FOR SALE: Desirable property with easy access to services in a popular, stable neighbourhood. Spectacular ocean and mountain views. Parking facilities available. Radio reception excellent (except during occasional solar flares); 0 beds. Would suit professional communicator. £400M.

If you were an estate agent selling hot properties in the Solar System, what features would catch your eye? The most obvious ones are planets, moons, asteroids and perhaps comets—the elements of the cosmic landscape that correspond to material objects. But the prime real estate in the solar system is far more nebulous: empty space. What makes agents drool when it comes to the cosmic property market—much as in the terrestrial one—is location. Not location in the conventional landscape of matter, but in the ever-shifting landscape of gravitation and time.

The desirability of certain routes or orbits has been known for some time. Right now hundreds of telecommunications satellites encircle the Earth in one specific orbit, roughly 36 000 kilometres above the equator. In this orbit—and only this one—the speed of the satellite matches the rotation of the Earth precisely, so the satellite remains fixed relative to the ground.

What is new is the rapidly increasing catalogue of ever more exotic orbits, fuelled largely by developments in the mathematical techniques needed to distinguish them from the apparently identical empty space nearby. The mere existence of many a cosmic des res was unsuspected even a few years ago, and without some state-of-the-art mathematical concepts and techniques their desirability would never have been recognised.

Raising the dead

The first time a truly exotic orbit showed up was around a decade ago, when a bunch of NASA engineers got a bright idea about recycling satellite ISEE-3, the third International Sun-Earth Explorer. The satellite was languishing in space, its stock of hydrazine fuel—used to adjust its angle and speed—so low that any serious manoeuvres were out of the question. This was a great pity, because comet Giacobini-Zinner was making its way back into the inner Solar System and ISEE-3 happened to have several instruments on board that would give useful information about the comet —if only the satellite were in the right place. The engineers needed to move an almost dead satellite millions of kilometres through space using hardly any fuel.

They did this by exploiting the "butterfly effect" from chaos theory, in which a tiny perturbation—the flap of a butterfly's wings, say—could cause a mighty hurricane. This effect is usually seen as a problem, but in fact it's an opportunity. If the butterfly can be persuaded to make the right flap, a small effort can give you the large response you desire.

In practice you can't do that for weather, because there are billions of butterflies and other competing influences. But for controlling a satellite, you can use the effect to advantage because now there's no unwanted competition. Either the engineers expel a small amount of hydrazine fuel from ISEE-3, or they don't. The only questions are: what difference does it make, and where should you do it? That's where chaos theory can help.

It's all down to the strange way gravity behaves when three or more objects are involved. Two objects always move around each other in ellipses (or parabolas or hyperbolas, if the speed is high enough), but three bodies can go all over the place—and tiny disturbances can have huge effects. The Earth, the Moon and ISEE-3 are three bodies, so chaos theory tells us that if the satellite is given the right nudge when its orbit is unusually sensitive to small changes, the resulting effect could be big. And the right moment to do this is when the satellite is close to the "neutral point" between Moon and Earth, where the Earth's gravity exactly cancels out that of the Moon.

So NASA's engineers flew the satellite past that point five times, emitting carefully calculated but very tiny squirts of precious hydrazine on each flyby, and persuaded the nearly defunct ISEE-3 to become the vibrant and exciting ICE, the International Cometary Explorer. The spacecraft successfully encountered Giacobini-Zinner, made the first ever direct measurements of a comet's tail, and confirmed the theory that comets are simply dirty snowballs. Not bad, considering.

This technique is now known as chaotic control, and it was put on a firm mathematical footing in 1990 by Edward Ott, Celso Grebogi and Jim Yorke, all at the University of Maryland at College Park. They devised a systematic technique—the OGY method—to make the butterfly work in your favour, by repeatedly tweaking a chaotic system in the right manner. The latest idea to emerge from OGY is a cheap way to send payloads from low Earth orbit to the Moon. The classical solution is called a Hohmann ellipse. At one end the ellipse passes close to the Earth, at the other end it grazes the Moon. The theory is that if you start with a craft in a circular low Earth orbit and use a burst of fuel to push it onto the path of a Hohmann ellipse, then all you need to do at the other end is to use another burst to slow it down and switch it into low lunar orbit.

A Hohmann ellipse may be neat, but unfortunately it's not fuel-efficient. So in 1996 Ott and his co-workers worked out a different orbit using much less fuel, which makes repeated chaotic swings near the Earth-Moon neutral point. One snag is that it would take about 10 000 years to follow that orbit. But with some careful squirts of fuel at crucial places the duration can be shortened to two years. The resulting trajectory involves 48 irregular circuits of the Earth followed by 10 circuits of the Moon, after which the payload settles neatly into parking orbit. It's not a hot piece of real estate yet, because we don't have any lunar colonies—but if we ever do, the new orbit would convey 83 per cent more goods to the colony than a Hohmann ellipse, for the same fuel. As long as the goods aren't perishable.

