Kepler's third law from a moon to a satellite
How one formula predicts the period of the Moon, a weather satellite and the planets, and what the semi-major axis really controls.
One law across wildly different scales
The striking thing about Kepler's third law is its reach. The same equation that gives the Moon a period near 27 days gives a low Earth orbit satellite a period near 92 minutes, purely because their orbits differ in size. Period scales with the semi-major axis to the power of three halves, so shrinking an orbit shortens its period sharply. That is why satellites close to Earth race around in minutes while distant ones creep.
What the central mass does
The denominator holds the product of the gravitational constant and the central mass. A heavier central body pulls harder, so for a given orbit size it drives a faster, shorter period. This is why an orbit of a certain radius around the Sun and the same radius around the Earth would not share a period: the Sun's far greater mass wins. The calculator makes this concrete by letting you keep the axis fixed and change the mass.
Turning a period into a mass
Because the law ties period, orbit size and central mass together, measuring any two reveals the third. Astronomers routinely invert it: watch a moon's orbit, note its radius and period, and solve for the mass of the planet it circles. This is one of the main ways the masses of distant worlds are found without ever landing on them. The tool computes the period direction, but the same relationship underpins those mass measurements.
Where the simple form applies
This version assumes the orbiting body is far lighter than the central mass, which is an excellent approximation for spacecraft, moons and planets. When two bodies are comparable in mass, such as a binary star, the full two-body form uses the sum of both masses instead. The law also ignores atmospheric drag and other perturbations, so a real low satellite slowly loses altitude even though the ideal period stays fixed for a given axis.