57.10: The Vis Viva Equation

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 $m$ :

$$\begin{align} K=\frac{1}{2} m v^{2} & \text { (kinetic energy) } \\[8pt] U=-\frac{G M m}{r} & \text { (potential energy) } \\[8pt] E=-\frac{G M m}{2 a} & \text { (total energy) } \end{align}$$

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,

$$\begin{align} E & =K+U \\[8pt] -\frac{G M m}{2 a} & =\frac{1}{2} m v^{2}-\frac{G M m}{r} . \end{align}$$

Solving for the orbit speed $v$, we find

$$v=\sqrt{G M\left(\frac{2}{r}-\frac{1}{a}\right)}$$

This result is known as the vis viva equation (Latin for "live force").