Satellite Speed Calculator

Circular orbital speed follows from setting the gravitational force GMm/r squared equal to the centripetal requirement mv squared over r. The satellite mass m cancels, which is why a loose bolt and a 420-tonne space station at the same altitude travel at exactly the same speed. The radius r is measured from the center of the central body, so an altitude of 408 km above Earth means r = 6,371 + 408 = 6,779 km. Mission planners normally use the standard gravitational parameter GM rather than G and M separately, because GM for Earth is known to roughly nine significant figures at 3.986004418 x 10^14 m^3/s^2 while G alone is known to only about four. Typical results: the ISS at 408 km moves at 7.66 km/s with a 92.5 minute period, GPS satellites at 20,200 km move at 3.87 km/s with a period near 11 hours 58 minutes, and geostationary satellites at 35,786 km move at 3.07 km/s. Escape velocity at any radius is exactly sqrt(2), about 1.414 times, the circular speed there. The model assumes an ideal two-body system, a perfectly circular orbit, and a spherically symmetric planet. Real orbits are elliptical and need the vis-viva relation, v squared = GM times (2/r minus 1/a), where a is the semi-major axis. Earth's equatorial bulge, captured by the J2 coefficient of about 1.0826 x 10^-3, precesses the orbital plane and is exploited deliberately to maintain sun-synchronous orbits. Below roughly 500 km, atmospheric drag steadily lowers the orbit, which is why the ISS requires periodic reboosts.

Johannes Kepler published his first two laws of planetary motion in Astronomia Nova in 1609 and the harmonic third law, that the square of the orbital period is proportional to the cube of the semi-major axis, in Harmonices Mundi in 1619. These were empirical fits to Tycho Brahe's observations of Mars rather than derivations from physical principles. Isaac Newton supplied the underlying cause in the Principia of 1687, showing that an inverse-square attraction reproduces all three of Kepler's laws, and illustrating orbital motion with his thought experiment of a cannon fired horizontally from a mountaintop at ever greater speed until the ball falls around the Earth instead of onto it. The practical era began when the Soviet Union launched Sputnik 1 on 4 October 1957 into an elliptical orbit of roughly 215 by 939 km with a period near 96 minutes, closely matching what the simple formula predicts. Geostationary orbit had already been described by Herman Potocnik in 1928 and popularized by Arthur C. Clarke in a 1945 Wireless World article; Syncom 3 became the first satellite placed there in 1964. Modern flight dynamics still begins with v = sqrt(GM/r) and then layers on perturbation models for the non-spherical gravity field, atmospheric drag, solar radiation pressure, and lunar and solar tides.