Planetary Motion and Kepler’s Laws

Kepler's first law — planets move in ellipses with the Sun at one focus. Most planetary orbits are very nearly circular, which is why treating them as circles works well.

Kepler's second law — a line joining a planet to the Sun sweeps out equal areas in equal times. A planet therefore moves fastest when closest to the Sun and slowest when furthest away. This is conservation of angular momentum, expressed geometrically.

Kepler's third law — the square of the orbital period is proportional to the cube of the mean orbital radius:

T² ∝ r³

Deriving the third law — gravity provides the centripetal force for a circular orbit:

GMm / r² = m v² / r

The orbiting mass cancels, giving v² = GM/r. Substituting v = 2πr/T and rearranging:

T² = (4π² / GM) r³

which is Kepler's third law, with the constant of proportionality depending only on the mass of the central body.

The orbiting mass cancels — so a satellite's orbital period depends on its altitude alone, not on whether it is a small probe or a large space station. Two satellites at the same radius orbit together whatever their masses.

Example: this cancellation is the same one that makes all objects fall at the same rate. The gravitational force is proportional to the orbiting mass, and so is the inertia resisting the resulting acceleration, so the mass drops out.

Orbital speed falls with radius — from v = √(GM/r), a more distant orbit is a slower one. Low-orbit satellites race around the Earth in about 90 minutes; the Moon, 60 times further out, takes 27 days.

Using the third law without knowing G or M — since the constant is the same for every body orbiting the same central mass, ratios can be taken:

T₁² / T₂² = r₁³ / r₂³

which lets you find one orbit from another without needing G, M, or any absolute values.