Devlin's Angle

February 1999

Stardust Equations

On Sunday, February 7, at 4:04 PM, a Lockheed Martin Delta II rocket lifted off from Cape Canaveral carrying an unmanned spacecraft about the size of an office desk, having the romantic name of Stardust.

If all goes well, on January 15, 2006, the spacecraft will return to Earth and release into the atmosphere a small capsule, measuring 30 inches across and 20 inches deep, which will parachute down onto the salt flats in the Utah desert, landing at 3:00 AM.

Inside the capsule, buried in two pieces of an ultra light, sponge-like, synthetic material called aerogel, scientists hope to find the precious substances the spacecraft was sent up to obtain: dust from distant stars (the "stardust") and minute particles from the tail of a comet. Expected total weight of the cargo: about one-thousandth of an ounce.

During its seven year journey, Stardust will travel some 3.1 billion miles through the solar system at an average speed of 48,000 miles per hour.

One major dust collection operation will take place over a period of a few minutes on January 2, 2004, when Stardust flies through the streaming tail of the comet Wild 2 (it's pronounced "vihlt"). In addition, between March and May 2000 and again between July and December 2002, while en route to the encounter with Wild 2, a second aerogel collector will collect interstellar dust from a recently discovered beam of particles streaming into the solar system from other stars in outer space.

[Click here to see an artist's illustration of Stardust's encounter with Wild 2, displayed on the NASA Stardust website.]

Apart from the two dust collections, the only other activity of note that the spacecraft will perform is to take close-up photographs of the comet and a few tourist snapshots during the journey.

With manned space flights getting most of the media attention -- especially when they involve septuagenarian American legends -- an unmanned mining operation costing a mere $200 million is unlikely to cause much of a stir. And yet, if you stop to think for a moment, a voyage of 3 billion miles culminating in the return to Earth at a specific location, at a specific time on a specific day seven years later, is a powerful reminder of the predictive power of mathematics. As much as being a triumph for engineering, a mission such as Stardust is a significant mathematical feat.

To mount the Stardust mission, both the orbit of the comet and the trajectory the spacecraft will follow on its seven year journey have to be calculated in advance, with great precision, so that the spacecraft will come within 75 miles of the comet's main body before heading back for Utah.

That would be difficult enough -- though essentially a routine application of Newtonian mechanics -- if the spacecraft followed the most direct route to its encounter with the comet. But that's not how NASA plans to carry out the mission. Although the Stardust-comet encounter will take place a mere 242 million miles from Earth, Stardust will have traveled 2 billion miles through space to get there, and will travel a further billion miles on the return journey. Clearly, the spacecraft will not be taking the most direct route, to say the least! Why?

Fuel economy, that's why. For Earthly travel, the longer the journey, the more fuel is required, but that's not the case for space travel. Without air resistance to slow it down, a spacecraft can make use of the gravitational pull of the Earth, the moon, or other planets, in order to propel it on its way. It's all a question of taking the right path.

The fuel-efficient stairways to heaven

In NASA-speak, Stardust will be making use of an "Earth Gravity Assist", or EGA, in order to meet up with the comet. The idea behind gravity assist is to use the gravitational force of a planet as a source of energy, either to change the direction of flight of a spacecraft or to increase its speed, or both.

For example, in 1970, the world watched breathlessly as NASA used a Moon gravity assist to rescue the Apollo 13 astronauts after an on-board explosion had severely damaged their spacecraft as it set off for the moon. By using a relatively small amount of fuel to put the spacecraft onto a suitable trajectory, the NASA engineers and the astronauts were able to use the Moon's gravity to turn around and head back for Earth. (A short -- but accurately timed -- rocket burn behind the moon ensured that the stricken craft did not go into permanent lunar orbit.)

In the case of Stardust, the Delta II rocket will launch the spacecraft into a wide, eccentric orbit around the sun, well outside the Earth's own orbit. That initial trajectory will bring the spacecraft back into close proximity with the Earth two years later. As Stardust approaches the Earth, it will swing around the Earth under the Earth's gravity. A small rocket burn will ensure that, when it starts to move away from the Earth, the spacecraft will follow a much wider heliocentric orbit (well beyond Mars and over half the distance to Jupiter) that will bring it to the Wild 2 orbit just as it (the spacecraft) completes its second circuit around the sun -- at precisely the moment when Wild 2 itself reaches that location. In other words, NASA will use the Earth's gravity to provide a kind of "slingshot" to fling the spacecraft in the right direction.

