Two bodies orbit like clockwork. Add a third and no formula can tell you where they'll be. Here's why, in real simulations, and how Orbit Slinger, a game about flinging planets, turns that problem into play.
Scroll, or watch the film first60 secondsEvery orbit simulatedOriginal score
The film runs through the whole argument in a minute. The rest of this page slows it down, one scene at a time, and you can scrub each one back and forth. Then it looks at how Orbit Slinger plays with the same physics.
One star, one planet: solved in 1687.
Newton showed that a lone planet's path is a fixed curve: an ellipse it retraces forever, or a one-way escape. Know the throw and you know where the planet will be on any date, even a million years out.
Too soft, too hard, just right.
It's the first thing the game teaches. Too soft and the planet falls into the star; too hard and it's gone. In between it circles, and rounder scores more: 1,000 bonus points for a perfect circle, 259 for this oval, none once the orbit is more oval than an eccentricity of 0.6.
Add a third, and the clockwork stops.
Burrau's problem, 1913: three stars of mass 3, 4 and 5, at rest on the corners of a 3-4-5 triangle. Let them go.
No pattern. No repeat.
Every close pass changes the next one, and the paths never settle into a shape you could write down. The flashes mark the closest passes.
Then one is thrown out.
The lightest star leaves for good while the other two pair off, as Szebehely and Peters found by computer in 1967. Our simulation, with a modern integrator, ends the same way.
Nudge one star by a millionth.
Same stars, same start, except the light one moves a millionth of the triangle's longest side. Solid: the original run. Rings: the nudged copy.
At first, nothing changes.
The two runs lie on top of each other while the gap between them quietly multiplies: 20 times wider by t 28, 2,000 times by t 41.
Then everything does.
In the original the light star escapes; in the copy all three are still tangled at the end. Tiny differences grow until they're everything. That's chaos, and it's why forecasts of a tangle like this run out: nothing is ever measured exactly.
Every planet pulls on the others.
In the game a planet feels the star and every other planet, so three planets around a star are a four-body problem, with no formula either. The game works it out step by step: 240 steps for every second of play.
At zero, clockwork.
Switch the planets' pull off and each one runs its own perfect circle forever. Nothing can go wrong, and nothing interesting happens.
At full strength, a pileup.
At real strength, neighbors drag each other out of their lanes: in this throw Jupiter hits Neptune after 35 seconds. We threw 100 random three-planet systems; only 39 lasted two minutes.
The game sets it to a tenth.
Orbit Slinger scales planet-on-planet gravity to 10%, so the star stays in charge. Of the same 100 throws, 86 lasted. That leaves enough pull for near misses and slingshots, which score, and little enough that a good throw holds.
Three suns on a figure-eight.
Orbit Slinger's newest campaign, added in version 2.3 this November, puts three equal suns on the figure-eight: one curve, three suns chasing each other, found by computer in 1993 and proved to exist in 2000. Your planets fly in the calm band outside it.
If planets pulled on the suns…
Same throw, but the suns feel the planets. Solid: the game's suns. Rings: the what-if. Here two suns touch after 3 minutes 52 seconds. Of 24 random throws we tried, 6 broke the eight within 15 minutes; the game's own tests found 19 of 40 within 30.
So in the game, they don't.
Suns pull planets; planets never pull suns. It's deliberate: the suns' dance is the same in every game, so each level can be designed against a known sky, proven winnable, and replayed exactly.
Where is it safe to orbit?
Around two suns, some lanes are calm and some fling planets out or into a sun. No formula draws the line between them. You find it by running it.
Every lane, eight ways.
Orbit Slinger's research put test planets on circles at every radius, from eight starting angles, and ran each for 30 minutes of game time: 432,000 steps. We did the same here, with the game's constants and two equal suns.
The line astronomers found.
From 2.3 times the suns' separation outward, all eight survive. Holman and Wiegert's 1999 study of binary stars put the edge at 2.39 times; the game's engine also finds 2.30, and its design rule puts planets' lanes beyond 2.4.
