The initial studies of the 3-body problem were aimed at modelling the motion of the Moon under the gravitational forces due to the Earth and Sun. The first periodic solutions were identified in the XVIII century by Leonhard Euler and Joseph Louis Lagrange but later attempts to find a closed form solution in the general case were not successful. The seminal work of Henri Poincaré in the XIX century showed that the reason for failure was intrinsic to the problem: besides well behaved regular solutions there are irregular and unpredictable solutions (deterministic chaos).
In the XX century names like Victor Szebehely, Michel Henon, Roger Broucke, John Hadjidemetriou and Carles Simó contributed with the identification and classification of several families of periodic solutions. The recent discoveries of exoplanets orbiting star binaries motivated searches for periodic orbits in these systems. While the 3-body composed of a star, planet and asteroid is widely studied due the abundance of applications in the solar system, new periodic orbits are still being identified. A particular interesting case occurs when there is a resonance (synchrony) between the orbital motions of the asteroid and planet as this provides a protection mechanism against collisions between the two objects.
The classical modelling of resonances was done for direct application in the solar system where most objects (planets and asteroids) orbit the Sun in the same direction (prograde motion) and approximately in the same plane. I will briefly show how to extend the modelling of resonances in the case of retrograde motion (2 objects orbiting the sun in opposite directions) and in the general case of configurations with arbitrary relative inclination, and I will describe new families of periodic orbits associated with resonances in retrograde and polar configurations.
The study of these new configurations allowed the identification of the first examples of such resonances in the solar system, namely: asteroids in retrograde resonances with Jupiter and Saturn; the first retrograde co-orbital of Jupiter (asteroid 2015 BZ509 – a result published in Nature); transneptunian Niku in a polar resonance with Neptune (the inclination between the orbits is near 90 degrees). Finally, I will describe large scale simulations of 2015 BZ509 and how these showed that the asteroid’s current orbital state is stable over the solar system’s age which implies, since it is retrograde, that it must have an extra-solar origin.