Then, after the brief, twelve-hour union of the two space travelers -- one of natural origins, the other "Made in America" -- Stardust will set off on its third and final wide loop around the sun, this time resulting in the reentry capsule's separation and high-speed entry into the Earth's atmosphere -- 28,000 miles per hour at first contact with the atmosphere, 70% faster than the space shuttle. (The main spacecraft itself will go into permanent solar orbit.)

If you click here, you'll see a map of Stardust's proposed trajectory, as shown on the NASA Stardust website. The map shows the three points where small guidance thrusters on the spacecraft will be fired to adjust the orbit, in March 2000, November 2001, and July 2003. These locations are carefully chosen so that a small force will lead to a significant change in the orbit. Apart from the guidance thrusters, gravity provides the craft's propulsion, with solar panels being used to generate the electricity to power the on board equipment.

All in all, with its use of gravity for propulsion and solar power to generate electricity, Stardust is a decidedly ecological spacecraft -- the NASA equivalent of organic farming. The same cannot be said of another planetary wanderer that will be in the heavens at the same time. Launched on October 15, 1997, Cassini was designed to explore Saturn and its moons. Besides being much larger (about the size of a 30-passenger school bus), and requiring a massive Titan IV rocket to send it on its way, Cassini is powered by nuclear energy. And it's the 72 pounds of plutonium fuel on board that has caused -- and continues to cause -- controversy about the enterprise.

Prior to the mission, a number of individuals and groups attempted to prevent it, concerned at the consequences of a mishap during the launch. (Certainly, the release of 72 pounds of highly radioactive and intensely toxic plutonium into the atmosphere could be deadly to a large segment of the Earth's population. Much of the debate centered on whether such an outcome were possible, given the design of the spacecraft and of its three nuclear power units.)

In the event, after two delays, the launch went ahead without problem. But that was not the end of the debate. It reemerged just after last Christmas, and is raging now, with the approach of the planned Earth-flyby of the spacecraft just 720 miles above the Earth's surface, that will take place this coming August.

For, as with Stardust, Cassini makes use of planetary gravitation in order to propel itself to its destination. (Unlike Stardust, Cassini has two large engines, but they are not used to propel the spacecraft to Saturn; rather, their function is to guide it around the Saturn system for four years of exploration when it gets there.) In the case of Cassini, however, gravity assist is used primarily to accelerate the spacecraft. This means that its trajectory is much more complicated than for Stardust.

Cassini's flight path to Saturn involves two swings by Venus -- one last April 21, the second on June 20 of this year -- followed by a close pass of Earth on August 16, and a close pass by Jupiter on December 30, 2000. Yes, that's right, to get to Saturn, far out in the Solar System, Cassini starts off by heading for Venus, which is closer to the Sun than is the Earth. In order to reach an outer planet, the spacecraft starts out traveling in the "wrong" direction.

NASA refers to Cassini's flightpath as a VVEJGA trajectory, for Venus-Venus-Earth-Jupiter Gravity Assist. [To see the entire trajectory, click here.] The spacecraft is scheduled to arrive at Saturn on July 1, 2004.

The current controversy concerning Cassini's upcoming Earth swingby centers on the risk of the spacecraft accidentally re-entering the Earth's atmosphere. According to the critics, with the spacecraft moving so fast (42,000 miles per hour), the 720 mile separation from the atmosphere is far too small, and a tiny error in computing the orbit or a sudden malfunction on the spacecraft could send it hurting toward Earth, with catastrophic consequences. NASA says that such an eventuality is so unlikely that there is no reason to call off the mission and destroy the spacecraft remotely as opponents are demanding. One thing both sides agree on however, is this: NASA has to make sure it gets its math right.

Once Cassini arrives in the vicinity of Saturn, it will begin a complicated sequence of orbits around the planet that will last four years, during which time various on board instruments will measure the atmosphere and magnetic field of Saturn and of several of its many moons. In addition, a separate probe, called Huyghens, will be released from the mother ship to carry out a special examination of Saturn's largest moon, Titan, a body similar in size to the Earth. The Huyghens probe will eventually land on Titan's surface, from where NASA hopes to receive pictures, relayed via Cassini.

In order to use gravity assist to increase the speed of a spacecraft, the idea is to use planets to provide "slingshots" that "hurl" the craft from one planet to another, its speed increasing with each slingshot. Put simply, here is how it works.

