Math · chapter 2 of 13
Two is easy
With two bodies you can write the answer down, once, and it is right forever. This is the only case that works that way.
Two rocks, nothing else. Watch the pair from the point that sits between them — the barycentre, the balance point. That point either sits still or coasts in a straight line at a steady speed, no matter what the rocks do, because the two pulls are equal and opposite and cancel in the total.
Sit on the balance point and there is only one thing left to track: the line from one rock to the other. Its length and its direction. Two moving bodies collapse into one moving arrow, and the arrow obeys the same inverse-square law with the two weights added together.
Solve that and you get a shape, not a list of positions. An ellipse, or a parabola, or a hyperbola — a slice through a cone. Each rock runs its own copy of the shape around the balance point, the lighter one on the bigger copy, and it repeats. Forever, exactly, with no stepping and no computer.
Kepler had the ellipse from Tycho Brahe's naked-eye records before Newton had the law that explains it. Newton then showed the inverse square and the ellipse are the same fact said two ways.
Two is the case where a formula exists. Everything hard about three bodies is the loss of this page.
Reading it: Track only the arrow from one body to the other. It speeds up toward the other end, at a rate set by the two weights added together.
Twelve numbers (two positions, two velocities, in space) become six. This is the whole trick, and there is no version of it for three.
Reading it: The distance at any angle around the orbit. One number sets the size, one sets how squashed it is.
e = 0 is a circle. e below 1 is an ellipse and closes. e = 1 is a parabola, e above 1 a hyperbola — those leave and do not come back.
Reading it: Square the time it takes to go around and you get the cube of the orbit's size, times a constant. Farther out means slower, and by exactly this much.
Kepler's third law. Written this way it is also a scale: the same equation weighs the Sun, weighs a black hole, and weighs a pair of stars nobody will ever visit.
This one runs in the browser, with JavaScript on. The words above it stand on their own.