The three laws
Johannes Kepler summarised planetary motion in three laws, based on careful observations by Tycho Brahe.
- Law 1 (Law of Orbits): Every planet moves in an ellipse with the Sun at one focus.
- Law 2 (Law of Areas): A line joining a planet to the Sun sweeps out equal areas in equal times. This means a planet moves faster when nearer the Sun and slower when farther away.
- Law 3 (Law of Periods): The square of a planet's orbital period T is proportional to the cube of the average orbital radius r: T² ∝ r³.
Key formula / 关键公式
T² / r³ = constant, i.e. T₁² / r₁³ = T₂² / r₂³.
From Newton's law this constant equals 4π² / GM, where M is the mass of the central body: T² = (4π² / GM) r³.
Why the laws work
For a planet in a near-circular orbit, gravity supplies the centripetal force: GMm/r² = mv²/r. Since the orbital speed v = 2πr/T, substituting gives T² = (4π²/GM) r³, which is exactly Kepler's third law. So the three laws are a natural consequence of the inverse-square gravitational force.
Worked example / 例题
Planet A orbits its star with period T₁ = 2.0 years at radius r₁. Planet B orbits the same star at radius r₂ = 4r₁. Find the period T₂ of planet B.
Using T² ∝ r³: T₂²/T₁² = (r₂/r₁)³ = 4³ = 64.
T₂² = 64 × (2.0)² = 256, so T₂ = 16 years.
Using the laws
- The constant in Kepler's third law is the same for every body orbiting the same central mass.
- Larger orbits always mean longer periods.
- Law 2 explains why Earth moves fastest at its closest approach to the Sun (perihelion).
Remember / 记住
- Orbits are ellipses, not perfect circles, but circular orbits are a good approximation.
- T² ∝ r³ — square the period, cube the radius.
- Equal areas in equal times means speed changes along the orbit.