Kepler's Laws of Planetary Motion

Definition

Kepler’s laws are three empirical principles describing the motion of planets around the Sun, later proven analytically by Newton using calculus and the law of universal gravitation.

Why It Matters

The heavens follow a clockwork logic. Kepler’s laws were the first great “bridge” between observation and physical law, providing the essential framework for all satellite navigation, planetary exploration, and our understanding of our place in the solar system.

Core Concepts

1. First Law (Ellipses): Orbits are ellipses with the Sun at one focus.

2. Second Law (Equal Areas): The radius vector sweeps out equal areas in equal times ($dA/dt = \frac{1}{2}r^2\dot{\theta} = \text{constant}$).

   • How to read: “The derivative of A with respect to t equals one half r squared theta dot, which equals a constant.”
   • Meaning: Angular momentum conservation makes the areal sweep rate fixed—planets move faster when closer to the Sun.

3. Third Law (Harmonic Law): $T^2 \propto a^3$. Specifically: $T^2 = \left( \frac{4\pi^2}{GM} \right) a^3$

   • How to read: “The square of T equals the quantity four pi squared divided by G M, all times a cubed.”
   • Meaning / when to use: Orbital period $T$ grows with semi-major axis $a$; compare planets or compute mass $M$ of the central body from one orbit’s $T$ and $a$.

Connected Concepts

• Graphs of Functions: Properties
• Library of Functions, Piecewise Functions
• Zeros of Polynomial Functions