Celestial Mechanics and Dynamical Astronomy

, Volume 125, Issue 2, pp 133–160 | Cite as

Secular resonances between bodies on close orbits: a case study of the Himalia prograde group of jovian irregular satellites

  • Daohai LiEmail author
  • Apostolos A. Christou
Original Article


The gravitational interaction between two objects on similar orbits can effect noticeable changes in the orbital evolution even if the ratio of their masses to that of the central body is vanishingly small. Christou (Icarus 174:215–229, 2005) observed an occasional resonant lock in the differential node \(\varDelta \varOmega \) between two members in the Himalia irregular satellite group of Jupiter in the N-body simulations (corresponding mass ratio \(\sim 10^{-9}\)). Using a semianalytical approach, we have reproduced this phenomenon. We also demonstrate the existence of two additional types of resonance, involving angle differences \(\varDelta \omega \) and \(\varDelta (\varOmega +\varpi )\) between two group members. These resonances cause secular oscillations in eccentricity and/or inclination on timescales \(\sim \)1 Myr. We locate these resonances in (aei) space and analyse their topological structure. In subsequent N-body simulations, we confirm these three resonances and find a fourth one involving \(\varDelta \varpi \). In addition, we study the occurrence rates and the stability of the four resonances from a statistical perspective by integrating 1000 test particles for 100 Myr. We find \(\sim \)10 to 30 librators for each of the resonances. Particularly, the nodal resonance found by Christou is the most stable: 2 particles are observed to stay in libration for the entire integration.


Irregular satellites Secular resonances Solar perturbations  Coorbital interactions Nodal resonance Kozai–Lidov mechanism Kozai cycle 



DL thanks Dr. Matjia Ćuk for correspondences on the evection phenomenon. We are grateful for the helpful comments from two anonymous referees, increasing the quality of the paper. The authors wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. Astronomical research at the Armagh Observatory is funded by the Northern Ireland Department of Culture, Arts and Leisure (DCAL). Figure 1 is produced using LibreOffice Draw; all the other figures are generated with gnuplot.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Armagh ObservatoryArmaghNorthern Ireland, UK
  2. 2.School of Mathematics and PhysicsQueen’s University BelfastBelfastNorthern Ireland, UK

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