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
We obtain the size and orbital distributions of near-Earth asteroids (NEAs) that are expected to be in the 1 : 1 mean motion resonance with the Earth in a steady state scenario. We predict that the number of such objects with absolute magnitudes H < 18 and H < 22 is 0.65 ± 0.12 and 16.3 ± 3.0, respectively. We also map the distribution in the sky of these Earth coorbital NEAs and conclude that these objects are not easily observed as they are distributed over a large sky area and spend most of their time away from opposition where most of them are too faint to be detected. c 2002 Elsevier Science (USA)
We re-derive the secular theory for Trojan-type motion from Morais (1999) using a Hamiltonian formulation and show how this methodology allows us to include the effect of an oblate central mass and the secular perturbations from additional bodies in a rigorous way. As an application of this work we locate secular resonances inside the co-orbital regions of the uranian satellites and the planets, and show that these are in good agreement with the behaviour observed in numerical integrations.
We test a crossing orbit stability criterion for eccentric planetary systems, based on Wisdom's criterion of first order mean motion resonance overlap (Wisdom 1980). We show that this criterion fits the stability regions in real exoplanet systems quite well. In addition, we show that elliptical orbits can remain stable even for regions where the apocenter distance of the inner orbit is larger than the pericenter distance of the outer orbit, as long as the initial orbits are aligned. The analytical expressions provided here can be used to put rapid constraints on the stability zones of multi-planetary systems. As a byproduct of this research, we further show that the amplitude variations of the eccentricity can be used as a fast-computing stability indicator.
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