Capture of interstellar objects I: the capture cross-section
  We study the capture of interstellar objects (ISOs) by a planet-star binary
with mass ratio $q\ll1$, 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 $\Delta
E\lesssim qv_c^2$, 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 $\Delta E\gtrsim qv_c^2$, 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 $\Theta=qa_p/R_p$. We also derive the cross-sections for
collisions of ISOs with planets or moons.
