Math · chapter 9 of 13

There is a formula. Nobody can add it up.

In 1912 Karl Sundman wrote a series that converges to the answer for almost every three-body start. One estimate of how many terms you would need to use it runs to a 1 followed by eight million zeros.

The last two chapters said there is no formula. That needs narrowing, because there is one, and the story of it is the best joke in the subject.

Karl Frithiof Sundman, 1912, working in Helsinki, proved that the three-body problem can be written as a power series that converges for all time — provided the total spin is not zero, which rules out only the triple collisions. Not an approximation. A convergent series, the answer, on paper.

The catch is in the fine print twice over. First, the series runs in powers of the cube root of time, not time — a fractional power, because a near-collision does something to the solution that whole powers cannot follow. Second, and fatally, it converges slowly. Slowly enough that in 1930 D. Beloriszky estimated the number of terms needed to compute a position at astronomical accuracy at around 10 to the eight-millionth power.

For scale: the observable universe holds something like 10⁸⁰ atoms. The exponent here is eight million, not eighty. A recipe that calls for more steps than there are particles is a recipe in name only.

Qiu-Dong Wang extended Sundman's result to any number of bodies in 1991, with the same catch. So the state of play is this: an exact expression exists and cannot be used; the bookkeeping route is proved closed; and every number anybody has ever actually used for a three-body system came out of stepping the equations forward and watching. That is the next chapter.

q(t) = Σ_n≥0 c_n τⁿ, τ ∝ t^1/3

Reading it: The positions written as an endless sum of powers — not of time, but of the cube root of time.

The cube root is forced by what happens near a two-body collision. Sundman's regularising change of variable is the same idea the numbers chapter uses to keep a computer from choking on a close pass.

terms needed ≈ 10^{8 000 000} vs atoms in the universe ≈ 10⁸⁰

Reading it: The number of terms you would have to add up, against the number of atoms there are.

Beloriszky's 1930 estimate, quoted in the standard references. A convergent series and a usable method are not the same thing, and this is the cleanest example anywhere of the difference.

Try it. Ask yourself what 'solved' should mean. A formula nobody can evaluate, or a stepping method that gives you twelve digits by lunchtime and tells you when to stop trusting it? Mathematicians and engineers answer that differently, and both are right about their own question.

Sources

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