How this instrument works
A circular orbit is a standing compromise. Gravity pulls inward at GM ⁄ r², while sweeping around a circle of radius r at speed v demands an inward acceleration of exactly v² ⁄ r. Set those two equal, watch the satellite's own mass drop out of both sides, and what remains is v = √(GM ⁄ r). Nothing holds the spacecraft up. It is falling the whole time, curving downward at precisely the rate the ground curves away beneath it.
Read that square root carefully, because its sign catches almost everyone: speed drops as radius grows. The International Space Station sweeps along at 7.67 km/s, a geostationary satellite six times further from the centre dawdles at 3.07 km/s, and the Moon — sixty Earth radii out — manages barely 1.02 km/s. Rearranged into a period, the same relation gives T = 2π√(r³ ⁄ GM), which is Kepler's harmonic law of 1619 wearing modern notation. He extracted it by fitting Tycho Brahe's planetary tables, sixty-eight years before Newton could say why it should hold.
Each assumption deserves naming. The path must be circular; ellipses need the vis-viva relation v² = GM(2 ⁄ r − 1 ⁄ a), of which this expression is only the special case where a = r. The central body must dominate the mass budget and be close to spherical, since Earth's equatorial bulge quietly rotates real orbital planes over weeks. Residual atmosphere is ignored, though it costs the station roughly two kilometres of altitude a month and forces periodic reboosts. There is one quiet consolation, too: the product GM is measured directly by tracking spacecraft, so it is pinned down far better than either factor alone.
Run the arithmetic backwards and it becomes a weighing machine. Solving for M = v²r ⁄ G turns any observed companion into a scale, which is how planetary masses were fixed long before probes flew past them.
- Enter Mass of the central body in kilograms or tonnes — Earth's 5.9722×10²⁴ kg comes preloaded.
- Give Orbital radius as the distance from that body's centre, not its surface: add the planet's own radius to whatever altitude you were quoted.
- Read Orbital speed in m/s, then flip that field's unit menu to km/h or mph for a figure you can picture.
- Divide 2πr by the result whenever you want the period; a 400 km orbit lands near 92 minutes.
Worked example — how fast the Space Station moves
Its orbit sits roughly 400 km up, and Earth's mean radius is 6371 km, so r = 6,771,000 m measured from the centre. Take M = 5.9722×10²⁴ kg. First the product: GM = 6.6743×10⁻¹¹ × 5.9722×10²⁴ = 3.98603×10¹⁴ m³/s². Dividing by r leaves 5.88691×10⁷ m²/s², and a square root finishes the job at v = 7672.62 m/s — 27,621 km/h, or 17,163 mph.
Check that against a clock. The circumference 2πr comes to 42,543 km, which at 7.6726 km/s takes 5545 seconds: 92.4 minutes, or a little over fifteen laps and sixteen sunrises in a day. Published figures land within seconds of this, the small gap owing to a path that is mildly elliptical and an altitude that sags between reboosts. Leaving Earth from the same radius would take √2 times as much, 10.85 km/s, so a low orbit already sits within 41% of departure.
Questions
Why do higher orbits travel more slowly?
Because gravity is weaker out there, so less speed is needed to bend the path into a circle. With r sitting in the denominator, quadrupling the radius halves the speed. No energy is going missing: a distant orbit still holds more total energy, since the gain in gravitational potential outweighs the kinetic loss. That is why climbing costs propellant even though you finish the manoeuvre moving slower than you started.
If I thrust forward, do I catch a target ahead of me?
No — you drop behind it. Firing prograde raises your orbit, a higher orbit carries a longer period, and so the target draws further ahead each lap. Real rendezvous inverts the instinct: brake, fall into a lower and quicker path, gain ground over several revolutions, then accelerate to climb back and meet. Gemini 4 burned most of its propellant in 1965 chasing a spent booster head-on and never closed; Gemini 6A used the patient method that December and made the first successful approach.
Does the satellite's own mass change the answer?
Not by a hair. It cancels when the gravitational and centripetal terms are balanced, so a shoebox cubesat and a 420-tonne station share one speed at equal radius. The formula does assume the central body dominates. For two comparable partners, Pluto and Charon among them, both swing about a shared barycentre and M has to become the combined mass of the pair.
What if the orbit is elliptical instead of circular?
Reach for the vis-viva equation, v² = GM(2 ⁄ r − 1 ⁄ a), with a the semi-major axis. Speed then varies continuously around the loop, peaking at periapsis and bottoming out at apoapsis. Setting a = r collapses it back to what this instrument computes. For scale, our own planet moves at about 30.3 km/s in early January and 29.3 km/s in early July — a 3% swing in something most people picture as a perfect circle.
Can this formula be used to weigh a planet or a galaxy?
Yes, and astronomers lean on it constantly. Rearranged to M = v²r ⁄ G, one measured companion gives the mass of whatever it circles, which is how Jupiter was weighed centuries before anything flew there. Point the method at spiral galaxies and it stops behaving: Vera Rubin's rotation curves in the 1970s showed outer stars moving far quicker than the visible matter permits, still among the strongest observational cases for dark matter.
How much does the poorly known value of G hurt the accuracy?
Hardly at all, because the result depends on the product GM rather than on the two factors separately. G remains the least precisely measured fundamental constant, uncertain to roughly 22 parts per million, while Earth's GM is fixed by satellite laser ranging to about one part in a billion at 3.986004418×10¹⁴ m³ s⁻². Substituting that better value shifts the Space Station answer from 7672.62 to 7672.60 m/s — a correction of two centimetres per second.