57.10: The Vis Viva Equation
( \newcommand{\kernel}{\mathrm{null}\,}\)
When an object of small mass m orbits a body of much larger mass M, we can use conservation of energy considerations to find the smaller body's velocity v at radial distance r. We have for the small body m :
K=12mv2 (kinetic energy) U=−GMmr (potential energy) E=−GMm2a (total energy)
where the quantity a is the radius for a circular orbit, the semi-major axis for an elliptical orbit, the negative of the semi-major axis for a hyperbolic orbit, or infinity for a parabolic orbit.
By conservation of energy,
E=K+U−GMm2a=12mv2−GMmr.
Solving for the orbit speed v, we find
This result is known as the vis viva equation (Latin for "live force").