Resonances in Nonaxisymmetric Gravitational Potentials
B. Sicardy
Abstract
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 2 π / 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.