A chaotic collaboration
15 July 2003
Mathematicians at Bristol's Department of Applied Mathematics teamed up with chemists at Utah State University in the USA to solve one of the outstanding problems of planetary science.
In the last couple of years many small moons have been found orbiting the giant planets in our Solar System. Rapid improvements in detection technology have resulted in ground-based searches that can cover a patch of sky roughly the size of the moon in a single exposure. In addition, improvements in computing mean that the data can be analysed in real time and faint, slow-moving objects automatically detected. As a result, Jupiter is now known to have 60 moons, more than 20 of which were found this year, Saturn has 30, Uranus 21, Neptune 11, Mars two, and the Earth just one. But these numbers are changing almost daily.
The large ‘regular’ moons like Jupiter’s Galilean moons of Io, Europa, Ganymede and Callisto are generally believed to have been formed at the same time as the planet they orbit. It was Galileo’s discovery of these in 1610 that helped him determine that the Earth was not the centre of the Universe. They have a roughly circular ‘prograde’ orbit that is in the same direction as the planet itself is rotating. The small, ‘irregular’ moons, on the other hand, are usually just a few miles across and have an orbit that is highly eccentric (cigar shaped), orbiting the planet at great distances. For example, those around Jupiter travel an average of 15 million miles and take about two years to do so. These moons are believed to have been asteroids that originally circled the Sun but were subsequently ‘captured’ by the planet they now orbit, early in the history of the Solar System. In the large majority of cases irregular moons have a ‘retrograde’ orbit, that is, in the opposite direction to that of the planet.
With the discovery of more and more of these moons, understanding the process by which they were captured remained one of the most outstanding problems in planetary science. What was the mechanism that allowed them to switch from one orbit to another? And why do so many of the captured moons have retrograde orbits? The answers to these questions eventually came from a rather unlikely source – theoretical chemists working with mathematicians.
Jupiter is now known to have 60 moons, more than 20 of which were found this year
In fact it is not quite so unusual as it might first appear for a group of chemists and mathematicians to be working on a project in astronomy. Initially the atom was viewed as a miniature Solar System and the theory of quantum mechanics grew out of an attempt to apply methods of celestial mechanics to atoms and molecules.
Professor Stephen Wiggins and Dr Andrew Burbanks, mathematicians at Bristol University, along with theoretical chemists Professor David Farrelly and his doctoral student Sergey Astakhov at Utah State University in the USA, were using chaos theory to try to understand the mechanics of chemical reactions so that they could design the outcome of a reaction. When the team heard about the discovery of the new moons and the problems associated with understanding how they were captured, they realised that the ‘chaotic’ approach they were using in chemistry might provide the solution. After all, breaking a molecule apart may be the opposite of capture but it is basically the same mathematics. What they hoped was that if they could solve the moons problem it might give them some insight into their chemical problems, since the equations governing motion are well known, while the inter and intra-molecular forces in chemistry are much more complicated and far less well understood.
They used a model called the ‘Restricted Three-Body Problem’ which makes the approximation that there are two large bodies (e.g. the Sun and Jupiter) moving on a circular orbit, with a third body (the moon) which is influenced by the two larger bodies, but they are not influenced by it. The results of this modelling showed the team that it was chaos that was the underlying mechanism that allowed the capture process to take place.
At certain points during its orbit around the sun, the moon passes through one of two open ‘bottlenecks’ into a central energy ‘bubble’ around a planet. Here it becomes caught up in a layer of chaos – a gravitational complexity – resulting from the gravitational effects of both the sun and the planet pulling on the moon. This chaotic layer slows it down sufficiently to prevent it from quickly passing out of the ‘pull’ of the larger planet. Eventually, if the moon is caught in the chaotic layer for long enough, friction caused by the planet’s extended atmosphere – believed to have existed in the early stages of formation of the Solar System – slowed it down further so that it became trapped in a permanent orbit around the planet.
The joint research team also explained the prevalence of retrograde moons by showing that the moons initially captured into prograde orbits have a tendency to approach the region very close to the planet. There they stand a greater chance of being destroyed by collisions with other moons, or the planet itself. Retrograde moons do not tend to get as close and so are more likely to survive, thereby explaining the far larger number of moons with retrograde orbits, especially around Jupiter.
Using the mathematical equations they developed to explain the capture mechanism, the Bristol and Utah research groups present an explanation which not only agrees well with the observed locations of the known irregular moons, but also predicts new regions where moons could be located. The ability to predict where new moons might be found should make life much easier for astronomers who, despite the advanced technology, still face the daunting task of searching huge regions of space for them. Astronomers believe that understanding the nature of these moons can reveal important clues about the early history of the planets. Such insights into understanding our own Solar System will help us understand how other solar systems came into being, and whether they might be favourable to life.
Insights into understanding our Solar System help us understand how other solar systems came into being
Since the approach described above can be applied to the problem of the interaction of atoms and molecules, the team is gearing up to make a big push in that direction this summer.