Periodic orbits of the retrograde coorbital problem

Abstract

Asteroid (514107) Ka'epaoka'awela is the first example of an object in the 1/1 mean motion resonance with Jupiter with retrograde motion around the Sun. Its orbit was shown to be stable over the age of the Solar System which implies that it must have been captured from another star when the Sun was still in its birth cluster. Ka'epaoka'awela orbit is also located at the peak of the capture probability in the coorbital resonance. Identifying the periodic orbits that Ka'epaoka'awela and similar asteroids followed during their evolution is an important step towards precisely understanding their capture mechanism. Here, we find the families of periodic orbits in the two-dimensional retrograde coorbital problem and analyze their stability and bifurcations into three-dimensional periodic orbits. Our results explain the radical differences observed in 2D and 3D coorbital capture simulations. In particular, we find that analytical and numerical results obtained for planar motion are not always valid at infinitesimal deviations from the plane.

A study of the 1/2 retrograde resonance: Periodic orbits and resonant capture

“…We continued our work on resonant capture in 3D (Namouni & Morais, 2015, 2018b and the relation with the PO families (Morais & Namouni, 2019). We obtained the families of POs associated with the 1/2 retrograde resonance at Jupiter to Sun mass ratio.…”

“…Therefore we restrict our search to POs of multiplicity 3. We compute the circular family of POs with the same multiplicity since from this there may be bifurcations to resonant planar POs (Hadjidemetriou, 1988;Kotoulas & Hadjidemetriou, 2002) and 3D POs (Hénon, 1973;Antoniadou & Libert, 2019;Morais & Namouni, 2019;Kotoulas & Voyatzis, 2020b).…”

“…The few searches which were extended to retrograde motion (inclination larger than 90 • ) were also based on continuation of planar prograde POs (Kotoulas & Voyatzis, 2005;Antoniadou & Libert, 2019). Morais & Namouni (2019) computed for the 1st time the families of stable POs for the planar retrograde 1/1 resonance and obtained the spatial families which bifurcate from them. Subsequently, Kotoulas & Voyatzis (2020a) computed the families of planar POs for interior retrograde resonances with Jupiter (i.e.…”

“…with semimajor axis larger than the planet's) based on continuation from planar prograde and retrograde POs (Kotoulas & Voyatzis, 2020b). Morais & Namouni (2019) explained how the families of POs for the retrograde 1/1 resonance determine resonant capture, in particular the 3-phase mechanism observed in the simulations (Namouni & Morais, 2015;Morais & Namouni, 2016a). The purpose of the current article is to extend the study of Morais & Namouni (2019) to the exterior 1/2 retrograde resonance, focusing on explaining the 3-phase capture mechanism and on unveiling the resonance structure associated with the PO families.…”

Dynamical structures of retrograde resonances: analytical and numerical studies

“…The dynamical system theory based on families of periodic orbits in the restricted three-body problem provides a new approach to understand the dynamics of retrograde MMRs. For example, in order to understand the capture mechanism of asteroid 2015 BZ509 and similar co-orbital objects, Morais & Namouni (2019) investigated families of periodic orbits of the retrograde co-orbital problem and studied their stability and bifurcations. Recently, in the Sun-Jupiter system, Kotoulas & Voyatzis (2020a) produced families of planar resonant retrograde periodic orbits in the planar circular and elliptic restricted three-body problems for the retrograde 2/1, 3/2, 4/3, 3/1, 5/3, 7/5, 4/1, 5/2 and 7/4 resonances.…”