Planetary Mass Calculator
Determine the mass of a star or planet in solar masses from a satellite's orbital period and semi-major axis. Useful for astronomers and students applying Kepler's third law to real or hypothetical systems.
About this calculator
This calculator applies a rearranged form of Kepler's Third Law, which relates the orbital period and semi-major axis of a body to the mass of the object it orbits. The governing formula is: M = (4π² × a³) / (G × T²), where M is the central body's mass in kilograms, a is the semi-major axis in metres, G is the gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), and T is the orbital period in seconds. The result is then divided by the solar mass (1.989 × 10³⁰ kg) to express it in solar masses. This method works for any two-body system where one mass dominates — such as a planet orbiting a star or a moon orbiting a planet. It underpins much of modern exoplanet science.
How to use
Suppose a planet orbits its star with a period of 365 days and a semi-major axis of 1 AU. Convert: T = 365 × 24 × 3600 = 31,536,000 s; a = 1 × 1.496 × 10¹¹ = 1.496 × 10¹¹ m. Then M = (4π² × (1.496 × 10¹¹)³) / (6.674 × 10⁻¹¹ × (31,536,000)²) ≈ 1.989 × 10³⁰ kg. Dividing by the solar mass gives exactly 1 M☉ — confirming the tool correctly recovers Earth's orbit around a one-solar-mass star. Enter your own orbital period in days and semi-major axis in AU to find the central body's mass.
Frequently asked questions
How does orbital period relate to planetary mass in Kepler's third law?
Kepler's Third Law states that the square of a satellite's orbital period is proportional to the cube of its semi-major axis, with the proportionality constant depending on the central mass. Rearranging the law as M = 4π²a³ / (GT²) lets us solve directly for mass. A shorter orbital period at a given distance implies a more massive central body, because stronger gravity is needed to maintain the orbit. This relationship holds for any gravitationally bound two-body system where one mass greatly exceeds the other.
What units should I use when calculating planetary mass from orbital data?
The calculator expects the orbital period in days and the semi-major axis in astronomical units (AU). Internally it converts days to seconds (× 86,400) and AU to metres (× 1.496 × 10¹¹) before applying Newton's form of Kepler's law. The result is output in solar masses for easy comparison with known stellar masses. Using the correct input units is critical — entering kilometres instead of AU, for example, will produce a result off by many orders of magnitude.
Can this calculator be used to find the mass of a planet from a moon's orbit?
Yes — the same formula applies to any central body and its satellite, provided the satellite's mass is negligible compared to the primary. To find a planet's mass, enter the moon's orbital period and semi-major axis. For example, using the Moon's period (27.32 days) and semi-major axis (0.00257 AU) returns Earth's mass. The technique is routinely used in planetary science to weigh planets from the orbits of their natural or artificial moons.