ABSTRACT We study the capture of interstellar objects (ISOs) by a planet–star binary with mass ratio q ≪ 1, semimajor axis ap, orbital speed vc, and eccentricity ep. Very close (slingshot) and wide encounters with the planet are amenable to analytical treatment, while numerically obtained capture cross-sections σ closely follow the analytical results even in the intermediate regime. Wide interactions can only generate energy changes $\Delta E\lesssim q{v}_{\mathrm{c}}^2$, when $\sigma \propto v_\infty ^{-2} |\ln \Delta E|^{2/3}$ (with v∞ the ISO’s incoming speed far away from the binary), which is slightly enhanced for ep > 0. Energy changes $\Delta E\gtrsim q{v}_{\mathrm{c}}^2$, on the other hand, require close interactions when σ ∝ (v∞ΔE)−2 hardly depending on ep. Finally, at $\Delta E\gtrsim {v}_{\mathrm{c}}^2$, the cross-section drops to zero, depending on the planet’s radius Rp through the Safronov number Θ = qap/Rp. We also derive the cross-sections for collisions of ISOs with planets or moons.