Kepler's Third Law Calculator

The Kepler's Third Law Calculator finds the orbital period of any orbiting body — a planet, moon, artificial satellite, or a star circling a black hole — from just two numbers: the semi-major axis of the orbit and the mass of the central body. It uses Newton's generalized form of Kepler's Third Law and shows every step, an animated orbit diagram, and a comparison against real solar-system orbits.

What Is Kepler's Third Law?

In 1619 Johannes Kepler discovered that the square of a planet's orbital period is proportional to the cube of its orbital distance from the Sun. Isaac Newton later derived this from his law of universal gravitation and made it exact and universal, replacing the proportionality with the mass of the central body. The result works for anything in a gravitationally bound orbit, anywhere in the universe.

Kepler's Third Law Formula

The Newtonian form used by this calculator is:

where T is the orbital period, a is the semi-major axis (the average orbital radius), M is the mass of the central body, and G is the gravitational constant, \( 6.674 \times 10^{-11}\ \text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2} \). Notice that the mass of the orbiting body and the eccentricity of the orbit do not appear.

The Simplified Solar-System Version

When the central body is the Sun, the constants collapse into a beautifully simple relationship if you measure period in years and distance in astronomical units (AU):

So Earth (a = 1 AU) has T = 1 year, Mars (a = 1.52 AU) has T ≈ 1.88 years, and Jupiter (a = 5.20 AU) has T ≈ 11.86 years. This calculator reports this shortcut automatically whenever you set the central mass to one solar mass.

Orbital Periods of the Planets

Why the Period Ignores Eccentricity

One of the most surprising consequences of Kepler's Third Law is that two orbits with the same semi-major axis have the same period, no matter how different their shapes are. A nearly circular orbit and a long, thin, cigar-shaped ellipse take exactly the same time to complete one lap, as long as their semi-major axes match. Eccentricity changes where the body speeds up and slows down (Kepler's Second Law), but not the total orbital time.

What Affects the Orbital Period?

How to Use This Calculator

1. Enter the semi-major axis: Type the orbit's semi-major axis and pick a unit — AU, million km, km, metres, or solar radii.
2. Enter the central mass: Type the mass of the body being orbited and pick a unit — solar masses, Earth masses, Jupiter masses, or kilograms.
3. Click Calculate: The tool applies \( T = 2\pi\sqrt{a^3 / GM} \) and returns the period instantly.
4. Review the results: See the period in every unit, the mean orbital velocity, an animated orbit, and how your orbit compares with real ones from the ISS to Neptune.

Worked Example: Earth Around the Sun

With a semi-major axis of 1 AU (\( 1.496 \times 10^{11} \) m) and a central mass of one solar mass (\( 1.989 \times 10^{30} \) kg), the formula gives \( T = 2\pi\sqrt{a^3/GM} \approx 3.156 \times 10^{7} \) seconds, which is 365.25 days — exactly one year, as expected.

Frequently Asked Questions

What is Kepler's Third Law?

Kepler's Third Law states that the square of a body's orbital period is proportional to the cube of the semi-major axis of its orbit. Newton generalized it to T = 2π√(a³ / GM), where a is the semi-major axis, M is the mass of the central body, and G is the gravitational constant. This lets you compute the period of any planet, moon, satellite, or star.

What inputs does the calculator need?

Only two: the semi-major axis of the orbit and the mass of the central body it orbits. The period does not depend on the mass of the orbiting object or on the eccentricity of the orbit, so those are not required.

Why does the period not depend on eccentricity?

Kepler's Third Law uses the semi-major axis, which is the average of the closest and farthest orbital distances. Two orbits with the same semi-major axis have exactly the same period even if one is a near-circle and the other is a long, thin ellipse. Eccentricity changes the shape and the speed at different points, but not the total time for one orbit.

What units can I use?

For the semi-major axis you can use astronomical units (AU), million km, km, metres, or solar radii. For the central mass you can use solar masses, Earth masses, Jupiter masses, or kilograms. The calculator converts everything to SI units internally and reports the period in seconds, minutes, hours, days, and years.

What is the simplified solar-system version?

When the central body is the Sun (one solar mass), Kepler's Third Law simplifies to T² (in years) = a³ (in AU). So the period in years is the semi-major axis in AU raised to the power 1.5. For Earth, a = 1 AU gives T = 1 year; for Jupiter, a = 5.2 AU gives about 11.9 years.

Can I use it for artificial satellites and moons?

Yes. Set the central mass to the Earth (or another planet) and enter the orbital radius as the semi-major axis. For a low Earth orbit near 6,791 km the calculator returns a period of about 93 minutes, matching the real orbital period of the International Space Station.

Additional Resources

• Kepler's Laws of Planetary Motion - Wikipedia
• Orbital Period - Wikipedia