Planetary ejection of a comet

A gravitational assist ejects a weakly bound comet when the increase in its specific orbital energy exceeds its binding energy. For near-parabolic coplanar encounters, the weak-deflection ejection impact parameter scales as $(M_p/M_{\star })Q$. Combining this with the phase-mixed comet surface density gives a one-encounter ejection time proportional to $Q^{1/2}$, within the local two-body encounter approximation.

For a particle initially having semimajor axis one and encountering a circular planet at radius one, the incoming stellar-frame speed is one. A conservative encounter at planet-relative speed $u$ changes the stellar-frame velocity by at most $2u$. Since stellar escape requires speed $\sqrt{2}$, a necessary bound is $u>(\sqrt{2}-1)/2$. Retaining the outgoing relative-velocity geometry sharpens the necessary bound to $u>\sqrt{2}-1$. Neither inequality by itself guarantees a realizable deflection at a specified impact parameter.