Orbital Velocity

Orbital velocity v_orb is the speed required for an object to maintain a stable circular orbit around a central body.

Set centripetal force = gravity: mv²/r = GMm/r² ⇒ v_orb = √(GM/r).

At Earth's surface (r ≈ R): v_orb = √(GM/R) = √(gR) ≈ 7.9 km/s (first cosmic velocity).

Period: T = 2πr/v = 2π·√(r³/GM) — Kepler's third law.

Higher orbits ⇒ slower velocity but longer period.

Geosynchronous orbit (T = 24 hours): r ≈ 42,164 km, v ≈ 3.07 km/s.

Total energy in circular orbit: E = −GMm/(2r). Negative ⇒ bound.

Escape velocity = √2 × orbital velocity at the same r.