Chaos is what you see. Underneath it are a few hard results, most of them more than a century old.
Three bodies need 18 numbers: a position and a velocity, in three directions, for each. Physics guarantees only 10 conserved quantities: energy, momentum (3), the center of mass's motion (3) and angular momentum (3). Heinrich Bruns proved in 1887 that no other algebraic ones exist, and Henri Poincaré ruled out other well-behaved ones in 1889.
Karl Sundman found a series that solves the problem exactly. It converges so slowly that, by David Beloriszky's 1930 estimate, a useful answer would take about that many terms.
Euler found three bodies in a line that stay in line (1767), and Lagrange an equilateral triangle that keeps its shape (1772); Jupiter's Trojan asteroids ride that triangle today. Cris Moore found the figure-eight by computer in 1993. Searches since 2013 have found thousands more, each needing exactly the right start.
In Orbit Slinger's tests, kicking one of the eight's suns by 3% of its typical speed still held for four hours of game time. A 5% kick broke it after 490 seconds, a 10% kick after 285.
Proved: mathematical results. Special cases: exact solutions that exist only for special starting points. Game tests: from Orbit Slinger's research notes.
Orbit Slinger is an iPhone and iPad game about flinging planets past a star and keeping them in orbit. Every choice below comes from its source code and its developer's research notes.
Every body moves by velocity Verlet, a method that keeps orbits stable for thousands of laps, in fixed steps of 1/240 of a second. No formula anywhere: just steps.
Guided mode (which scores nothing) draws your planet's path a quarter of an orbit ahead by running a copy of the whole system, and a test checks that forecast and flight match bit for bit. The computer knows every number exactly. Your finger doesn't, and that's the game.
Planet-on-planet gravity is scaled to 10% so the star stays in charge: enough for slingshots and near misses, not so much that every level turns into a pileup.
In the multi-star campaigns, suns pull each other at full strength and pull every planet, but planets never pull suns. The developer's notes call it the restricted problem: the figure-eight can't break, and replays rebuild the sky exactly.
A scripted player throws planets onto random clean lanes, retrying until enough attempts win. Tests run it on every level and check that the resident bodies hold on their own. With no formula, playing is the proof.
Clockwork pays for order: round orbits, alignments, resonances. Chaos pays for nerve: near misses (up to 2,500), slingshots (1,000 when a flyby changes a planet's orbital energy by 6% or more), wild orbits, and knocking a resident out of the system.
Orbit Lab, a separate sandbox, lets everything pull on everything, with starter systems that include the figure-eight, Jupiter's Trojans, a Tatooine-style world around two suns, and a slingshot.
Orbit Slinger is on the App Store for iPhone and iPad. Sol and our solar system are free; the other star systems, including the three Exotic Stars campaigns, and Orbit Lab are one-time unlocks. No ads, no tracking. It's made by Justin Martin, who runs this site.
Burrau's problem and its nudged copy (the hero, the three-body and millionth scenes, and the film) were integrated with SciPy's DOP853 at a relative tolerance of 10−12, in units with G = 1; energy held to about a part in a billion.
Every other orbit on this page runs on Orbit Slinger's own physics, rebuilt line for line from its source: G = 800, planet-on-planet gravity ×0.1, velocity Verlet at 1/240 s, collisions at 0.9 times the summed radii, escape beyond 500 units. The throws shown are examples; the 100-throw and 24-throw counts are from our runs. The 19-of-40 count, the kick tests and the 2.30 edge come from the game's research notes, and our stability map reproduced the 2.30.
The stability map: suns of mass 150 and 150, 60 units apart on a circular orbit, test planets at 1.6 to 3.2 times that separation, 8 starting angles each, 30 minutes of game time. The animation shows the first 45 seconds; the colored rings show the full 30 minutes.
The film was made with HyperFrames, which renders HTML to video, from the same simulations; its score was synthesized from scratch and timed to the close passes. Facts are current as of November 8, 2026.