Abstract
We continue the investigation of the dynamics of retrograde resonances initiated in Morais and Giuppone (Mon Notices R Astron Soc 424:52–64, doi:10.1111/j.1365-2966.2012.21151.x, 2012). After deriving a procedure to deduce the retrograde resonance terms from the standard expansion of the three-dimensional disturbing function, we concentrate on the planar problem and construct surfaces of section that explore phase-space in the vicinity of the main retrograde resonances (2/\(-\)1, 1/\(-\)1 and 1/\(-\)2). In the case of the 1/\(-\)1 resonance for which the standard expansion is not adequate to describe the dynamics, we develop a semi-analytic model based on numerical averaging of the unexpanded disturbing function, and show that the predicted libration modes are in agreement with the behavior seen in the surfaces of section.
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Notes
A retrograde orbit with inclination \(I>90^\circ \) can be obtained from a prograde orbit with inclination \(180^\circ -I\) by inverting the direction of motion which implies a swap between ascending and descending nodes.
D’Alembert rule is not obeyed because the canonical transformations described in Sect. 2.1 imply that angles for the test particle are measured in the opposite direction of the binary’s motion.
Here, we define \(\lambda =M+\omega +\varOmega \). If we define \(\lambda ^\star =M+\varpi ^\star \) with \(\varpi ^\star =\omega -\varOmega \) then the retrograde resonant angle is \(\phi =\lambda ^\star -\lambda ^\prime -2\,\varpi ^\star \) in agreement with the conclusions of the previous section.
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Acknowledgments
We thank both reviewers for helpful suggestions that improved the article’s clarity. We acknowledge financial support from FCT-Portugal (PEst-C/CTM/LA0025/2011). The surfaces of section computations were performed on the Blafis cluster at the University of Aveiro.
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A part of this paper was presented as Paper DDA 303.02 at the 44th Annual Meeting of the AAS Division of Dynamical Astronomy, 2013, Paraty, Brazil.
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Morais, M.H.M., Namouni, F. Retrograde resonance in the planar three-body problem. Celest Mech Dyn Astr 117, 405–421 (2013). https://doi.org/10.1007/s10569-013-9519-2
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DOI: https://doi.org/10.1007/s10569-013-9519-2