Capture of interstellar objects I: the capture cross-section

Abstract

We study the capture of interstellar objects (ISOs) by a planet-star binary with mass ratio $q\ll 1$, semi-major axis $a_p$, orbital speed $v_c$, and eccentricity $e_p$. Very close (slingshot) and wide encounters with the planet are amenable to analytic treatment, while numerically obtained capture cross-sections $\sigma$ closely follow the analytical results even in the intermediate regime. Wide interactions can only generate energy changes $\Delta E\lesssim q{v}_c^2$, when $\sigma \propto v_{\infty }^{-2}|\ln \Delta E{|}^{2/3}$ (with $v_{\infty }$ the ISO's incoming speed far away from the binary), which is slightly enhanced for $e_p>0$. Energy changes $\Delta E\gtrsim q{v}_c^2$, on the other hand, require close interactions when $\sigma \propto (v_{\infty }\Delta E{)}^{-2}$ hardly depending on $e_p$. Finally, at $\Delta E\gtrsim v_c^2$, the cross-section drops to zero, depending on the planet's radius $R_p$ through the Safronov number $\Theta =q{a}_p/R_p$. We also derive the cross-sections for collisions of ISOs with planets or moons.

16 pages, to be published in MNRAS

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