Going ballistic: the many varieties of free fall

Natural History, Nov, 2002 by Neil deGrasse Tyson

The most extreme example of an elongated orbit is the famous case of the hole dug all the way to China. Contrary to the expectations of our geographically challenged fellow citizens, China is not opposite the United States on the globe. The southern Indian Ocean is. To avoid emerging under two miles of water, we should dig from Shelby, Montana, to the isolated Kerguelen Islands.

Now comes the fun part. Jump in. You now accelerate continuously in a weightless, free-fall state until you reach Earth's center--where you vaporize in the fierce heat of the iron core. But let's ignore that complication. You zoom past the center, where the force of gravity is zero, and steadily decelerate until you just reach the other side, at which time you have slowed to zero. Unless a Kerguelenian instantly grabs you, though, you will fall back down the hole and repeat the journey indefinitely. Besides making bungee jumpers jealous, you have executed a genuine orbit, taking about an hour and a half--the same amount of time as the space shuttle's.

Some orbits are so eccentric that they never loop back around again. At an eccentricity of exactly one you have a parabola, and for eccentricities greater than one the orbit traces a hyperbola. To picture these shapes, aim a flashlight at a nearby wall. The emergent cone of light will form a circle. Now gradually angle the flashlight upward, and you create ellipses of higher and higher eccentricities. When your light cone points straight up, the light that still falls on the nearby wall takes the exact shape of a parabola. Tip the flashlight away from the wall a bit more, and you have made a hyperbola. (Now you have something different to do when you go camping.) Any object with a parabolic or hyperbolic trajectory moves so fast that it will never return. If astronomers ever discover a comet with such an orbit, we will know that it has emerged from the depths of interstellar space and is on a one-time tour through the inner solar system.

Newtonian gravity describes the force of attraction between any two objects anywhere in the universe, no matter where they are found, what they are made of, or how large or small they may be. For example, you can use Newton's law to calculate the past and future behavior of the Earth-Moon system. But add a third object--a third source of gravity--and you severely complicate the system's motions. More generally known as the three-body problem, this manage a trois yields richly varied trajectories whose tracking usually requires a computer.

Some clever solutions to this problem deserve attention. In one case, called the restricted three-body problem, you simplify things by assuming the third body has so little mass compared with the other two that you can ignore its presence in the equations. With this approximation, you can reliably follow the motions of all three objects in the system. And we're not cheating. Many cases like this exist in the real universe--the Sun, Jupiter, and one of Jupiter's itty-bitty moons, for instance. In another case drawn from the solar system, an entire family of rocks move in stable orbits around the Sun, a half-billion miles ahead of and behind Jupiter. These are the Trojan asteroids, each one locked in its solar orbit (as if by Star Trek's tractor beams) by the gravity of Jupiter and the Sun.