The Lagrange point phenomenon exists in any system wherein the gravitational fields of two large bodies interact to create five “sweet spots”, within which lies the exact centripetal force necessary for a smaller object to remain in a constant orbit with the larger objects. There are five Lagrange points within the Sun-Earth orbit, as well as five Lagrange points within the Earth-Moon orbit. These points will vary according to the mass of the two larger objects, as well as the distance between them. The first three of these points (commonly referred to as L1, L2, and L3) were discovered by Swiss physicist Leonard Euler; the remaining two were discovered a few years later by Italian physicist Joseph Louis Lagrange, who first published his findings in a 1772 publication entitled “Essay on the three-body problem.” The first three of these points are considered “unstable” while the remaining two are “stable,” based on their positions within the two gravitational fields. Taking advantage of these points enables a small object, like a satellite built by NASA, to remain in orbit within a two-body system in order to make observations from a constant position relative to those two bodies. Robert Forward makes use of this concept quite frequently in Rocheworld to describe the activities of the human space travelers as they explore the binary planet system of Roche and Eau, and the unique qualities of these binary planets raise the possibility for Lagrange points that are quite unlike those we observe in our own Solar System.
The first Lagrange point, called L1, is located between the two larger masses. For example, in the Sun-Earth system, anything orbiting between the Sun and Earth will experience the gravitational forces of both masses pulling it in opposite directions (toward
each respective mass). The closer an object rotates to the sun, the faster it will move, because it has to maintain a higher velocity to remain in orbit, counteracting the gravitational pull of the sun which increases as it gets closer. If an object lies in precisely the right distance between the Earth and the Sun, the Earth’s gravitational pull will slow the object down enough to to make it rotate in tandem with Earth, even though it is closer to the sun than Earth. Therefore an object in L1 will remain in the same position relative to the Earth and the Sun as long as it remains in this exact range. In practice, a satellite in the L1 range can be used to monitor the Sun’s solar winds before they reach earth. L2 is located beyond the Earth on its “dark side”; although it is further from the Sun’s gravitational pull, the Earth’s gravitational pull accelerates its orbit to the extent that it can “keep up” with Earth’s orbit and remain on Earth’s dark side as long as it maintains the L2 position. L3 is located on the direct opposite side of the Sun, slightly beyond Earth’s orbit, out of Earth’s view at all times as it is blocked, of course, by the Sun. These three Lagrange points are “unstable” because a slight alteration in the satellite’s course would shift it out of balance and closer to one of the two masses, much like a ball would be considered “unstable” at rest on top of a hill, because a slight push in any direction would send it rolling down one of the sides.
In the Earth-Sun system, the L4 and L5 lagrange points lie 60 degrees ahead and 60 degrees behind the Earth relative to the direction of its orbit around the sun. Their positions mark the apex of equilateral triangles where the other two vertices are the Earth’s center and the Sun’s center. The L4 and L5 areas form a “bowl-like” gravitational field wherein a slight push would merely send the object into orbit around that Lagrange point because of their unique geometric arrangement; they are virtually “free” from the gravitational pull of of the Earth and the Sun and they are essentially getting “swept along” by both bodies over a larger area rather than getting pulled in opposite directions along a narrow path like L1, L2, and L3.
In Forward’s hypothetical Rocheworld, the Lagrange points function similarly, but the stable points are different. As explained by Dr. Philipson later in the book, the stable points between these two bodies of equal mass are at plus and minus 90 degrees, rather than at a more acute angle as can be found in a system where one of the two larger masses is smaller than the other. Additionally, because these two planets are the same mass, L1 and L2 (relative to the planet you are coming from) exist exactly between the two planets, to the effect that a small object could essentially “float” between the two masses within this range. Unlike L1 in the Earth-Sun system, L1 between Roche and Eau exists at an equal distance between the two planets because both of them have the same gravitational pull, due to their
equal masses. Therefore, the unstable Lagrange points can be found along three points: one on the “outside” of Roche, one between the two planets, and one on the “outside” of Eau, and the distance from the center point is the same to either “outside” point. Because of this symmetry between the two planets, any time the crew passes through a Lagrange point, they experience a zero-gravity effect, since the equal gravity from both planets of equal mass cancel each other out in these precise areas. But beyond creating a unique effect, the “stable” Lagrange points in this system prove incredibly useful to the Terminator crew. As they discover the properties of these two planets as well as the interaction of their symmetrical gravitational fields, they calculate the precise locations of the “stable” L4 and L5 Lagrange points and take advantage of them in order to launch communication satellites. This allows for better fuel efficiency as less added force would be needed to remain within a stable path, unlike the L1, L2, and L3 points where much more frequent propulsion would be necessary to maintain a stable position. They use these communication satellites to remain in contact with each other as they split up and explore, even when they are on the furthest opposite sides of the opposite planets. To this end, Robert
Forward demonstrates a practical and vital use of Lagrange points in space exploration, and in many ways demonstrates what modern scientists already consider when launching satellites to monitor space within our own Solar System. Satellites in the Earth-Sun L1 monitor the Sun’s solar winds and provide advance warning as to the degree of radiation that the Earth will encounter from the Sun. Satellites at L2 can observe deep space and avoid the constant light-dark heating-cooling cycle of ordinary Earth satellites, which is ideal for sensitive equipment. A satellite in L3 could monitor the opposite side of the Sun, tracking developments on the Sun’s surface that could effect Earth later on and give even more advance warning than satellites in L1. L4 and L5, like in Rocheworld, provide a stable and fuel-efficient way to remain in orbit with Earth and provide a constant vantage point into space. Certainly, for a long-term interstellar mission like that described in Rocheworld, knowledge of Lagrange points and their uses in exploratory ventures are incredibly important for conservation of energy and maximum efficiency.
