From HomoExcelsior

Three Body Problem

In 1772, Josef Louis Lagrange was working on the infamous 3 body problem when he discovered an interesting little quirk. Originally, he had set out to discover a way to easily calculate the gravitational interaction between arbitrary numbers of bodies in a system, because Newtonian mechanics conclude that such a system results in the bodies orbiting chaotically until there is a collision, or a body is thrown out of the system so that equilibrium can be achieved. The logic behind this conclusion is that a system with one body is trivial, as it is merely static relative to itself; a system with two bodies is very simple to solve for, as the bodies orbit around their common center of gravity. However, once more than two bodies are introduced, the mathematics go crazy. A situation arises where you would have to calculate every gravitational interaction between every object at every point along its trajectory. Lagrange, however, wanted to make this simpler. He did so with a simple conclusion: "the trajectory of an object is determined by finding a path that minimizes the action (which is the Lagrangian) over time. This is found by subtracting the potential energy from the kinetic energy. With this way of thinking, Lagrange re-formulated the classical mechanics (introduced by Sir Isaac Newton) to give rise to Lagrangian mechanics.

With his new system of calculations, Lagrange's work led him to hypothesize how a third body of negligible mass would orbit around two larger bodies which were already orbiting one another. This model is one which closely describes a Sun-Earth-Space Debris system (for sake of simplicity, this is the system that will be continually referenced throughout this paper). The interesting quirk that Lagrange discovered about the two larger bodies was that there would be five points of gravitational equilibrium (Lagrange points) where the gravitational forces of the bodies would precisely equal the orbital motion necessary to rotate with them. This means that bodies of negligible mass which are positioned in a Lagrange point are stationary relative to the other two (relatively more massive) bodies if there is no gravitational interference from a body outside the system.

Since it is not intended for the reader to merely accept the fact that there are Lagrange points, and they just happen to be the result of interacting gravitational fields, the "what," "why," and "how" of Lagrange point origin will now be covered for each point. Also, it would be helpful to reference back to the above picture if there is any confusion concerning the placement of the Lagrange points.

Descriptions of Individual Lagrange Points

L1

Lies on a line between the Sun and Earth, but is positioned closer to Earth.

As an object orbits the Sun closer than the Earth, it naturally tends to have a shorter orbital period than Earth; however this scenario does not include the gravitational influence of Earth. If the object is directly between the two bodies, the resulting effect is Earth's gravity weakening that of the Sun's, thereby increasing the object's orbital period. As the object gets closer to Earth, the effect increases until it reaches a certain point (L1) where it's orbital period equals that of Earth.

L2

Lies on a line defined by the Sun and Earth, but beyond the position of Earth.

An effect similar to that which causes the L1 point occurs on the other side of the Earth (further away from the Sun). Normally, the object would have an orbital period which is greater than that of Earth. However, when the object gets closer to Earth the gravitational influence of Earth causes the object's orbital period to decrease until it reaches the L2 point. It is then that the object's orbital period equals that of Earth's.

L3

Lies on a line defined by the Sun and Earth, but beyond the position of the Sun.

On the opposite side of the Sun, just outside the orbit of Earth, the combination of the Sun's and Earth's gravity causes the object's orbital period to equal that of Earth. Since the position of this Lagrange point lies behind the Sun, any objects which may be orbiting in the L3 point are outside the range of Earth based observations.

L4

Lies at the apex of an equilateral triangle with the base defined by the line between the Sun and Earth.

L5

Lies at the apex of an inverted equilateral triangle with the base defined by the line between the Sun and Earth.

The L4 and L5 points lie at 60 degrees ahead of, and behind the Earth in its orbit. Unlike the other Lagrange points, L4 and L5 are essentially resistant to gravitational perturbations. It's because of this stability that objects tend to accumulate in these points.

Stability

On the matter of Lagrange point stability, the fact that there are more than 3 objects in the Universe lets us know that there can't be true stability in a Lagrange point. Objects located at points L1, L2, and L3 are comparable to a ball that's sitting on a hill. If there's some sort of outside force that acts on the ball, it will set the ball in motion down the hill where it will stay until there is another force that places the ball back on the hill. Prime factors for instability in the Sun-Earth system's Lagrange points are Venus, Jupiter, and the Asteroid Belt.

On the other hand, when we look at the L4 and L5 points, we see that there are a few forces which can aid in holding an object in orbit around a Lagrange point. For argument's sake, let's assume that an object forms at the center of the L4 point. As it feels the gravitational influences from objects outside the Sun-Earth system, it begins to wander away from the center of L4. As it gets further away from the center, forces which resemble those that cause the Coriolis effect here on Earth bend the object into a kidney shaped (from the perspective of Earth) orbit around the L4 point. This situation is similar to that of a marble inside a bowl. As long as there are no outstanding forces acting on the marble that would cause it to get flung out of the bowl, it will merrily remain in its ceramic gravity well.

The stability of Lagrange points, besides contributing to the build-up of asteroids, allows humans to place man-made satellites in orbit around them. Two such satellites are: the SOHO (Solar and Heliospheric Observatory) which is located in the Sun-Earth system's L1 point, and WMAP (Wilkinson Microwave Anisotropy Probe) which is in the Sun-Earth system's L2 point.

The intent of SOHO is to study the internal structure of the Sun, examine its massive atmosphere, and determine the origin of solar winds. SOHO has allowed scientists to get the best understanding of the interactions between the Sun and Earth than has been possible before. Even still, many are hoping that SOHO will help solve several of the more puzzling riddles, such as: How the solar corona is heated; how solar wind accelerates; and physical characteristics of the solar interior. All of this is possible due to the unobstructed viewpoint that SOHO has, since it is nestled in its home away from home, the Sun-Earth system's L1 point.

WMAP is a venture to map the cosmic background's temperature differences. This light is the oldest in the Universe, and is, in a sense, a baby picture of our Universe. The most recent data collected is of the Universe when it was 380,000 years old (13 billion years ago). This is not unlike taking a picture of someone who is 80 years old on the day they were born. All of this is possible because of its convenient position in the Sun-Earth system's L2 point. This vantage point allows WMAP to have full view of space without having to worry about the Sun, Earth, or Moon being in the way.

Conclusion

In an attempt to make things simpler for his calculations, Josef Louis Lagrange also developed an idea that would make things easier for experimentalists many years down the road. From a mathematical hypothesis, to man-made satellites exploiting their convenient placement, Lagrange points have proven to be a wonderful tool for scientists and mathematicians alike.

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