Celestial Mechanics and Dynamical Astronomy

, Volume 129, Issue 1–2, pp 1–23 | Cite as

Secular resonances between bodies on close orbits II: prograde and retrograde orbits for irregular satellites

  • Daohai LiEmail author
  • Apostolos A. Christou
Original Article


In extending the analysis of the four secular resonances between close orbits in Li and Christou (Celest Mech Dyn Astron 125:133–160, 2016) (Paper I), we generalise the semianalytical model so that it applies to both prograde and retrograde orbits with a one-to-one map between the resonances in the two regimes. We propose the general form of the critical angle to be a linear combination of apsidal and nodal differences between the two orbits \( b_1 \Delta \varpi + b_2 \Delta \varOmega \), forming a collection of secular resonances in which the ones studied in Paper I are among the strongest. Test of the model in the orbital vicinity of massive satellites with physical and orbital parameters similar to those of the irregular satellites Himalia at Jupiter and Phoebe at Saturn shows that \({>}20\) and \({>}40\%\) of phase space is affected by these resonances, respectively. The survivability of the resonances is confirmed using numerical integration of the full Newtonian equations of motion. We observe that the lowest order resonances with \(b_1+|b_2|\le 3\) persist, while even higher-order resonances, up to \(b_1+|b_2|\ge 7\), survive. Depending on the mass, between 10 and 60% of the integrated test particles are captured in these secular resonances, in agreement with the phase space analysis in the semianalytical model.


Irregular satellites Secular resonances Solar perturbations Coorbital interaction N-body simulation 



The authors are grateful for the constructive comments from two anonymous referees, increasing the quality of the paper. We 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 for Communities (DfC). Figure 13 is produced using LibreOffice Draw and Inkscape; all the other figures are generated with gnuplot.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Armagh ObservatoryCollege Hill, ArmaghUK
  2. 2.School of Mathematics and PhysicsQueen’s University BelfastBelfastUK

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