On retrograde orbits, resonances and stability
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On retrograde orbits, resonances and stability M. H. M. Morais1 · F. Namouni2
Received: 15 April 2015 / Revised: 12 October 2015 / Accepted: 1 November 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract We start by reviewing our previous work on retrograde orbital configurations and on modeling and identifying retrograde resonances. Then, we present new results regarding the enhanced stability of retrograde configurations with respect to prograde configurations in the low mass ratio regime of the planar circular restricted 3-body problem. Motivated by the recent discovery of small bodies which are in retrograde resonance with the Solar System’s giant planets we then explore the case with mass ratio 0.001 and show new stability maps in a grid of semi-major axis versus eccentricity for the 2/1 and 1/2 retrograde resonances. Finally, we explain how the stability borders of the 2/1 and 1/2 retrograde resonances are related to the resonant orbits’ geometry. Keywords
Resonance · Stability · Retrograde
Mathematics Subject Classification
70F15 · 70F07
1 Introduction A retrograde orbital configuration consists of two or more bodies moving around a central mass in opposite directions or, more precisely, such that the relative inclination1 180◦ ≥ I > 90◦ . When I = 180◦ the orbits are retrograde and coplanar.
1 The relative inclination between two orbits is the angle between the respective angular momentum vectors.
Communicated by Elbert E. N. Macau, Antônio Fernando Bertachini de Almeida Prado and Cristiano Fiorilo de Melo.
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M. H. M. Morais [email protected]; [email protected]
1
Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista (UNESP), Av. 24-A, 1515, 13506-900 Rio Claro, SP, Brazil
2
Université de Nice, CNRS, Observatoire de la Côte dAzur, CS 34229, 06304 Nice, France
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M. H. M. Morais, F. Namouni
Examples of retrograde orbital motion in the Solar System include: many comets, 63 small bodies2 including Mars crossing asteroids, a subset of Centaurs called Damocloids, a TNO (2008KV42), and distant objects on nearly parabolic orbits. Many irregular satellites of the giant planets and Neptune’s moon Triton also have retrograde orbital motion. Extrasolar planetary systems are detected through indirect methods and the relative inclinations are usually not known.3 Retrograde configurations could be achieved, e.g., through capture of a free-floating planet in a star cluster which has isotropic probability (Perets and Kouwenhoven 2012; Varvoglis et al. 2012). Analysis of radial velocity data for the star ν-Octantis A by Ramm et al. (2009) led to the claim that there is a planet about half-way between ν-Octantis A and its binary companion, νOctantis B. This result was puzzling since a planet at such location in a prograde configuration would be unstable due to the strong perturbation from ν-Octantis B and led to alternative hypothesis for this system (Morais and Correia 2012). Numerical integrations by Eberle and Cuntz (2010) and Go´zdz
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