Coupling between corotation and Lindblad resonances in the presence of secular precession rates

Abstract

We investigate the dynamics of two satellites with masses \(\mu _s\) and \(\mu '_s\) orbiting a massive central planet in a common plane, near a first order mean motion resonance \(m+1{:}m\) (m integer). We consider only the resonant terms of first order in eccentricity in the disturbing potential of the satellites, plus the secular terms causing the orbital apsidal precessions. We obtain a two-degrees-of-freedom system, associated with the two critical resonant angles \(\phi = (m+1)\lambda ' -m\lambda - \varpi \) and \(\phi '= (m+1)\lambda ' -m\lambda - \varpi '\), where \(\lambda \) and \(\varpi \) are the mean longitude and longitude of periapsis of \(\mu _s\), respectively, and where the primed quantities apply to \(\mu '_s\). We consider the special case where \(\mu _s \rightarrow 0\) (restricted problem). The symmetry between the two angles \(\phi \) and \(\phi '\) is then broken, leading to two different kinds of resonances, classically referred to as corotation eccentric resonance (CER) and Lindblad eccentric Resonance (LER), respectively. We write the four reduced equations of motion near the CER and LER, that form what we call the CoraLin model. This model depends upon only two dimensionless parameters that control the dynamics of the system: the distance \(D\) between the CER and LER, and a forcing parameter \(\epsilon _L\) that includes both the mass and the orbital eccentricity of the disturbing satellite. Three regimes are found: for \(D=0\) the system is integrable, for \(D\) of order unity, it exhibits prominent chaotic regions, while for \(D\) large compared to 2, the behavior of the system is regular and can be qualitatively described using simple adiabatic invariant arguments. We apply this model to three recently discovered small Saturnian satellites dynamically linked to Mimas through first order mean motion resonances: Aegaeon, Methone and Anthe. Poincaré surfaces of section reveal the dynamical structure of each orbit, and their proximity to chaotic regions. This work may be useful to explore various scenarii of resonant capture for those satellites.