Circular Orbital Velocity

Orbital velocity: v = √(GM/r)

In a circular orbit, gravity does exactly the job of the centripetal force keeping the object on its path. Set the two equal and you get v = √(GM/r). Here G = 6.674·10⁻¹¹ N·m²/kg², M is the mass of the central body, and r is measured from its center. Take the ISS at roughly 400 km up: r works out to 6,781 km, which gives v ≈ 7.67 km/s and a period near 92 min. Push out to 35,786 km and a geostationary satellite settles at 3.07 km/s with T = 24 h, hanging over the same point on the equator. Escape velocity follows from v_esc = √2·v_orb, so about 1.414 times the orbital speed.

Applications

Space agencies lean on it for mission planning. So do the GPS and Galileo navigation constellations, and the teams deploying Starlink and OneWeb. It shows up in Hohmann transfer maneuvers and gravitational slingshots — the trick Voyager 1 and 2 used to pick up speed off Jupiter and Saturn — as well as in working out launch windows and backing planetary masses out of satellite observations.

FAQ

Why measure r from the center? Newton's shell theorem shows that a spherically symmetric mass pulls as though all of it sat at the center. That's why you add the altitude to the body's radius rather than using altitude alone.

Does the satellite's mass matter? It doesn't. Orbital velocity comes down to M and r, nothing else. A loose bolt and a full space station travel at the same speed if they're at the same altitude.

What about elliptical orbits? For those you use the vis-viva equation v² = GM(2/r − 1/a), where a is the semi-major axis. The object moves fastest at perihelion and slows to its minimum at aphelion.