About this orbital period calculator
Every satellite, from the International Space Station to the Moon itself, is obeying the same simple rule, and an orbital period calculator lets you apply that rule to any orbit you can imagine. Give it the mass of the central body and the radius of the orbit, and it returns how long one full circuit takes, together with the speed the object must travel to stay in that orbit. The answers are often surprising: the space station laps the Earth roughly every ninety minutes, a geostationary satellite takes a full day so that it appears to hover over one spot, and the Moon needs nearly a month. All of those numbers come from one equation, Kepler's third law, and this tool puts it in your hands.
Kepler's third law, in its Newtonian form, says that the square of the orbital period is proportional to the cube of the orbit's radius, with the constant of proportionality set by the mass of the central body and the gravitational constant. In symbols, T equals two pi times the square root of r cubed divided by G times M, where T is the period, r is the distance from the centre of the central body, G is the universal gravitational constant, and M is the central mass. The orbital speed follows just as directly: v equals the square root of G times M divided by r. Notice what is missing from both formulas: the mass of the satellite itself. A feather and a space station at the same altitude would orbit in exactly the same time, which is one of the most elegant consequences of the equivalence of gravitational and inertial mass.
Using the tool is a matter of two choices. Pick the central body from the dropdown, which sets its mass, or imagine your own by comparing with the listed values, and enter the orbit radius measured from the centre of the body, not from its surface, which is the detail most people get wrong at first. The preset buttons cover the orbits people actually ask about: the space station at about 6,771 kilometres from Earth's centre, geostationary orbit at 42,164 kilometres, Earth's own path around the Sun, and a low orbit around the Moon. Results give the period in seconds, hours, and days, plus the orbital speed, and the history lets you line up several orbits and watch the period stretch as the radius grows.
The space station example is worth working through, because it shows the arithmetic and the physics at once. Earth's mass is about 5.972 times 10 to the 24 kilograms, and the station orbits about 400 kilometres above the surface, which is 6,771 kilometres from Earth's centre. Cubing that radius, dividing by G times Earth's mass, taking the square root, and multiplying by two pi gives a period of about 5,546 seconds, or roughly 92.4 minutes, which matches the station's real schedule of about sixteen sunrises a day. Its speed comes out at about 7.66 kilometres per second, which is why reaching orbit is mostly about going fast sideways, not merely going high: at that speed, the station is perpetually falling towards Earth but missing it, which is what an orbit is.
Geostationary orbit is the other classic case, and it explains a piece of modern life that most people use without noticing. At a radius of 42,164 kilometres from Earth's centre, the same formula gives a period of about 86,164 seconds, or about 23.93 hours, essentially one sidereal day, the time Earth takes to rotate once relative to the stars. A satellite there circles the equator at exactly the rate Earth turns beneath it, so it appears fixed in the sky, which is why a satellite television dish can point at one spot permanently instead of tracking a moving target. That special radius is no coincidence: it is the direct consequence of Kepler's law, and it is the reason communications, weather, and broadcast satellites cluster in that particular ring above the equator.
The same law scales far beyond Earth. The Moon, orbiting at about 384,400 kilometres from Earth's centre, takes about 27.3 days, which is why months and moons have been linked since antiquity. Earth, orbiting the Sun at about 149.6 million kilometres, takes a year by definition, and the law lets you check that the Sun's mass is consistent with that fact. Jupiter's great moons, from Io's swift 1.8-day circuit to Callisto's stately 16.7 days, all obey the same relationship with Jupiter's mass, and it was exactly this pattern that let Kepler, and later Newton, see that one law governs the heavens as well as falling apples. Exoplanet hunters use the same idea in reverse, inferring a star's mass or a planet's distance from the period they observe.
A few cautions keep the tool honest. The formula assumes a circular orbit, while real orbits are ellipses, so for an eccentric orbit you should use the semi-major axis as the radius to get the period, which remains correct even though the speed varies around the orbit. It also ignores the pull of other bodies and the slight flattening of the central body, which matter for precise satellite operations but not for the understanding this tool is built for. And remember the radius trap: altitudes quoted in news reports are measured from the surface, so add the body's radius, about 6,371 kilometres for Earth, before entering the number. Get that right, and the calculator becomes a reliable pocket orrery, ready to time any orbit from a backyard satellite pass to a moon of Jupiter.
A second worked example, the Moon, shows the same law at a very different scale. The Moon orbits about 384,400 kilometres from Earth's centre, and the same formula gives a period of about 2.36 million seconds, which is about 27.3 days, matching the sidereal month, the time the Moon takes to return to the same position against the stars. The familiar calendar month is slightly longer, about 29.5 days, because Earth has moved along its own orbit in the meantime, so the Moon must travel a little further to line up with the Sun again. The distinction between the two months is a good reminder that an orbit's period is defined against the stars, while the phases we see depend on the Sun's position as well.
Kepler himself never wrote the law in the form used here. His third law, published in 1619, stated that the square of a planet's period is proportional to the cube of its mean distance from the Sun, a pattern he found by years of labour over Tycho Brahe's precise observations. It was Newton, more than sixty years later, who showed that this pattern follows from universal gravitation and the laws of motion, and who supplied the constant of proportionality involving the central mass. That is why the calculator uses the Newtonian form: it generalises Kepler's discovery from the Sun's planets to any central body, from the Moon to Jupiter to a star in a distant galaxy, and it is one of the clearest examples in science of a pattern becoming an explanation.
One more idea is worth carrying away from this calculator: the period depends on the cube of the radius, so small changes in altitude produce surprisingly large changes in period. Doubling an orbit's radius multiplies its period by about 2.83, not by two, which is why low orbits are fast and high orbits are slow in a strongly nonlinear way. That single relationship explains the whole architecture of near-Earth space, from the ninety-minute dash of the space station to the stately day-long circuit of a geostationary satellite, and it is the reason mission planners treat altitude as a precious resource rather than a mere number.