Hyperbolic Kepler orbit

A positive-energy inverse-square orbit is a hyperbola with two scattering asymptotes.

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.

The gravitational scattering angle is the change between incoming and outgoing asymptotic velocity directions.

For a local two-body hyperbolic encounter with central parameter $\mu$ and allowed centre separation $r_{\min }\ge r_0$, conservation of energy gives equal asymptotic speeds. The gravitational scattering angle therefore impliesPut $z=v_{\infty }\sqrt{r_{\min }/\mu }$ and use $2z/(1+z^2)\le 1$. The optimizing grazing orbit has $z=1$, orbital eccentricity two and deflection $\pi /3$. For finite spherical bodies $r_0$ is the sum of their radii; excluding actual contact makes the upper bound a supremum. Without a finite exclusion radius the ideal point-mass problem has no finite bound.

In a gravitational assist, a small body's speed relative to a moving massive body is deflected with nearly unchanged magnitude. Transforming back to the inertial frame can transfer orbital energy between the bodies.