Kepler's Laws of Planetary Motion

Kepler's three laws describe how planets orbit the Sun. They were derived empirically and later explained by Newton's law of gravitation.

Concept

1. Law of orbits: Every planet moves in an elliptical orbit with the Sun at one focus.
2. Law of areas: The radius vector sweeps equal areas in equal times (a consequence of angular momentum conservation).
3. Law of periods: $T^2 \propto a^3$ where $a$ is the semi-major axis.

Derivation

For a circular orbit, gravity provides centripetal force: $\frac{GMm}{r^2}=\frac{mv^2}{r}\Rightarrow v=\sqrt{GM/r}$. The period is $T=\frac{2\pi r}{v}=2\pi\sqrt{r^3/GM}$, giving $T^2=\frac{4\pi^2}{GM}r^3$ — Kepler's 3rd law for circular orbits.

Worked Example

Mars has $a=1.524\,\mathrm{AU}$. Its period is $T=\sqrt{1.524^3}\approx 1.88$ years.

Common Confusions

• Law of areas is angular momentum conservation in disguise — not a separate principle.
• Kepler's 3rd law works for elliptical orbits too with $a$ replacing $r$.

Key Takeaways

• Orbits are ellipses, not circles.
• $T^2 \propto a^3$ is universal for any central-force inverse-square system.