= Non-colliding gravitational velocity-kick bound
{title2=$|\Delta\boldsymbol v|_{\max}$}
For a local two-body hyperbolic encounter with central parameter $\mu$ and allowed centre separation $r_{\min}\geq r_0$, <conservation of energy> gives equal asymptotic speeds. The <gravitational scattering angle> therefore implies
$$
|\Delta\boldsymbol v|=\frac{2v_\infty}{1+r_{\min}v_\infty^2/\mu}
\leq\sqrt{\frac{\mu}{r_0}}=\frac{v_{\rm esc}(r_0)}{\sqrt2}.
$$
Put $z=v_\infty\sqrt{r_{\min}/\mu}$ and use $2z/(1+z^2)\leq1$. 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.
Back to article page
