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 << 1, semimajor axis a(p), orbital speed v(c), and eccentricity e(p). Very close (slingshot) and wide encounters with the planet are amenable to analytical 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 less than or similar to q v(c)(2), when sigma proportional to v(infinity)(-2) vertical bar ln Delta E vertical bar(2/3) (with v(infinity) the ISO's incoming speed far away from the binary), which is slightly enhanced for e(p) > 0. Energy changes Delta E greater than or similar to q v(c)(2), on the other hand, require close interactions when sigma proportional to (v(infinity)Delta E)(-2) hardly depending on e(p). Finally, at Delta E greater than or similar to 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.