Circular Orbital Velocity

Orbital velocity: v = √(GM/r)

For a circular orbit, gravitational attraction equals centripetal force, giving v = √(GM/r), where G = 6.674·10⁻¹¹ N·m²/kg², M is the central body's mass and r is the orbital radius (from the center, not the surface). Example: the ISS orbits at ~400 km altitude, so r = 6,771 km from Earth's center, giving v ≈ 7.67 km/s and a period of ~92 minutes. A geostationary satellite at r = 42,164 km has v = 3.07 km/s and T = 23 h 56 min (sidereal day), staying fixed above a meridian. The Moon orbits Earth at v ≈ 1.022 km/s; Earth orbits the Sun at ~29.78 km/s. Escape velocity is v_esc = √2·v_orb. Near supermassive black holes, stars reach relativistic speeds — S2 around Sgr A* at the Milky Way's center exceeds 2% of c at perihelion.

Applications

Launch window calculation, Hohmann transfer maneuvers, gravitational slingshots (Voyager 1 and 2), low-Earth-orbit constellations (Starlink, OneWeb), space telescopes (Hubble at 540 km, James Webb at L2 Lagrange point), GPS and Galileo navigation, and inferring central-body masses from orbital observations.

FAQ

Why is r measured from the center, not the surface? Newton's shell theorem proves a spherically symmetric mass acts gravitationally as if concentrated at its center. So altitude above the surface must be added to the body's radius.

Does the satellite's mass matter? No — orbital velocity depends only on the central mass M and radius r. A bolt and a space station orbit at the same speed at the same altitude.

What if the orbit is elliptical? Velocity varies: fastest at perihelion, slowest at aphelion. The vis-viva equation v² = GM(2/r − 1/a) generalizes the formula for any conic orbit, where a is the semi-major axis.