Kepler's Third Law Formula
Kepler's Third Law relates a planet's orbital period to its distance from the Sun.
Includes formula, examples, and applications to exoplanets.
The Formula
Kepler's Third Law states that the square of a planet's orbital period is proportional to the cube of its semi-major axis (average orbital distance). Johannes Kepler derived this empirically in 1619 from Tycho Brahe's observations — it was one of the most powerful unifying laws in the history of astronomy, later explained by Newton's law of gravitation.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| T | Orbital period (time for one complete orbit) | seconds (s) |
| a | Semi-major axis (average orbital distance) | meters (m) |
| G | Gravitational constant = 6.674 × 10⁻¹¹ | m³ kg⁻¹ s⁻² |
| M | Mass of the central body (e.g. the Sun) | kilograms (kg) |
Simplified Form for Solar System Objects
When T is measured in years and a is measured in Astronomical Units (AU), this relationship becomes elegantly simple. Earth: T = 1 year, a = 1 AU → 1² = 1³. Mars: a = 1.524 AU → T = √(1.524³) ≈ 1.88 years.
Example 1 — Earth's Orbital Period (Verification)
Earth: semi-major axis a = 1.496 × 10¹¹ m, Sun's mass M = 1.989 × 10³⁰ kg
T² = (4π² / (6.674×10⁻¹¹ × 1.989×10³⁰)) × (1.496×10¹¹)³
T² = (39.478 / 1.327×10²⁰) × 3.348×10³³
T² = 2.975×10⁻¹⁹ × 3.348×10³³ = 9.958×10¹⁴
T = √(9.958×10¹⁴) ≈ 3.156×10⁷ s ≈ 365.25 days ✓
Example 2 — Jupiter's Orbital Period
Jupiter: semi-major axis = 5.203 AU, using simplified form
T² = a³ = (5.203)³ = 140.85
T = √140.85 ≈ 11.87 years
Jupiter's orbital period ≈ 11.87 years (actual: 11.86 years ✓)
Solving for Other Variables
Rearranging for the semi-major axis allows astronomers to determine a planet's orbital distance if only its period is known — crucial for exoplanet discovery. By measuring how long a planet takes to transit its star, we can calculate exactly how far away it orbits.
When to Use It
- Calculating orbital periods of planets, moons, and artificial satellites
- Finding orbital distances from measured periods (exoplanet detection)
- Determining the mass of a star or planet from a satellite's orbit
- Comparing orbital periods of different bodies around the same central mass