A Perturbative Treatment of the Retrograde Co-orbital Motion

An important area of research in celestial mechanics is the analysis of dynamics in mean-motion resonances (MMR). In this paper, we consider a retrograde MMR 1:1 corresponding to a co-orbital motion, in which the asteroid and the planet revolve around the Sun in opposite directions. The motivation was provided by the recent discovery of real celestial bodies moving in such resonances with giant planets (the most famous example being 514107 Ka'epaoka'awela, a retrograde co-orbital asteroid of Jupiter). Our study is conducted in the context of a spatial restricted circular three-body problem. Applying double numerical averaging, we construct the equations describing the secular evolution of the asteroid's orbit and generate phase portraits in an e - omega space. In some cases, the averaging procedure is complicated by the coexistence of several modes of resonant motion. It leads to different scenarios of the long-term behavior and, consequently, to the splitting of the phase portrait into several sheets.