How to use it
- Choose whether to find the period or the orbit size.
- Pick the central body, or Custom mass and enter its mass.
- Choose whether the orbit size is a semi-major axis or an altitude above the surface.
- Enter the orbit size in km or AU, or the period in minutes, hours, days or years.
Formula
T = 2π × √(a³ ÷ μ), where μ = G × M is the gravitational parameter of the central body, and a = ∛(μ × T² ÷ 4π²) for the reverse.
The listed bodies use the same GM values and mean radii as the escape velocity calculator, for example μ = 1.32712440018 × 10²⁰ m³/s² for the Sun and 3.986004418 × 10¹⁴ m³/s² with a mean radius of 6371 km for Earth. 1 AU = 149,597,870,700 m; a year here is 365.25 days. The mean orbital speed is the orbit’s circumference 2πa divided by the period.
Worked example
Earth around the Sun, a = 1 AU: T = 2π × √((1.495978707 × 10¹¹)³ ÷ 1.32712440018 × 10²⁰) = 31,558,196 s = 365.2569 days, at a mean speed of 29.785 km/s.
A satellite 420 km above Earth (a = 6,791 km) goes round in 92.82 minutes. The reverse, a period of one sidereal day, 23.9344696 h, gives a = 42,164.2 km: the geostationary orbit, 35,793.2 km above the mean radius.
Tips
- Elliptical orbits: the period depends only on the semi-major axis, half the longest diameter of the ellipse.
- Altitude needs a radius. For a custom body enter its radius, or give the semi-major axis instead.
- Units: a period of 1 year at 1 AU around 1 solar mass is a quick check of the units.