Hyperbolic Kepler orbit

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

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.