Secular Dynamics of an Exterior Test Particle: The Inverse Kozai and Other Eccentricity-Inclination Resonances

The behavior of an interior test particle in the secular 3-body problem has been studied extensively. A well-known feature is the Lidov-Kozai resonance in which the test particle’s argument of periapse librates about $\pm {90}^{\circ }$ and large oscillations in eccentricity and inclination are possible. Less explored is the inverse problem: the dynamics of an exterior test particle and an interior perturber. We survey numerically the inverse secular problem, expanding the potential to hexadecapolar order and correcting an error in the published expansion. Four secular resonances are uncovered that persist in full $N$-body treatments (in what follows, $\varpi$ and $\mathrm{\Omega }$ are the longitudes of periapse and of ascending node, $\omega$ is the argument of periapse, and subscripts 1 and 2 refer to the inner perturber and outer test particle): (i) an orbit-flipping quadrupole resonance requiring a non-zero perturber eccentricity $e_1$, in which ${\mathrm{\Omega }}_2-{\varpi }_1$ librates about $\pm {90}^{\circ }$; (ii) a hexadecapolar resonance (the “inverse Kozai” resonance) for perturbers that are circular or nearly so and inclined by $I\simeq {63}^{\circ }\mathrm{/}{117}^{\circ }$, in which ${\omega }_2$ librates about $\pm {90}^{\circ }$ and which can vary the particle eccentricity by $\mathrm{\Delta }{e}_2\simeq 0.2$ and lead to orbit crossing; (iii) an octopole “apse-aligned” resonance at $I\simeq {46}^{\circ }\mathrm{/}{107}^{\circ }$ wherein ${\varpi }_2-{\varpi }_1$ librates about $0^{\circ }$ and $\mathrm{\Delta }{e}_2$ grows with $e_1$; and (iv) an octopole resonance at $I\simeq {73}^{\circ }\mathrm{/}{134}^{\circ }$ wherein ${\varpi }_2+{\varpi }_1-2{\mathrm{\Omega }}_2$ librates about $0^{\circ }$ and $\Delta e_2$canbeaslargeas0.3forsmall$e_1 \neq 0$. The more eccentric the perturber, the more the particle’s eccentricity and inclination vary; also, more polar orbits are more chaotic. Our inverse solutions may be applied to the Kuiper belt and debris disks, circumbinary planets, and stellar systems.