Celestial Mechanics & Timekeeping

The five points in a two-body system where a small third body can remain fixed relative to the two, its motion balanced by their combined gravity. The James Webb Space Telescope orbits the Sun–Earth L2 point, and the Trojan asteroids sit at Jupiter's L4 and L5.

When the orbital periods of two bodies form a simple whole-number ratio, their repeated close alignments deliver gravitational kicks that add up. Such resonances carve the Kirkwood gaps in the asteroid belt and lock Jupiter's inner moons into a precise chain.

The gravitational motion of many bodies at once. Beyond two bodies there is no general closed-form solution, so the positions of the planets and moons are found by integrating the equations of motion numerically — the basis of modern ephemerides.

The set of numbers — typically six — that fully specify an orbit and a body's position along it: its size and shape, its orientation in space, and where the body is at a given time. They are the compact language in which every ephemeris and mission trajectory is expressed.

The small departures of a real orbit from a perfect two-body ellipse — caused by the pull of other bodies, a planet's equatorial bulge, atmospheric drag, and radiation. They accumulate over time, which is why orbits must be tracked and recomputed continually.

A slow resonance not between orbital periods but between the rates at which orbits themselves precess. Acting over millions of years, secular resonances reshape the eccentricities and inclinations of orbits and help clear certain regions of the Solar System.

The locking of a body's rotation to its orbit by tides. The Moon keeps one face toward the Earth in a 1:1 lock, while Mercury rotates three times for every two orbits of the Sun — a 3:2 spin–orbit resonance.

The region around a body within which its own gravity dominates over that of the more massive body it orbits. It sets how far out moons and rings can stably orbit a planet — anything beyond a planet's Hill sphere is pulled away by the Sun.

The motion of a small body under the gravity of two larger ones orbiting each other. It has no general closed-form solution, yet its study yields the Lagrange points, the concept of orbital stability, and much of modern spacecraft trajectory design.

The distance within which a body held together only by its own gravity is pulled apart by the tidal forces of the body it orbits. It is why planetary rings lie close in — where a moon could not survive — and why comets passing too close to a planet break into fragments.

The slow change of orbits and rotations caused by tidal friction. It is why the Moon is receding from the Earth and the day is lengthening, and why close-in moons and planets end up rotating in step with their orbits.