Hyperbolic excess speed ($v_{\infty }$)

The hyperbolic excess speed is the limiting relative speed far from an attracting body on an unbound hyperbolic Kepler orbit. Its specific orbital energy is $\epsilon =v_{\infty }^2/2>0$. At radius $r$ from a body of gravitational parameter $\mu$, conservation of energy gives $v^2=v_{\infty }^2+2\mu /r$. The outgoing velocity direction as well as its magnitude matters when this orbit is matched to another attracting body's frame in a patched conic approximation.