What is Kepler's Third Law?
The 17th-century German astronomer Johannes Kepler made a number of astronomical observations. Upon analysing them, he found that every planet in the Solar System follows three rules. Here we focus on the third:
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
This means that the relationship between a planet's distance from the Sun and how long it takes to orbit is perfectly predictable—and, because the laws of physics are universal, the same relationship holds for any planetary system, any binary star system, or any moon orbiting a planet.
Kepler's Third Law Equation
We can derive Kepler's third law from Newton's Law of Gravitation by equating the centripetal force to the gravitational force:
m · r · ω² = G · m · M / r²
where:
- m — mass of the orbiting body (planet)
- r — orbital radius (semi-major axis a for a circular orbit)
- ω — angular velocity (ω = 2π / T)
- G — gravitational constant, G = 6.674 × 10⁻¹¹ m³ / (kg · s²)
- M — mass of the central star
Substituting ω = 2π / T and rearranging gives the simplified Kepler's third law:
T² = (4π² / G·M) · a³ → a³ / T² = G·M / 4π² = constant
For the general (elliptical) case, including the planet's own mass m:
a³ / T² = G·(M + m) / 4π²
The planet's mass correction is tiny: the Sun (M = 1.989 × 10³⁰ kg) outweighs Earth (m = 5.972 × 10²⁴ kg) by a factor of ~333,000. Enable the planet mass option in the calculator only when you need maximum precision.
How to Use the Kepler's Third Law Calculator
The calculator works in three modes—select what you want to find from the Find dropdown:
- Orbital Period (T) — provide the semi-major axis a and the star mass M; the calculator returns the orbital period.
- Semi-Major Axis (a) — provide the orbital period T and the star mass M; the calculator returns the orbital radius.
- Star Mass (M) — provide both a and T; the calculator returns the mass of the central star.
Unit Systems
| Quantity | Astronomical (default) | Metric | American |
|---|---|---|---|
| Distance | AU (astronomical unit) | km, m, ly | miles, ft |
| Mass | M☉ (solar masses) | kg, M⊕ | lb, M⊕ |
| Time | yr (years) | s, h, d | d, h, yr |
Kepler's Third Law Example
Try entering M = 1 M☉ (one Solar mass) and T = 1 yr (one year), then solve for the semi-major axis a. The result is exactly 1 AU—Earth's situation. This is the built-in self-check for the calculator.
The table below shows all Solar System planets and verifies that a³/T²≈ 1 for each:
| Planet | Semi-Major Axis (AU) | Orbital Period (yr) | a³ / T² |
|---|---|---|---|
| Mercury | 0.387 | 0.241 | 1.000 |
| Venus | 0.723 | 0.615 | 1.000 |
| Earth | 1.000 | 1.000 | 1.000 |
| Mars | 1.524 | 1.881 | 1.000 |
| Jupiter | 5.203 | 11.862 | 1.000 |
| Saturn | 9.537 | 29.457 | 1.000 |
| Uranus | 19.191 | 84.011 | 1.000 |
| Neptune | 30.069 | 164.800 | 1.000 |
All planets satisfy Kepler's third law to high precision, confirming that a³/T² = constant throughout the Solar System.
Frequently Asked Questions
What is the "semi-major axis"?
Planetary orbits are ellipses, not perfect circles. The semi-major axis a is half the longest diameter of the ellipse—it equals the average distance from the planet to the star over one orbit. For nearly circular orbits (like Earth's), a is very close to the average orbital radius.
What is an astronomical unit (AU)?
One AU is the average Earth-Sun distance: 1 AU ≈ 149,597,870.7 km ≈ 92,955,807 miles. It is the natural unit of length for Solar System astronomy.
What is a solar mass (M☉)?
One solar mass equals the mass of the Sun: M☉ ≈ 1.989 × 10³⁰ kg ≈ 333,000 Earth masses. It is the natural unit of mass in stellar astronomy.
Does Kepler's third law apply outside the Solar System?
Yes. The law applies to any gravitationally bound two-body system: exoplanets orbiting other stars, moons orbiting planets, binary stars, and even artificial satellites. The constant G·M / 4π² changes with the central mass, but the relationship a³ ∝ T² is universal.
What is the difference between the simplified and general formula?
The simplified formula assumes the planet's mass m is negligible compared to the star mass M. The general formula uses G·(M + m). The difference is less than 0.0003% for Earth-Sun, but may matter for large gas giants, binary stars, or other massive companions.
How does the calculator determine star mass from a period and distance?
By rearranging Kepler's third law: M = 4π²·a³ / (G·T²) – m. This is how astronomers estimate stellar masses from observed planetary orbits—even for stars hundreds of light-years away.