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
Key Notes
- Formula: T² = (4π²/GM) × a³: T is the orbital period, a is the semi-major axis, G is the gravitational constant, and M is the mass of the central body. In solar units: T² (years) = a³ (AU) — Earth's orbit (1 AU, 1 year) verifies this immediately.
- Proportionality: T² ∝ a³: The ratio T²/a³ is constant for all objects orbiting the same body. Neptune's orbital period can be predicted from its orbital radius alone, using Earth's orbit as a reference — no knowledge of G or the Sun's mass required.
- Weighing celestial objects: Rearranging gives M = 4π²a³/(GT²). By measuring an object's orbital period T and semi-major axis a, you can calculate the mass of what it orbits. This is how we know the Sun's mass, the mass of black holes, and the mass of exoplanet host stars.
- Geostationary orbit: A satellite with a 24-hour period stays fixed above one point on Earth. Kepler's third law gives the required orbital radius: r = (GMT²/4π²)^(1/3) ≈ 42,164 km from Earth's center (about 35,786 km above the equator).
- Applications: Kepler's third law is used to predict planetary positions, design satellite constellations, estimate exoplanet orbital parameters from transit timing, and calculate spacecraft injection trajectories for interplanetary missions.