Kepler's Laws
Johannes Kepler derived three laws of planetary motion between 1609 and 1619 from Tycho Brahe's precise astronomical measurements. They are empirical — Kepler found them by fitting curves to data, not by deriving them from a physical theory. That derivation came later, when Newton showed all three follow from the inverse-square law of gravitation.
First Law: Planets move in ellipses, with the Sun at one focus.
Second Law: The line segment connecting a planet to the Sun sweeps equal areas in equal intervals of time. A planet moves faster when it is closer to the Sun and slower when farther away, in exactly the ratio that keeps the swept area constant.
Third Law: The square of a planet's orbital period is proportional to the cube of the semi-major axis of its ellipse: T² ∝ a³. For the solar system, T² = a³ when T is in years and a is in astronomical units (AU, the Earth-Sun distance).
The Role of Tycho Brahe
Feynman opens his discussion of gravity with Brahe, not Kepler or Newton. Brahe spent decades measuring the positions of planets with instruments far more accurate than any previous observer's — not because he had better theories but because he believed measurement mattered more than argument. He determined Mars's position to within 1/50 of a degree. This precision was what made Kepler's pattern-finding possible, and it eventually falsified Kepler's original models. The first law — ellipses — arose because circular orbits didn't fit Brahe's Mars data.
What Newton Explained
Newton's derivation of Kepler's laws from the inverse-square gravitational force was the first major demonstration of the power of mathematical physics. He showed:
- First law: the only closed orbits under an inverse-square attractive force are ellipses (and in special cases, circles as a limiting case).
- Second law: this is just conservation of angular momentum. Since gravity acts along the line connecting the planet to the Sun, it exerts no torque. Angular momentum L = r × mv is therefore constant, which requires the swept-area rate to be constant.
- Third law: the period scales as a^(3/2) because the circumference of the orbit scales as a, and the speed scales as 1/√a (from the balance between gravity and centripetal acceleration). The ratio gives T ∝ a^(3/2).
Predictive Power
Kepler's third law allows astronomers to calculate the semi-major axis of any planet's orbit from its period alone, using the Sun's mass as a calibration. This is still how the distances of newly discovered exoplanets are estimated from their transit timing.
Connections
- universal-gravitation — Newton's law that explains all three of Kepler's laws
- galilean-inertia — the first law of motion underlies the equal-areas result
- gravitational-constant — Kepler's laws give GM for the Sun but not G alone