Fix the mistake: Universal Gravitation and Planetary Motion

Kepler's Three Laws

Kepler's three laws describing planetary motion are empirical laws derived solely from observational data in the 17th century. They can be derived theoretically from the law of universal gravitation.

Basic Knowledge

First law (law of elliptical orbits): A planet moves along an elliptical orbit with the Sun at one focus.

Second law (law of equal areas): The area swept out per unit time by the line segment (radius vector) connecting a planet and the Sun is constant.
→ The planet moves faster at perihelion (the point closest to the Sun) and more slowly at aphelion (the point farthest from the Sun).

Third law (harmonic law): The orbital period T and the orbital radius (semi-major axis) a of a planet satisfy:
T² ∝ a³ (T²/a³ = constant)
Specifically: T²/a³ = 4π²/(GM) (G: gravitational constant, M: mass of the Sun)

Further Exploration (Background and Significance)

Kepler analyzed his teacher Tycho Brahe's enormous collection of observational data for 20 years and discovered these laws. At that time, no one knew why they held; they were not explained until Newton theoretically derived them all from the law of universal gravitation.
The second law is equivalent to conservation of angular momentum. Because the gravitational force on a planet always points toward the center (the Sun), the external torque is zero → angular momentum is conserved → areal velocity is constant.
Proof of the third law (for a circular orbit): From gravity = centripetal force, GMm/r² = m(2πr/T)²/r → T² = 4π²r³/(GM) → T²/r³ = 4π²/(GM) = constant.
The distance ratio can be calculated from the period ratio of Earth and Mars: Mars's orbital period T_M ≈ 1.88 years → orbital radius a_M = a_E × (T_M/T_E)^(2/3) ≈ 1.52 AU.

Points of Caution (Common Confusions)

① The third law states proportionality between T² and a³ (not between T and a). ② For an elliptical orbit, use the semi-major axis a, not the “orbital radius.” ③ Constant areal velocity means the planet is faster at perihelion, not that its speed is constant. ④ Use the mass of the Sun in GM (not the mass of the planet).

Practice

1. Taking Earth's orbital period as T_E = 1 year and its orbital radius as a_E = 1AU, find the orbital radius of a planet with an orbital period of 8 years.
2. At which point is a planet's speed greater, perihelion or aphelion? State the reason as well.
3. Assuming a circular orbit, derive Kepler's third law T²/a³ = 4π²/(GM).