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5 Gifs of n-body Orbits [Animation]


"What I do is I have code that minimizes things," says Robert Vanderbei, a Professor of Operations Research at Princeton University. He can minimize the light coming into a telescope from stars — to better see if there are more-softly lit planets traveling in their wakes. He also uses the code to analyze climate data.

But it works for other things, too.

"I was surprised to know that what I'm good at is useful for n-body problems," he tells me.

The n-body problem is a classic in physics. Its solutions are configurations in which objects – planets, say — can travel around in each other’s gravity in a stable manner. (The moon and Earth perform a modified version of a 2-body solution — they travel in each other’s gravity, but also that of the Sun, and other planets. The n-body problem assumes bodies that are of equal mass, and in an environment where there is no other gravity. I’ll keep calling them planets, but understand that they are not ordinary planets.)

To find a solution to the n-body problem is to minimize something called an action functional, which describes the energy of group of planets — the accumulated difference between the kinetic and potential energy, to be more specific about it — as they swing and pull and dance around each other. The result is the path of least resistance, the way that the planet’s motion will naturally fall into place, without an external forces – a purely gravitational path. Out of these paths, the ones that are periodic — that is, the planets swing around in the same motion for at least some extended period of time — are the solutions to the n-body problem.

Vanderbei first learned of a solution to the n-body problem called the figure 8, discovered by physicist Cris Moore, the first physicist who used a computer to solve the n-body problem.

"I said wow, that is amazing," he tells me. And then he went home and tried to figure it out with his minimizing code. In a couple hours, he had reproduced Moore’s figure 8:

He modified the computer code a little, so that it would run through cases with different starting positions and velocities. Some of these starting positions — most of them, actually — end up just sending the planets immediately flying off in different directions, or crashing into each other. But run through enough configurations, and you can find solutions. This is how he found “Ducati” -- which he refers to as his favorite.

This one is what would happen if the Sun, Earth, and Moon all had the same mass, and there were only two months in a year. (Like Ducati, others experts had found this orbit before Vanderbei did — unbeknownst to him.)

Vanderbei says this one, the 5 point star, which was discovered by Zhifu Xie and Tiancheng Ouyang, surprised him the most. It’s the kind of thing you would draw mindlessly on a napkin. (The lines of the start aren’t technically straight, though that’s not apparent in the animation.)

He found this one in June, which is in the same family — which is to say, a tweak of starting positions and velocities — as Xie and Ouyang’s star:

(Many, many more animations of n-body orbits — in the form of applets — can be found here.

“I take other people’s ‘aha’ moments and run with them,” says Vanderbei, who works on the n-body problem sometimes when he gets home in the evenings, coming up with animations for solutions that are out there, and making tweaks of his own.

“Its totally for fun. Well, 99%.”

When I speak to him, he’s in the middle of putting together a presentation for conference at Princeton on the payload for a pair of former-spy telescopes —which could include instruments that hunt for planets, a la Kepler.

“The solutions to the n-body problem could inform what we might find out there. I always worry that [our] minds aren't open enough."

The views expressed are those of the author and are not necessarily those of Scientific American.

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