Dynamical structures of retrograde resonances: analytical and numerical studies

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 Poincar\'e surfaces of section. Some modifications are introduced in the production of Poincar\'e 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 Poincar\'e 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.