Vis-Viva Equation

The vis-viva equation is energy conservation in orbital form. At every point the total specific mechanical energy is ε = ½v² − GM/r = −GM/(2a) — a constant set only by the semi-major axis a. Two orbits with the same a but different eccentricities carry identical energy and have the same period; eccentricity shapes the trajectory without changing the energy budget. Speed is highest at periapsis (minimum potential energy) and lowest at apoapsis. The escape trajectory is the limiting case a → ∞, giving v_esc = √(2GM/r) = √2 · v_c — exactly √2 times the circular velocity at the same radius.

The vis-viva equation says an orbiting object's speed depends on just two things: how far it currently is from the central body (r) and the overall size of its orbit, the semi-major axis a. Plug both into v = √(GM(2/r − 1/a)) and you get the speed. Notice the object's own mass never appears — a paperclip and a space station in the same orbit travel at exactly the same speed.

Vis viva is Latin for 'living force', the term Leibniz coined in the 1680s for the quantity mv² — the ancestor of kinetic energy. It sparked a decades-long dispute with Newton's followers, who insisted momentum mv was the true 'measure of motion'. Both sides were partly right, and the v² that survives in this equation is a direct echo of Leibniz's living force.