Going ballistic: the many varieties of free fall

Natural History, Nov, 2002 by Neil deGrasse Tyson

Another special case of the three-body problem was discovered in recent years. Take three objects of identical mass and have them follow each other in tandem, tracing a figure eight in space. Unlike those automobile race-tracks where people go to watch cars smashing into each other at the intersection of two ovals, this setup takes better care of its participants. The forces of gravity require that the system "balances" for all time at the point of intersection, and, unlike the complicated general three-body problem, all motion occurs in one plane. Alas, this special case is so odd and so rare that there is probably not a single example of it among the hundred billion stars in our galaxy, and perhaps only a few examples in the entire universe, making the figure-eight three-body orbit an astrophysically irrelevant mathematical curiosity.

Beyond one or two other well-behaved cases, the gravitational interaction of three or more objects eventually makes their trajectories go bananas. To see how this happens, simulate Newton's laws of motion and gravity on your computer. Now nudge every object according to the force of attraction between it and every other object in the simulation. Recalculate all forces and repeat. The exercise is not simply academic. The entire solar system is a many-body problem, with asteroids, moons, planets, and the Sun in a state of continuous mutual attraction. Newton worried greatly about this problem, which he could not solve with pen and paper. Fearing the entire solar system was unstable and would eventually crash its planets into the Sun or fling them into interstellar space, Newton postulated that God might step in every now and then to set things right.

The eighteenth-century French astronomer and mathematician PierreSimon de Laplace presented a solution to the many-body problem of the solar system more than a century later in his treatise Mecanique Celeste. But to do so, he had to invent a new form of mathematics known as perturbation theory. The analysis begins by assuming that there is only one major source of gravity and that all the other forces are minor, though persistent--exactly the situation that prevails in our solar system. Laplace then demonstrated analytically that the solar system is indeed stable, and that you don't need new laws of physics to show this.

Or is it? Modern analysis demonstrates that on timescales of hundreds of millions of years--periods much longer than the ones considered by Laplace--planetary orbits are chaotic. That leaves Mercury vulnerable to falling into the Sun, and Pluto vulnerable to getting flung out of the solar system altogether. Worse yet, the solar system might have been born with dozens of other planets, most of them now long lost to interstellar space. And it all started with Copernicus's simple circles.

If you could somehow rise above the plane of the solar system, you would see each star in our Sun's neighborhood moving to and fro at ten to twenty kilometers a second. But collectively those stars orbit the galaxy in wide, nearly circular paths, at speeds in excess of 200 kilometers a second. Most of the hundred billion stars of the Milky Way lie within a broad, flat disk, and--like the orbiting objects in all other spiral galaxies--the clouds, stars, and other constituents of the Milky Way thrive on big, round orbits.