As the spacecraft approaches the planet, the planet's gravity accelerates the space vehicle. If the spacecraft's initial velocity is too low, or if it is heading too close to the planet, then the planet's gravitational pull will simply suck it down to the planet's surface. But if the initial speed is great enough, and if its orbit does not bring it too close to the planet, then the gravitational pull will just bend the spacecraft's trajectory around, and the accelerated spacecraft will shoot right past the planet and start to head away from the planet.

If there were no other gravitational sources around, then the gravity from the planet just passed would start to slow down the spacecraft as it moved away. If the planet were stationary, the slow-down effect would be equal to the initial acceleration, so there would be no net gain in speed. But the planets are themselves moving through space at high speeds, and this is what gives the "slingshot" effect. Provided the spacecraft is traveling through space in the same direction as the planet (say, counterclockwise around the sun), the spacecraft will emerge from the gravity assist maneuver moving faster than before. (In theory, the speed of the spacecraft can be increased by the speed of the planet.)

In fact, the same planet can be used two or more times in succession, with the speed of the spacecraft being increased on each flyby. This is what NASA does with Cassini, sending it on two Venus flybys before it swings by Earth and then out towards the edge of the solar system.

Cassini provides a good example of a multi-planet gravity assist trajectory. By choosing a trajectory that brings the spacecraft successively into one gravitational field after another, it is possible to fling the vehicle across space at ever increasing velocities, much like the skier who goes from one mogul to another, the hang glider who soars from one air current to the next, or the surfer who rides the ocean waves, riding first one crest and then, as it starts to lose force, moving onto another.

For this to work, the planets have to line up in just the right fashion, so NASA has to time its gravity assist missions with considerable care. Some missions may only be possible once every hundred years or more. (It would be 175 years before another Cassini mission could be mounted.)

The only fuel required to surf space using gravity assist is the modest amount required for rocket propulsion to change direction at suitable moments. Gravity provides the main thrust. And once you are in space, gravity is free. You just have to find it. Like the secret to success in (some) business(es), the trick is to be in the right place at the right time, doing the right thing. Everything else follows automatically with no further effort.

NASA has been using gravity assist on a regular basis since the 1973 Mariner 10 mission, which flew past Venus on the way to Mercury. Among other missions that have made successful use of the technique since then were Pioneer 11 to Saturn, Voyagers 1 and 2 to the outer planets, Galileo to Jupiter, and Ulysses to the Sun. The most complicated gravity assist trajectory was that of Voyager 2, which swung by Jupiter, Saturn, and Uranus on its way to Neptune in August 1989, at which point it used a swing by of Neptune itself to help propel it out of the solar system and into deep space.

Wild in the heavenly streets

Even wilder than the idea of flinging spacecraft across the heavens using planets as slingshots (gravity assist) is a technique known as chaotic control, which NASA first used in the early 1980s. Gravity assist flights depends on the fact that, in the vicinity of a planet, mathematicians can write down and solve equations that describe the path of a spacecraft that goes there, enabling them to calculate the trajectory of the craft with considerable accuracy. All they need to know are the masses of the planet and the spacecraft and the direction and speed of the spacecraft's initial approach toward the planet. The rest is straightforward. (In most cases you can get a good result using college calculus and Newtonian mechanics.)

Chaotic control, on the other hand, depends upon the fact that when a spacecraft is midway between two or more planets -- more accurately, in the region where the gravitational forces from those planets cancel each other out -- it can be virtually impossible to solve the mathematical equations that describe the spacecraft's trajectory.

A simple way to think of the situation is in terms of taking a canoe trip. At the start of the trip (the launch), you have to expend some energy to get the canoe out into the river. But once you are in the middle of the river and pointing in the right direction, you can sit back and let the current take you along, with little more than an occasional stroke of the oars to keep you in the middle of the river (a course-correction thruster burn).

But then you come to a place where several rivers come together and several more flow out. Suddenly you find yourself in the middle of the rapids: a seething cauldron of intersecting currents, where you are buffeted from all sides by conflicting pressures. There's no way that you can work out in advance how to negotiate these rapids to make your way to the outflowing river you want to take. Nevertheless, for a skilled canoeist, all it takes is one or two strokes of the oars at just the right moment, and the canoe heads in precisely the right direction. The expert canoeist is able to take advantage of the chaos to maneuver the boat in the desired direction. Practically all of the propulsive power comes from the water; all the canoeist does is provide the occasional nudge in order to take advantage of the chaos.

One thing to note is that, when your canoe is traveling down the middle of a fairly fast flowing river, it can require a lot of effort to change direction and go against the flow in some way. Though it might seem paradoxical at first, the chaos of the rapids can provide an opportunity to change direction using skill -- brainpower -- rather than raw physical power. This is precisely what NASA has learned to do to steer its spacecraft across the solar system, using mathematics rather than rocket fuel.

