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Fix $M_{\star },{\rho }_p,t_{\star }$ and an initial-orbit ratio $\kappa =a_0/a_p$; use $\kappa =1$ for the figure. The qualitative finite-age planetary scattering regimes have two useful boundaries.

First, the order-unity escape-to-orbital-speed boundary $\chi =1$ isIt slopes down with logarithmic slope $-3/2$. Above it a planet can give large fractional binding-energy changes, favouring planetary ejection of a comet; below it weak kicks and repeated collision opportunities favour accretion.

Second, equating the characteristic comet energy diffusion time to the age givesThis boundary slopes up with logarithmic slope $3/4$. Above it the many-encounter diffusion estimate fits within the available age; below it that estimate exceeds the age.

Above both boundaries, rapid ejection is favoured. Above the age boundary but below the escape-speed boundary, repeated encounters can act during the age while individual kicks remain weak, so collision or accretion is commonly favoured. Below the age boundary, the diffusion model predicts incomplete ejection; the escape-speed boundary still distinguishes strong from weak individual kicks.

These labels are statistical expectations under the stated encounter model. In particular, the given diffusion coefficient supplies no collision rate, so it does not prove that every object in the low-$\chi$ region is accreted within $t_{\star }$. Nor does a long ejection time exclude faster collisions or other loss mechanisms. Very massive planets with $N_{\mathrm{ej}}\lesssim 1$ require a few-encounter treatment rather than an extrapolation of the diffusion formula.