Resonances in non-axisymmetric gravitational potentials

Abstract

We study sectoral resonances of the form $j\kappa =m(n-\Omega )$ around a non-axisymmetric body with spin rate $\Omega$, where $\kappa$ 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 resonance order. This describes $n/\Omega \sim m/(m-j)$ resonances inside and outside the corotation radius,as well as prograde and retrograde resonances. Results are: (1) the kinematics of a periodic orbit depends only on $(m^{\prime },j^{\prime })$, the irreducible (relatively prime) version of $(m,j)$. In a rotating frame, the periodic orbit has $j^{\prime }$ braids, $|m^{\prime }|$ identical sectors and $|m^{\prime }|(j^{\prime }-1)$ self-crossing points; (2) thus, Lindblad resonances (with $j=1$) are free of self-crossing points; (3) resonances with same $j^{\prime }$ and opposite $m^{\prime }$ have the same kinematics, and are called $twins$; (4) the order of a resonance at a given $n/\Omega$ depends on the symmetry of the potential. A potential that is invariant under a $2\pi /k$-rotation creates only resonances with $m$ multiple of $k$; (5) resonances with same $j$ and opposite $m$ have the same kinematics and same dynamics, and are called $truetwins$; (6) A retrograde resonance ($n/\Omega <0$) is always of higher order than its prograde counterpart ($n/\Omega >0$); (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.

arXiv comment20 pages, 2 figures

References in corpus (7)

• A ring system detected around the Centaur (10199) Chariklo
• The size, shape, density and ring of the dwarf planet Haumea from a stellar occultation
• On the Migration of Jupiter and Saturn: Constraints from Linear Models of Secular Resonant Coupling with the Terrestrial Planets
• Stability of prograde and retrograde planets in circular binary systems
• Size and shape of Chariklo from multi-epoch stellar occultations
• Ring dynamics around non-axisymmetric bodies with application to Chariklo and Haumea
• The dynamics of rings around Centaurs and Trans-Neptunian Objects

Cited by in corpus (4)

• Dynamics around Non-Spherical Symmetric Bodies: I. The case of a spherical body with mass anomaly
• Rapid formation of binary asteroid systems post rotational failure: a recipe for making atypically shaped satellites
• Rings around irregular bodies I. Structure of the resonance mesh, applications to Chariklo, Haumea and Quaoar
• Rings around irregular bodies. II. Numerical simulations of the 1/3 spin-orbit resonance confinement and applications to Chariklo