= Solution
Let $\chi=v_{\rm esc}/v_k$, with $v_k=\sqrt{GM_\star/a_p}$. The <Safronov number> is $\Theta=\chi^2/2$. In the instantaneous <planet> frame let the asymptotic vectors be $\boldsymbol u_{\rm in}$ and $\boldsymbol u_{\rm out}$; their magnitudes agree. In the stellar frame the corresponding <specific orbital energy> change during a short encounter is
$$
\Delta\varepsilon_\star=\boldsymbol V_p\cdot
(\boldsymbol u_{\rm out}-\boldsymbol u_{\rm in}),\qquad
|\Delta\varepsilon_\star|\leq\frac{v_kv_{\rm esc}}{\sqrt2}.
$$
The quadratic relative-speed terms cancel. A <comet> with <semi-major axis> comparable to $a_p$ has stellar binding <energy> per unit <mass> of order $v_k^2/2$. Consequently $\chi$ measures whether a single strong encounter can change a substantial fraction of that binding.
\b[Large escape-to-orbital-speed ratios favour ejection; small ratios favour collision or <accretion> during repeated encounters]. For $\chi\gg1$, a suitably oriented <gravitational assist> can eject the <comet> before it strikes the <planet>. For $\chi\ll1$, most individual kicks are too weak; many close passages may be needed, providing repeated chances of collision.
The crossover is of order unity, not a sharp theorem about each <orbit>. The initial binding <energy>, encounter speed and geometry matter: an already nearly parabolic <comet> may be ejected by a low-$\chi$ <planet>, while an unfavourably directed kick can make a <comet> more tightly bound even when $\chi$ is large.
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