Dynamical structures of retrograde resonances: analytical and numerical studies

Abstract

In this work, retrograde mean motion resonances (MMRs) are investigated by means of analytical and numerical approaches. Initially, we define a new resonant angle to describe the retrograde MMRs and then perform a series of canonical transformations to formulate the resonant model, in which the phase portrait, resonant centre and resonant width can be analytically determined. To validate the analytical developments, the non-perturbative analysis is made by taking advantage of Poincaré surfaces of section. Some modifications are introduced in the production of Poincaré sections and, in particular, it becomes possible to make direct comparisons between the analytical and numerical results. It is found that there exists an excellent correspondence between the phase portraits and the associated Poincaré sections, and the analytical results agree well with the numerical results in terms of the resonant width and the location of resonant centre. Finally, the numerical approach is utilized to determine the resonant widths and resonant centres over the full range of eccentricity. In particular, seven known examples of retrograde asteroids including 2015 BZ509, 2008 SO218, 1999 LE31, 2000 DG8, 2014 AT28, 2016 LS and 2016 JK24 are found inside the libration zones of retrograde MMRs with Jupiter. The results obtained in this work may be helpful for understanding the dynamical evolution for asteroids inside retrograde MMRs.

A Systematic Study about Orbit Flips of Test Particles Caused by Eccentric Von Zeipel–Lidov–Kozai Effects

“…For such a kind of non-integrable system, Poincaré surface of section provides a powerful tool to numerically explore the global structures in the phase space. The technique of Poincaré sections has been used in different contexts Li et al 2014a;Naoz et al 2017;Malhotra & Zhang 2020;Lei & Li 2021;Lei 2021b). In this section, our purpose is to analyze the surfaces of section at different levels of Hamiltonian to find libration islands causing orbital flips.…”

A systematic study about orbit flips of test particles caused by eccentric von Zeipel-Lidov-Kozai effects

“…According to the discussions made by Shevchenko (2016) and Lei & Li (2021), the longitude of pericenter is generally defined by = Ω + sign(cos i)ω. Thus, the critical argument σ 1 adopted in this work is equal to the longitude of pericenter for both prograde and retrograde configurations.…”

“…For such a kind of non-integrable systems, Poincaré surface of section provides a powerful tool to numerically explore the global structures in the phase space. The technique of Poincaré sections has been used in different contexts Li et al 2014a;Naoz et al 2017;Malhotra & Zhang 2020;Lei & Li 2021;Lei 2021b). In this section, our purpose is to analyse the surfaces of section at different levels of Hamiltonian to find libration islands causing orbital flips.…”

“…2, we use ∆H to stand for the size of libration islands causing orbit flips 5 . By analysing the Poincaré sections (Malhotra & Zhang 2020;Lei & Li 2021;Lei 2021b), it is possible to generate the size of flipping regions as a function of the Hamiltonian ∆H(H). According to the definition of H, we can produce the boundaries of flipping regions in the space (e 0 , i 0 ), as shown in Fig.…”

Bridges and gaps at low-eccentricity first-order resonances

“…These results were reproduced with an analytical Hamiltonian approach by Lei & Li (2020), who also extended the study of the libration widths (based on an analysis of phase portraits) to first-order exterior MMRs. Later on, Lei & Li (2021) also studied retrograde orbits and coorbital motion.…”