High inclination orbits in the secular quadrupolar three-body problem

The Lidov-Kozai mechanism (Kozai 1962; Lidov 1962) allows a body to periodically exchange its eccentricity with inclination. It was first discussed in the framework of the quadrupolar secular restricted three-body problem, where the massless particle is the inner body, and later extended to the quadrupolar secular nonrestricted three body problem (Harrington 1969; Lidov & Ziglin 1976; Ferrer & Osacar 1994). In this paper, we propose a different point of view on the problem by looking first at the restricted problem where the massless particle is the outer body. In this situation, equilibria at high mutual inclination appear (Palacián et al. 2006), which correspond to the population of stable particles that Verrier & Evans (2008, 2009) find in stable, high inclination circumbinary orbits around one of the components of the quadruple star HD 98800. We provide a simple analytical framework using a vectorial formalism for these situations. We also look at the evolution of these high inclination equilibria in the non restricted case.