Orbital Period Calculator
Calculate the orbital period and speed of a satellite or planet with Kepler's third law, from the orbit's size and the mass it goes around.
Cómo usar esta calculadora
- 1Enter the orbit's semi-major axis — the average orbital distance — in AU, kilometres, or metres.
- 2Enter the mass at the centre of the orbit, in solar masses, Earth masses, or kilograms.
- 3Read the orbital period and the mean orbital velocity.
Cómo funciona
Kepler's third law
T = 2π √(a³ ÷ (G × M)) T = orbital period (seconds) a = semi-major axis of the orbit (metres) M = mass being orbited (kg) orbital velocity ≈ √(G × M ÷ a)
Kepler observed that the square of a planet's period is proportional to the cube of its orbital distance, and Newton derived exactly why from his law of gravitation. The full form includes the central mass, so the same equation predicts the period of a moon around a planet, a planet around a star, or a star around a galaxy — you just supply the right mass and distance. Larger orbits take longer both because the path is longer and because the orbiting body moves more slowly.
Ejemplo resuelto
For Earth's orbit — semi-major axis 1 AU (1.496 × 10¹¹ m) around 1 solar mass (1.989 × 10³⁰ kg): T = 2π √(a³ ÷ GM) ≈ 3.156 × 10⁷ seconds, which is 365.3 days, or one year. Earth's mean orbital velocity comes out at 29.8 km/s.
Orbital Period Calculator: la guía completa
The law that measures the universe
Kepler's third law is one of the most useful equations in astronomy because it links two things we can often measure — the size of an orbit and its period — to something we usually cannot measure directly: mass. Rearranged, it becomes the primary way astronomers weigh the cosmos. Watch how long a moon takes to circle a planet, or a star to circle the galactic centre, and the period and distance reveal the mass doing the pulling.
It is how we know the Sun's mass, the mass of Jupiter from its moons, and the mass of the supermassive black hole at the centre of the Milky Way from the orbits of stars whipping around it. The same short formula spans fourteen orders of magnitude in size.
Why farther means slower and longer
A planet twice as far from the Sun does not simply take twice as long to orbit — it takes about 2.83 times as long, because period grows as the three-halves power of distance. There are two reasons stacked together: the orbital path is longer, and the orbiting body also moves more slowly because gravity is weaker farther out.
This is why Mercury races around the Sun in 88 days while Neptune plods through a 165-year orbit. It is also why low-orbiting satellites must move faster than high ones: the International Space Station, only 400 km up, laps the Earth every 92 minutes at nearly 8 km/s, while a geostationary satellite 36,000 km up takes a full 24 hours.
Geostationary orbit, worked backwards
One of the most valuable orbits in engineering comes from running this law in reverse. A satellite that stays fixed above one point on Earth must have an orbital period of exactly one day. Setting the period to 24 hours and solving for the semi-major axis gives an altitude of about 35,786 km — the geostationary belt where communications and weather satellites sit.
Enter Earth's mass and try different distances above and you can find that altitude yourself: the orbit whose period matches Earth's rotation is the one that appears motionless in the sky, which is what lets a satellite dish point at a single spot and never move.
Preguntas frecuentes
What is Kepler's third law?
It states that the square of an orbital period is proportional to the cube of the orbit's semi-major axis. Newton's version adds the central mass: T = 2π √(a³ ÷ GM). It lets you find an orbital period from the orbit's size and the mass being orbited, or the mass from the period and size.
What is a semi-major axis?
It is half the longest diameter of an elliptical orbit — the average of the closest and farthest points from the central body. For a circular orbit it is simply the radius. It is the distance the law depends on, not the instantaneous distance, which changes as the body moves along its ellipse.
Why does the ISS orbit so much faster than the Moon?
Because it is far closer to Earth. Orbital velocity falls off with distance, so the ISS at 400 km altitude moves at about 7.7 km/s and orbits every 92 minutes, while the Moon at 384,000 km moves at about 1 km/s and takes 27 days. Both obey the same law.
Can this weigh a planet or star?
Yes — that is its most powerful use. Rearranged to solve for mass, M = 4π²a³ ÷ (GT²), it turns an observed orbit into the mass of whatever is being orbited. This is how astronomers measure the mass of stars, planets, and even black holes.