Orbital Period Calculator
Enter any two of the three orbital parameters - semi-major axis, central body mass, and orbital period - and the calculator finds the third using Kepler's Third Law. Switch between satellite mode (one massive central body) and binary system mode (two comparable masses), choose from multiple unit systems, and see the full step-by-step derivation with your actual numbers. Results update instantly as you type.
Formula
Worked example
Earth orbits the Sun at 1 AU = 1.496 x 10^11 m with M_Sun = 1.989 x 10^30 kg. T = 2pi * sqrt((1.496e11)^3 / (6.674e-11 * 1.989e30)) = 2pi * sqrt(3.348e33 / 1.327e20) = 2pi * sqrt(2.522e13) = 2pi * 5.022e6 = 3.156e7 s = 365.25 days.
What is the orbital period?
The orbital period is the time a body takes to complete one full revolution around another body along its orbit. For planets it is called the sidereal period (measured against fixed stars), which differs from the synodic period (the time between successive alignments as seen from a third body, such as Earth). This calculator computes the sidereal period directly from Kepler's Third Law and the physical parameters of the orbit.
Kepler's Third Law - the formula explained
Johannes Kepler published his Third Law in 1619: the square of the orbital period is proportional to the cube of the semi-major axis. In modern form, T = 2pi * sqrt(a^3 / (G * M)), where T is the period in seconds, a is the semi-major axis in meters, G = 6.674 x 10^-11 m^3 kg^-1 s^-2 is the gravitational constant, and M is the mass of the central body in kilograms. For a binary system where both bodies have comparable mass, M is replaced by the total mass M1 + M2. Isaac Newton later derived this result from his law of universal gravitation, showing it holds for any two bodies in a bound gravitational orbit.
Satellite mode vs. binary system mode
In satellite mode the orbiting body's mass is assumed negligible compared to the central body, so only the central mass appears in the formula. This is accurate for planets orbiting stars, moons orbiting planets, and artificial satellites - even the most massive planet (Jupiter) is only 0.1% of the Sun's mass, a negligible correction. In binary mode the full mass of both bodies appears as the sum M1 + M2. Binary star systems - including spectroscopic binaries, visual binaries, and the Pluto-Charon system - require this treatment because neither object's mass can be ignored. Astronomers also apply it to measure the masses of supermassive black holes from the orbits of stars near galactic centers.
Solving for semi-major axis or central mass
Kepler's Third Law can be rearranged algebraically to solve for any one of the three quantities T, a, or M when the other two are known. To find the axis: a = cuberoot(G * M * T^2 / (4pi^2)). To find the central mass: M = 4pi^2 * a^3 / (G * T^2). Astronomers use the mass form routinely - by measuring a moon's orbital period and distance from a planet, they can determine the planet's mass with high precision. The same technique works for extrasolar planets, binary pulsars, and even galaxy clusters. Each solve mode in this calculator uses the exact rearrangement with full unit conversion.
Solar System orbital periods and semi-major axes
| Body | Semi-major axis (AU) | Orbital period | Orbital velocity (km/s) |
|---|---|---|---|
| Mercury | 0.387 | 87.97 days | 47.36 |
| Venus | 0.723 | 224.70 days | 35.02 |
| Earth | 1.000 | 365.25 days | 29.78 |
| Mars | 1.524 | 686.97 days | 24.13 |
| Jupiter | 5.203 | 11.86 years | 13.07 |
| Saturn | 9.537 | 29.45 years | 9.69 |
| Uranus | 19.19 | 84.02 years | 6.81 |
| Neptune | 30.07 | 164.80 years | 5.43 |
| Moon (around Earth) | 0.00257 (384,400 km) | 27.32 days | 1.02 |
| ISS (around Earth) | 0.0000043 (408 km alt) | 92.68 min | 7.66 |
Reference values from NASA planetary fact sheets. Periods are sidereal (relative to fixed stars).
Frequently asked questions
What units does this calculator accept for distance and mass?
For the semi-major axis you can choose meters, kilometers, astronomical units (AU), or light-years. For mass you can choose kilograms, solar masses (1.989 x 10^30 kg), Earth masses (5.972 x 10^24 kg), or Jupiter masses (1.898 x 10^27 kg). For the period output you can select seconds, minutes, hours, days, or years. All inputs are converted to SI units internally before the calculation.
What is the semi-major axis and how is it different from the orbital radius?
The semi-major axis is half the longest diameter of the elliptical orbit. For a circular orbit it equals the constant radius. For an elliptical orbit it is the average of the closest approach (periapsis) and the farthest point (apoapsis). Kepler's Third Law uses the semi-major axis, not the instantaneous distance, because that average distance determines the average gravitational energy and thus the period.
Can this calculate orbital periods for artificial satellites around Earth?
Yes. Set the central body mass to 1 Earth mass (5.972 x 10^24 kg), set the semi-major axis to the orbital radius (Earth's radius of 6,371 km plus altitude), and choose the satellite mode. For example, the International Space Station orbits at roughly 408 km altitude, so a = 6,371 + 408 = 6,779 km, giving a period of about 92.7 minutes, which matches real ISS data closely.
Why does a longer orbital period mean a larger orbit?
A body in a larger orbit moves more slowly and covers a longer path, so each of those effects individually would increase the period. Combined, the period scales as a^1.5, meaning a doubling of the semi-major axis increases the period by 2^1.5 = 2.83 times. This is why Neptune takes 164.8 years while Earth takes just 1 year despite both orbiting the same Sun.
How accurate is Kepler's Third Law?
For most purposes the accuracy is very high - errors only become significant for very fast-moving objects (where special-relativistic corrections apply), very strong gravitational fields (general relativity), or systems with three or more comparable-mass bodies (the restricted three-body problem). For Solar System planets, the law is accurate to well within 0.1% of observed values. For Mercury, the general-relativistic precession of its orbit amounts to only 43 arcseconds per century beyond the Newtonian prediction - a tiny but famously observed discrepancy that was one of the earliest tests of Einstein's theory.
What is the difference between sidereal and synodic orbital period?
The sidereal period is the time for one orbit measured relative to distant stars - the true orbital period calculated by this tool. The synodic period is the time between successive identical alignments as seen from another body (for example, the time between successive full moons as seen from Earth). They are related by: 1/T_synodic = |1/T_inner - 1/T_outer|. Earth's sidereal year is 365.25 days; the synodic period of Mars as seen from Earth is 780 days, much longer than Mars's actual 687-day sidereal period.