Orbital Period & Escape Velocity

Every formula here depends on the product GM (the standard gravitational parameter), never on G and the mass separately — which is fortunate, because G is the least precisely known of the physical constants while GM is measured very accurately from the orbits themselves. The page takes GM either from a mass you enter or from the surface gravity and radius, since GM = g·R². Give it a surface gravity of 9.807 m/s² and a radius of 6 371 km and everything below follows; give it 1.62 and 1 737 km and the same page describes a different world.

Kepler's third law in Newton's form is T = 2π√(a³ ÷ GM) — the period depends only on the semi-major axis, not on the eccentricity, so a circular orbit and a long ellipse with the same a take exactly as long. The circular speed at radius r is √(GM ÷ r) and the escape velocity is √(2GM ÷ r), exactly √2 times it. For an eccentric orbit the speeds at periapsis and apoapsis come from the vis-viva equation v² = GM(2/r − 1/a). A synchronous orbit is the a whose period matches the body's rotation, a = ∛(GM·T² ÷ 4π²).

Two point masses, no atmosphere, no third body, no radiation pressure, no relativity. A real low orbit decays from drag; a real synchronous satellite needs station-keeping against the Moon, the Sun and the body's own lumpiness; and only an equatorial, circular, prograde synchronous orbit actually stays over one spot — anything else traces a figure of eight. The altitude is measured from the surface unless you switch to a semi-major axis from the centre, a distinction worth a second look whenever an orbit number seems 6 371 km out. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.