Although most objects in the solar system have nearly coplanar motion with the same orbital direction as the planets some exceptions exist, especially amongst comets, Centaurs and a few Transneptunian objects. These objects on high inclination orbits may be captured in mean motion resonances with the planets. Explaining these configurations is important from a dynamical point of view and also to help constrain solar system formation models.
The classical disturbing function widely used in planetary dynamics studies is a series expansion of the three-body problem’s gravitational interaction with respect to zero eccentricity and zero inclination.
This restricts its validity to nearly coplanar prograde orbits and thus renders the expansion useless to model the dynamics of small bodies with high inclinations with respect to the planet’s orbital plane. To overcome this limitation we derived series expansions of the gravitational interaction with respect to reference inclinations of 180 degrees — to model nearly coplanar retrograde orbits (Morais&Namouni 2013, CMDA 117: 405-421) and 90 degrees — to model nearly polar orbits (Namouni&Morais 2017, MNRAS 471: 2097-2110). Finally, we generalized this work for an arbitrary reference inclination (Namouni&Morais 2018, MNRAS 474: 157-176).
Our expansions are essential to identify the relevant resonant terms which are dominant when the orbital frequencies of the planet’s and small body on a high inclination orbit are commensurable (mean motion resonance). As real life applications I will show that asteroid (514107) 2015 BZ509 is in a
very stable retrograde 1/1 mean motion resonance with Jupiter (Wiegert et al 2017, Nature 543: 687-689; Morais&Namouni 2017, Nature 543: 635-636 ) and that TNO (471325), also known as Niku, is in the 7/9 polar mean motion resonance with Neptune (Morais&Namouni 2017, MNRAS-Letters 472: L1-
Finally, I will discuss how large scale simulations allowed us to conclude that Jupiter’s retrograde coorbital, (514107) 2015 BZ509, must have been captured from the interstellar medium (Namouni&Morais 2018, MNRAS-Letters 477: L117-L121).