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Bruns' paper contained an error that PoincarĂ© later corrected (Barrow-Green (1997), p. 164). Many generalizations of Bruns' and Poincare's results have appeared, and the existence of integrals for the n-body problem is still a topic of mathematical research (see, for example, Julliard-Tosel (2000)). At the beginning of the twentieth century, F. R. Moulton wrote: 'The practical importance of the theorems of Bruns and Poincare have often been overrated by those who have forgotten the conditions under which they have been proved to be true' (Moulton (1970), Article 147).

different way. In particular, it does not follow that Weierstrass' question asking for a solution in terms of a convergent power series cannot be answered. In fact, it was shown by Paul Painleve in 1895 that such a solution was possible. Painleve was investigating singularities in differential equations, and established that the only possible singularities in the three-body problem were due to collisions between the masses. Given some initial conditions that do not lead to a collision in the future, a convergent series solution must exist.55

The existence of a solution is one thing, finding it is another. In the final volume of his Celestial Mechanics (1896) Tisserand wrote:

The rigorous solution of the three-body problem is no further advanced today than during the time of Lagrange, and one could say that it is manifestly impossible.

This was overly pessimistic. A complete theoretical solution to the three-body problem was developed during the first decade of the twentieth century by Karl Sundmann from the Helsinki observatory, and published in 1912. He obtained a series solution in powers of t1/3 that converged provided the total angular momentum of the system was nonzero. Sundmann's method did not work in the case of a triple collision, but he showed that this could only happen if the angular momentum was zero. However, when only two of the masses collide -a binary collision - he was able analytically to extend his solution beyond the singularity, effectively allowing the bodies to bounce off one another. This process is called 'regularization'. An improved and simplified theory was developed by the Italian mathematician Tullio Levi-Civita in 1920.

Given the nature of the problem Sundmann was solving - one that had defeated the greatest mathematicians and astronomers since its formulation by Newton - his method was remarkably simple, depending only on well-established results in the theory of differential equations. It may appear something of a paradox that Sundmann's solution is not particularly well known. After all, he achieved what Poincare had not: he actually answered one of the

55 Painleve could not extend his result to the case of more than three bodies, and conjectured that singularities not due to collisions (he called them 'pseudocollisions') do exist in the n-body problem when n > 3. In 1908, the Swedish mathematician and astronomer Edvard Hugo von

Zeipel showed that a pseudocollision corresponds to a situation in which one of the masses disappears off to infinity in finite time (McGehee (1986)). The first example of a noncollision singularity was constructed as part of his Ph.D. thesis by Zhihong (Jeff) Xia in 1987 (Xia (1992)) for the case n = 5, and his solution can be generalized to any n > 4. The ideas underlying Xia's construction are described in Saari and Xia (1995). The existence of pseudocollisions when n = 4 is still an open question (Diacu and Holmes (1996)). Quoted from Barrow-Green (1997), p. 190.

For a brief sketch of Sundmann's ideas, see, for example, Saari (1990), Hall and Josic (2000). Sundmann's method did not apply to the case of more than three bodies, and it took almost 70 years before a solution to the general case was found by a Chinese research student at the University of Cincinnati, Quidong Wang (Wang (1991)).

questions set for Oscar II's prize. But the reason is simple; the series in the solution are so slowly convergent that they provide no qualitative information about the behaviour of the system and, from a practical point of view, the solution is totally useless. It has been estimated that, in order to achieve the accuracy required to compare with observations, one would need to take 108000000 terms in Sundmann's series!59

### Chaos in the Solar System

Poincare saw that the equations of celestial mechanics allow for some very strange types of solutions. But do any celestial bodies actually exhibit such behaviour? The answer to this question is: Yes, though our understanding of the extent to which Solar System dynamics is chaotic is by no means complete.60 What is clear is that the phenomenon of chaos is tied closely to the dynamics of resonances.

The most commonly cited example of chaos in the Solar System is the motion of Hyperion, a small and irregularly shaped (about 380 x 290 x 230 km) satellite of Saturn. Little was known about Hyperion until the Voyager 2 spacecraft passed close by it in 1981, but it turns out that its spin rate and orientation change markedly during the course of a few orbital periods (each about 21 days). Given sufficient time, one would expect any aspherical satellite to present the same face to its parent planet due to the action of tidal forces, just as in the case of our Moon (see p. 319). Most major satellites in the Solar System display such a 1: 1 spin-orbit resonance, though there are exceptions. In the case of Pluto and Charon, where the masses of the two bodies are much closer than in other planet-satellite systems, both keep the same face toward the other. They are said to be 'totally tidally despun'.

The process by which synchronous rotation is reached is very slow and, in the case of Hyperion - where the moon is far from its host planet - the tidal forces are very weak and the appropriate timescale is the age of the Solar System. A phase plane analysis of the spin-orbit resonance problem shows that the synchronous state is an island of stability surrounded by a chaotic region, the size of which depends on numerous factors, including the eccentricity of

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