two-body problem
Take two objects in empty space — a star and its planet, the Earth and the Moon — pulling on each other by gravity and nothing else. How do they move forever after? This is the two-body problem, and its beauty is that it can be solved exactly, with pen and paper, giving a complete and tidy answer. It is the one piece of celestial mechanics where nature is fully tractable.
The trick is to watch the right things. Each body orbits the system's center of mass, the balance point between them. By describing the motion as the separation between the two bodies plus the drift of that balance point, the messy two-body problem collapses into a single, simpler 'one-body' problem: a single fictitious particle moving in a fixed gravitational well. Out of that come exactly Kepler's results — each body traces an ellipse (or a parabola or hyperbola if it is unbound), with the center of mass at the shared focus, and the period set by the total mass M_total via P^2 proportional to a^3 / M_total.
Because it is exactly solvable, the two-body problem is the bedrock of orbital prediction and the workhorse of measuring masses: time a binary star or a planet and its moon, measure the orbit, and the equations hand you the masses. Its limitation is severe and important — add just one more body and there is no general closed-form solution at all, which is the leap to the three-body problem and to chaos.
A binary star is the perfect two-body lab: by tracking how the two stars circle their shared center of mass over years, astronomers read off both stellar masses directly — almost the only way we know how much a star weighs.
An exactly solvable problem becomes a precision scale for stars.
The exact solution assumes only two point masses and pure inverse-square gravity. Real systems have other planets, non-spherical bodies, and relativistic corrections, so the clean two-body answer is the first approximation, not the final orbit.