Chaos - James Gleick [70]
A system like a globular cluster is far too complex to be treated directly as a many-body problem, but its dynamics can be studied with the help of certain compromises. It is reasonable, for example, to think of individual stars winging their way through an average gravitational field with a particular gravitational center. Every so often, however, two stars will approach each other closely enough that their interaction must be treated separately. And astronomers realized that globular clusters generally must not be stable. Binary star systems tend to form inside them, stars pairing off in tight little orbits, and when a third star encounters a binary, one of the three tends to get a sharp kick. Every so often, a star will gain enough energy from such an interaction to reach escape velocity and depart the cluster forever; the rest of the cluster will then contract slightly. When Hénon took on this problem for his doctoral thesis in Paris in 1960, he made a rather arbitrary assumption: that as the cluster changed scale, it would remain self-similar. Working out the calculations, he reached an astonishing result. The core of a cluster would collapse, gaining kinetic energy and seeking a state of infinite density. This was hard to imagine, and furthermore it was not supported by the evidence of clusters so far observed. But slowly Hénon’s theory—later given the name “gravothermal collapse”—took hold.
Thus fortified, willing to try mathematics on old problems and willing to pursue unexpected results to their unlikely outcomes, he began work on a much easier problem in star dynamics.
This time, in 1962, visiting Princeton University, he had access for the first time to computers, just as Lorenz at M.I.T. was starting to use computers in meteorology. Hénon began modeling the orbits of stars around their galactic center. In reasonably simple form, galactic orbits can be treated like the orbits of planets around a sun, with one exception: the central gravity source is not a point, but a disk with thickness in three dimensions.
He made a compromise with the differential equations. “To have more freedom of experimentation,” as he put it, “we forget momentarily about the astronomical origin of the problem.” Although he did not say so at the time, “freedom of experimentation” meant, in part, freedom to play with the problem on a primitive computer. His machine had less than a thousandth of the memory on a single chip of a personal computer twenty-five years later, and it was slow, too. But like later experimenters in the phenomena of chaos, Hénon found that the oversimplification paid off. By abstracting only the essence of his system, he made discoveries that applied to other systems as well, and more important systems. Years later, galactic orbits were still a theoretical game, but the dynamics of such systems were under intense, expensive investigation by those interested in the orbits of particles in high-energy accelerators and those interested in the confinement of magnetic plasmas for the creation of nuclear fusion.
Stellar orbits in galaxies, on a time scale of some 200 million years, take on a three-dimensional character instead of making perfect ellipses. Three-dimensional orbits are as hard to visualize when the orbits are real as when they are imaginary constructions in phase space. So Hénon used a technique comparable to the making of Poincaré maps. He imagined a flat sheet placed upright on one side of the galaxy so that every orbit would sweep through it, as horses on a race track sweep across the finish line. Then he would mark the point where