An important extension of the idea of a gravitational neutral point is the set of Lagrange points discovered by the great French 18th-century mathematician Joseph-Louis Lagrange. In a system with two orbiting objects where one is much more massive than the other—the Sun and Earth, say—the light body will go round the heavy one in an ellipse. If you ignore effects of speed, there will be a single neutral point between the bodies where their gravitational forces cancel each other out. But there's also the accelerating "centrifugal" force associated with moving in a circle. Lagrange discovered five neutral points where all three forces—gravity from the heavy body, gravity from the light one, and "centrifugal" force—cancel. They are called Lagrange points, L1, L2, L3, L4 and L5 (see Diagram).

In principle a satellite or probe placed at a Lagrange point will stay there forever. In practice, though, tiny disturbances may cause it to move. If the heavy body is sufficiently massive compared with the light one, then L4 and L5 are stable positions, but L1, L2 and L3 are always unstable—a probe placed there will gradually wander away. However, the whole point of control engineering is to stabilise the unstable, and it turns out that regular, but relatively small, expenditures of fuel can keep a probe close to an unstable Lagrange point for decades.

Cool properties

Several recent and projected space missions are making good use of Lagrange points. For instance the billion-dollar Solar and Heliospheric Observatory, a joint project between NASA and the European Space Agency, is hovering near the L1 point, about 1.6 million kilometres from Earth in the direction of the Sun—ideal for observing all sorts of solar activity while remaining in touch with the ground.

The successor to the Hubble Space Telescope will also lurk at a Sun-Earth Lagrange point, this time L2. The Next Generation Space Telescope (NGST) will orbit the Sun about 1.6 million kilometres further out than Earth and will remain in much the same relative position throughout its working life. Hubble, which was launched in 1990, orbits the Earth, and its structure includes a heavy tube which keeps out unwanted light that would interfere with the images it produces. It's a lot darker out near L2, so that cumbersome tube can be dispensed with for NGST, saving launch fuel. In addition, L2 is a lot colder than low Earth orbit, and that makes infra-red telescopy much more effective. L2, then, is not so much a hot property as a real cool one.

The prize for the weirdest orbit of all, though, has to go to the planned Genesis mission, which will collect samples of the solar wind—a vast stream of charged particles emitted by the Sun—and return them to Earth. NASA's Jet Propulsion Laboratory plans to launch the probe in January 2001. It must hover in space for long enough to collect samples of the solar wind and bring them back to low Earth orbit. They will then be returned to the ground in a capsule with a parachute.

This is a demanding task requiring a complex orbit, but the budget of $216 million is relatively low, so the more obvious, fuel-hungry trajectories are ruled out. The chosen trajectory is based on the work of a team including mathematician Jerrold Marsden and computer scientist Wang-Sang Koon at Caltech, and Martin Lo at JPL. Genesis will wander near the Sun-Earth L1 point, collecting its samples and making other observations of the solar wind, and then return via the L2 point before sinking back into parking orbit by way of a few close encounters with the Moon (see Diagram).

But why must the spacecraft divert to L2 rather than returning directly to Earth? It turns out that sending Genesis directly home from L1, inside Earth's orbit around the Sun, takes more fuel than sending it back from L2, which is outside. But to take advantage of this, what you really need is a free detour from L1 to L2.

That's just what Marsden and colleagues found. They discovered a "heteroclinic connection"—a constant-energy path linking L1 to L2. It works like this. Imagine two neighbouring hills, of equal height. The equilibrium points are the hill tops. Place a ball on top of one hill, and roll it down into the valley in just the right manner for it to climb to the top of the second hill. That's an example of a heteroclinic connection. In the absence of friction, the ball moves from one hill to the other using only gravity, which costs nothing since it is present anyway. Along the way, gravitational potential energy is converted into kinetic energy as the ball accelerates downhill, and is then converted back again as it climbs the far hill—but the total energy never changes.

To find exactly what path the spacecraft needed to take to exploit this heteroclinic connection between L1 and L2, Marsden and his colleagues had to look for something called an unstable manifold. Remember the ball balancing on the top of the hill. Give it a push and all directions are down—that's an unstable manifold. But now imagine a ball balancing on a mountain pass—some directions are up, or stable, and some down, or unstable. In an ordinary landscape like this, it's easy to spot the unstable direction. But finding it in space is much harder. For Genesis, the trick has to be played in an 18-dimensional landscape—six coordinates each for Genesis, Earth and Sun. The unstable manifold now is not a single curve, but a highly complex multidimensional surface. So until recently, calculating its shape and position was virtually impossible, taking far too much computer time.

However, in 1996 Michael Dellnitz of Hamburg University developed efficient mathematical techniques for performing such calculations (see "Get down"), and this was the breakthrough needed to find the unstable manifold of the Genesis orbit. Since it is a multidimensional surface, many different trajectories can use it to leave L1: the one proposed for the Genesis mission is chosen to fly past L2 in a manner that makes the final return to Earth especially straightforward.

Sounds complicated? Not really. With today's mathematical techniques it isn't much harder to find such an orbit than it is for a prospective purchaser to spot a seaside cottage with a really nice view. You just need to know what's important and how to find it. When you know how to locate those crucial but invisible features of the celestial landscape, you can be confident of picking up the best bargains on the sky street.

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