Kepler's Third Law
Kepler's third law relates a planet's orbital period to its distance from the star.
Calculate orbits for any celestial body.
The Formula
Kepler's third law states that the square of the orbital period is proportional to the cube of the semi-major axis. This applies to any object orbiting a larger body — planets, moons, or satellites.
Variables
| Symbol | Meaning |
|---|---|
| T | Orbital period (seconds) |
| a | Semi-major axis of the orbit (meters) |
| G | Gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²) |
| M | Mass of the central body (kg) |
| π | Pi (approximately 3.14159) |
Example 1
Find Earth's orbital period around the Sun
a = 1.496 × 10¹¹ m (1 AU)
M = 1.989 × 10³⁰ kg (mass of the Sun)
T² = (4π² / (6.674 × 10⁻¹¹ × 1.989 × 10³⁰)) × (1.496 × 10¹¹)³
T² = (39.478 / 1.327 × 10²⁰) × 3.348 × 10³³
T ≈ 3.156 × 10⁷ seconds ≈ 365.25 days
Example 2
Find the orbital period of the Moon around Earth
a = 3.844 × 10⁸ m (average Earth-Moon distance)
M = 5.972 × 10²⁴ kg (mass of Earth)
T² = (4π² / (6.674 × 10⁻¹¹ × 5.972 × 10²⁴)) × (3.844 × 10⁸)³
T ≈ 2.361 × 10⁶ seconds ≈ 27.3 days
When to Use It
Use Kepler's third law when:
- Calculating how long a planet or satellite takes to complete one orbit
- Finding the distance of an orbiting body from its central mass
- Comparing orbital periods of different planets or moons
- Designing satellite orbits at specific altitudes