Orbital altitude from period and body calculator

Enter the period in seconds, the body's mass in kilograms, and its radius in meters. The calculator applies Newtonian gravity to find the orbit's center-to-center radius, then subtracts the supplied body radius to report altitude. It is useful for spacecraft studies, classroom exercises, and consistent data checks where a deterministic SI-unit result is more useful than a simulation.

How period determines circular-orbit altitude

A circular orbit has one radius at every point, so its period and the central body's gravitational strength are directly connected. The calculator begins with the standard two-body relation for a negligible orbiting mass: the cube of orbital radius equals the gravitational constant multiplied by body mass and period squared, divided by four times pi squared. Taking the cube root gives the distance from the body's center to the orbit. Altitude is a different quantity, so the final step subtracts body_radius_m from that center-to-center distance. This distinction matters around large planets and moons: an orbital radius is not a height above the surface. The calculation uses the fixed gravitational constant 6.67430 × 10^-11 in SI units. It assumes a spherical reference radius and a perfectly circular Newtonian orbit. Because every operation is deterministic scalar arithmetic, identical inputs always produce identical outputs without a network request, clock, random value, numerical integration, or hidden body database.

Enter compatible SI values and read the result

Provide period_seconds as a positive finite number and body_mass_kg as a positive finite number. Provide body_radius_m as a finite value that is zero or greater. Seconds, kilograms, and meters must be used together; mixing minutes, kilometers, or grams into these fields changes the physical scale and can produce a plausible-looking but incorrect answer. The response repeats normalized inputs, gives orbital_radius_m from the center of the body, and gives altitude_m relative to the supplied radius. A negative altitude is intentionally preserved rather than converted into an error. It means the requested mathematical circular orbit lies inside the reference surface, so it is not a viable free-space orbit for that body. The solver rejects a zero or negative period because no circular orbit is defined by such a duration, and it rejects zero or negative mass because the requested gravitational model would not define the bound orbit. Each successful API item costs $0.002.

Assumptions, precision, and practical limits

Use the result as an idealized orbital estimate, not as a complete mission design. The equation treats the central body as spherically symmetric and the orbiting object's mass as negligible compared with body_mass_kg. It ignores atmospheric drag, oblateness, terrain, tides, third-body perturbations, radiation pressure, propulsion, and relativistic effects. Real bodies may have equatorial, polar, mean, or datum radii, so choose body_radius_m consistently with the altitude definition you need. Likewise, use a sidereal orbital period when comparing an inertial orbit; an apparent or synodic period can describe a different observation. Floating-point results are rounded to twelve significant digits for stable machine-readable output, but that presentation does not make uncertain inputs exact. The gravitational constant and measured body properties carry uncertainty, and a body's effective gravitational parameter may be known more accurately than mass multiplied by G. This calculator deliberately accepts mass because that is the stated input contract. For operational trajectories, validate the estimate with authoritative body constants and a higher-fidelity dynamics model.