Although we can't see gravitational forces with our eyes, mathematicians can "see" them by writing down equations (and graphing their solutions on a computer screen if they wish). What they find, more or less, is that most of the time the gravitational forces exerted by the planets in the solar system are like a vast system of rivers. (A better picture would be a vast ocean, with many currents, but rivers provide a more familiar setting for our canoeist example.) But there are some places where the gravitational rivers from different sources (planets) intersect, and in these regions you can get gravitational rapids. The theoretical existence of such points was first discovered by the 18th Century French mathematician Joseph-Louis Lagrange, and are nowadays known as Lagrange points. By directing a spacecraft to a Lagrange point, NASA can -- by using a lot of mathematics but only a tiny amount of fuel -- change the craft's direction to send it toward the intended target. (Not all Lagrange points are like rapids. At some of them you get a region of calm. If you find yourself at such a point, it requires quite a lot of energy to escape. I'll say more about Lagrange points in a moment.)

Before we go any further, I should say what mathematicians (and space scientists) mean by the word "chaos". A more accurate term would be "unpredictable." Both rivers and gravitational forces obey the laws of physics, and as such are entirely deterministic, i.e. not random. However, in both examples, situations can arise in which small causes can give rise to major effects. In such a situation, if the consequences of two or more of those small initial causes interact, the result can be virtually impossible to predict. You get what mathematicians call "chaos".

The most oft-cited illustration of (mathematical) chaos is the "butterfly effect," where the flapping of a butterfly's wings in China can cause a hurricane in Florida two months later. In principle, this is indeed possible. In systems where different factors interact and effects can build up, such as the weather, a small initial action can indeed lead to a significant effect. In practice, however, so many factors influence the weather that it would be impossible to cause a hurricane intentionally, by, say, clapping your hands. When it comes to the weather, nature chooses which, if any, butterfly will have an effect.

In outer space, however, there are far fewer factors involved, and that opens up the possibility of being able to turn things around, and to make use of chaos to serve our own ends. In particular, a tiny nudge given to a spacecraft's trajectory at just the right moment (e.g., at a chaotic Lagrange point) could result in a major -- and planned -- change in where it ends up some weeks or even months later.

The irony is, chaotic control is made possible by what mathematicians used to regard as an insurmountable problem: their inability to solve the equations of motion for a system of three or more objects moving freely in space under gravitational forces -- the so-called "three body problem." For two bodies, the solution is easy: the two bodies move around each other in elliptical orbits -- or follow parabolic or hyperbolic paths in some special cases. But with three or more bodies, the relative motion can be highly erratic, with a tiny perturbation to the motion of one of the objects giving rise to wildly different behavior of all three.

In the case where one of the three bodies is a tiny spacecraft and the other two are large planets, the planets' gravitational forces will dominate that of the spacecraft, of course, and hence the movement of the spacecraft relative to the two planets can be very unstable. The instability is particularly high near the "neutral points," where the gravitational forces of the two planets cancel each other out. This enables space scientists to make use of the neutral points for navigational purposes.

In fact, things are a bit more complicated than that, since the relative rotation of the two planets produces a "centrifugal" force that also contributes to the position of any neutral points. This is the situation that Lagrange examined. He found that in a system where one planet orbits around a much larger one (such as the moon around the earth or the earth around the sun), there will be five points where all three forces cancel out (the two gravitational forces and the "centrifugal" force). These points are the Lagrange points, generally labeled L1, L2, L3, L4, and L5.

The L1, L2, and L3 points all lie on the straight line connecting the two planets, with L1 and L3 inside the smaller planet's orbit around the larger and L2 outside that orbit. L4 and L5 lie on the orbit, located symmetrically one to each side of the line connecting the others. Click here for diagrams and further details about Lagrange points, displayed on NASA's website.

[If you want to see some of the mathematics involved in locating Lagrange points, there are two excellent web sites you can consult, one at Stanford and the other at The Geometry Center. These sites are not for the mathematically faint-hearted.]

Because the gravitational and "centrifugal" forces all cancel out at a Lagrange point, you might expect that a small object such as a spacecraft that was brought to a halt at a Lagrange point would remain there indefinitely. But things are a bit more complicated than that.

