Retrograde resonance in the planar three-body problem

Abstract

We continue the investigation of the dynamics of retrograde resonances initiated in Morais & Giuppone (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.Comment: Celestial Mechanics and Dynamical Astronomy, in pres

The disturbing function for polar Centaurs and transneptunian objects

“…We remark that the presence of the twodimensional Laplace coefficients is related to the fact that the two angles f + ω and Ω − λ ′ enter the expression of ρ −(2i+1) independently in contrast to the expansions of nearly coplanar orbits where these angles enter as the sum f +ω±(Ω−λ ′ ) where the ± signs are for prograde and retrograde orbits respectively (Morais & Namouni 2013a). The two-dimensional Laplace coefficients also satisfy the following relations:…”

“…The second term, that we denoteRi, is the indirect perturbation that comes from the reflex motion of the star under the influence of the mass m ′ as the standard coordinate system is chosen to be centered on the star. In the following we use the notations and steps of the literal expansions for nearly coplanar prograde orbits (Murray & Dermott 1999) and nearly coplanar retrograde orbits by Morais & Namouni (2013a) so that the reader is able to see the similarities and differences of the three expansions.…”

“…The classical series can also be used for nearly coplanar retrograde orbits after having applied the procedure devised by (Morais & Namouni 2013a) that allows one to get retrograde resonant terms from prograde ones. In essence, the retrograde series is an expansion in terms of e and cos 2 (I/2) where the latter vanishes for I = 180…”

“…The only inclination-free force amplitude is proportional to e 7 . When the algorithm of (Morais & Namouni 2013a) is applied to the classical disturbing function (of prograde orbits) to produce a series for nearly coplanar retrograde orbits, the force amplitude of a p:q retrograde resonance to lowest order in e and cos(I/2) is proportional to e |p+q|−2k cos(I/2) 2k where 0 2k |p + q|. This gives even to a first order resonance (i.e.…”

“…The retrograde disturbing function helped to identify the first Centaurs and Damocloids in retrograde resonance with Jupiter and Saturn (Morais & Namouni 2013a).…”