How this instrument works
The harmonic law says something faintly outrageous: how long an orbit takes depends on the size of its ellipse and on nothing else. Squash a circular path into a long thin one without altering its semi-major axis, and the period will not shift by a second. The body tears through periapsis and then crawls near apoapsis, and those two effects cancel to the last digit. That is why a semi-major axis is written into this expression rather than a radius. For a circle the two agree; for Halley's comet, only its semi-major axis of 17.83 AU predicts a return every 75 years.
Kepler landed on the relation on 15 May 1618 — an arithmetic slip in March had made him throw it away once already — and printed it a year later in Harmonices Mundi as a bare proportion: squares of periods stand in the ratio of cubes of mean distances. No masses appear in that sentence, and no constant either. It compares one planet against another and stays silent about what holds either in place. Bolting 4π² ⁄ GM onto the front, which the Principia supplied in 1687, turned a comparison into an absolute measurement and, with it, a weighing machine. Watch anything go round anything, time one lap, measure the ellipse, and the mass at the centre falls straight out. Almost every stellar and planetary mass in the literature traces back to that inversion.
Read as an equation rather than a proportion, it assumes the orbiting body's mass is a rounding error, that the central body is spherical or point-like, and that nothing else is pulling. The first assumption is the easiest to relax and the most often forgotten — swap M for M + m and carry on. It matters more than people expect. Predicting the Moon's month from Earth's mass alone returns 27.45 days against an observed 27.32; adding the Moon's own 1.2% share drops the figure by four hours and lands within an hour of the truth. Beyond that, Earth's equatorial bulge pulls the several definitions of period apart for low satellites, Jupiter's repeated tugs carve the Kirkwood gaps out of the asteroid belt at resonant periods, and general relativity rotates an ellipse without much disturbing how long a lap takes.
- Enter Mass of the central body in kilograms or tonnes — Earth's 5.9722×10²⁴ kg is preloaded.
- Give Orbital semi-major axis in metres or kilometres: half the sum of the closest and furthest distances from that body's centre, or simply the radius when the path is circular.
- Read Orbital period in seconds, then switch that field's unit menu to minutes, hours, days or years to suit the scale you are working at.
- Sanity-check against something familiar — 92 minutes just above the atmosphere, a day at geostationary height, a year at one astronomical unit.
Worked example — the orbit that hangs still overhead
A geostationary satellite has to circle once per rotation of the planet beneath it, so try the radius that is supposed to do exactly that: M = 5.9722×10²⁴ kg, a = 42,164,000 m from Earth's centre. Take the product first, GM = 3.986025×10¹⁴ m³/s². Cube the axis: a³ = 7.4959281×10²² m³. Divide, and 1.8805520×10⁸ s² remains; its square root is 13,713.32 s. Multiply by 2π and the answer is T = 86,163.34 seconds — 23 hours, 56 minutes, 3.3 seconds.
That sits three-quarters of a second below the sidereal day of 86,164.09 s, and the shortfall is worth reading rather than dismissing. Roughly 0.52 s of it comes from the rounded axis: the true value is nearer 42,164.17 km, and period climbs as the axis to the power one and a half, so 170 metres of radius buys half a second. The remaining 0.23 s comes from building GM as a product of two separately measured quantities. Laser ranging pins Earth's GM directly, far tighter than G and M can ever be multiplied together, and that measured product runs a whisker low against the one used here. Both gaps are smaller than the station-keeping tolerance any operator actually flies to.
Questions
Do I enter the orbit's radius or its semi-major axis?
The semi-major axis, which for a circular path is just the radius. For an ellipse, take half the sum of the closest and furthest distances from the central body's centre — never one or the other on its own. Eccentricity itself never appears in the formula: a path skimming the atmosphere at one end and reaching far out at the other keeps precisely the period of a circle with the same a. Typing in an apoapsis distance is the commonest way to end up wrong by a large factor.
Does the orbiting body's own mass change the period?
Not detectably while it stays negligible beside the central mass — a cubesat and a 400-tonne station at equal a share one period exactly. The strict version divides by G(M + m). Pluto and Charon show why that matters: Charon carries about 12% of the pair's mass, and leaving it out would misstate their 6.39-day period by nearly 6%. Eclipsing binary stars are handled the same way, and the inversion is how stellar masses came to be known at all.
Why does the geostationary answer fall short of a full day?
Because a satellite parked over one spot has to match the planet's spin against the stars, not against the Sun. A sidereal day runs 86,164.09 s, some 3 minutes 56 seconds shy of the 86,400 s on a wall clock, since Earth must turn a little past one full rotation to bring the Sun back to the same meridian. Design for 86,400 s instead and the orbit sits slightly too high, drifting roughly a degree of longitude westward every day — enough to walk off a fixed dish inside a fortnight.
Can I work in years and astronomical units instead?
For anything circling the Sun, yes, and the whole thing collapses to T² = a³ with no constant whatsoever. One AU returns one year, Jupiter's 5.204 AU returns 11.87 years, Neptune's 30.07 AU returns 164.9. That choice of units quietly folds the Sun's GM into the number one, which is exactly the shape Kepler had and the reason he never needed a gravitational constant. This instrument runs on SI, so feed it kilograms, metres and seconds, then let the Orbital period field convert.
How do I get an orbit's size from a period I already know?
Invert it: a = ∛(GM·T² ⁄ 4π²). Constellation designers do this constantly. GPS satellites are built to lap twice per sidereal day, a period of 43,082 s, which fixes the semi-major axis at 26,562 km and puts them about 20,190 km above the surface. Sun-synchronous imaging orbits and the 12-hour Molniya ellipses get chosen the same way. Here, adjust Orbital semi-major axis by hand until Orbital period reads what you want.
Where does this law stop being accurate?
Wherever a third body intrudes or the central mass departs from a sphere. Earth's oblateness splits period into several distinct definitions for low satellites — anomalistic, nodal and sidereal differ by tens of seconds. Resonant tugging by a large neighbour reshapes orbits over long spans rather than single laps. Relativity turns Mercury's ellipse by 43 arcseconds a century while barely touching its 88-day year. Near a neutron star or a black hole, the Newtonian framing gives out entirely.