Very little correction is required to maintain an object at either of the L4 and L5 Lagrange points. These points are stable: an object brought to rest there would, in theory, remain there forever unless subjected to a major new force. (In practice, the forces resulting from the movement of other bodies in the universe will mean that an occasional correction may be required.) As a result, the Earth-Moon L4 and L5 points have been suggested as possible locations for a future space station.

But the other three Lagrange points -- L1, L2, and L3 -- are unstable: a tiny perturbation is all it takes to send a small object located there off onto a journey that can be all but impossible to predict.

The difference between the stable and unstable Lagrange points is a bit like the difference between standing at the bottom of a (sharp pointed) crater and the top of a (sharp pointed) mountain. Both places are "stable," in that in your immediate vicinity you can walk around on the level. At the crater bottom, it can be very difficult to walk too far in any one direction, because gravity keeps dragging you back down. (Crater bottoms are like L4 and L5 Lagrange points.) But if you walk too far in any one direction from the mountain top, pretty soon you will find yourself tumbling downhill, away from the summit. (Mountain tops are like the L1, L2, and L3 Lagrange points.)

The mountain comparison is in fact quite a good one on two other counts. For one thing, in the case of a mountain summit, notice that a small difference in direction when you leave the summit can result in a huge difference in where you end up when you tumble down the mountain side. A few feet to the left or to the right of your intended direction and you can end up miles off course when you reach the valley bottom below. This means that a mountain top is a good place to change direction of travel, with little expenditure of energy. You just need to make sure your first few steps are in the right direction! This is the basis of chaotic control.

Another lesson we can learn from the mountain example comes from the observation that, if the summit is so sharp that you cannot stand there in comfort, then one way to achieve a good measure of stability there, with a minimal expenditure of energy, is to keep on circling around the summit, just a few feet below it. Since you are not descending, you remain close to the summit. But since you are not climbing, you are not expending much energy. This is exactly what NASA does when it wants to station a satellite at one of the unstable Lagrange points. It puts the satellite into a small orbit around the Lagrange point. Such orbits are called "halo orbits." (The mathematics behind halo orbits is quite a bit more complicated than for regular orbits around planets. If you really want to know, you should consult an expert.)

For instance, Vice President Gore recently proposed that a satellite be put into orbit at an Earth-Moon Lagrange point to beam back live photographs of the Earth for constant display on the Internet. Neither of the two stable Lagrange points could be used, because they are too far away from Earth. In fact the only viable possibility is L1, the Earth-Moon Lagrange point closest to Earth. To make Gore's suggestion a reality, NASA will have to put the satellite into a small halo orbit around L1.

To get back to chaotic control now, the idea is to find a way to use the instability of an L1, L2, or L3 Lagrange point change the direction of a spacecraft without expending much fuel. On its own, mathematics can't do that -- the very nature of an unstable Lagrange point is that the behavior of an object there is chaotic. But mathematics combined with some heavy duty computing can sometimes work. The first demonstration of the power of the math-computing alliance to achieve chaotic control was when NASA used it to coax a second, unplanned mission out of the International Sun-Earth Explorer 3 (ISEE-3) spacecraft, the third in a series of three missions to study the solar wind and the solar-terrestrial relationship at the boundaries of the Earth's magnetosphere. Launched on August 12, 1978, ISEE-3 was placed into orbit around the Sun-Earth L1 point, some 235 Earth radii from Earth.

After four years of excellent service measuring plasmas, energetic particles, waves, and fields, NASA decided to try to recycle ISEE-3 to take a close-up look at the comet Giacobini-Zinner, which was making its way back into the inner Solar System. The problem was, hardly any of the spacecraft's hydrazine fuel remained, so there was no possibility of any major rocket maneuvers to put it into a new trajectory. NASA scientists decided to see if they could make use of the gravitational instability in the region of, first the Sun-Earth L1 point and then the Earth-Moon L1 point to persuade ISEE-3 to go where they wanted it to. On June 10, 1982, they began a series of what turned out to be 15 small fuel burns to gradually nudge the satellite onto a trajectory toward the Earth-Moon L1 point.

Once they had the satellite in the Earth-Moon system, the NASA engineers then flew it past the Earth-Moon L1 point five times, giving it a tiny nudge on each lunar flyby, until they had put it onto a path that would lead it eventually to an encounter with Giacobini-Zinner. The fifth and final lunar flyby took place on December 22, 1983, when the satellite passed a mere 119 km above the Apollo 11 landing site. At that point, the spacecraft was renamed the International Cometary Explorer (ICE).

