Introduction to the N-body problem

Introduction

The N-body problem dates back to Newton, who formulated the law of gravitational attraction. It is the problem of describing the motion of N bodies under their mutual influence. For example the Earth moves around the Sun while at the same time the Moon revolves around the Earth. In a nutshell, this constitutes the three-body problem.

This seemingly simple problem has fascinated mathematicians past and present. Indeed for 3 bodies or more there are no general solutions! The trajectories of the bodies depend on many things, namely their masses, coordinates and velocities at the beginning. In fact the motion can look very different if those inital conditions are modified, even by the slightest change.

Because of this we need to perform computer simulations to describe the motion reliably. This is done by numerical integrations in which the accuracy of the computer and the calculations are crucial to the reliability of the N-body code.

The essential point of such an orbit integration is to advance the solutions by small time intervals. At each point of the simulation the instantaneous particle forces are calculated and the particles take a small step under that force. By the time they have moved, the forces of the particles on each other have changed so they are calculated again and the next step is taken.

The motion is essentially cut into very small segments. In this way, the motion of each body can be represented accurately by a continuous curve. The art of scientific computing is to employ an efficient method for obtaining reliable solutions over long times.

Watching the simulations, we can admire the beautiful celestial dance carried out in new and intricate patterns until one of the bodies finally acquires sufficient velocity to escape from the system in a similar manner to a space rocket leaving the Earth.

3-body Method:

The chaotic nature of the three-body problem is mainly due to close encounters which produce large deflections. Thereby errors on small scales are magnified to larger errors in subsequent encounters and the final result, the trajectory of the bodies, can be completely different. These interactions are particularly troublesome for standard integration methods in spite of appropriate reductions of the time intervals between each new force calculation.

Instead we employ the concept of two-body regularization. The close approach of two point-mass particles is treated by a mathematical transformation which converts the solution to a well-behaved form. This treatment is generalized to two such solutions in parallel with respect to the third body, denoted as the reference body. Depending on the configuration, a different body is chosen as reference body so that the two relative motions form the most dominant interactions.

The property of the solutions is such that even three bodies close together can usually be handled in a satisfactory way. The only exception to that is a very small region in the neighbourhood of a triple collision. The resulting code is quite sophisticated but also highly accurate and surprisingly fast. It can be downloaded freely (see links).

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