Vis-Viva Equation

Celestial Mechanics · 1687

Equation: v² = GM·(2/r − 1/a)
Field: Celestial Mechanics
Named After: Isaac Newton
Year: 1687

Symbols

v orbital speed at distance r from the central body · a semi-major axis of the orbit · G gravitational constant · M central mass · for a circular orbit r = a and v = √(GM/r)

Significance

Speed anywhere along any orbit from energy bookkeeping alone. 'Vis viva' is Leibniz's name for mv² — twice the kinetic energy, a century and a half before Coriolis put the half in — and this equation is no more than the conserved sum of that and the gravitational potential, rearranged for v. The everyday tool of mission design: every Hohmann transfer, gravity assist and orbital insertion burn starts from this line.