The Three-Body Problem

The Three-Body Problem , Mauri Valtonen and Hannu Karttunen , Cambridge U. Press, New York, 2006. $85.00 (345 pp.). ISBN 978-0-521-85224-1

Mauri Valtonen and Hannu Karttunen have chosen a fine title for their unusual book, The Three-Body Problem , which covers a wide range of both theory and applications of an issue central to basic physics. In the late 1600s, Isaac Newton tested gravity among three massive bodies by studying the Moon’s motion in the combined field of Earth and the Sun. But it was only in 1878 that astronomer George W. Hill found a method yielding results that matched the ancient Gréek observations.

Three-body problems are also fundamental in atomic and high-energy physics. The negative hydrogen ion, which consists of one proton and two electrons, determines the Sun’s surface temperature to be 6000 K; we here on Earth can count on an average temperature of about 300 K. The proton consists of three quarks, and yet its magnetic moment is still not adequately explained. Overall, three-body problems are important but difficult to understand.

Valtonen and Karttunen work at the Väisälä Institute for Spacscodee Physics and Astronomy at the University of Turku in Finland. Their introduction offers a long and surprising list of examples from modern astrophysics, all based on classical mechanics. Except in the book’s final chapter, the three bodies are treated as point particles, where the mass is concentrated in a mathematical point without such further properties as angular momentum. The systems without a stable configuration are emphasized—for example, an asteroid whose motion is determined by the Sun and Jupiter. In the first five chapters, the authors provide a classical mechanics course that goes well beyond the ordinary graduate textbooks.

The Kepler motion, including the hyperbolic trajectories, is the necessary starting point. But three-body systems have four degrees of freedom that cannot easily be separated by a canonical transformation, so one cannot avoid some careful formal development. Valtonen and Karttunen have carefully written down every step in the arithmetic and included helpful diagrams. Each chapter ends with problems, some involving numbers from astronomy.

The authors’ main achievement is their coverage in the last five chapters of the many different applications in astrophysics in which one of the three bodies is much lighter than the other two. Those applications are based on an unperturbed two-body system that rotates in a fixed plane. A rotating reference system for the third body then becomes most natural and introduces a Coriolis force, in addition to the gravitational field. This feature causes all kinds of marvelous things to happen, like the neighborhood of the Lagrangian points, where the third body can remain at rest. Depending on the initial condition, however, the third body can scatter and thus change the orbital elements of the two heavy bodies. There is a whole catalog of disasters, such as capture, exchange, flyby, collision, and even ionization, that are important to comets from the Oort cloud. The authors describe such phenomena with approximate arguments and numerical results.

In the last chapter, Valtonen and Karttunen step outside the solar system and consider three bodies of comparable mass, but with known volumes like our nearest neighbor in spacscodee, Alpha Centauri, where all bodies are self-sustaining stars. Slow or fast three-body encounters with more or less dramatic consequences may exist. Sometimes those encounters resemble phenomena in atomic and molecular physics or even nuclear and particle physics. Although the chapter’s title, “Some Astrophysical Problems,” sounds innocent, it starts immediately with a section on binary black holes in the centers of galaxies. The reader now advances to the post-Newtonian formalism, which includes the effect of gravitational radiation. The mathematics is greatly simplified, and the arguments are kept within the bounds of reasonable models without undue speculations. The reader gets involved in whole galaxies colliding and melting together so that individual stars find binary companions for life. The chapter concludes with comets—fuzzy balls on a haphazard journey through the solar system.

In The Three-Body Problem , readers will find the necessary theoretical ingredients and will also enjoy the great variety of technical explanations for phenomena in the solar system and beyond. The book would be useful for a graduate course in modern astrophysics and makes interesting reading for an amateur who has some background in classical mechanics.