2019A&A...621A.101B

Several planets have been found that orbit their host star on retrograde orbits (spin-orbit angle φ>90°). Currently, the largest measured projected angle between the orbital angular momentum axis of a planet and the rotation axis of its host star has been found for HAT-P-14b to be~=171°. One possible mechanism for the formation of such misalignments is through long-term interactions between the planet and other planetary or stellar companions. However, with this process, it has been found to be difficult to achieve retrogradely orbiting planets, especially planets that almost exactly counter-orbit their host star (φ~=180°) such as HAT-P-14b. By contrast, orbital misalignment can be produced efficiently by perturbations of planetary systems that are passed by stars. Here we demonstrate that not only retrograde fly-bys, but surprisingly, even prograde fly-bys can induce retrograde orbits. Our simulations show that depending on the mass ratio of the involved stars, there are significant ranges of planetary pre-encounter parameters for which counter-orbiting planets are the natural consequence. We find that the highest probability to produce counter-orbiting planets (~=20%) is achieved with close prograde, coplanar fly-bys of an equal-mass perturber with a pericentre distance of one-third of the initial orbital radius of the planet. For fly-bys where the pericentre distance equals the initial orbital radius of the planet, we still find a probability to produce retrograde planets of ~=10% for high-mass perturbers on inclined (60°<i<120°) orbits. As usually more distant fly-bys are more common in star clusters, this means that inclined fly-bys probably lead to more retrograde planets than those with inclinations <60°. Such close fly-bys are in general relatively rare in most types of stellar clusters, and only in very dense clusters will this mechanism play a significant role. The total production rate of retrograde planets depends then on the cluster environment. Finally, we briefly discuss the application of our results to the retrograde minor bodies in the solar system and to the formation of retrograde moons during the planet-planet scattering phase.