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The three-body problem

Given three masses, their positions and their velocities, where will they be at any later time? Newton answered this for two bodies in 1687: they trace ellipses, parabolas or hyperbolas forever, and a formula tells you exactly where they will be. Add a third body and there is no such formula.

Why it's hard to solve

• Too many unknowns, too few rules. Three bodies in space have 18 numbers that change over time (3 position and 3 velocity components each). Conservation of energy, momentum and angular momentum pins down only 10 of them. In 1887 Heinrich Bruns proved there are no other algebraic conserved quantities to make up the difference.
• No general closed-form answer. Henri Poincaré showed around 1890 that the motion generally can't be reduced to a neat formula at all. Karl Sundman did find an infinite series that solves it (1912), but it converges so slowly that it is useless in practice.
• So we simulate. Like this page, astronomers step the bodies forward in small time increments with a computer. That works, but every step adds a tiny error.

Why it's unstable

• Chaos. Poincaré also discovered that the system is chaotic: a difference in starting positions far smaller than you could ever measure grows exponentially over time. Two nearly identical setups soon look nothing alike, so long-term prediction is impossible even though the laws are exact.
• Close encounters. When two bodies pass near each other they swap a lot of energy very quickly. One can be flung onto a wide orbit, or out of the system.
• Ejection is the usual ending. Most three-body systems eventually throw one body out. The other two settle into a tight pair carrying away the leftover energy.

The rare stable dances

Special starting positions do repeat forever. Euler (1767) found three bodies in a line, and Lagrange (1772) found them at the corners of an equilateral triangle; Jupiter's Trojan asteroids ride that triangle. In 1993 Cris Moore found three equal masses chasing each other around a figure-eight. But most of these orbits are balanced on a knife-edge: nudge them and chaos takes over.

About this simulation

Helios runs in 2D with equal masses. It softens gravity at very close range so bodies pass through each other instead of reaching infinite speeds. It also adds a weak pull toward the center of the screen so bodies return instead of escaping. Click to place bodies, press Play, and watch how small changes in placement lead to completely different outcomes.