The pulling power of chaos
Our Science Essay ponders the riddle of the wandering stars. Starting with Poincaré, complex maths i
What is the most efficient way to get a space probe to its target? When Apollo 11 went to the moon in 1969 it followed a conventional Hohmann transfer orbit. Imagine an egg-shaped outline, with the earth at the bottom. As the spacecraft comes up the left-hand side, it burns fuel to accelerate, and swings into orbit around the moon.
This was the quickest route – aside from the impractical one of flying straight out by burning fuel the whole time – and, in a manned mission, speed was of the essence. However, we now know that when efficient use of fuel is the main objective, and time is unimportant, less direct routes can be much better. When Nasa sent the Cassini probe to Saturn, it first went inwards in the solar system, undergoing two close encounters with Venus. Then it swung back past the earth and on to Jupiter before making a sharp turn to meet Saturn.
Trajectories such as this exploit the slingshot effect, in which the spacecraft steals energy from a planet. The tiny spacecraft speeds up considerably, pulled towards the planet by gravity; the massive planet slows down very slightly, but not enough to notice. Yet there is another, subtler effect of orbital dynamics which is also being used to get spacecraft to their targets using as little fuel as possible: chaos.
The technique was first used in 1991. A Japanese space probe, Hiten, had been surveying the moon. Having completed its mission and returned to orbit the earth, it had pretty much run out of fuel. Edward Belbruno, an orbital analyst at Nasa’s Jet Propulsion Laboratory in Los Angeles, came up with an idea that sounded impossible. He wanted to extend its useful life and enhance its scientific value by sending it back to the moon. Then it would visit the moon’s Trojan points – the points in space 60 degrees ahead of and behind the moon in its orbit where gravity and centrifugal forces cancel each other out. There it could search for cosmic dust that might have become trapped.
It sounded crazy, but Belbruno knew a way to do it. Mathematicians and physicists had realised that the motion of bodies under gravity can be chaotic – highly irregular, despite obeying entirely deterministic laws. Chaotic orbits are sensitive to very small disturbances. Normally, this feature is seen as an obstacle to prediction, but Belbruno realised that it could be used to advantage. Very small changes in position or speed, which use very little fuel,
can cause large changes to the trajectory. That makes it easy to redirect the spacecraft in a fuel-efficient, though possibly slow, manner.
One place where chaotic orbits can arise is somewhere called the “L1 Lagrange point” between the earth and the moon, where the net gravitational force is zero (essentially, objects are “suspended” between the two bodies because of the forces generated by each). Belbruno designed a new orbit that took Hiten close to the L1 point, where a short, carefully calculated burst of its rockets would loop it out to where he wanted it to go. He faxed his proposals, unsolicited, to the Japanese team; they loved the idea. When the probe arrived at L1, it found there was no more dust than you’d expect; after a few years orbiting the moon, Hiten was crashed into its surface in 1993. Still, it had ushered in a new era of space travel. A similar trick was used for Nasa’s Genesis mission to bring back samples of the solar wind.
The first Oscar
Our fascination with the planets goes back to prehistoric times, when human eyes watched the star-spangled splendour of the night sky and human minds were awed by the cosmic spectacle. Countless stars moved across the sky, pinpricks of light on a gigantic, rotating velvet-black bowl.
A few of those pinpricks of light, however, did not obey the rules. They went walkabout. The Greeks called them planetes – wanderers; we call them planets. Their paths are complicated and sometimes loop back on themselves. It is not surprising that the ancients attributed their movements to the caprices of supernatural beings.
Ptolemy, a Roman who lived in Egypt around AD120, began the lengthy process of taming the solar system, proposing that we live in an earth-centred universe in which everything revolves around humanity in complex combinations of circles supported by giant crystal spheres. Around 1300, the Persian Islamic philosopher Najm al-Katibi proposed a heliocentric (sun-centred) theory, but changed his mind. The big breakthrough came in 1543 when Nicolaus Copernicus published On the Revolutions of the Celestial Spheres. He was clearly influenced by al-Katibi, but he went further, setting out an explicitly heliocentric system. Among its implications was the novel thought that human beings were not at the centre of things. To the Christian Church, this suggestion was contrary to doctrine, and explicit heliocentrism was heresy.
The riddle of the wandering stars was finally answered in 1609 by Johannes Kepler, an assistant to the astronomer Tycho Brahe. When his employer died unexpectedly, Kepler took over as court mathematician to Emperor Rudolph II. His main role was casting imperial horoscopes, but he also had time to analyse the orbit of Mars. For years, he tried without success to fit the planet’s orbit to an egg-shaped curve, sharper at one end than the other. In 1605 he decided to try an ellipse, equally rounded at both ends. He discovered that this shape fitted the observations, and declared: “Ah, what a foolish bird I have been!”
