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 $(m',j')$, the irreducible (relatively prime) version of $(m,j)$. In a rotating frame, the periodic orbit has $j'$ braids, $|m'|$ identical sectors and $|m'|(j'-1)$ self-crossing points; (2) thus, Lindblad resonances (with $j=1$) are free of self-crossing points; (3) resonances with same $j'$ and opposite $m'$ 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 $true~twins$; (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.