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Unit 13

The Concepts of Chaos

13.7 Fly Me to the Moon


Lorenz's discovery of sensitive dependence in the 1960s occurred at the time of the golden era of space exploration in both the United States and the USSR. These two competing superpowers utilized the best of deterministic, Newtonian thinking to send human beings into space and to the moon. Achieving this required huge, expensive, rockets and enormous amounts of fuel. Most of the fuel required for a space flight was needed to escape the grip of Earth's gravity and to allow different types of orbits. Additional fuel was required to enable spacecraft to move between different orbits, including orbits that coincided with the path of the moon.

path of the moon

To compute these orbits, engineers used classic linear thinking, sticking to paths that they knew would be forgiving. They knew that small changes would result in small movements, and this helped to minimize error and maximize control. The problem with this strategy is that the opposite is also true: large movements require large changes, and large changes require large amounts of fuel. Exploring the solar system, or just our closest neighbor, the moon, in this manner is effective and relatively safe, but it is extremely expensive.

Fast-forward thirty years to the 1990s and the space race was in decline. After the breakup of the USSR, the United States' chief competitor for space dominance was out of the game. With the chief impetus for space exploration out of the picture, the United States space program had slowly declined from its ambitious projects of the 60s, 70s, and early 80s. No longer could they justify expensive missions, such as those that landed humans on the moon. In this political/social climate, a new paradigm of space exploration began to take shape.

In the 1990s, scientists at NASA's Jet Propulsion Laboratories began to wonder whether some of the ideas from chaos theory might be useful in designing a way to travel around the solar system using very small amounts of fuel. They thought that perhaps they could use nonlinearities to their advantage to get large accelerations for relatively little amounts of fuel. To get a better idea of how this would work, let's return to the example of falling leaves from the introduction to this unit.

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Recall that in our opening example, the two falling leaves started out in almost, but not quite exactly the same position. By the time they reached the ground, they ended up in very different locations. This is an example of the sensitive dependence that is the hallmark of chaos theory.


If we imagine the two leaves to be spacecraft and the branch to be the Earth's orbit, then we get some sense for how this new paradigm of space exploration works. Two spacecraft could start out in minutely different positions and be carried throughout the solar system to very different locations. A very small adjustment at the beginning of a journey could determine whether a spacecraft ends up orbiting the moon or Pluto. The mechanism that would make all this possible came to be called the Interplanetary Superhighway (IPS).


To understand how the IPS works and what it has to do with chaos theory, let's look a little more closely at how the gravitational fields of different planetary bodies interact.

Sun-Earth System Item 1713/NASA/WMAP Science Team, LAGRANGE POINTS 1-5 OF THE SUN-EARTH SYSTEM (2001). Courtesy of NASA/WMAP Science Team. Note the Lagrange points of the Earth-Moon system, L1, L2, L3, L4, L5

We normally envision an orbit to be an elliptical path that results when mutual gravitation between two bodies acts to keep one (the satellite) circling around the other without flying off into space or crashing into its surface. Other types of orbits are possible, however. One alternative type of orbit is characterized by instability, and it is highly susceptible to small changes of course. These orbits are known as halo orbits, and they are the nodes of the IPS network.

Halo orbits take advantage of what are known as Lagrange points. These are points in space where two or more different gravitational fields are exactly balanced. An object situated at a Lagrange point will be able to remain motionless in space, like the rope in a stalemated tug-of-war.

Lagrange point

Just a minimal applied force is enough to send an object hurtling away from the Lagrange point in much the same way that a mere touch is sufficient to send a delicately balanced grape rolling off of the top of an upside-down bowl. If you knew exactly how and where to nudge the grape, you could control where it ends up (for a perfectly spherical grape). Furthermore, your small exertion would result in a large effect on the grape's position. This is the essence of how sensitive dependence can be harnessed and used to help us explore our solar system.

Objects can sit at Lagrange points, albeit tentatively. They can also orbit them in a manner similar to how they would orbit a planet, except that orbits around Lagrange points are extremely unstable. The IPS is a very precise path that connects the different Lagrange points across our solar system. It can be visualized as a system of tubes whose surfaces represent paths that naturally tend toward Lagrange points. By staying on the surface of one of these tubes, a spacecraft can basically surf the gravitational landscape of the solar system using very little fuel. Imagine our grape being nudged off of the first overturned bowl and onto the pinnacle of another overturned bowl, where the process is repeated—a theoretically perpetual system of motion with very little input energy.

system of motion

In this system course corrections or alterations require very little fuel compared to the amount required in the more Newtonian paradigm of powering one's way through space along deterministic orbits. By taking advantage of the sensitive dependence of Lagrange points in the IPS, spacecraft can travel farther more economically, and can devote more of their payload to mission equipment as opposed to the equipment and materials related to propulsion. NASA began to design missions using these concepts in the late 1990s and early 2000s. The IPS is both an exciting development in the field of space exploration and a triumph of using the mathematics of nonlinear systems and chaos.

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