In 1609, Kepler published A New Astronomy, stating two basic laws of planetary motion. First law: all planets move in ellipses with the sun as a focus. Second law: a planet moves along its orbit in such a manner that it sweeps out equal areas in equal times. In 1619 he returned to planetary orbits in The Harmony of the World. The book contained many curious ideas – for example, that planets emit musical sounds as they roll round the sun. But it also contained his third law: the squares of the time taken for planets to orbit are proportional to the cubes of their distances from the sun.
This work led to one of the greatest scientific discoveries of all time. In his Mathematical Principles of Natural Philosophy of 1687, Isaac Newton proved that Kepler’s three laws are equivalent to a single universal law of gravitation. Two bodies attract each other with a force that is proportional to their mass and inversely proportional to the square of the distance between them. Newton’s law of gravity had a huge advantage over Kepler’s ellipses: it applies to any system of bodies, however many there might be. The price to be paid is the way the law prescribes the orbits – not as geometric shapes, but as solutions of a mathematical equation. The problem is to solve it.
Newton achieved that for two bodies – a planet plus the sun – and the answer is what Kepler had already discovered: the bodies move around their common centre of gravity in elliptical orbits. But some questions involve more than two bodies. If you want to predict the motion of the moon with high precision, you have to include both the sun and the earth in your equations. So, fresh from Newton’s success with the motion of two bodies under gravity, mathematicians and physicists moved on to three bodies. Their initial optimism rapidly dissipated; the three-body problem turned out to be very different from the two-body problem. In fact, it defied solution.
Only in the late 19th century did its true complexity become apparent, however, when Henri Poincaré tried to win a scientific prize. The 60th birthday of Oscar II, king of Norway and Sweden, happened in 1889. The Norwegian mathematician Gösta Mittag-Leffler persuaded the king to mark the occasion by announcing a prize for calculating the motion of any number of bodies under gravity and finding out whether the solar system is stable.
Poincaré decided to start with the simplest case: two bodies (say the sun and a planet) moving in perfect circles, with the third body being a dust particle of negligible mass. Even that version proved too ambitious and he failed to solve it, but he made so much progress that he was awarded the prize anyway.
In particular, Poincaré proved that sometimes the orbit of the dust particle became extraordinarily messy. He deduced this from some highly original ideas that made it possible to infer features of the solutions without actually solving the equations, saying: “One is struck by the complexity of this figure that I am not even attempting to draw.” We now recognise Poincaré’s discovery as a sign that the dynamics of such a system are chaotic. The equations are not random, but their solutions can be very irregular, sharing features with properly random processes. This idea is colloquially known as chaos theory, and it all goes back to Poincaré and his Oscar award.
Well, that’s the story that historians of mathematics used to tell. Around 1990, however, June Barrow-Green found a copy of Poincaré’s prize-winning memoir in the depths of the Mittag-Leffler Institute in Sweden. She realised that when he submitted his work he had overlooked the chaotic solutions. He spotted the error before the memoir was published, and paid to have the original version destroyed and a corrected version printed. His initial oversight lay undiscovered for a century.
Building on Poincaré’s discovery, we now know that the three-body problem does not have simple solutions. Even so, vast progress has been made on the many-body problem in special cases; for example, when all of the bodies have the same mass. This is seldom a realistic assumption in celestial mechanics, but it is sensible for some models of elementary particles, such as electrons. In 1993, Cristopher Moore at the Sante Fe Institute found a solution to the three-body problem in which the bodies play follow-my-leader along the same orbit. Even more surprising is the shape of the orbit – a figure of eight.
Stranger than imagination
In 2000, the Spanish mathematician Carles Simó used a computer to show that this configuration is stable: it persists after small disturbances. Indeed, it remains stable even when the three masses are slightly different, so, somewhere in the universe, there might be three stars of almost identical mass, chasing each other along a figure-of-eight path.
The same year, Douglas Heggie of Edinburgh University estimated that the number of such triple stars lies somewhere between one per galaxy and one per universe. The figure-of-eight orbit is a planetary dance in which the bodies return to the same positions but swap their identities, each occupying the location that the body in front of it has vacated. This kind of orbit is called a choreography. Using a computer, Simó has found a huge number of choreographies, which can involve a large number of bodies.
