vis viva equation
v² = GM(2/r − 1/a) — gives the orbital speed at any distance r for an orbit with semi-major axis a.
Definition
The vis-viva equation is the master formula of orbital mechanics. It combines conservation of energy and the geometry of conic sections into a single expression: v² = GM(2/r − 1/a), where r is the current distance from the central body and a is the semi-major axis of the orbit. For a circular orbit (r = a everywhere), it reduces to v = √(GM/a). At perihelion the speed is maximum; at aphelion, minimum.
The name comes from Leibniz's 'vis viva' (living force), his term for what we now call kinetic energy. The equation encodes the total orbital energy: E = −GMm/(2a). Negative energy means a bound elliptical orbit. Zero energy means a parabolic escape trajectory. Positive energy means a hyperbolic flyby. Every Hohmann transfer, every gravity assist, every satellite orbit change is computed from this equation.