What the Orbital Period Calculator does and when to use it
This calculator finds how long one orbit takes for a small object circling a much heavier body, along with its speed and how many laps it completes per day. Typical uses include satellites around Earth, moons around planets, and planets around a star. You provide the mass of the central body in kilograms and the orbital radius in kilometres, measured from the centre of the central body.
The result assumes a circular orbit, or the semi-major axis of an elliptical one, and treats the orbiting object's mass as negligible next to the central mass. That is an excellent approximation for satellites and planets, but not for two bodies of similar mass such as a binary star.
Formula and method
Kepler's third law, in Newton's form, gives the orbital period as T = 2π × sqrt(a³ / (G × M)). The circular-orbit speed follows from v = sqrt(G × M / a). The calculator uses G = 6.6743 × 10⁻¹¹ N·m²/kg² and converts kilometres to metres internally.
Notice that the period depends only on the central mass and the radius. The satellite's own mass cancels out, which is why a bolt and a space station share the same orbit at the same altitude.
- T orbital period in seconds.
- a orbital radius or semi-major axis in metres (entered in km).
- G gravitational constant, 6.6743 × 10⁻¹¹ N·m²/kg².
- M mass of the central body in kilograms.
- v speed of a circular orbit at that radius.
Worked example
A satellite orbits Earth (M = 5.972 × 10²⁴ kg) at about 408 km altitude, similar to the International Space Station. Earth's mean radius is about 6371 km, so a = 6371 + 408 = 6779 km = 6.779 × 10⁶ m.
- Compute a³ = (6.779 × 10⁶)³ = 3.115 × 10²&sup0; m³.
- Compute G × M = 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ = 3.986 × 10¹⁴.
- Divide: 3.115 × 10²&sup0; / 3.986 × 10¹⁴ = 7.815 × 10⁵, and its square root is about 884.
- T = 2π × 884 = about 5555 s, which is 92.6 minutes.
The calculator returns 5554.8 s, 92.58 min, a speed of 7.668 km/s and 15.554 orbits per day. As a check on the method, using the Sun's mass (1.989 × 10³&sup0; kg) with a = 149,600,000 km gives about 365.2 days, matching Earth's year.
Common mistakes and how to interpret the result
- Entering altitude instead of radius. The radius is measured from the body's centre, so add the body's own radius to the altitude above the surface.
- Using the wrong units. Kilometres are expected for the radius and kilograms for the mass; scientific notation such as 5.972e24 is accepted.
- Assuming the result covers strongly elliptical orbits. For an ellipse, enter the semi-major axis for the period, but the speed shown is only the circular value and will differ along the path.
- Ignoring perturbations. Atmospheric drag, the Moon, the Sun and the central body's shape all nudge real orbits, so predictions over long times need more detailed models.
Related calculators
- Earth Orbit Calculator — orbit details for objects around Earth.
- Escape Velocity Calculator — speed needed to leave a body's gravity.
- Hohmann Transfer Calculator — move between two circular orbits.
- Gravitational Force Calculator — attraction between two masses.