The solar system is, was, and will be, far stranger than we imagine. Consider the comet Oterma. A century ago, Oterma’s orbit was well outside that of Jupiter. After a close encounter with the giant planet, its orbit shifted inside that of Jupiter. After another close encounter,
it switched back to outside again. We can confidently predict that Oterma will continue to switch orbits in this way every few decades, not because it breaks Newton’s law, but because it obeys it. Oterma’s gyrations are a far cry from Kepler’s tidy ellipses. The explanation is straight out of science fiction. In Pandora’s Star, Peter Hamilton portrays a future where people travel to planets encircling distant stars by train, running the railway lines through a wormhole, a short cut through space-time.
In his Lensman series, Edward Elmer “Doc” Smith came up with the hyperspatial tube, which malevolent aliens used to invade human worlds from the fourth dimension. Although we don’t have wormholes or aliens from the fourth dimension, the planets and moons of the solar system are tied together by a network of multidimensional mathematical tubes that provide energy-efficient routes from one world to another. If we could visualise the ever-changing gravitational landscape that controls the planets, we would see these tubes, swirling along with the planets as they orbit the sun.
Oterma’s orbit lies inside two tubes, which meet near Jupiter at a Lagrange point. One tube lies inside Jupiter’s orbit, the other outside.
At the Lagrange point the comet can switch tubes, or not, depending on chaotic effects of Jovian and solar gravity; once inside a tube, however, Oterma is stuck there until the tube returns to the junction. Like a train that has to stay on the rails, but can change its route to another set of rails if someone switches the points, Oterma has some freedom to change its itinerary, but not a lot.
As such, the way to plan an efficient mission profile is to work out which tubes are relevant to your choice of destination. Then you route your spacecraft along the inside of the first inbound tube, and when it gets to the associated Lagrange point you fire a quick burst on the motors to redirect it along the most suitable outbound tube. That tube naturally flows into the corresponding inbound tube of the next switching point . . . and so it goes on.
Plans for future tubular space missions are already being drawn up. In 2000, Wang Sang Koon, Martin Lo, Jerrold Marsden and Shane Ross used the tube technique to plot what they described as a “Petit Grand Tour” – an energy-efficient route – around the moons of Jupiter, ending in orbit around Europa. In 2005, Michael Dellnitz, Oliver Junge, Marcus Post and Bianca Thiere used tubes to plan an energy-efficient mission from the earth to Venus.
Their route would use one-third of the fuel required by the European Space Agency’s Venus Express mission, which has observed Venus since 2006.
Past, present, future The influence of tubes may go further. Dellnitz has discovered evidence of a natural system of tubes connecting Jupiter to each of the inner planets. This remarkable structure, known as the Interplanetary Superhighway, hints that Jupiter, long known to be the dominant planet of the solar system, also plays the role of a celestial Grand Central Station. In the past, its tubes may well have organised the formation of the entire solar system, determining the spacings of the inner planets.
So, is the solar system stable? The answer is a definite “maybe”. Two research groups, run by Jack Wisdom of the Massachusetts Institute of Technology and Jacques Laskar of the Observatoire de Paris, have pioneered highly accurate computational methods to understand the probable future of the solar system. Wisdom’s group has found that Pluto behaves chaotically over timescales of several hundred million years.
In 1999, Norman Murray of the Canadian Institute for Theoretical Astrophysics and Matthew Holman of the Smithsonian Astrophysical Observatory discovered that the orbit of Uranus can also change chaotically, so that it occasionally gets close to Saturn, with the possibility that Uranus would then be ejected from the solar system. However, it will probably take about one quintillion years for this to happen. (The sun will blow up into a red giant much sooner, about five billion years from now. The earth will move outwards and might just escape being engulfed, even though tidal interactions will probably pull it into the sun. In any case, our planet’s oceans will boil away long before that. And, anyway, the typical lifetime of a species is no more than five million years.)
It’s not just the future that is chaotic; the same methods can be used to investigate the solar system’s past. In 1993, Renu Malhotra of the University of Arizona realised that the early solar system must have been far more dynamic than had been assumed. As the planets were condensing from the primal gas cloud surrounding the sun, there came a time when Jupiter, Saturn, Uranus and Neptune were nearly complete. Among them circulated huge numbers of rocky and icy “planetesimals”, small bodies about ten kilometres across. Many of these were ejected into the wider solar system, reducing the energy of the four giant planets. Neptune migrated outwards. So did Uranus and Saturn. Jupiter, the big loser in the energy stakes, moved inwards.
So, our solar system’s apparently stable plan arose through an intricate dance of the giants, in which they threw the smallest bodies at each other in a riot of chaos.
Is the solar system stable? Probably not, but don’t worry: we won’t be around to find out.
Ian Stewart is emeritus professor of mathematics at the University of Warwick. His latest book is “17 Equations That Changed the World” (Profile, £15.99)