Resonances in Nonaxisymmetric Gravitational Potentials

Abstract We study sectoral resonances of the form jκ = m(n − Ω) around a nonaxisymmetric body with spin rate Ω, where κ and n are the epicyclic frequency and mean motion of a particle, respectively, where j > 0 and m (<0 or >0) are integers, j being the order of the resonance. This describes n/Ω ∼ m/(m − j) resonances inside and outside the corotation radius, as well as prograde and retrograde resonances. Results are as follows: (1) the kinematics of a periodic orbit depends only on (m′, j′), the irreducible (relatively prime) version of (m, j). In a rotating frame, the periodic orbit has j′ braids, identical sectors, and self-crossing points; (2) thus, Lindblad resonances (with j = 1) are free of self-crossing points; (3) resonances with the same j′ and opposite m′ have the same kinematics, and are called twins; (4) the order of a resonance at a given n/Ω depends on the symmetry of the potential. A potential that is invariant under a 2π/k-rotation creates only resonances with m multiple of k; (5) resonances with the same j and opposite m have the same kinematics and same dynamics, and are called true twins; (6) A retrograde resonance (n/Ω < 0) is always of higher order than its prograde counterpart (n/Ω > 0); (7) the resonance strengths can be calculated in a compact form with the classical operators used in the case of a perturbing satellite. Applications to Chariklo and Haumea are made.