On June 5, 1985, ICE was maneuvered into Giacobini-Zinner's tail, some 26,550 km behind the comet itself, and its instruments began to analyze the tail's composition, eventually confirming the theory that comets are essentially just "dirty snowballs."

Unexpectedly, ICE survived its passage through the comet's tail, and in 1986 it was used to make distant observations of Halley's Comet. The spacecraft will return to the vicinity of the Earth in 2014, when it may be possible to capture it and bring it back to Earth. NASA has already donated it to the Smithsonian Institute in case it is indeed recovered.

Following NASA's initial, somewhat "trial and error" approach with ISEE-3, the mathematics of chaotic control began to be developed properly in 1990, starting with some work of Edward Ott, Celso Gregobi, and Jim Yorke of the University of Maryland. Their method, known after their initials as the OGY technique, involves the calculation of a sequence of "butterfly wing flaps" that will produce the desired effect -- something not possible with weather on Earth but which can be achieved when it comes to maneuvering a spacecraft in space.

If we were ever to establish a space station on the Moon, the OGY technique could be used to get materials from Earth orbit to lunar orbit and back again in a relatively cheap fashion. The "standard" way to do this is to use a rocket burn to put the spacecraft into an elliptical trajectory that just touches the Earth at one end and the Moon at the other, called a Hohmann ellipse. In addition to the initial rocket burn to move the spacecraft out of, say, Earth orbit and onto the Hohmann ellipse, a second rocket burn is required to slow it down at the other end and put it into lunar orbit.

In 1995 Erik Bollt of the United States Naval Academy and James Meiss of the University of Colorado discovered a different way that is much more fuel efficient. To send a cargo from Earth orbit to lunar orbit (the same approach works in the other direction), the idea is to give the spacecraft a hefty nudge that knocks it out of Earth orbit and puts it into the first of a chaotic series of wider Earth orbits that each pass through the Earth-Moon L1 point. After the craft has gone through the L1 point a number of times (48 according to one calculation), with the occasional tiny nudge from the rockets from time to time, it slips into a chaotic series of wide orbits around the Moon, still passing through the L1 point. After a number of lunar orbits (10 by the calculation that gives 48 Earth orbits), it ends up automatically in a stable, close orbit around the moon.

It takes much longer to get from Earth orbit to lunar orbit (or back again) using this technique -- about two years as opposed to three days the "standard" way. But it's much more fuel efficient: you can carry 80% more materials this way. That's not much use if you want to ship perishable food supplies to the crew of a lunar space colony, but fine for transporting building materials to the Moon and bringing mined minerals back to Earth.

NASA will use the latest version of chaotic control on the Genesis mission, scheduled for launch in January 2001 to carry out a two-and-a-half year "trawling" operation to collect charged particles in the solar wind and return them to Earth. Genesis will do its collection work while it orbits the Sun-Earth L1 point. But by the time it has finished, it will not have enough fuel for a direct return to Earth. Instead, it will first be sent onto a long detour to the L2 point, outside the Earth's heliocentric orbit. Because of the flow of the "gravitational rivers," from the Sun-Earth L2 point it can be brought back very economically to the Earth-Moon L1 point, where a few, cheap chaotic orbits of the Moon will eventually put it into a stable orbit of the Earth, from which its cargo capsule will be released to parachute down onto the Utah salt flats in August 2003.

What makes this possible is the existence of a "free ride" path from L1 to L2. (It's a bit like finding a freeway from New York to Los Angeles that you can drive without using any gasoline.) It took a large dose of advanced mathematics to discover this path, but the resulting cost savings achieved by making the L1-to-L2 detour are huge. NASA will run the Genesis mission on a total budget of $216 million, a paltry amount by space-exploration standards.

To the outsider, modern space exploration might appear to be largely a matter of engineering and rocket science. But a key factor is the clever use of mathematics to make nature's own gravitational forces do most of the work for us. The raw muscular power that sent the Apollo missions to the moon has given way to delicate movements more reminiscent of jujitsu. In the early days, rocket science was a mixture of propellant chemistry, engineering, and a lot of luck. These days, it's a mixture of propellant chemistry, engineering, and a lot of mathematics.

- Keith Devlin

NOTE: Keith Devlin discussed the subject of this column in a nationally broadcast radio interview with Scott Simon on NPR's Weekend Edition on Saturday January 30. Click here to listen to the interview. (This requires that your computer has RealAudio installed.)
Devlin's Angle is updated at the beginning of each month.
Keith Devlin ( is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book The Language of Mathematics: Making the Invisible Visible, has just been published by W. H. Freeman.