Using Kepler’s third law, this tool finds how long one orbit takes from the central mass and the orbit’s semi-major axis. It works for satellites around a planet, moons, or planets around a star — anything dominated by a single central gravitational body.
How it works
The period comes from Kepler’s third law in Newtonian form:
T = 2π · √(a³ ÷ GM)
where a is the semi-major axis in metres, M is the central mass in kilograms, and G is the gravitational constant, 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻². The tool converts your axis from kilometres to metres, applies the formula to get the period in seconds, then also displays it in days (÷ 86,400) and years (÷ 31,557,600). Presets fill in the mass of the Sun, Earth, Jupiter or the Moon.
Reference examples
Earth orbiting the Sun (the defaults): M = 1.989 × 10³⁰ kg and a = 149,597,870 km (1 AU = 1.496 × 10¹¹ m).
- T = 2π · √((1.496×10¹¹)³ ÷ (6.6743×10⁻¹¹ × 1.989×10³⁰))
- T ≈ 3.156 × 10⁷ seconds ≈ 365.25 days ≈ 1 year
| Central body | Semi-major axis | Period |
|---|---|---|
| Sun | 1 AU (Earth) | ≈ 365 days |
| Sun | 1.52 AU (Mars) | ≈ 687 days |
| Sun | 5.2 AU (Jupiter) | ≈ 11.9 years |
| Sun | 9.54 AU (Saturn) | ≈ 29.5 years |
| Earth | 6371 km (surface) | ≈ 84 minutes |
| Earth | 6771 km (LEO, 400 km up) | ≈ 92 minutes |
| Earth | 42,164 km (GEO) | ≈ 24 hours |
| Earth | 384,400 km (Moon) | ≈ 27.3 days |
Geostationary orbit at 42,164 km gives exactly one day — that is what keeps communication and weather satellites fixed over one point on the ground.
Understanding the inputs
Semi-major axis is half the longest diameter of an elliptical orbit. For a circular orbit it equals the orbital radius. For an elliptical orbit it is the average of the closest approach (periapsis) and the farthest point (apoapsis):
a = (periapsis + apoapsis) / 2Central mass dominates the system. Kepler’s third law in this form ignores the orbiting body’s mass, which is valid whenever the satellite is much lighter than the central body — true for all planets around the Sun and for spacecraft around planets.
Practical uses
- Satellite mission planning. Match orbital period to a communications window or a ground-track repeat cycle by adjusting the semi-major axis.
- Astronomy homework. Check your worked solutions for planetary orbits by entering the planet’s actual semi-major axis.
- Space game design. Generate realistic orbital periods for fictional star systems by setting a custom central mass.
- Checking existing orbits. Enter a known satellite’s altitude to verify its published period matches Kepler’s prediction.
All values are computed in your browser; nothing is uploaded.