Orbital Period Calculator — Kepler's Third Law
Per the NASA Planetary Fact Sheet (GSFC/NSSDC) and NIST's CODATA values for the gravitational constant G = 6.674 × 10⁻¹¹ N·m²/kg², Kepler's Third Law relates orbital period to semi-major axis: T² = 4π²a³ / (GM), where a is the semi-major axis, G is the gravitational constant, and M is the mass of the central body. Used in US classrooms following Next Generation Science Standards (NGSS HS-ESS1-4), in NASA mission planning at JPL, and by amateur astronomers tracking exoplanets discovered by Kepler/TESS missions.
When to use this calculator
- Calculate orbital periods of planets and satellites.
- Solve orbital mechanics problems.
- Verify Kepler's Third Law with real data.
- Calculate exoplanet orbits.
- Understand why distant planets have longer years.
Real example: Earth orbiting the Sun
- Data: a = 1.496 × 10¹¹ m (1 AU), M = 1.989 × 10³⁰ kg.
- Formula: T² = (4π² / GM) · a³ = (4π² / (6.674e-11 × 1.989e30)) · (1.496e11)³.
- Result: T ≈ 3.156 × 10⁷ s ≈ 365.25 days ≈ 1 year.
- Interpretation: matches the length of an Earth year, as expected.
How it works
1 min readKepler's Third Law
Kepler's Third Law in its Newtonian form relates the orbital period to the semi-major axis: T² = (4π² / GM) · a³, where G is the gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), M is the mass of the central body, and a is the semi-major axis.
Orbital Periods of the Planets (Around the Sun)
| Planet | Semi-major Axis (AU) | Period (Years) |
|---|---|---|
| Mercury | 0.387 | 0.24 |
| Venus | 0.723 | 0.62 |
| Earth | 1.000 | 1.00 |
| Mars | 1.524 | 1.88 |
| Jupiter | 5.203 | 11.86 |
| Saturn | 9.537 | 29.46 |
| Uranus | 19.19 | 84.01 |
| Neptune | 30.07 | 164.8 |
1 AU = 1.496 × 10¹¹ m; mass of the Sun = 1.989 × 10³⁰ kg.
When to Use and Common Mistakes
Frequently asked questions
What is Kepler's Third Law?
The square of a planet's orbital period is proportional to the cube of its semi-major axis: T² ∝ a³. Planets farther from their star have longer orbital periods.
What is a semi-major axis?
The semi-major axis is half the longest diameter of an elliptical orbit. For nearly circular orbits, it's approximately the orbital radius.
How long is a year on Jupiter?
About 11.86 Earth years (4,333 days). Jupiter's semi-major axis is 5.2 AU from the Sun.
Does Kepler's Third Law work for artificial satellites?
Yes. Replace M with Earth's mass. The ISS (a ≈ 6,771 km from Earth's center) has an orbital period of about 92 minutes.
What is an Astronomical Unit (AU)?
An AU is the average Earth-Sun distance: 1 AU = 1.496 × 10¹¹ m ≈ 93 million miles or 149.6 million kilometers.
Can I use Kepler's law to study exoplanets?
Yes. If you know the star's mass and measure the planet's orbital period (via transit detection), you can calculate the exoplanet's orbital distance.
Does the planet's mass matter in the calculation?
For planets much less massive than their star, the planet's mass is negligible. For binary systems with comparable masses, use M₁ + M₂.
What units should I use for Kepler's Third Law?
Always use SI units: meters for distance, kilograms for mass, and seconds for time. Mixing units will give incorrect results.