The Three-Body Problem, with Epicycles

Kepler found the relationship between areas encompassed by elliptical orbits, back when epicycles were cool. Galileo discovered the consistent force acting upon falling bodies, and epicycles were the norm. Newton showed that gravitation fit observations of elliptic orbits of planets, and epicycles fell out of fashion. Astronomers have relied upon the elliptic model ever since. Epicycles were dead.

Yet…

Newtonian mechanics relied upon the fact that the sun is enormous, compared to all its orbiting bodies. This difference in mass simplified the problem of orbital motion into what is called a ‘two-body’ problem: the HUGE sun, and a comparatively tiny planet. Influences from other planets could be ignored.

Not all orbits can be simplified that way, and the elegant ellipse cannot capture the motion of orbiting bodies when three of them influence each other. THAT is the ‘three-body’ problem, and it is unsolved for the general case. (Certain special cases have been solved, such as the orbits of Earth’s Trojan asteroid groups at the Lagrangian points…)

I offer that, to solve three-body orbital mechanics, we must revivify epicycles… in an unusual way.

The Set of Epicycles, Mapped as a Manifold:

Epicycles work a bit like a Spirograph. A larger circle is used as the rim for the movement of a smaller circle, which rotates as it progresses along the edge of the larger circle. Together, they form florets, like an ellipse that tilts while it orbits. More complex epicycles can be formed, by placing more circles within the smaller circle; each circle completes its own rotation at a different pace. Together, they account for the ‘wobble’ of the moon, Mercury, and comets.

As astronomical observations improved, more epicycles were needed, to account for all the subtle wobbles in the heavens. Newtonian astronomers considered epicycles overly complex and redundant. Our computer-era may allow us to tackle that complexity, and finally solve three-body orbits.

Each orbit, even with many bodies, does have an associated epicyclic description. We just need a map, to understand their relationship. To do so: given three particular masses orbiting each other, you find the epicycles which describe their motion. As you vary the mass, position, or velocity of any one of those bodies, the epicycles of all three must be changed. With a map from one particular instance of [mass, position, velocity] to [epicycles], we can elaborate to a map of other epicycles continuously. (That is, an infinitesimal change in mass, position, or velocity will create an infinitesimal change in the circles used to describe the orbits; they are both without discontinuity, and form a smooth map, one to the other.) This continuous map is a manifold.

Why can’t we just find a function for the orbits of three bodies?

Functions describe a lot of things, but many behaviors cannot be reduced to a function. A circle does not pass the ‘vertical line test’, and must be expressed as a set of parameterizations — it requires two functions, together. Three-body orbits are worse.

Instead, a map from [mass, position, velocity] to [epicycles] can be found experimentally, using many particular orbits, until the shape of that manifold can be predicted. My guess is that the manifold itself does have a functional description, but many of the orbits we would study require enormous numbers of epicycles. Newton couldn’t have done it by hand. (Ask Leibniz!)

If such a manifold is revealed by computer experimentation, and enough points on its surface are found for us to fit a function to the shape of that manifold, then we will have a map of solutions to the three-body problem. With that map, you could take the [mass, position, velocity] coordinates of your three objects, and find the associated epicycles which predict their orbits exactly. No one function can describe all orbits, but we would have a function that indexes Spirograph-solutions. So… can I borrow your supercomputer?