N-Body Problem

The N-body problem is one ideally suited to a computer—a machine that can repeat a prescribed string of commands with blazing speed. So the task now seems simple: plug Newton's law of gravitation into a gigahertz computer, specify the initial positions and velocities of the objects under study, hit return, and sit back with a cool drink to watch the fun. Of course, nothing is ever that easy.

Even the fastest computer in the world cannot overcome the fact that direct numerical computation of the evolution of a large system of objects is not feasible. Consider a star cluster. Because every star acts gravitationally on every other star, the number of calculations required to determine all the gravitational force components in a system of N objects goes roughly as N squared. For a modest-sized globular cluster of 500,000 stars, the computer would need to do 25 billion calculations—and that's just for the first force computation. Remember that after a certain time interval, usually called the timestep, the forces must all be recalculated. Globular clusters are long-lived objects; the ones orbiting the Milky Way Galaxy are probably 10 billion years old. Millions of timesteps would be required to trace the entire history of the cluster, and the total number of calculations runs into the quintillions.

Even worse, early N-body calculations showed that a very common occurrence in star clusters was the formation of close binary stars—two stars orbiting one another so closely that in some cases they share a common atmosphere. The timestep for the force calculation for these stars has to be very short, because the stars change their positions relative to one another quite rapidly. Additionally, the very small distance between these stars leads to singularities—places where the computations produce values going to infinity. (This happens because when stars are nearly touching, the distance between them is almost zero relative to the size of the entire cluster or galaxy, and this value appears in the denominator of the force equation.)

The manifold difficulties with performing explicit N-body calculations have led researchers since 1960 to construct increasingly complex computer codes to deal with them. Codes have been developed that can apply different time steps to different objects, prevent the formation of close binaries, and effectively reduce the number of particles in a calculation by determining the average force imparted by groups of particles, rather than calculating them all explicitly. Many codes get rid of singularities by applying what is usually called "softening" to the system; this effectively means changing the force equation so that the troublesome zero never appears. Still other methods leave the realm of direct N-body calculations altogether and apply statistical methods to estimate the behavior of the entire system. Although these approximate methods have represented the best compromise available in the face of limited computing power, they are clearly not fully satisfactory, since different methods applied to the same N-body system do not always yield the same results. Since reproducibility of results is a cornerstone of science, the various methods summarized above have sometimes been sharply criticized, and there is widespread agreement that so far, no method yet devised is fully capable of correctly addressing the evolution of stellar and galactic systems.

This ongoing difficulty is the source of the question mark in the title of this part of this essay. It is possible to write a complex computer program that runs without crashing and generates page after page of results—and yet have those results bear no resemblance at all to reality. Computers are very fast, but very literal-minded: they will do exactly what you tell them to, even if what you tell them is riddled with errors. Most (but not all) researchers realize that, and qualify their work with appropriate statements about the limitations of their code and their machines. Any aspiring scientist would do well to follow this lead, which is almost nowhere more important than in the arena of N-body calculations.