lloyd Carpenter, working together.
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Last Update: 01 January 2020
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2 thoughts on “Home, Home in Lagrange”
I’m sad to tell you Dr. John Stockton’s website doesn’t seem to exist any more. See https://groups.google.com/forum/#!topic/jsmentors/FQ0ry-VnRMo Perhaps you could update your link to an archived copy of his work.
Earth Moon L2 (EML2) is an interesting location. In terms of delta V it’s close to Mars and other destinations in our solar system. Using Farquhar’s route it is closer to LEO than EML1. I did a blog post on EML2: http://hopsblog-hop.blogspot.com/2015/05/eml2.html
It took me awhile to find an archived copy of the Merlyn pages, but I’ve now updated the link.
Hopefully, the archive will not subject to academic abnegation.